Important formulas in combinatorics

Question

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;

Answer 1

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

Answer 2

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.

Answer 3

$$\zeta(3)={5\over2}\sum_{n=1}^{\infty}{(-1)^{n-1}\over n^3{2n\choose n}}$$ 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.

Answer 4

$$\Theta (C_5)=\sqrt 5.$$

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 $\Theta (G) \le \theta (G)$.

Answer 5

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.

Answer 6

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.

Answer 7

$$\max (|A+A|,|A\times A|) \ge \frac12 |A|^{4/3} (\log |A|)^{-1/3}.$$

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.

Answer 8

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.

Answer 9

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.

Answer 10

(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.

Answer 11

$$\def\multichoose#1#2{{\left(\kern-.3em\left(\genfrac{}{}{0pt}{}{#1}{#2}\right)\kern-.3em\right)}} \binom{n}{k} = (-1)^{k} \multichoose{-n}{k}$$

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.

Answer 12

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.

Answer 13

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 $(1+qx),\ x\in X$).

Answer 14

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.

Answer 15

$$1-H(\delta) \le R(\delta) \le H\left(\frac{1}{2} -\sqrt {\delta (1-\delta)}\right).$$

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. (*)

Answer 16

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: $$A_n(p,q)=\left[\frac{(pq)!}{1^12^2\cdots p^p(p+1)^p\cdots q^p(q+1)^{p-1}(q+2)^{p-2}\cdots (p+q-1)^1}\right]^2.$$

Answer 17

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 $I$.

Answer 18

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 $m_k > m_{k-1} >\cdots > m_j \ge j >0$.

Answer 19

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.)

Answer 20

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).

Answer 21

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: $$ \frac{1}{(q)_\infty} = \sum_{n\geq 0} \frac{q^{n^2}}{(q)_n (q)_n} \ . $$

Answer 22

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.

Answer 23

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 $\beta>0$.

Answer 24

Read's 1958 beautiful formula for the asymptotic number of 3-regular graphs with n vertices

$$g_3(n) \sim \frac {(3n)! e^{-2}}{(3n/2)!288^{n/2}}.$$

Answer 25

$$(1-\lambda_2)/2\le h(G)\le \sqrt{2(1-\lambda_2)},$$ 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.

Answer 26

$$\frac {\alpha (G)}{n}\le \frac {\lambda_{\min}}{d-\lambda_{\min}}$$

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.

Answer 27

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.

Answer 28

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.

Answer 29

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: $$A = \#\mbox{int}(P \cap \mathbb{Z}^2) + \frac{\partial(P \cap \mathbb{Z}^2)}{2} -1.$$

Answer 30

Davis-Slepian-Polya formula for the number of simple graphs on $n$ nodes $$ \frac{1}{n!} \sum_{j_1+2j_2+\cdots+n j_n=n}\frac{n!}{\prod\limits_{k=1}^n k^{j_k} j_k!} 2^{\displaystyle \frac{1}{2}\left( \sum_{k=1}^n k j_k^2 - \sum_{\text{ $k$ odd}} j_k \right) + \sum_{k=1}^n \sum_{i=1}^{k-1} (k,i) j_k j_i}. $$