All Questions
Tagged with enumerative-combinatorics reference-request
90 questions
32
votes
5
answers
9k
views
How many binary operations are associative?
Let $X$ be a finite set of $n$ elements, and consider a binary operation $\odot: X \times X \rightarrow X$. There are $n^{n^2}$ such binary operations, as the $n \times n$ table entries can each
be ...
- 151k
18
votes
2
answers
1k
views
A combinatorial interpretation for $n$-ary trees for negative $n$
The ordinary generating function $T_n=T_n(x)$ for the $n$-ary trees satisfies the functional equation
$$
T_n=1+xT_n^n.
$$
This is usually defined for $n\ge 0$, but the functional equation can be ...
- 1,190
15
votes
4
answers
3k
views
Collecting alternative proofs for the oddity of Catalan
Consider the ubiquitous Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$. In this post, I am looking for your help in my attempt to collect alternative proofs of the following fact: $C_n$ is odd if and ...
- 43.1k
15
votes
2
answers
1k
views
A rather curious identity on sums over triple binomial terms
While exploring the Baxter sequences from my earlier MO post, I obtained a rather curious identity (not listed on OEIS either). I usually try to employ the Wilf-Zeilberger (WZ) algorithm to justify ...
- 43.1k
15
votes
2
answers
363
views
Generating functions for objects with irrational sizes
A problem I'm investigating concerns a combinatorial class in which the 'atoms' have irrational sizes. It seems likely that this is something that has been considered before, but I haven't been able ...
- 674
14
votes
3
answers
1k
views
On the finite sum of reciprocal Fibonacci sequences
I want to check if $$\left\lfloor \left( \sum_{k=n}^{2n}{\frac{1}{F_{2k}}} \right)^{-1} \right\rfloor =F_{2n-1}~~(n\ge 3) \tag{$*$}$$ where $\lfloor x \rfloor$ is th floor function.
The Fibonacci ...
- 137
14
votes
0
answers
270
views
A symmetry of lattice paths
The number of $n$-step NSEW lattice paths from $(0,0)$ to $(a,b)$ that intersect the line $y=k$ precisely $t$ times is independent of $k$, for $0\leq k\leq b$, where we assume $b\geq0$ for simplicity.
...
- 2,896
13
votes
2
answers
803
views
Two interpretations of a sequence: an opportunity for combinatorics
The sequence that is addressed here is resourced from the most useful site OEIS, listed as A014153, with a generating function
$$\frac1{(1-x)^2}\prod_{k=1}^{\infty}\frac1{1-x^k}.$$
In particular, look ...
- 43.1k
13
votes
0
answers
323
views
Reference request: exponential growth rates of subword-closed languages are integers
For a language $L$ over the finite alphabet $\Sigma$, let $L_n$ denote the set of words in $L$ of length $n$. The word $u$ is a subword of $w$ if $u$ can be obtained from $w$ by deleting letters (...
- 2,339
12
votes
1
answer
594
views
Equality of two $q$-series. Proof?
Recall the notation $(z;q)_n=(1-z)(1-zq)(1-zq^2)\cdots(1-zq^{n-1})$.
My earlier MO question did not find enough interest or yield an answer. Perhaps the modulo $2$ part might have thrown people off. ...
- 43.1k
12
votes
1
answer
405
views
Looking for a "clever" argument for a $q$-series identity
Consider the below $q$-series identity. One of the things I like about this expansion is how nicely the difference on the left hand side factors to the right hand side of the equation.
$$\prod_{k\geq1}...
- 43.1k
11
votes
3
answers
557
views
In search of a $q$-analogue of a Catalan identity
Let $C_n=\frac1{n+1}\binom{2n}n$ be the all-familiar Catalan numbers. Then, the following identity has received enough attention in the literature (for example, Lagrange Inversion: When and How):
\...
- 43.1k
11
votes
1
answer
884
views
And, yet, another evaluation to Catalan numbers
Construct the $n$-tuple Cartesian product of the ternary set $X_n=\{0,1,2\}\times\cdots\times\{0,1,2\}=\{0,1,2\}^n$. Define its subset $W_n$ according to the rule (here $y=(y_1,\dots,y_n)$ is made use ...
