Important formulas in combinatorics Motivation:
The poster for the conference celebrating Noga Alon's 60th birthday, fifteen formulas describing some of Alon's work are presented. (See this post, for the poster, and cash prizes offered for identifying the formulas.) This demonstrates that sometimes (but certainly not always) a major research progress, even areas, can be represented by a single formula. Naturally, following Alon's poster, I thought about representing other people's works through formulas. (My own work, Doron Zeilberger's, etc. Maybe I will pursue this in some future posts.) But I think it will be very useful to collect major formulas representing major research in combinatorics.  
The Question
The question collects important formulas representing major progress in combinatorics.
The rules are:
Rules
1) one formula per answer
2) Present the formula explicitly (not just by name or by a link or reference), and briefly explain the formula and its importance, again not just link or reference. (But then you may add links and references.) 
3) Formulas should represent important research level mathematics. (So, say $\sum {{n} \choose {k}}^2 = {{2n} \choose {n}}$ is too elementary.) 
4) The formula should be explicit as possible, moving from the formula to the theory it represent should also be explicit, and explaining the formula and its importance at least in rough terms should be feasible.
5) I am a little hesitant if classic formulas like $V-E+F=2$ are qualified. 
An important distinction:
Most of the formulas represent definite results, namely these formulas will not become obsolete by new discoveries. (Although refined formulas are certainly possible.) A few of the formulas, that I also very much welcome, represent state of the art regarding important combinatorial parameters. For example, the best known upper and lower bounds for diagonal Ramsey's numbers. In the lists and pictures below an asterisk is added to those formulas.  
The Formulas (so far)
In order to make the question a more useful source, I list all the formulas in categories with links to answer (updated Feb. 6 '17). 

Basic enumeration: The exponential formula; inclusion exclusion; Burnside and Polya; Lagrange inversion; generating function for Fibonacci; generating function for Catalan; Stirling formula; Enumeration and algebraic combinatorics: The hook formula; Sums of tableaux numbers squared, Plane partitions;  MacMahon Master Theorem; Alternating sign matrices;  Erdos-Szekeres; Ramanujan-Hardy asymptotic formula for the number of partitions; $\zeta(3)$; Shuffles; umbral compositional identity; Jack polynomials; Roger-Ramanujan; Littlewood-Ricardson; 

Geometric combinatorics: Dehn-Somerville relations; Zaslavsky's formula; Erhard polynomials; Minkowski's theorem.  Graph theory: Tutte's golden identity; Chromatic number of Kneser's graph; (NEW)Tutte's formula for rooted planar maps ; matrix-tree formula; Hoffman bound; Expansion and eigenvalues; Shannon capacity of the pentagon; Probability: Self avoiding planar walks; longest monotone sequences (average); longest monotone sequences (distribution); Designs: Fisher inequality; Permanents: VanderWaerden conjecture; Coding theory: MacWilliams formula; Extremal combinatorics: Erdos-Sauer bound; Ramsey theory: Diagonal Ramsey numbers (); Infinitary combinatorics: Shelah's formula (); A formula in choiceless set theory. Additive combinatorics: sum-product estimates (*); Algorithms: QuickSort. 

(Larger formulas): Series multisection; Faà di Bruno's formula; Jacobi triple product formula; A formula related to Alon's Combinatorial Nullstellensatz; The combinatorics underlying iterated derivatives (infinitesimal Lie generators) for compositional inversion and flow maps for vector fields (also related to the enumerative geometry of associahedra); some mysterious identities involving mock modular forms and partial theta functions; 

Formulas added after October 2015:
Hall and Rota Mobius function formula; Kruskal-Katona theorem; Best known bounds for 3-AP free subsets of $[n]$ (*);

After October 2016: Abel's binomial identity; Upper and lower bounds for binary codes; 
 A: Either Knop and Sahi's formula for the integer Jack polynomials,
$$J^{(\alpha)}_\mu(x) = \sum_{T\text{ admissible of shape } \mu} d_T(\alpha) x^T$$
where the sum is over a certain set of tableaux, and $d_T$ is a weight,
or the more general formula for the modified Macdonald polynomials,
$$\tilde{H}_\mu(x;q,t) = \sum_{T \text{ shape } \mu} q^{inv(T)} t^{maj(T)}x^T$$
which is a really amazing formula.
A: Fisher's inequality
$$b \ge v.$$
Asserts that the number of blocks in every 2-design is at least the number of elements. A design is a collection of $k$-elements subsets (called blocks) of a set $V$ with $v$ elements such that every pair of elements of $V$ belong to the same number of blocks. This fundamental relation is closely related to the Erdos-DeBruijn theorem in extremal combinatorics, and the linear theoretic proof by Bose is an important starting point for linear algebra methods in combinatorics. 
A: I'm confused about why no one has mentioned Stirling's formula for the factorial function $n!$, clearly the most famous and important formula in asymptotic combinatorics, and easily one of the most important and recognizable in all of mathematics. It states that the number $n!$ of permutations of $1,\ldots,n$ is given asymptotically as $n\to\infty$ by
$$
n! \sim \sqrt{2\pi n} (n/e)^n.
$$
The weaker statement that $n! \sim C\sqrt{n} (n/e)^n$ for some number $C>0$ was first proved by De-Moivre. James Stirling found the value of $C$ shortly afterwards, around 1730 apparently.
Note that Gil was asking for formulas that represent "important research level mathematics" and was vague about whether classical results are allowable. Of course nowadays the principles behind how to prove Stirling's formula and similar results are well-understood, but this was undoubtedly cutting-edge research back in the 1700s. In addition, many new proofs of the formula have been published throughout the ages, including quite recently (see here for a non-subscription download). Furthermore, the standard proof uses the saddle-point method and can be thought of as a prototype for a large number of new and very modern results in asymptotic combinatorics that are being published all the time. So, Stirling's formula is not just "old" mathematics -- it is a true gem that contains important ideas at the forefront of current research. And this is not even mentioning its importance as a fundamental result that plays a role in the derivation of many basic estimates and bounds in virtually all areas of mathematics, as well as computer science, physics, statistics, etc.
A: Tutte's formula (circa 1963) for the number of rooted planar maps with $n$ edges:
$$\#M_n = \frac{2}{n+3}3^nC_n$$
where $C_n = \frac{1}{n+1}\binom{2n}{n}$ is the $n$th Catalan number. This is a surprisingly simple formula. Moreover, this formula is the beginning of an important story about universal $2$-dimensional random structures because the limit of the uniform random planar map is the so-called "Brownian map" which has seen a lot of attention in the last ~10 years. As such it is related to topics like quantum gravity. Note, however, that Tutte used generating function techniques to prove the above formula whereas the scaling limit phenomena are based off bijective techniques that came later (80s-90s).
See these notes for some more details: https://arxiv.org/abs/1101.4856.
Tutte's theory of counting planar maps and triangulations is a true festival of formulas. Here is the original formula (given above in a slightly different form) of Tutte for rooted planar maps and another one from the paper A new branch of enumerative graph theory

