Questions tagged [enumerative-combinatorics]
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160
questions with no upvoted or accepted answers
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"Special" meanders
One of the open problems in combinatorics is enumeration of meanders.
Here on MO I only could find them under the heading not-especially-famous-long-open-problems-which-anyone-can-understand.
Since my ...
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14
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266
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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.
...
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13
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314
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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 (...
- 2,329
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515
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$q$-analogue of the multinomial theorem?
The $q$-binomial theorem states that
$$
\prod_{k=0}^{n-1}(1+q^kt) = \sum_{k=0}^n q^{\binom k2}{n\brack k}_q t^k.
$$
This identity is a $q$-analogue of the binomial theorem
$$
(1+t)^n = \sum_{k=0}^n \...
- 5,594
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327
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The number of labeled pairs of edge disjoint trees and related questions
I wonder what is known on the following:
1) What is the number $T_k(n)$ of $k$-tuples of (pairwise) edge-disjoint trees $(T_1,T_2,\dots, T_k)$ with $n$ labelled vertices?
2) (harder, it seems) What ...
- 24.2k
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627
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Wilf's conjecture: complementary Bell numbers
The complementary Bell numbers or Uppuluri–Carpenter numbers, denoted $\tilde{B}_n$, can be delivered by
$$G(x):=\sum_{n\geq0}\tilde{B}_n\frac{x^n}{n!}=e^{1-e^x}.$$
Definition. Fix an integer $m\geq0$....
- 41.8k
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266
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Number of updown sequences of $1,1,2,2,\cdots,n,n$
I would like to count the updown sequences of the set $\{1,1,2,2, \cdots, n,n \}.$
Sequence $a_1, a_2, a_3, \ldots$ is an updown sequence if the sequence satisfies the following: $ a_1 \lt a_2 \gt a_3 ...
- 121
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Two remarkable weighted sums over binary words
This question builds off of the previous MO question Number of collinear ways to fill a grid.
Let $A(m,n)$ denote the set of binary words $\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_{m+n-2})$ consisting ...
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133
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Smallest counterexample to Stein's conjecture?
An equi-$n$-square is an $n$ by $n$ array of cells filled with the symbols $1,2,\dots,n$ so that each symbol occurs exactly $n$ times.
(Every Latin square of order $n$ is an equi-$n$-square, but the ...
- 2,330
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Number of rectangles in an n-by-n grid of points
I am seeking a formula for the number of rectangles with vertices belonging to an $n \times n$ square grid of points. This is sequence A085582 in the OEIS. Note that the grid in question has $n^2$ ...
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190
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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 ...
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277
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The $n$ queens problem with no three on a line
The $n$ queens problem asks if we can place $n$ queens on an $n\times n$ chessboard such that no two queens attack one another. For example, when $n=8$, here are two solutions (images taken from ...
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321-avoiding and parity-alternating permutations
It is classical that 321-avoiding permutations are enumerated by the Catalan numbers.
A permutation is parity-alternating if it sends even integers to even integers, and odd integers to odd. I am ...
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Number of tautologies of a given size?
Fix some complete set of $L$ logical connectives such as $\{ \wedge, \neg \}, \{\Rightarrow, \neg \}, \{\ \vee, \wedge, \neg \}, \{ \uparrow\}, \{\wedge, \vee, \neg, \Rightarrow \}$ - I'll assume all ...
- 1,065
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Number of sets $S$ for which number of permutations in $S_n$ with descent set $S$ is odd
The descent set $D(w)$ of a permutation $w=a_1 a_2\cdots a_n\in\frak{S}_n$ is
defined by $D(w)=\{ 1\leq i\leq n-1\,:\, a_i>a_{i+1}\}$.
Given a set $S$, let
$\beta_n(S)$ denote the number of ...
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Why does Loday call the permutohedra "zylchgons"?
Today I was reading Jean-Louis Loday's classic paper, "Realization of the Stasheff polytope", in which he produces a simple and very pretty realization of the associahedra as convex polytopes. He ...
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275
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pattern-avoiding permutations vs multi-core partitions
Let $\mathfrak{S}_n$ be the permutation group on $[n]$. Given the pattern $\sigma=k(k-1)\cdots321$, let $I_n(\sigma)$ be the number of involutions in $\mathfrak{S}_n$ that avoid the pattern $\sigma$. ...