- 43.1k
10
votes
0
answers
191
views
What is known about the number of permissible simplicial complexes given the number of k-cells?
Motivation: I am working on a problem that reduces to finding simplicial complexes given some data (details are unnecessary), but all I have managed to wrangle from my input is the number of cells of ...
- 1,488
9
votes
2
answers
1k
views
Extracting constant terms: is there a direct way?
$\DeclareMathOperator\CT{CT}$
Let $\CT_t(f(t))$ denote the constant term of the Laurent polynomial of $f(t)$.
Define the two functions $F(x_1,\dots,x_n)$ and $G(y)$ by
$$F:=\prod_{i=1}^nx_i^{-1}(1-x_i)...
- 43.1k
9
votes
1
answer
299
views
in need of a direct combinatorial/bijective proof
The following are very familiar and basic items, individually.
(1) The number $a(n)$ of rectangles (parallel to axes) in an $n\times n$ square grid.
(2) The number $b(n)$ of cubes (parallel to axes) ...
- 43.1k
8
votes
1
answer
725
views
Number of simplicial polytopes with a given f-vector
Plenty of very nice literature is available on the characterization of f-vectors of simplicial complexes of diverse sorts (results by Billera, Bjoerner, Kalai, Stanley, among others). I mention, as an ...
- 902
7
votes
2
answers
712
views
Total sum of squares of characters of the symmetric group $\mathfrak{S}_n$
In my earlier MO post, I proposed the double sum $\sum_{\mu\vdash n}\sum_{\lambda\vdash n}\chi_{\mu}^{\lambda}$ regarding characters of the symmetric group $\mathfrak{S}_n$. Soon after, I started ...
- 43.1k
7
votes
1
answer
223
views
Result attribution for eigenvalues of a matrix of Pascal-type
A few years ago, I wanted to cite a result in a paper, for which I could not find a reference. I ended up not using the full strength of it, and the part that I needed could be easily proved. Still, I'...
- 1,190
7
votes
0
answers
98
views
Pattern avoidance and P-recursiveness
A sequence $\{a_n\}_{n \geq 0}$ is said to be P-recursive if there exist polynomials $p_0(n), p_1(n), \dots , p_k(n)$ such that
$$
\sum_{i=0}^k p_i(n) a_{n+i}=0
$$
for all $n \in \mathbb N$.
Let $ P$ ...
- 1,608
6
votes
3
answers
444
views
Enumeration of lattice paths of a specific type
One of the approaches to "Special" meanders led (in particular) to the following question:
What is the number $a_{m,n}(\ell)$ of $\ell$-step paths from $(1,1)$ to $(m,n)$ using the ...
- 18.4k
6
votes
2
answers
432
views
Plane partitions as sums of determinants
Consider the Vandermonde's determinant computed by
$$V(x_1,\dots,x_m):=\det(x_j^{i-1})_{i,j=1}^m=\prod_{1\leq i<j\leq m}(x_i-x_j).$$
The number of plane partitions in an $n\times m\times m$ box (...
- 43.1k
6
votes
2
answers
612
views
Counting $\pm 1$ and $0$'s in the character tables of $\frak{S}_n$
Let $\chi_{\mu}^{\lambda}$ denote a value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu, \lambda\vdash n$. When $\mu=(n)$, then it's known that
$$\sum_{\lambda\vdash n}\...
- 43.1k
6
votes
1
answer
330
views
A generalization of derangement number
What is the number of $n \times n$ binary matrices with row and column sums 2 and with only zeros on the diagonal? This simple problem must have been treated somewhere, but I couldn't find any ...
- 1,608
6
votes
1
answer
260
views
Intuitive explanations of the Carlitz-Scoville-Vaughan theorem
Crossposted from MSE:
I recently came across Ira Gessel's slides on a theorem he says should "be considered one of the fundamental theorems of enumerative combinatorics."
The Carlitz-...
- 113
6
votes
1
answer
422
views
Constant term extraction using combinatorial Nullstellensatz
$\DeclareMathOperator\CT{CT}$Given a Laurent polynomial $g$, let $\CT(g)$ denote its constant term.