Answer by Sam Hopkins
A: An elegant product-sum identity, involving the symmetric groups $S_n$ on $n$ letters, and parametrized by a real number $r$, is Florentino's identity:
$$
1+\sum_{n\geq1}\sum_{\sigma\in S_{n}}\frac{y^{n}}{n!}{1 \over \det(I_{n}-xM_{\sigma})^{r}}=\ \prod_{k\geq0}(1-yx^{k})^{-\binom{k+r-1}{k}},
$$
where $M_{\sigma}$ is the $n\times n$ permutation matrix of $\sigma\in S_{n}$, and $I_n$ the identity matrix of the same size. It is valid on the formal power series ring ${\mathbb R}[[x,y]]$.
This imples Molien's formula for the Hilbert-Poincaré series of the ring of $S_n$ invariants polynomials in $n$ variables:
$$
\frac{1}{n!}\sum_{\sigma\in S_{n}}\frac{1}{\det(I_{n}-xM_{\sigma})}=\prod_{k=1}^{n}\frac{1}{1-x^{k}},
$$
as this is the coefficient of $y^n$ in the former series, when $r=1$.
There are also multivariable versions of Florentino's identity. One of these gives a new expression for the basic hypergeometric series $_{2}\phi_{1}$ in the form:
$$
 _{2}\phi_{1}(\frac{1}{x_{1}},\frac{1}{x_{2}};\,y;\,q;\,yx_{1}x_{2}) \, = \,\, 1+\sum_{n\geq1}\frac{y^{n}}{n!}\sum_{\sigma\in S_{n}}\frac{\det[(I_{n}-x_{1}M_{\sigma})(I-x_{2}M_{\sigma})]}{\det(I_{n}-qM_{\sigma})}
$$
The proofs of these identities are in the article Plethystic exponential calculus and characteristic polynomials of permutations,
and use the so-called plethystic exponential.
A: $$\zeta(3)={5\over2}\sum_{n=1}^{\infty}{(-1)^{n-1}\over n^3{2n\choose n}}$$ was the starting point for Apéry's proof of the irrationality of $\zeta(3)$. [OK, so it's Number Theory, not combinatorics --- but, look! it has a binomial coefficient in it!]. Here is Alf van der Poorten's report. 
A: An elegant and rather unexpected formula for the powers of the generating function for Catalan numbers $C_n = \frac{1}{n+1}\binom{2n}{n}$:
$$\left(\sum_{n=0}^{\infty} \frac{1}{n+1}\binom{2n}{n} \cdot x^n\right)^m = 
\sum_{n=0}^{\infty} \frac{m}{n+m}\binom{2n+m-1}{n} \cdot x^n.$$
The formula can be further continued as $\ldots =\left(\frac{1-\sqrt{1-4x}}{2x}\right)^m$, but this would spoil the beauty of the above identity.
A: $$\Theta (C_5)=\sqrt 5.$$
This is the formula by Lovasz for the Shannon capacity of the cycle of length 5.
The Shannon capacity of a graph $\Theta (G)= \lim_{n \to \infty}(\omega(G^n))^{1/n}$, where $\omega (G)$ is the largest size of an independent set of  vertices in $G$, and $G^n$ is the $n$-fold strong product of $G$. A key to Lovasz' proof was the introduction of   a new spectral parameter $\theta (G)$, and a proof that $\Theta (G) \le \theta (G)$.   
A: The MacWilliams identity 
$$W(C^\perp;x,y) = \frac{1}{\mid C \mid} W(C;y-x,y+x).$$ 
This identity connects the weight enumerator of a linear binary code $C$ with that of the dual code $C^\perp$. Here, $C$ is a linear subspace of   $ \mathbb{F}_2^n$, $C^{\perp}$ is the dual space, and $W(C,x,y)$ is the weight enumerator defined as follows: Let $C_t$ be the number of code-words of weight $t$ (namely, vectors in $X$ with $t$ '1's), $$W(C;x,y)=
\sum A_t x^t y^{n-t}.$$
The identity extends also to codes over other fields and to non-linear codes. It is very important in coding theory and has various other applications. 
A: Burnside Lemma and Redfield-Polya Theorem are celebrated results in combinatorics. They allow to enumerate objects modulo group actions. One of the classic (and simplest) examples is enumeration of necklaces modulo rotations. Other examples include various graphs, chemical compounds, etc.
A: Since infinitary combinatorics were allowed and nobody has done it let me introduce some noise with Shelah´s celebrated cardinal arithmetic inequality:$$\aleph_\omega^{\aleph_0} < \max\{\aleph_{\omega_4},(2^{\aleph_0})^+\},$$
which shows that cardinal exponentiation is not as wild as it was thought to be.
People may want to read Shelah´s paper Cardinal Arithmetic for Skeptics, Bulletin of the AMS, Vol.26, Num. 2, 1992.
A: $$\max (|A+A|,|A\times A|) \ge \frac12 |A|^{4/3} (\log |A|)^{-1/3}.$$
This sum product-relation by Solymosi is one of the highlights of additive combinatorics. Sum-product theorems have many recent applications. Here $A$ is an arbitrary set of positive real numbers and $A+A=\{a+a': a,a' \in A\}$.
$A \times A=\{a \cdot a': a,a' \in A\}$.  
Let me add two other important formulas from additive combinatorics. 
The Cauchy-Davenport relation 
$$|A+A| \ge 2|A|-1.$$
And the Plünnecke relation 
$$|A+A| \le C|A| \implies |kA-\ell A| \le C^{k+\ell} |A|. $$
Solymosi's record was broken several times and the current world record is by George Shakan. See this paper and this Quanta Magazine's article. 

A: I'm not sure if the Garsia–Haiman $n!$ conjecture (now a theorem) counts as a formula, but it feels like a formula to me.  Let $\mu$ be a partition of $n$, and let the coordinates of the cells in the Ferrers diagram of $\mu$ be $\{(p_1,q_1), \ldots, (p_n,q_n)\}$, where $p$ is the row coordinate and $q$ is the column coordinate, indexed from zero, so that the corner cell is $(0,0)$. Define
$$\Delta_\mu(x_1,\ldots,x_n,y_1,\ldots,y_n) := \det
\pmatrix{x_1^{p_1}y_1^{q_1} & x_2^{p_1} y_2^{q_1} & \cdots & x_n^{p_1} y_n^{q_1} \cr
         x_1^{p_2}y_1^{q_2} & x_2^{p_2} y_2^{q_2} & \cdots & x_n^{p_2} y_n^{q_2} \cr 
         \vdots & \vdots & \ddots & \vdots \cr    
         x_1^{p_n}y_1^{q_n} & x_2^{p_n} y_2^{q_n} & \cdots & x_n^{p_n} y_n^{q_n} \cr}$$
Then the dimension of the space $\mathscr{H}_\mu$ spanned by all partial derivatives of all orders of $\Delta_\mu$ is $n!$.
The $n!$ conjecture lies at the center of a web of fascinating algebraic combinatorics that features Macdonald polynomials, diagonal harmonics, and Hilbert schemes.  Garsia has said that when they were first mapping out a conjectural picture of this corner of the mathematical universe, they fairly quickly realized that the $n!$ conjecture was a crucial linchpin.  Although they couldn't immediately prove it, they initially assumed that because the formula was so simple, it would be easy to prove, and they focused their attention on other conjectures.  One by one, the other conjectures were proved, and the $n!$ conjecture was left as the surprisingly stubborn nut.  It was eventually proved by Haiman using surprisingly delicate arguments from commutative algebra and algebraic geometry.  I believe that even today, the only proof is essentially a dimension-counting argument, and that in general there is no known explicit basis for $\mathscr{H}_\mu$ that is indexed by $n!$ combinatorially defined objects.
A: A beautiful and deep formula I learned about from Ken Ono is the
Nekrasov-Okounkov-Westbury formula:
$$
\prod_{n\ge 1} (1-q^n)^{z-1}=
\sum_{\lambda} q^{|\lambda|}\ \prod_{\square\in\lambda}
\left(1-\frac{z}{h(\square)^2}\right)\ .
$$
Here, $\lambda$ is summed over all integer partitions of arbitrary weight. The product over $\square$ is over all boxes of the Ferrers diagram corresponding to $\lambda$. Finally, $h(\square)$ denotes the hook length at the position given by the box $\square$. The article by Nekrasov-Okounkov is here. The one by Westbury is here.
I am not aware of an elementary proof for this identity. The case $z=2$ is Euler's Pentagonal Number Theorem.
A: $$\def\multichoose#1#2{{\left(\kern-.3em\left(\genfrac{}{}{0pt}{}{#1}{#2}\right)\kern-.3em\right)}}
 \binom{n}{k} = (-1)^{k}  \multichoose{-n}{k}$$
Here $\binom{n}{k}$ is the binomial coefficient "$n$ choose $k$": the number of ways to select $k$ items from a set of $n$. And $\multichoose{n}{k}$ is ``$n$ multichoose $k$'': the number of ways to select $k$ items from a set of $n$, where you are allowed to select the same item multiple times.
Of course, to interpret the above formula you must see each side as a polynomial in $n$. Then it is a straightforward exercise to show that the formula holds. However, this formula is the ur-combinatorial reciprocity result. For more on combinatorial reciprocity, see Stanley's paper: Combinatorial reciprocity theorems.
A: I propose one of the combinatorial formulas for the Littlewood-Richardson coefficients,
$c^\lambda_{\mu\nu} =$ the number of skew semi-standard Young tableaux with shape $\lambda/\mu$ and weight $\nu$ with the Yamanouchi condition.
A: (Initially I had this in my other answer, then I followed a suggestion by S. Carnahan and made it a separate one)
There are several impressive combinatorial proofs of some mysterious identities involving  mock modular forms and partial theta functions. Just to give an example - here is my favorite (I actually mentioned it in a comment to one of my MO questions),
\begin{multline*}
\frac q{1+q}+2\frac{(1-q)q^2}{(1+q)(1+q^2)}+3\frac{(1-q)(1-q^2)q^3}{(1+q)(1+q^2)(1+q^3)}+...\\+\frac{(1-q)(1-q^2)(1-q^3)\cdots}{(1+q)(1+q^2)(1+q^3)\cdots}\left(\frac q{1-q}+\frac{q^3}{1-q^3}+\frac{q^5}{1-q^5}+...\right)\\=2q-4q^4+6q^9-8q^{16}+...
\end{multline*}
It appears in Combinatorial Proofs of q-Series Identities by Robin Chapman.
A: Shuffles, stuffles and other dual laws
Mother Formula
All what follows is around the same recursive formula/pattern. 
\begin{equation}
au*bv=a(u*bv)+b(au*v)+\varphi(a,b)(u*v)\quad (0)
\end{equation}
The shuffle product appears in many contexts (representation theory, iterated integrals, Hecke algebras, symmetric functions, decomposition of polytopes, theory of languages, of codes, of automata). 
It turns out that it can be better understood as a law dual to a comultiplication. These co-operations were introduced, in combinatorics, by a seminal paper of Joni and Rota (S.A. Joni and G.-C. Rota, Coalgebras and bialgebras in combinatorics, Stud. Appl. Math.
61 (1979) 93–139.).    
Considering two (non empty) words as card decks $au,bv$ the top cards being respectively $a,b$, the shuffle product of $au$ and $bv$ reads (I do not know how to write the Cyrillic ``Sha'', which is the standard sign for the shuffle, in MathJax, so I use $\sqcup$) 
\begin{equation}
au\sqcup bv=a(u\sqcup bv)+b(au\sqcup v)\quad (1)
\end{equation}
which is the sum of all possible shuffles between $au$ and $bv$ (two disjoint cases $a$ or $b$ on top).
Formula $(1)$ together with the initialization making neutral the empty word i.e. 
\begin{equation}
w\sqcup 1=1\sqcup w=w
\end{equation}
defines perfectly the shuffle product.
Now, this law is better understood as "dual". I mean, if you define the natural pairing on the words by $\langle u\mid v\rangle:=\delta_{u,v}$ you get 
\begin{equation}
\langle u\sqcup v\mid w\rangle=\langle u\otimes v\mid \Delta(w)\rangle
\end{equation}
with 
\begin{equation}
\Delta(w)=\sum_{I+J=[1..|w|]}w[I]\otimes w[J]\quad (2)
\end{equation}
where $|w|$ stands for the length of $w$ and, for $I=\{i_1,i_2,\cdots i_k\}$ a choice of places (indexed in increasing order $i_1<i_2<\cdots <i_k$), $w[I]$ is the subword 
\begin{equation}
w[I]=w[i_1]w[i_2]\cdots w[i_k]
\end{equation} 
(therefore $\Delta(w)$ is sometimes called the ``unshuffling'' of $w$). 
The miracle is that the unshuffling is a morphism i.e. it can be defined letter by letter
\begin{equation}
\Delta(a_1a_1a_2\cdots a_n)=\Delta(a_1)\Delta(a_2)\cdots \Delta(a_n)
\end{equation}
with $\Delta(a)=a\otimes 1+1\otimes a$ for a single letter. 
Many other combinatorial deformations/perturbations of the shuffle product follow a similar scheme, let me exemplify two of them. 
Stuffle (also called Hoffman's shuffle, quasi-shuffle, sticky shuffle) which appears in many contexts (harmonic sums, lambda rings, quasi-symmetric functions). This time, the set of cards is infinite, more precisely, you have an alphabet $\{y_i\}_{i\in \mathbb{N}_{>0}}$ indexed by non-zero integers. The stuffle law is defined recursively as 
\begin{equation}
w*1=1*w=w\ ;\ y_iu*y_jv=y_i(u* y_jv)+y_j(y_iu*v)+y_{i+j}(u*v)\quad (3)
\end{equation}
the term $y_{i+j}(u*v)$ is the reason why certain physicists call it ``sticky shuffle'' because, in this case, the cards $y_i,y_j$ stick together.
Here the dual law is again a morphism defined on the letters by 
\begin{equation}
\Delta(y_k)=y_k\otimes 1+1\otimes y_k+\sum_{i+j=k}y_i\otimes y_j
\end{equation}
Infiltration and $q$ infiltration Infiltration products can be traced back to the paper of 