- 41.8k
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Cycles of length $2^n - 2$ in the De Bruijn graph
It is well known that the number of (cyclic) De Bruijn sequences is $2^{2^{n-1}-n}$. This number may also be interpreted as the number of cycles of length $2^n$ in the De Bruijn graph of order $n$.
...
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149
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Partial order on graphs induced by homomorphism counts
For graphs $F$ and $G$, let $\hom(F,G)$ denote the number of homomorphisms (adjacency preserving maps) from $F$ to $G$. Define a relation $\le_{\hom}$ on (isomorphism classes of) graphs as $G \le_{\...
- 2,289
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Are most semigroups nilpotent of degree 3?
A semigroup is nilpotent of degree 3 if every product of 3 elements gives the same result. In 2012, Andreas Distler and James D. Mitchell wrote that:
It is part of the folklore of semigroup theory ...
- 21.5k
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130
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How many simplicial spheres with $n$ vertices and $N$ facets?
Let $s_d(n,N)$ be the number of different $d$-dimensional simplicial spheres on $n$ labelled vertices and $N$ facets (= $d$-simplices). I am in search for the best know upper bounds, especially for $d\...
- 12.6k
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Why are these two determinants equal?
This question is a follow up on Mark Wildon's comment from an earlier MO question.
As usual, let $(q)_k=(1-q)(1-q^2)\cdots(1-q^k)$ with $(q)_0:=1$. Also, define the Gaussian polynomials by
$$\binom{n}...
- 41.8k
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92
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$ ...
- 1,454
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144
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Inequality of product of discrete cosines
Let $k,a,b,c$ be odd positive integers. Consider the following inequality:
$$
\sum_{x,y \in [k]} \cos^a\bigg(\frac{2\pi}{k}\cdot x\bigg) \cdot \cos^b\bigg(\frac{2\pi}{k}\cdot y\bigg) \cdot \cos^c\bigg(...
- 71
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Factoring a function from a finite set to itself
Let $S$ be a finite set and $f: S \to S$ be a function. Let $k = |f(S)|$ and let $\alpha$ be the partition of $S$ into $f$-fibers, i.e. $\alpha = \{ \alpha_t \}_{t \in f(S)}$ where $\alpha_t = f^{-1}(\...
- 675
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Explaining $\left(a-1\right)^n \cdot n! \mid a^{n-1} \prod_{i=1}^n \left(a^i-1\right)$ by a free $S_n$-action
Here is an olympiad-level problem on elementary number theory:
Let $a$ be an integer and $n$ a positive integer. Prove that
\begin{align}
\left(a-1\right)^n \cdot n! \mid a^{n-1} \prod_{i=1}^n \left(...
- 33.2k
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692
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How many solutions $\pm1\pm2\pm3...\pm n=0$
As the title said, I'm very interested how many variants to choose $n$ signs from all $2^n$ variants when expression lead to zero. I tried to get recurrent formula but nothing happened.
- 171
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in search of intepretations and connections for $k$-central binomials
Fix a positive integer $k$. Then, the sequences
$$c(n,k)=\frac{k^n}{n!}\prod_{m=1}^{n-1}(1+km)=[x^n]\left(\frac1{1-k^2x}\right)^{1/k}$$
are referred to as "$k$-central binomial coefficients",...
- 41.8k
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343
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How does the number of self-avoiding paths between two points scale, in a square/cubic lattice?
Consider two different infinite graphs, whose vertices are drawn from $\mathbb Z^2$ or $\mathbb Z^3$. Let $P_d : \mathbb Z^d \times \mathbb N \to \mathbb N$ for $d \in \{2,3\}$ be the function such ...
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364
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Is this just a numerical accident or what?
In a complementary proof for a matrix determinant of $a_{i,j}=\binom{n-1+i}j$, raised by BillyJoe, I showed the more general evaluation
$$\det\left(\binom{i+p}{j+k-1}\right)_{1\leq i,j\leq m}
=\prod_{...
- 41.8k
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274
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Power law correction factor in tree enumeration via naïve division
It is a theorem of Otter, building on fundamental work of Pólya, that the number of unlabeled trees on $n$ vertices is $\approx C \alpha^{n} n^{-5/2}$, where $C = 0.534\ldots$ and $\alpha = 2.955\...
- 22.9k
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475
views
Darkness in the lamplighter group
Consider paths through the lamplighter group $\mathbb{Z}_n\wr\mathbb{Z}$ with steps consisting of moving left, moving right, and toggling the lamp at the current position. How many paths of length $m$ ...