Consider the specific Laurent polynomial
$$f_n(x_1,\dots,x_r)=\left(1+\prod_{j=1}^r(1+x_j)+\prod_{j=...
- 43.1k
6
votes
1
answer
259
views
Toroidal alternating sign matrices
Consider $n\times k$ matrices with entries from $\{0,1,-1\}$ such that the sum in each row and each column is 0 and the non-zero numbers in each row/column alternate in sign (so, they alternate if we ...
- 109k
6
votes
0
answers
365
views
Is this just a numerical accident or what?
In a complementary proof for a matrix determinant of $a_{i,j}=\binom{n-1+i}j$, raised by BillyJoe, I showed the more general evaluation
$$\det\left(\binom{i+p}{j+k-1}\right)_{1\leq i,j\leq m}
=\prod_{...
- 43.1k
5
votes
2
answers
1k
views
Stirling numbers of the second kind with maximum part size
The stirling number of the second kind $S(n,k)$ counts the number of partitions of the set $[n]$ into $k$ non-empty parts. I found a definition for the numbers called the $r$-associated stirling ...
- 1,835
5
votes
1
answer
327
views
Reference request: enumeration under group action
Is there a reference for the following lemma (which is useful in counting unlabeled k-trees)? It seems to me that it should be known, but I haven't been able to find it anywhere.
Let $G$ be a finite ...
- 17k
5
votes
1
answer
650
views
Counting Problems where Labeled is Known but Unlabeled is Not
Cayley's formula states that the number of labeled trees on $n$ vertices is $n^{n-2}$. There are many nice proofs of this compact formula.
To contrast, counting unlabeled trees is considerably harder....
- 51
4
votes
2
answers
343
views
Number of partitions of $n$ and number of different integers in 1-avoiding partitions
Consider the number of integer partitions of $n$, usually denoted by $p(n)$ and generated by
$$\sum_{n\geq0}p(n)x^n=\prod_{k\geq1}\frac1{1-x^k}.$$
I have encountered an interesting enumeration.
Take ...
- 43.1k
4
votes
1
answer
698
views
Total sum of characters of the symmetric group $\frak{S}_n$
Let $\chi_{\mu}^{\lambda}$ denote a value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu, \lambda\vdash n$. When $\mu=(n)$, then it's known that
$$\sum_{\lambda\vdash n}\...
- 43.1k
4
votes
2
answers
285
views
Is this a known symmetry of lattice paths?
I recently came across the fact that NE lattice paths from $(0,0)$ to $(m,n)$ in aggregate pass through each row and column an equal number of times (which also has a corresponding binomial identity); ...
- 43
4
votes
1
answer
349
views
The fraction $\frac{g_{\mu}}{f_{\lambda}}$ is an integer
Let $\lambda=(\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_{\ell(\lambda)}>0)$ be an integer partition of $n\in\mathbb{N}$; i.e., $\lambda_1+\cdots+\lambda_{\ell(\lambda)}=n$.
One may now associate $...
- 43.1k
4
votes
1
answer
329
views
Enumerating subsets with no triple appearing together more than once
This question is motivated by a real-world application related to an art project that involves displaying images, but my search hit a dead end after finding the wikipage about Kirkman systems (other ...
- 7,831
4
votes
1
answer
225
views
Integer-valued polynomials from Pólya counting
Let finite group $G$ act on a finite set $X$ and hence on colorings $Y^X$, where $Y=\{1,2,\ldots,k\}$ is a set of colors. The Burnside-Pólya-Redfield-etc. counting theorem says that the number of ...
- 24.2k
4
votes
1
answer
298
views
Enumeration of dominated Dyck paths
Using horizontal steps $(1,0)$ and vertical steps $(0,-1)$, consider the lattice paths starting from $(0,q)$ and reaching $(p,0)$ with $p$ horizontal and $q$ vertical steps. The set of such paths $\...
- 43.1k
4
votes
3
answers
1k
views
Polya's theory of counting and commutative algebra
Do you know if there exist algebraic studies of the ring of the power series which emerge when using the theory of Polya for enumeration of sets with certain symmetries? For instance if some ideals ...