Chen, Fox and Lyndon, Free differential calculus, IV - The quotient groups of the lower central series" (Ann. Math 65, 163-178, 1958) 

and later (1981) to Oschenschläger's work about binomial coefficients 

Ochsenschläger, P., Binomialkoeffizenten und Shuffle-Zahlen, Technischer Bericht, Fachbereich Informatik, T. H. Darmstadt. (1981)

Let's define directly the $q$ infiltration (for $q=0$, you get the shuffle and $q=1$ the infiltration)
\begin{equation}
w*1=1*w=w\ ;\ au*bv=a(u* bv)+b(au*v)+\delta_{a,b}\,q\,a(u*v)\quad (4)
\end{equation} 
Again, the dual law is a morphism defined on the letters by 
\begin{equation}
\Delta(a)=a\otimes 1+1\otimes a+q\,a\otimes a
\end{equation}
and there is a beautiful analogue of formula $(2)$
\begin{equation}
\Delta(w)=\sum_{I\cup\, J=[1..|w|]}q^{|I\,\cap\, J|}w[I]\otimes w[J]\quad (5)
\end{equation}
A bit more
Endowed with these dual laws (shuffle, stuffle), the free algebra (with coefficients within a $\mathbb{Q}$-algebra) becomes an enveloping bialgebra (and hence a Hopf algebra) whereas, except when $q$ is nilpotent, the infiltration coproduct turns the free algebra into a bialgebra which is NOT a Hopf algebra (due to the presence of non-invertible group-like elements as $(1+qx),\ x\in X$).  
A: I don't believe anyone has yet mentioned the Möbius function formula of Hall and Rota, although several answers have referred to formulas in which this formula is an element in a proof.
Specifically, if $x<y$ are elements in a poset $P$, then the formula is
$$\mu([x,y]) = \tilde{\chi}(\Delta[x,y]).$$
Here $\Delta[x,y]$ is the order complex of $[x,y]$, the simplicial complex consisting of all chains of elements in $P$ strictly between $x$ and $y$.  Of course $\mu$ is the Mobius function (which controls inclusion-exclusion over $P$), and $\tilde{\chi}$ is the reduced Euler characteristic.
This formula was proved by Hall in a 1936 paper.  As far as I understand, it was Rota who noticed the connection with Euler characteristics in the paper "On foundations of combinatorial theory I. Theory of Mobius functions".  It has since become a cornerstone of topological and poset combinatorics.  
Möbius number calculations can often be significantly simplified and/or viewed in a more general context by use of homotopy type and other tools from topology.  Two favorite examples of mine:  If some element of a lattice $L$ has no complement, then $\Delta L$ is contractible and hence $\mu(L)=0$ (a result of Bjorner and Walker).  And Rota's Crosscut Theorem is a special case of the Nerve Theorem.
A: The Hook Formula. If $\lambda$ is a partition of $n$ then the number of standard Young tableaux of shape $\lambda$ is 
$$f^\lambda = \frac{n!}{\prod_{\alpha \in [\lambda]} h_\alpha} $$
where $h_\alpha$ is the hook-length of the box $\alpha$ in the Young diagram $[\lambda]$ of $\lambda$, as shown below for $(5,4,2,1)$. The special case $\lambda = (n,n)$ gives the Catalan numbers:
$$f^{(n,n)} = C_n = \frac{(2n)!}{(n+1)!n!} = \frac{1}{n+1} \binom{2n}{n}. $$
If $m_k>m_{k-1}>\dots>m_1$ are hook-lengths in the first column of Young diagram $\lambda$, i.e. lengths of rows are $0<m_1\leqslant m_2-1 \leqslant m_3-2\leqslant \dots \leqslant m_k-(k-1)$, then equivalent form is
$$
f^{\lambda}=\frac{n!}{\prod m_i!}\prod_{1\leqslant i<j\leqslant k} (m_j-m_i).
$$
This formula for $f^{\lambda}$ was established by G. Frobenius (Uber die charaktere der symmetrischer gruppe, Preuss. &ad. Wk. sitz. (1900), 516–534.) and A. Young (Quantitative substitutional analysis II, Proc. London Math. Sot., Ser. 1, 35 (1902), 361–397). Equivalence follows from the observation that product of hook lengths in $j$-th row equals $m_j!/\prod_{i<j} (m_j-m_i)$.
The Hook Formula was first proved by Frame, Robinson and Thrall. It is important as a unifying result in enumerative combinatorics. It also gave another early indication (after Nakayama's Conjecture) of the importance of hooks, $p$-cores and $p$-quotients to the representation theory of the symmetric group.