- 1,793
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Does the likelihood of these tables exist?
Probably it does, and may be a number near $e^{-3/2}$ for 2-deficient tables. First some background.
Early on in my studies of universal algebra, I encountered a result of Vadim Murskii, with the ...
- 13k
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How to represent the even signed permutations by Young tableaux?
The well-known RSK correspondence established the connection between table pair (P,Q) and the permutations in symmetry group Sn(Coxeter group of type A). Also, there is a similar correspondence for ...
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What is known about the $q$-analogue of the simplex?
I am interested in the field with one element. I am thus interested in combinatorial interpretations of the Gaussian binomial coefficients. Richard Stanley's "Enumerative combinatorics" mentions ...
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When can we determine an $f$-vector or rank-generating function from its unordered list of coefficients?
Let $f_i$ be the number of $i$-dimensional faces in a $d$-dimensional simplicial complex $\Delta $. Recall that the $f$-vector of $\Delta $ is the vector $(f_{-1},f_0,f_1,\dots ,f_d)$ where $f_{-1}=...
- 3,482
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Counting Selections of Entries such having an Extremal Permutation of length n^2+1
Let $S_{n^2+1}$ be permutations of length $n^2+1$. By Erdos-Szekeres Theorem. any $s \in S_{n^2+1}$ would have a monotone subsequence(increasing or decreasing)of length $n+1$.
Say a permutation $s$ of ...
- 393
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149
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Combinatorial classes where not almost all objects are asymmetric
Let $\mathcal{C} = \bigcup_{n=0}^{\infty}\mathcal{C}_n$ be a class of finite (labeled) combinatorial objects, where $\mathcal{C}_n$ is the set of objects on $[n] = \{1,2,\dotsc,n\}$. For example, $\...
- 22.9k
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Identity for classes of plane partitions
There are several classes of plane partitions in the literature.
Among these, let's look at the enumeration of three of them: the symmetric (SPP), totally symmetric (TSPP) and totally symmetric and ...
- 41.8k
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174
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A close cousin of involutions?
If $\mathfrak{S}_n$ denotes the permutation group on $n$ letters, then $Inv(n)=\{\pi: \pi^2=1\}\subset\mathfrak{S}_n$ is the set of involutions or
self-inverse permutations. The latter is enumerated ...
- 41.8k
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Counting perfect matchings with integrals
Has anyone used the Joni-Rota-Godsil integral formula (see details below) to count perfect matchings of square-grid graphs, Aztec diamond graphs, hexagon-honeycomb graphs, etc.? (Or even just to ...
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204
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Sum $f(n_1,n_2,\ldots,n_k) 1^{n_1} 2^{n_2} \ldots k^{n_k}$ over partitions
Use the notation $(n_1,n_2,\ldots,n_k) \vdash n$ to denote that $(n_1,n_2,\ldots,n_k)$ is a partition of the positive integer $n$, that is, $n_1+n_2+\ldots+n_k = n$ and $n_1 \ge n_2 \ge \ldots \ge n_k ...
- 261
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"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 ...
- 41.8k
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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{...
- 41.8k
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How can we prove this combinatorial identity?
This is a follow up on my earlier MO post. Let's recall the sets
$$\mathbf{K}_n=\{\mathbf{k}\in\mathbb{Z}^n: k_i\geq0, k_1+\cdots+ k_n=n, k_1+\cdots k_i\leq i, \text{for all $1\leq i\leq n$}\}$$
and $\...
- 41.8k
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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 ...
- 61
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An atlas for the enumeration of planar maps
The theory of planar map enumeration was started by Tutte and iniciated the theory of map enumeration when trying to solve the 4-colour theorem by enumerative arguments. Nowadays a wide diversity of ...
- 1,543
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152
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How many choices for $(f,g)$ such that $f \circ g = h$?
For $a,b,c \in \mathbb{N}$ let $[a] := \{1,...,a\}$ and suppose a map $h: [a] \rightarrow [c]$ is given. How many choices $(f,g)$ for maps $f: [a] \rightarrow [b]$ and $g: [b] \rightarrow [c]$ are ...
- 1,087
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197
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Non-crossing and crossing bijection in higher genus
This is a follow-up question of my SO post I'll briefly mention it here.
So given a $n$ cycle say $(1,2,\ldots,n)$, what are the monotonic 2 -tuples, of the form $(a,b)(c,d)$, monotonicity in on the ...
- 685
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157
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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-...
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