- 902
4
votes
1
answer
197
views
On a double sum involving binomial coefficients
For natural $n$, let
\begin{equation}
p_n:=2^{1-n}\sum_{v=1}^l \binom l{(v+l)/2}1(v\equiv l)
\sum_{u=1-v}^{v-1}\binom k{(u+k)/2}1(u\equiv k), \tag{1}\label{1}
\end{equation}
where $k:=\lfloor(n+1)/...
- 128k
4
votes
1
answer
263
views
A refinment of Beck's conjecture
Let $\mathcal{O}(n)$ and $\mathcal{D}(n)$ denote the set of all integer partitions of $n$ into odd parts and distinct parts, respectively. Let $o(n)=\#\mathcal{O}(n)$ and $d(n)=\#\mathcal{D}(n)$. ...
- 43.1k
4
votes
1
answer
221
views
Reference for a definition of Catalan numbers
The $l$-th Catalan number ${2l\choose l}\frac{1}{l+1}$ is equal
to the number of sequences $s_0,\ldots,s_{l+1}$ of length $l+2$ with the following
properties:
(1) $s_0=s_{l+1}=1$ and $s_1,\ldots,s_l$ ...
- 17.9k
4
votes
1
answer
216
views
Counting inversions in a certain patterned matrix
Let $p$ and $q$ be relatively prime. Consider the $p\times q$ matrix $A$ containing the entries $1, 2, 3, \dots, pq$, which is formed via $a_{11} = 1, a_{22} = 2, \dots, a_{p-1,q-1} = pq-1, a_{pq} = ...
- 41
4
votes
0
answers
97
views
"Convolving" a general Catalan with classical Catalan
Consider what is sometimes known as generalized Catalan sequence
$$\mathcal{{\color{red}C}}_{a,b}:=\frac{2b+1}{a+b+1}\binom{2a}{a+b}.$$
Observe that $\mathcal{{\color{red}C}}_{n,0}$ reduces to the ...
- 43.1k
4
votes
0
answers
181
views
Fuss-Catalan: how does equality of these determinants hold?
There are many ways that the Catalan numbers seemed to have been generalized, one among them is through what Graham-Knuth-Patashnik (in Concrete Mathematics) dubbed as the Fuss-Catalan numbers
$\frac1{...
- 43.1k
4
votes
0
answers
135
views
Permutations avoiding a family of consecutive patterns
Let $B=\{1324,14325,154326,1654327,\ldots\}$ be the set of permutation patterns of the form $1(m-1)(m-2)\cdots 2m$ for $m\geq 4$. I'm interested in the set $\mathcal P$ of permutations that avoid all ...
- 135
4
votes
0
answers
163
views
An identity for Schur polynomials
Given a partition $\lambda$, the Schur polynomials can be defined, among many other ways, as
$$S_{\lambda}(\xi_1,\dots,\xi_a)=\frac{\det\left(\xi_i^{\lambda_j+a-j}\right)_{i,j=1}^a}{\det\left(\xi_i^{a-...
- 43.1k
4
votes
0
answers
205
views
Dimension of a certain space of symmetric functions: Part I
Let $s_{\lambda}$ denote the Schur polynomial associated to a partition $\lambda$. A partition $\lambda$ is called a $t$-core if none of its hook lengths are multiples of $t$.
QUESTION. Consider the ...
- 43.1k
4
votes
0
answers
68
views
Asymptotics of the number of minimal strongly connected digraphs
Is anything known about the number of minimal strongly connected digraphs on $n$ labeled nodes? (``Minimal’’ meaning that on the deletion of any arc, strong connectivity is lost.) Some values are ...
- 1,112
4
votes
0
answers
213
views
Counting the polytopes of the translates of the resonance hyperplane arrangement inside the unit hypercube
Let $n$ be a positive natural number. For all $\emptyset \subset S \subseteq \{1, \ldots, n\}$ and $k \in \mathbb{Z}$, define the hyperplane $H(S,k)$ in $\mathbb{R}^n$ given by the equations
$$H(S,k):=...
- 283