A: Let $P_n$ be a permutation of $1,2,\ldots,n$. Also, suppose $is(P_n)$ and $ds(P_n)$ shows the longest increasing and decreasing subsequences of permutation $P_n$, respectively.
So, the Erdős–Szekeres inequality is:
$$is(P_n)\times ds(P_n)\geq n\,.$$
Also, the above inequality is best possible and if $n=pq$, we have the equality.
Now, suppose $A_n(p,q)$ shows the total number of permutation of $1,2,\ldots,n$ for which we have $is(P_n)=p$ and $ds(P_n)=q$. then $A_n(p,q)$ is equal to the sum of all $(f^{\lambda})^2$, where $\lambda$ is a partition of $n$ satisfying $\ell(\lambda)=p$ and $\lambda_1=q$. Also, $f^{\lambda}$ can be evaluate by hook-length formula, which is appeared in one of the above answers.
For special case, if $n=pq$, we have:
$$A_n(p,q)=\left[\frac{(pq)!}{1^12^2\cdots p^p(p+1)^p\cdots q^p(q+1)^{p-1}(q+2)^{p-2}\cdots (p+q-1)^1}\right]^2.$$
A: The Sauer–Shelah-Vapnik-Chervonenkis lemma:
$$ |C| \le \sum_{i=0}^d \binom{n}{i} ,$$
where $C\subseteq\{0,1\}^n$ and its VC-dimension $d$ is the size of the largest shattered index set $I\subset[n]$, where we say that $I$ is shattered if the restriction of $C$ to $I$ is all of $\{0,1\}^{|I|}$. The bound is tight, achievable, e.g., by $C$ corresponding to all subsets of $n$ of size at most $d$.
For $n\ge d$, we have the simple estimate $\sum_{i=0}^d \binom{n}{i}\le(en/d)^d$. The actual lemma has several proofs, the shortest being via Pajor's lemma, which states that $C$ shatters at least $|C|$ distinct index sets $I$.
A: The Kruskal-Katona inequality:   $$|\partial{\cal F}| \ge {{m_k} \choose {k-1}} + {{m_{k-1}} \choose {k-2}}+ \cdots + {{m_j} \choose {j-1}}.$$ 
Here ${\cal F}$ is a family of $k$-sets, and $\partial {\cal F}$ is its shadow, namely all sets of size $(k-1)$ contained by some set in $\cal F$. And, 
$$|{\cal F}|=m= {{m_k} \choose {k}} + {{m_{k-1}} \choose {k-1}}+ \cdots + {{m_j} \choose {j}},$$ where $m_k > m_{k-1} >\cdots > m_j \ge j >0$.
A: $$1-H(\delta) \le R(\delta) \le H\left(\frac{1}{2} -\sqrt {\delta (1-\delta)}\right).$$
This formula describes the state-of-the-art lower and upper bound for the rate of binary codes of length $n$, as $n$ tends to infinity, and minimal distance $\delta n$, $0 < \delta < 1/2$. The lower bound is due to Gilbert (1952). No better lower bound is known today.  The upper bound is by McEliece, Rudemich, Rumsey, and Welsh (MRRW) (1977), who described also an improved upper bound for $\delta \ge 0.273$.
No better upper bounds than those discovered by MRRW are known today.
Van Lint's book on the theory of error-correcting codes is a good source. (*)
A: The BEST Theorem (https://en.wikipedia.org/wiki/BEST_theorem) for the number of Eulerian circuits of an Eulerian directed graph $G$:
$$ec(G) = t_w(G) \cdot \prod_{v\in V} (\mathrm{deg}(v)-1)!,$$
where $t_w(G)$ is the number of arborescences rooted at any fixed vertex $w\in G$. The number $t_w(G)$ can be computed as a determinant thanks to (a directed graph version of) the matrix-tree theorem, already mentioned in another answer.
This is a remarkable formula because, like many other formulas mentioned in answers to this question, it is right "on the border" of what is computationally tractable. For instance, as mentioned in the Wikipedia article above, the problem of counting Eulerian circuits in an undirected graph is by contrast #P-complete. 
(Another very similar "on the border" result in enumeration in graph theory is the Kasteleyn method for computing perfect matchings of a planar graph, compared to the difficulty of computing perfect matchings of an arbitrary graph, which should be an answer if it is not already.)
A: A parking function is a sequence $(a_1, \ldots, a_n)$ of positive integers such that, if $b_1 \le b_2 \le \cdots \le b_n$ is the increasing rearrangement of the sequence $(a_1, \ldots, a_n)$, then $b_i\le i$.

Theorem. The number of parking functions of length $n$ is $(n+1)^{n-1}$.

Parking functions are related to a host of seemingly unrelated combinatorial objects, such as labeled trees (there is a close connection with Cayley's formula $n^{n-2}$ for the number of labeled trees on $n$ vertices), noncrossing partitions, and hyperplane arrangements (the Shi arrangement in particular).  There is even a connection with the $n!$ conjecture mentioned in another answer, in that the action of the symmetric group on the space of parking functions is isomorphic to a certain action on a space of coinvariants.  More information may be found in many places, e.g., these slides by Richard Stanley.
A: The Lindström–Gessel–Viennot lemma is a very powerful tool for proving all kinds of combinatorial product formulas. It says that if $G$ is a directed acyclic edge-weighted graph, and $M$ is the square matrix whose rows are indexed by some set of vertices $\{u_1,\ldots,u_n\}$ of $G$ and whose columns are indexed by some other set of vertices $\{v_1,\ldots,v_n\}$, and whose $(u,v)$th entry is the sum $\sum_{p\colon u \to v}\omega(p)$ of the weights of all paths connecting $u$ to $v$, then the determinant of $M$ is
$$ \mathrm{det}(M)= \sum_{(P_1,\ldots,P_n)\colon A\to B} \mathrm{sign}(\sigma(P))\prod_{i=1}^{n}\omega(P_i),$$
where the sum is over all non-intersecting (i.e., vertex-disjoint) tuples $(P_1,\ldots,P_n)$ of paths, where $P_i$ is a path from $u_i$ to $v_{\sigma(i)}$ for the corresponding permutation $\sigma(P)$. The lemma is especially useful when one can argue (e.g., because of planarity) that the only such non-intersecting tuples must connect $u_i$ and $v_i$ (i.e., $\sigma(P)$ must be the identity).
This lemma can be used to prove that the combinatorial and determinantal definitions of the Schur functions agree. It can also be used to give a very nice proof of MacMahon's product formula for the number of plane partitions in a box (an earlier answer to this question).
A: The Durfee square formula for the generating function of the number of partitions doesn't seem to be in this list yet, I find it rather appealing:
$$
\frac{1}{(q)_\infty} = \sum_{n\geq 0} \frac{q^{n^2}}{(q)_n (q)_n} \ .
$$
A: The exponential formula Can be phrased as 
All = exp(Connected)
In a more precise way, if you have a class $\mathcal{C}$ of labelled graphs which is locally finite i.e. for every finite set $F$ and $k\in \mathbb{N}$
$$
S(n,k)=\operatorname{card}(\mathcal{C}(F,k))<+\infty
$$
where $\mathcal{C}(F,k)$ stands for the subclass of graphs with $F$ as labels and $k$ connected components ($S(n,k)$ is supposed to depend only on $n=\operatorname{card}(F)$). If, moreover, the class $\mathcal{C}$ is closed by  


*

* relabeling 

* connected components (i.e. $\Gamma\in \mathcal{C}$ iff all connected components 
of $\Gamma$ are in $\mathcal{C}$)   

* disjoint union
   

then 
$$
\sum_{n,k\geq 0}S(n,k)\frac{x^n}{n!}y^k=e^{y(\sum_{n\geq 1}S(n,1)\frac{x^n}{n!})}\qquad (1).
$$
This formula has many applications and variants in combinatorics as the computation of the GF of the Bell, Stirling numbers, number of cycles, graphs of endofunctions (with or without constraints), set partitions and the analog for unlabelled graphs to cite only a few. 
All the matrices $S(n,k)$ possess the Sheffer property i.e. the EGF of the k-th column is (up to a scalar) the k-th power of the EGF of the first (for $k=1$). It is equivalent to formula (1).
Matrices having the Sheffer property (not only provided by classes of labelled graphs) form an infinite dimensional Lie group generated by vector fields on the line (see Tom Copeland's answer). Connections of this group can be seen in combinatorial physics, statistics on graphs and over categories.  
A usual, useful and (almost) immediate generalisation. In fact, we have
$$
S(n,k)=\operatorname{card}(\mathcal{C}(F,k))=\sum_{\gamma\in \mathcal{C}(F,k)} \mathbf{1}(\gamma)
$$ 
where $\mathbf{1}$ is the constant (equal to $1$) function on the class $\mathcal{C}$, and one can, for free (i.e. with the same proof), replace $\mathbf{1}$ by any $\mathbb{Q}$-algebra valued multiplicative statistics, "$c$" i.e. such that 
$$
c(\gamma_1\sqcup \gamma_2)=c(\gamma_1)c(\gamma_2);\ c(\mathcal{C}_\emptyset)=1 
$$
(where $\mathcal{C}_\emptyset$ is the empty graph and $\sqcup$ stands for the disjoint union).  
Then, with
$$
S_c(n,k)=\sum_{\gamma\in \mathcal{C}(F,k)} c(\gamma)
$$ 
(again, the sum is supposed to depend only on $n=\operatorname{card}(F)$), one still has
$$
\sum_{n,k\geq 0}S_c(n,k)\frac{x^n}{n!}y^k=e^{y(\sum_{n\geq 1}S_c(n,1)\frac{x^n}{n!})}
$$
Examples with the polynomial algebra $\mathbb{Q}[X]$ ($X$ is the alphabet $X=\{x_i\}_{i\geq 1}$) 


*

* With the class of permutation graphs and the $\mathbb{Q}[X]$-valued multiplicative statistics
$$
c(\pi)=\prod_{k\geq 1} x_k^{c_k(\pi)}
$$ 
(where $c_k(\pi)$ is the number of $k$-cycles in $\pi$), one gets the cycle index formulas for the symmetric groups
$$
\sum_{n=0}^\infty \frac{z^n}{n!} \sum_{\pi \in \mathfrak{S}_n} c(\pi) = \prod_{j=1}^\infty \exp \bigl( \frac{x_j}{j}z^j \bigr)\ .
$$
(see Mark's comment below). 


* With the class of set partitions (the set of graphs of equivalences on $[1,n]$ will be denoted $\mathcal{P}_n$ and $\mathcal{P}=\cup_{n\geq 0}\mathcal{P}_n$) and the $\mathbb{Q}[X]$-valued multiplicative statistics
$$
c(\pi)=\prod_{k\geq 1} x_k^{c_k(\pi)}
$$ 
(where $c_k(\pi)$ is the number of $k$-blocks in $\pi\in \mathcal{P}$), one gets  similar formulas  
$$
\sum_{n=0}^\infty \frac{z^n}{n!} \sum_{\pi \in \mathcal{P}_n} c(\pi) = \prod_{j=1}^\infty \exp \bigl( \frac{x_j}{j!}z^j \bigr)
$$


One can also use the same machinery for classes of graphs of endofunctions with constraints (as idempotent endofunctions and the like). 

The analytic part of the exponential formula can be viewed as a particular case of Faà di Bruno's formula which itself can be traced back to the work of Arbogast (Louis-François-Antoine) and Newton–Girard's formulas. It is equivalent to Witt's formulas. Modern achievements are Riddell's formulas for labelled and unlabelled graphs.

A: MacMahon's formula for the number $M(a,b,c)$ of plane partitions that fit in an $a \times b \times c$ box:
$$ M(a,b,c) = \prod_{i=1}^{a} \prod_{j=1}^{b} \prod_{k=1}^{c} \frac{i+j+k-1}{i+j+k-2} $$
For more details, see https://en.wikipedia.org/wiki/Plane_partition#MacMahon_formula. This is arguably one of the most unexpected and beautiful formulas in algebraic combinatorics. Note that nothing like this holds when we move from plane partitions to "solid partitions" or beyond.
A: $$\frac {\alpha (G)}{n}\le \frac {\lambda_{\min}}{d-\lambda_{\min}}$$
This is Hoffman's bound for the independence number $\alpha (F)$ (namely, the largest number of vertices in an independent set of vertices in $G$), of a $d$-regular graph with $n$ vertices. Here $\lambda_{\min}$ is the smallest eigenvalue of the adjacency matrix of $G$.  For more details see e.g. this paper.
A: $$(1-\lambda_2)/2\le h(G)\le \sqrt{2(1-\lambda_2)},$$  is the "discrete Cheeger-Buser inequality", relating the spectral gap of the discrete Laplace operator to the discrete Cheeger constant of a graph.  In particular, it gives the spectral characterization of expander families.
Here $h(G)$ is the expansion of a graph $G$, and $\lambda_2$ refers to the second smallest eigenvalue of the Laplacian of $G$. The  inequality on the right is due to Alon-Millman and Tanner. The inequality on the left is by Alon.
A: Let $r_k(n)$ denotes the size of the largest cardinality of a subset $A$  of $\{1,2,\dots,n\}$, such that $A$ does not contain a k-term arithmetic progression. The following formula describes the state of knowledge for $k=3$. For references and related bounds see this Wikipedia article.
$$ 2^{-8\sqrt {\log n}} \le  \frac {r_3(n)}{n} \le C \frac { (\log \log n)^4}{\log n}$$
Update: the upper bound were improved. The current identity (Kelley and Meka, 2023) is:
$$ 2^{-8\sqrt {\log n}} \le  \frac {r_3(n)}{n} \le 2^{-(\log n)^\beta},$$ for some
$\beta>0$.
A: $$
\left[\prod_{i=1}^n x_i^{d_i}\right]f(x_1,\dots,x_n)=\sum_{a_i\in A_i}\frac{f(a_1,\dots,a_n)}{\prod_{i=1}^n\prod_{b\in A_i\setminus a_i} (a_i-b)},\, \deg f\leq \sum d_i,\, |A_i|=d_i+1.
$$
This is the formula which proves Alon's Combinatorial Nullstellensatz. Here $f(x_1,\dots,x_n)$ is polynomial of degree at most $d_1+\dots+d_n$, $A_i$ are subsets of the ground field of given sizes $|A_i|=d_i+1$. CN claims that when $f$ does vanish on $\prod A_i$, coefficient of $\prod x_i^{d_i}$ of $f$ also vanishes. This claim follows from the formula immediately.
As well as MacMahon Master Theorem it allows to get a quick proof of Dixon's identity
$$
[x^{2n}y^{2n}z^{2n}](x-y)^{2n}(y-z)^{2n}(z-x)^{2n}=(-1)^n\frac{(3n)!}{n! n! n!}.
$$
Just apply the formula to the polynomial
$$
f(x,y,z)=\prod_{i=-(n-1)}^n (x-y-i)(y-z-i)(x-z-i)
$$
(the trick is that it has the same coefficient of $x^{2n}y^{2n}z^{2n}$ as $(x-y)^{2n}(y-z)^{2n}(z-x)^{2n}$) and sets $A_1=A_2=A_3=\{0,1,\dots,2n\}$. The only non-zero summand in RHS corresponds to $x=0, y=n, z=2n$ and may be easily calculated.
As for the history, this formula was rediscovered several times. It appeared in recent papers by Schauz (Algebraically solvable problems:
describing polynomials as equivalent to explicit solutions, Electron. J. Combin. 15 (2008)), Lason (A generalization of Combinatorial Nullstellensatz, Electron. J. Combin. 17 (2010)) and Karasev and Petrov (Partitions of nonzero elements of a finite field into pairs, Israel J. Math. 192 (2012)). But as I've learnt from Vladislav Volkov it actually goes back even to C. G. Jacobi (Theoremata nova algebraica circa systema duarum aequationem inter duas variabiles propositarum, J. Reine Angew. Math. 14 (1835), 281-288), it is a special case of Euler-Jacobi formula for complete intersections (grid $\prod A_i$ is a typical complete intersection). My hope is that other cases of Euler-Jacobi (and beyond) formula also may have applications in combinatorics.
A: Let $f_i$ be the components of the $f$-vector of a simplicial polytope in $d$ dimensions:
$f_i=$ the number of faces of dimension $i$.
The Dehn-Sommerville equations express linear relations
among the $f_i$. The equations can be phrased in several forms.
Here is one:
$$ 
f_{k-1}
 = \sum_{i=k}^d (-1)^{d-i} \binom{i}{k} f_{i-1}\;.
$$
The usual convention is that $f_{-1}=f_d=1$.
For $k=0$ and $d=3$, the equation becomes
$$
f_{-1} = -f_{-1} + f_0 - f_1 + f_2 \;,
$$ 
i.e., $V-E+F=2$. For arbitrary $d$ and $k=0$, the equation yields
the Euler characteristic $\chi$.
For $k=1$ and $d=3$, the equation evaluates to
$$f_0 = f_0 - 2f_1 + 3 f_2 \;,$$
i.e., $2E = 3F$, because "simplicial" means the faces are triangles.
History. According to the Wikipedia article, 

"For polytopes of dimension 4 and 5, [the equations] were found by Max Dehn in 1905. Their general form was established by Duncan Sommerville in 1927."

A: Lagrange Inversion (Lagrange–Bürmann formula) plays an important role in combinatorics. There is a dedicated MO discussion of its applications (in combinatorics and beyond). In particular, see my answer there for its application for generating functions.
A: The umbral compositional identity
$$ (x)_{\frac{}{n}}= Lah_n((x)_{\frac{\bullet}{}})$$
and the inverse relation
$$ (-1)^n(x)_{\frac{n}{}}= Lah_n(-(x)_{\frac{}{\bullet}})$$
as explained combinatorially by Joni, Rota, and Sagan in From Sets to Functions: Three Elementary Examples, where $((x)_{\frac{}{\bullet}})^n=(x)_{\frac{}{n}}$ is a rising factorial polynomial; $((x)_{\frac{\bullet}{}})^n = (x)_{\frac{n}{}}$, a falling factorial polynomial; and $Lah_n(x)= n! \; \sum_{k=0}^n \binom{n-1}{k-1} \; \frac{x^k}{k!}$, a Lah polynomial. For a nice, short, geometric tutorial on the Whitney numbers Rota alludes to in the paper (coined by Rota), see Josh Cooper's math webpage.
This illustrates Rota and associates' influential program to interpret what are essentially identities of finite operator calculus / umbral calculus using combinatorial constructs--posets, lattices, cycles, Mobius inversion, etc. (Rota's combinatorial interpretation of the Dobinski formula is also an early illustration.)
A further umbral composition with the Bell / Touchard polynomials, $\phi_n(x)$, gives $GS2_{n,j}(y)$, the generalized Stirling numbers of the second kind for positive integer values of $y$ that underlie the normal ordering representation of the Lie derivative rep for the Witt-Lie algebra:
$$ [-y\;(-\phi.(x)/y)_{\frac{\bullet}{}}]^n =  \sum_{k=0}^n S1_{n,k} \; (-y)^{n-k} \sum_{j=0}^k  S2_{k,j} \; x^j = \sum_{j=0}^n \;GS2_{n,j}(y) \; x^j = RT_n(y,x)\;  \; $$
with
$$(x^{1+y}D)^n = x^{ny}\; RT(y,:xD:)$$
where $(:xD:)^n = x^nD^n$ and $D=d/dx \, .$ These coefficients can be interpreted as enumerating forests of m-ary trees as well as other combinatorial structures. $y=-1,0,1$ give generators for $SL2$.
The generalized Stirling numbers of the first kind comprise the inverse matrix for that of $GS2_{n.j}(y)$ and are readily deduced from either the matrix rep of $GS2$ or through direct umbral compositional inversion, noting that $(\phi.(x))_{\frac{n}{}}= x^n = \phi_n((x)_{\frac{\bullet}{}}) \; ,$
$$(-y\phi.(-x/y) )_{\frac{n}{}} = \sum_{k=0}^n S1_{n,k}\; (-y)^k \sum_{j=0}^k  S2_{k,j} \; (-x/y)^j = \sum_{j=0}^n \;GS1_{n,j}(y) \; x^j \; .$$
The generalized Dobinski relation immediately leads to other formulas for these numbers.
This rich interplay between umbral calculus and analytic combinatorics has a long history stretching from Scherk, Blissard, Cayley, and Sylvester to modern times with Rota and Flajolet, among many others.
A: I guess this can be counted as infinitary combinatorics. And it is a fundamental formula in choiceless set theory.
$$\aleph_0\leq^*\mathfrak p\iff\aleph_0\leq2^\frak p$$
Namely, given a set $X$ of cardinality $\frak p$, there is a surjection from $X$ onto $\Bbb N$ if and only if there is an injection from $\Bbb N$ into $\mathcal P(X)$. This is a theorem of Kuratowski, and using it we can deduce all sort of things, e.g. if there is an infinite set without a countable subset, then there is one which can be mapped onto $\Bbb N$ 
Proof. If $A$ is an infinite set without a countable subset, either $A$ can be mapped onto $\Bbb N$, else $\mathcal P(A)$ can be mapped onto $\Bbb N$; however from the formula above, $\mathcal P(A)$ has no countable subset. $\square$
One might wonder what use are sets which have no countably infinite subset. But due to their inherent [Dedekind-]finiteness, they can be used for certain "odd" combinatorial constructions and counterexamples to many theorems which appeal to the axiom of choice.
A: What about the Erhart's polynomial? Is the statement that, for a $d$-dimensional polytope in $\mathbb{R}^n$ and $t > 0$:
$$\#(tP \cap \mathbb{Z}^n) = \sum_{i=0}^d a_i t^{i},$$
for some $a_i \in \mathbb{Q}$, considered a "formula"? This yields Pick's formula for a $2$-d integer polygon:
$$A = \#\mbox{int}(P \cap \mathbb{Z}^2) + \frac{\partial(P \cap \mathbb{Z}^2)}{2} -1.$$
A: Davis-Slepian-Polya formula for the number of simple graphs on $n$ nodes
$$
\frac{1}{n!} \sum_{j_1+2j_2+\cdots+n j_n=n}\frac{n!}{\prod\limits_{k=1}^n k^{j_k} j_k!} 2^{\displaystyle \frac{1}{2}\left( \sum_{k=1}^n k j_k^2 - \sum_{\text{ $k$ odd}} j_k \right) + \sum_{k=1}^n \sum_{i=1}^{k-1} (k,i) j_k j_i}.
$$
A: Read's 1958 beautiful formula for the asymptotic number of 3-regular graphs with n vertices
$$g_3(n) \sim \frac {(3n)! e^{-2}}{(3n/2)!288^{n/2}}.$$
A: This is a little bit of a lark, but I would argue it still marks important progress in combinatorics: $T=1+T^2\implies T^7=T$. Specifically, this is the (loose) initial justification for the "Seven Trees In One" (categorically) natural bijection between the set of binary trees and the set of 7-tuples of binary trees. It's an exhibition of the still-being-explored connections between combinatorial species, types, and category theory.
A: Given a complex matrix $A=(a_{i,j})_{m \times m}$ and an $m$-tuple of non-negative integers $(k_1,\cdots,k_m)$, denote by $G(k_1,\cdots,k_m)$ the coefficient of $\prod_{i=1}^{m} x_i^{k_i}$ in the product $\prod_{i=1}^{m} (\sum_{j=1}^{m} a_{i,j}x_j)^{k_i}$.
The MacMahon Master Theorem is a useful, deep and elegant formula for the generating function of the $G(k_1,\cdots,k_m)$'s:
$$\sum_{(k_1,\dots,k_m)} G(k_1,\dots,k_m) \, t_1^{k_1}\cdots t_m^{k_m} \, = \,
\frac{1}{\det (I_m - TA)},$$
where $T=\text{Diag}(t_1,\cdots,t_m)$.
Its importance was initially in the field of enumerative combinatorics, where it was used to count permutations and other functions (it got its name due to trivializing many such counting problems). For example, taking $a_{i,j} = 1-\delta_{i,j}$, the coefficient $G(1,\cdots,1)$ counts permutations with no fixed points. The formula can also be used to establish combinatorial identities, such as Dixon's beautiful one.
Although I. J. Good proved the formula using a generalization of Lagrange Inversion (which deserves to appear in this thread also), this is not merely some special case. It has a nicer theory and several quantum generalizations.
Just to emphasize the combinatorial aspect, I will mention that there are combinatorial proofs of the MacMahon Master Theorem. One such proof uses a Möbius function on permutations to prove that $\det (I_m - TA) \left( \sum_{(k_1,\dots,k_m)} G(k_1,\dots,k_m) \, t_1^{k_1}\cdots t_m^{k_m} \right) = 1$. It is due to Foata and can be found as a guided exercise in "The Art Of Computer Programming, Vol III", subsection 5.1.2, exercise 20.
Another (somewhat similar) proof uses a sign-reversing involution (a very powerful idea) and a graph-theoretic interpretation. A reference is section 4.19 of these notes.
A: Not sure if this fits, but I find a proof of the Jacobi triple product formula in the form
$$
\prod_{n>0}(1+q^{n-\frac{1}{2}}z)(1+q^{n-\frac{1}{2}}z^{-1})=\left(\sum_{l\in\mathbb{Z}}q^{l^2/2}z^l\right)\prod_{n>0}(1-q^n)^{-1}
$$ 
based on the idea of Dirac sea extremely significant and thought-provoking. Wikipedia cites (13.3) of Peter J. Cameron's Combinatorics: Topics, Techniques, Algorithms where it is attributed to Borcherds.
A: $$\frac {1!~4!~7! \dots (3n-2)!}{n! (n+1)! (n+2)!\dots (2n-1)!}$$
This remarkable formula counts the number of alternating sign matrices of order $n$ as well as, monotone triangles, descending plane partitions whose parts do not exceed $n$, and various other important combinatorial entities. (See also this item on the online encyclopedia of integer sequences.) 
Alternating sign matrices are $n$ by $n$ matrices with entries $+1$ $-1$ and $0$  such that each row and column the non-zero entries alternate in signs, and first non zero entry is $+1$.  They were defined by  Mills, Robbins, and Rumsey who conjectured their number. The formula was first proved by Zeilberger. 
A: 
The asymptotic formula of the average number of comparisons used by the Quick Sort algorithm.


\begin{align*}
Q_n=2n(\ln n + \gamma -2)+2\ln n+2\gamma+1+O\left(\frac{1}{n}\right)\tag{1}
\end{align*}

Volume 3 of Knuth's classic The Art of Computer Programming is titled Sorting and Searching. It presents a wealth of applications of these two fundamental combinatorial themes and one gem is C.A.R. Hoare's Quicksort algorithm.
Quicksort is the standard sorting procedure in UNIX systems and has been cited as we can read in  this paper by J.A. Fill as one  of the ten   algorithms     with the  greatest   influence on     the   development  and practice of science and engineering  in the  $20$th century.
Here's the algorithm according to Quick sort - Average complexity by J. Cichon.

Algorithm: Let $Q_n$ denote the average number of comparisons over all permutations of $n$ pairwise different elements stored in an array of size $n$. Since $Q_0$ and $Q_1$ are clearly zero, we may assume that $n\geq 2$. We select a pivot element and need $n-1$ comparisons with all remaining elements. The pivot divides the array in a left and a right part. The left part can be of any size $k$ from $0$ to $n-1$, with $k$ equiprobable. This leads to the recursion formula (Quick Sort Equation):


\begin{align*}
Q_n=(n-1)+\frac{1}{n}\sum_{k=0}^{n-1}\left(Q_k+Q_{n-1-k}\right)
\end{align*}
or due to symmetry
\begin{align*}
Q_n=(n-1)+\frac{2}{n}\sum_{k=0}^{n-1}Q_k
\end{align*}


A nice closed formula of $Q_n$ in terms of Harmonic numbers is stated in J. Cichon's paper:


\begin{align*}
Q_n=(n+1)(4H_{n+1}-2H_n-4)
\end{align*}

The Quick Sort algorithm and the asymptotic formula (1) showing order $n\ln n$  can be seen as one  representative of a whole class of related sorting algorithms all of them with great importance and wide applicability.
A: Two more:
The number of positive integer solutions to $x^2 + y\leq z$ (here $x, y$ are the unknown), with $y$ prime, is
$$r(z) \sim \frac{2}{3}\frac{z^{3/2}}{\log z}.$$
See here.
Let $S$ be an infinite subset of positive integers, and the number of elements of $S$ less or equal to $x$ is asymptotically equal to $a x^b / (\log x)^c$ with $0<a, 0<b<1, c\geq 0$. Then the number of positive integer solutions to $x + y\leq z$ (here $x, y\in S$ are the unknown) is asymptotically equal to
$$r(z)\sim\frac{a^2b z^{2b}}{(\log z)^{2c}}\cdot \frac{\Gamma(b)\Gamma(b+1)}{\Gamma(2b+1)}.$$
This covers both sums of two primes and sums of two squares. See here.
A: Matrix-tree Theorem. Let $G$ be a graph of size $n$, let $\lambda_{1}\geq \ldots\geq \lambda_{n}$ be the eigenvalues of its Laplacian matrix. Then the number of spanning trees of $G$ is
$$ \frac{\lambda_{1}\lambda_{2}\ldots \lambda_{n-1}}{n} $$
A: For a permutation $\sigma \in S_n$, let $\ell(\sigma)$ denote the maximal length of an increasing subsequence in $\sigma$. Define
$$
\ell_n = \frac{1}{n!} \sum_{\sigma \in S_n} \ell(\sigma),
$$
the average value of $\ell(\sigma)$ for a $\sigma$ chosen uniformly at random from $S_n$.
The problem of finding the asymptotic value of $\ell_n$ for large $n$ was proposed by the famous mathematical and nuclear physicist Stanislaw Ulam in 1961. It was further popularized by John Hammersley in 1970, and solved (to a first order of approximation) in 1977 by Anatoly Vershik and Sergei Kerov, and independently by Ben Logan and Larry Shepp, who proved the following remarkable formula, which is the subject of my answer:

The Vershik-Kerov-Logan-Shepp formula:
$$
\ell_n \sim 2 \sqrt{n}.
$$

To give a proper explanation of why this formula is considered by many to be extremely "important research level mathematics" would require a book-length exposition. Fortunately, someone has written a book about this precise subject. As a brief summary, I will mention that:


*

*The proof of this result is extremely clever and nontrivial (e.g., it takes up pages 5-68 in the book I mentioned above).

*The proof requires a combination of combinatorial techniques, in particular a use of the hook length formula (another Important Formula in Combinatorics, in fact it's currently the most highly voted answer to this Math Overflow question), and difficult analytic techniques (complex analysis, Hilbert transforms, the calculus of variations). A nice illustration of the principle that "no (area of) math is an island."

*The result and its proof by Vershik-Kerov-Logan-Shepp are just the beginning of a long story involving the discovery of a much deeper structure underlying such asymptotic behavior. Research on this and closely related subjects has been flourishing for the last twenty years and providing occupation for a large number of researchers, graduate students, postdocs, etc. It was also implicated in various awards and honors to several well-known mathematicians (e.g., Andrei Okounkov's Fields Medal).
A: The Tutte golden identity
$$ {\chi}_T({\phi}+2)=({\phi}+2)\; {\phi}^{3\,V(T)-10}\,
  ({\chi}_T({\phi}+1))^2 $$
relates the value of the chromatic polynomial $\chi$ of any planar triangulation $T$ at $\phi +2$ and the square of the value at $\phi +1$, where $\phi =\frac{1+\sqrt 5}{2}$ is the golden ratio. $V(T)$ in the formula denotes the number of the vertices of the triangulation.
Tutte used this identity to give an elegant proof that ${\chi}_T({\phi}+2)>0$, a fact interesting in connection to the $4$-color theorem. [Reference: W.T. Tutte,
More about chromatic polynomials and the golden ratio. 1970 Combinatorial Structures and their Applications (Proc. Calgary Internat. Conf., Calgary, Alta., 1969) pp. 439–453 Gordon and Breach, New York]
   Recently this identity has been shown to fit in the framework of quantum topology.
A: The Rogers-Ramanujan identities are partition identities, i.e., statements that equate the number of integer partitions of an integer $n$ belonging to two different partition classes. There are two identities, so I recognize that posting both in one answer bends Rule (1) of Gil's question slightly, but I'm guessing any reasonable person will agree that the two identities belong and deserve to be stated together as one conceptual result.
Of course, I also recognize that according to the rules of the question the identities need to be stated as formulas. There are two ways to do this, which are equivalent to each other but quite different in presentation. First, the "pure" combinatorial statement of the identities is
$$
A(n) = B(n), \ \ \ \ \ \ C(n)=D(n), \qquad (n=1,2,3,\ldots),
$$
where: 
$A(n)$ denotes the number of partitions of an integer $n$ not containing two consecutive parts (also known as minimal difference 2 partitions); 
$B(n)$ denotes the number of partitions of $n$ into parts all of which are congruent to $1$ or $4$ mod $5$; 
$C(n)$ denotes the number of partitions of $n$ not containing two consecutive parts and not having any parts equal to $1$; 
and $D(n)$ denotes the number of partitions of $n$ into parts all of which are congruent to $2$ or $3$ mod $5$.
The problem with the above "formulas" is that most of the logical statement of the result is pushed down to the above verbal definitions that are much longer than the formulas themselves. This may cast some legitimate doubt about whether the R-R identities deserve to be considered proper formulas. Fortunately there is a second algebraic formulation using generating functions that encodes the entire statement of the identities into two self-contained equations, namely
$$
\prod_{m=0}^\infty \frac{1}{(1-x^{5m+1})(1-x^{5m+4})} = \sum_{n=0}^\infty \frac{x^{n^2}}{(1-x)(1-x^2)\ldots(1-x^n)},
$$
$$
\prod_{m=0}^\infty \frac{1}{(1-x^{5m+2})(1-x^{5m+3})} = \sum_{n=0}^\infty \frac{x^{n(n+1)}}{(1-x)(1-x^2)\ldots(1-x^n)},
$$
where: ... well, where nothing! In this formulation, no extra verbiage needs to be added.
The fact that the above equations really encode the same statement as the combinatorial statement above is not difficult to see.
The R-R identities were proved by the British mathematician Leonard James Rogers in 1894, in the algebraic form above. I believe Rogers did not recognize the elegant combinatorial content, which is probably why his paper was largely ignored. They were then rediscovered by Ramanujan around 1913. See here for a bit more of their fascinating history.
The importance of the R-R identities is that they are extremely simple to state, and yet highly surprising as well as nontrivial to prove. They have had a large influence on research on partition bijections, bijective proofs, and algorithms in combinatorics, and are still inspiring new proofs and other new research (see for example the papers "Partition Bijections, a Survey" by Igor Pak and "A combinatorial proof of the Rogers-Ramanujan identities" by Pak and Boulet). 
Perhaps most amazingly in my opinion, the R-R identities played a key role in Baxter's 1980 solution of the hard hexagon model in statistical physics.
A: Let $c_n$ be the number of self-avoiding walks of length $n$ on the
hexagonal lattice starting at a fixed vertex. An amazing result of
H. Duminil-Copin and S. Smirnov in 2010 asserts that
$$ \mu_c \mathrel{\mathop:}= \lim_{n\to\infty}c_n^{1/n} =
    \sqrt{2+\sqrt{2}}. $$
This is the a big breakthrough in the subject of self-avoiding walks.
Moreover, Flory (1948) and Nienhuis (1982) conjectured that for some
constant $A$ we have $c_n\sim An^{11/32}\mu_c^n$. See
https://arxiv.org/pdf/1007.0575.pdf.
A: $$(2/e) (1+o(1)) k2^{k/2} \le  R(k+1, k+1) \leq k^{- C {\log k}} \textstyle \binom{2k}{k}.$$
Best lower and upper bound for diagonal Ramsey numbers. The Ramsey number $R(k,\ell)$ is the smallest integer $n$ such that any two-coloring of the edges of the complete graph on $n$ vertices $K_n$ by red and blue, there either is a red $K_k$ (namely, a complete graph on $k$ vertices all of whose edges are colored red), or a blue $K_{\ell}$. The lower bound is an improvement, by a constant factor, using the Lovasz local lemma, of Erdos' original 1947 lower bound. The upper bound is an (update:) a 2020 improvement by Sah of an (end update) improvement by a quasipolynomial (in $k$) factor of Erdos's bound by Conlon from 2006. (See this paper.)
A: The Ramanujan-Hardy asymptotic formula for the number of partitions $p(n)$ of $n$ is the following: 
$$p(n) \sim \frac{1}{4n\sqrt{3}}\exp\left(\sqrt{\frac{2n}{3}}\right), \quad n \to \infty$$
The proof of this formula led Ramanujan and Hardy to discover the circle method. 
The circle method and related techniques have led to founding the subject of "analytic combinatorics". See the text by Flajolet and Sedgewick.
A: $N({\cal A})= \sum _{x\in L({\cal A})}(-1)^{r(x)}\mu (0,x)$
This is Zaslavsky's formula for the number of regions in an arrangement of hyperplanes.
The details: Given an arrangement of hyperplanes $\cal A$ in ${\mathbb R}^d$, $N({\cal A})$ is the number of regions of the arrangement, namely, connected components in the complement of the union of all hyperplanes. A remarkable fact is that this number depends only on the combinatorics of the lattice of flats determined by $\cal A$, namely the set of all intersection of hyperplanes in the family ordered by inclusion. The formula gives a simple description on the number of regions in terms of the Möbius function of such flats.
The importance: This is an extremely useful formula and a starting point to much research and important questions. For example, if you replace hyperplanes by subspaces of various dimensions then there is a formula by Goresky and Macpherson giving the Betti numbers of the complement in terms of the lattice of flats.
A: Faà di Bruno's formula generalizes the chain rule to higher order derivatives. Most compact form of Faà di Bruno's formula involves Bell polynomials $B_{n,k}\left(x_1,x_2,\dots,x_{n-k+1}\right)$ and illustrates its combinatorial nature:
$${d^n \over dx^n} f(g(x)) = \sum_{k=1}^n f^{(k)}(g(x))\cdot B_{n,k}\left(g'(x),g''(x),\dots,g^{(n-k+1)}(x)\right).$$
A: Van der Waerden's conjecture (1926): If $A$ is a doubly stochastic matrix of size $n\times n$ then
$$\text{per}(A)\ge\frac{n!}{n^n}$$
Moreover, equality holds if and only if all the entries of $A$ are $\frac{1}{n}$.
The conjecture was proved by B. Gyires (1980), G. P. Egorychev (1981) and D. I. Falikman (1981). Egorychev and Falikman won the Fulkerson Prize for this.

A square matrix with non-negative entries is said to be doubly stochastic if every row and every column sums up to $1$.
The permanent of a matrix $A=(a_{ij})$ of size $n\times n$ is
$$\text{per}(A)\overset{\text{def}}{=}\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i\sigma(i)}$$
i.e. the only difference from the definition of determinant is that we don't multiply by the sign of $\sigma$. Turns out that while computing the determinant is easy (e.g. using Gaussian elimination), computing the permanent is  P#-complete even for 0/1 matrices, which makes it an important problem in complexity theory.
You are welcome to edit the post and elaborate on the importance of the theorem.
A: The Kneser graph $KG_{n,k}$ is the graph on $k$-subsets of $\{1, \dots, n\}$ with two subsets made adjacent when they are disjoint. The formula $$\chi(KG_{n,k}) = n - 2k + 2$$ was proved by Lovász in 1978 using topological methods, which gave birth to  the area of topological combinatorics.
Some references:

*

*Kneser's conjecture, chromatic number, and homotopy.

*Using the Borsuk-Ulam theorem.

A: For a Young diagram $\lambda$ of size $n$, let $f^\lambda$ denote the number of standard Young tableaux of shape $\lambda$ (discussed above in Mark Wildon's answer about the hook length formula). Then
$$
\sum_{\lambda} (f^\lambda)^2 = n!,  \qquad\qquad\qquad\qquad\qquad (*)
$$
where the sum ranges over all Young diagrams of size $n$.
This formula is interesting because it is a meeting point of ideas from combinatorics and representation theory. One proof is bijective and involves a way of mapping permutations (enumerated by the right-hand side) to pairs of standard Young tableaux of similar shape (enumerated by the left-hand side). This mapping is known as the Robinson-Schensted correspondence (a special case of the RSK correspondence).
A second proof of $(*)$ interprets the formula as a special case of the representation theoretic fact that the sum of the squared dimensions of the irreducible representations of a finite group is equal to the order of the group. One then relies on the classification of irreducible representations of the symmetric group $S_n$ (they are in one-to-one correspondence with Young diagrams $\lambda$ of size $n$) and the fact that the dimension of the irreducible representation associated with a Young diagram $\lambda$ is equal to $f^\lambda$ (both of these facts being themselves quite nontrivial, and extremely interesting in their own right). To me it's fascinating that $(*)$ has two such completely different proofs, both using combinatorial ideas but the second one being much more algebraic and intricate. (There is also another purely combinatorial proof due to Greene, Nijenhuis and Wilf based on the very pretty idea of the "hook walk," which is a kind of planar random walk.)
A third reason why $(*)$ is interesting is that it has proved extremely important in the study of longest increasing subsequences of permutations (described in my answer about the Vershik-Kerov-Logan-Shepp formula and in Richard Stanley's answer elaborating on the same subject). In this context it is often rewritten as
$$
\sum_{\lambda} \frac{(f^\lambda)^2}{n!} = 1,  \qquad\qquad\qquad\qquad\qquad (**)
$$
and interpreted as the statement that the function assigning weight $(f^\lambda)^2/n!$ to a Young diagram $\lambda$ is a probability measure, known as Plancherel measure. This measure is extremely natural and interesting and its analysis is the subject of a huge literature.
As a final comment, I find it regretful that $(*)$ doesn't have a standard name in the literature. Any suggestions for what would be a good name to refer to it by?
A: Series multisection is a folklore formula (Riordan called it an "ancient vintage" in his 1968 book "Combinatorial identities"), which from a given analytical generating function for some numerical sequence allows one to obtain a generating function for a subsequence with indices forming an arithmetic progression. In particular, it leads to a closed-form expression for sums of binomial coefficients taken with a certain step $c$:
$${q\choose d} + {q\choose d+c} + {q\choose d+2c} + \cdots = \frac{1}{c}\cdot \sum_{k=0}^{c-1} \left(2 \cos\frac{\pi k}{c}\right )^q\cdot \cos \frac{\pi(q-2d)k}{c}.$$
A: No doubts, the inclusion-exclusion principle generates most common type of formulae used in enumerative combinatorics. Examples include explicit formulae for derangements, Striling numbers, rook polynomials, Euler's totient function, and so on.
$$\left\lvert \bigcup_{i=1}^n A_i \right\rvert = \sum_{\emptyset\ne J\subseteq\{1,2,\dots,n\}} (-1)^{|J|-1} \left\lvert \bigcap_{j\in J}^n A_j \right\rvert$$
A: Expanding on Dan Romik's posting of the Vershik-Kerov-Logan-Shepp
formula for the expected length of the longest increasing subsequence
of a random permutation $\sigma\in S_n$, there is the fantastic
formula of Baik, Deift, and Johansson for the limiting distribution
of the length $\mathrm{is}(\sigma)$ of the longest increasing
subsequence of $\sigma\in S_n$: for random (uniform) $\sigma\in S_n$
and all $t\in\mathbb{R}$ we have
  $$ \lim_{n\rightarrow\infty} \mathrm{Prob}                                          
  \left(\frac{\mathrm{is}(\sigma)-2\sqrt{n}}{n^{1/6}}\leq t\right)                    
    = F(t), $$
where $F(t)$ is the Tracy-Widom distribution. The Tracy-Widom
distribution $F(t)$ is defined as follows. Let $u(x)$ be the solution
to
   $$ \frac{d^2}{dx^2}u(x) = 2u(x)^3+xu(x) $$
with certain initial conditions. Then
 $$ F(t) = \exp\left( -\int_t^\infty (x-t)u(x)^2\,dx                                  
   \right). $$
A: Abel's identity (also referred to as Abel's generalization of the binomial formula)
$$x^{-1}(x+y+n)^n=\sum_{k=0}^n{{n} \choose {k}}(x+k)^{k-1}(y+n-k)^{n-k}.$$ 
A: Minkowski theorem (of 1896, by Hermann Minkowski) is sometimes mentioned as the cornerstone of the geometry of numbers. It gives an upper bound on the volume of closed convex sets (symmetric w.r.t. origin) in $\mathbb{R}^n$ that do not contain an arrow with integer coordinates.
Considering the real vector space $\mathbb{R}^n$, a lattice of points $L$ in $\mathbb{R}^n$ of determinant $\mathrm{det}(L)$, and a closed convex set $S$ in $\mathbb{R}^n$ that is symmetric w.r.t. the zero vector and of volume at least $2^n \mathrm{det}(L)$, the theorem states that $S$ contains at least one nonzero point of $L$.
A: In a wider context, there is a well-known list of 17 formulas (selected by Ian Stewart) that changed the course of history, see https://www.businessinsider.com/17-equations-that-changed-the-world-2014-3?IR=T
