Questions tagged [enumerative-combinatorics]
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480
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A Near Closed-Form Expression of Strict Partition Function Inquiry [closed]
I am an independent researcher working in various fields of mathematics and sciences. I am working on a strict partition problem. I believe I have found a very fast exact solution that is a near-...
0
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0
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20
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Algorithm of finding and counting cycles of varying lengths in dynamic or evolving graphs?
In this paper Alon, N., Yuster, R. & Zwick, U. Finding and counting given length cycles. Algorithmica 17, 209–223 (1997)., the authors present various methods for efficiently locating and tallying ...
5
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1
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181
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Bijective proof for an identity concerning Stirling numbers of second kind
Let $\genfrac{\{}{\}}{0pt}{}{n}{k}$ the Stirling number of second kind, where $k$ is the number of parts in the partition.
If we take the identity that transforms the polynomial base $x^k$ into the ...
5
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3
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394
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Applying $\sum_i \partial_{x_i}$, $\sum_i x_i \partial_{x_i}$ and $\sum_i x_i^2 \partial_{x_i}$ to Schur polynomials
The operators $L_k=\sum_i x_i^k\frac{\partial}{\partial x_i}$, with integer $k$, take symmetric polynomials into symmetric polynomials.
Is it known how to write the result of the application of $L_0$, ...
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How many rigid 4-regular graphs are there?
I am interested in any formulas for the number of globally rigid 4-regular graphs, or Laman graphs of degree at most 4, on $n$ vertices. The bound can be for graphs with labeled or unlabeled vertices.
7
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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\...
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Combining (generalized) Polya enumeration with invariant properties
Let's say we want to enumerate maps $f$ between two finite sets $X$ and $Y$ modulo the action of groups $G$ on $X$ and $H$ on $Y$. Additionally we want $f$ to satisfy a certain property $P$ that is ...
5
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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, $\...
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Ordered combinatorial classes and partitions
Let $\mathcal{C}$ be a combinatorial class and let $\leq$ be a partial order on $\mathcal{C}$. We say that $(\mathcal{C},\leq)$ is an ordered combinatorial class if for all $x,y\in\mathcal{C}$, $$x&...
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Estimation of a combinatoric formula
Assume $n\ge m$, what is the estimation of
$$\sum_{k_1+\dots +k_m\,=\,n,\\ k_1\ge 1,\,\dots,\,k_m\ge 1} C_n^{k_1,\dots,k_m} \left(\frac{1}{k_1}+\frac{1}{k_2}+\dots +\frac{1}{k_m} \right)$$
where $C_n^{...
3
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Factorization of symmetric polynomials
Let $\Lambda_n$ be the algebra of all symmetric polynomials in $n$ variables, which we also consider as an infinite-dimensional vector $\mathbb{Q}$-space, whose basis is the Schur polynomials.
The ...
2
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222
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Distribution of peaks in Dyck paths
A Dyck word is a sequence of open and closed brackets such that the brackets come in correctly matched pairs. For example $(()(()))()$ is a Dyck word, while $())(()$ is not. A Dyck path is a visual ...
9
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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 ...
2
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1
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Can you confirm the positivity of a quantity involving the Stirling numbers of the first kind
Let $s(m,n)$ denote the Stirling numbers of the first kind. For $m,n\in\mathbb{N}$, define
\begin{equation}
\mathcal{Q}(m,n)=(-1)^n\sum_{\ell=0}^{2n} \binom{m+\ell-1}{m-1} s(m+2n-1,m+\ell-1)\biggl(\...
7
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Combinatorial consequences of de Branges's Theorem?
I'm usually not a proponent of the mentality “here is a tool, what results can we prove with it?” (as I prefer to start at the other end with a well-motivated problem), but this famously entertaining ...
3
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Ordinary partitions vs partitions into odd parts
Let $\mathcal{P}(n)$ be the set of all unrestricted partitions of $n$ while $\mathcal{O}(n)$ stand for the set of all partitions of $n$ into odd parts. We adopt the power notation for partitions $\...
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Enumerating all inequivalent planar embeddings of a planar graph
Graph $G$ can be embedded (or has an embedding) in the space if $G$ can be drawn in the space if $G$ can be drawn in such a way that no two edges cross except at an end-vertex in common. A Graph $G$ ...
3
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208
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Number of partitions of set restricted by sum of square of part size
Let $p_1^{a_1}p_2^{a_2}\cdots$ denotes the integer partition of $n$, i.e. $a_1p_1+a_2p_2+\cdots=n$. Or equivalently $m_1+m_2+\cdots=n$. It is known that the number of partitions of set $\{x_1,x_2,\...
14
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2
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875
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Sequences that don't count algebraic structures on finite sets
People count $n$-element groups, $n$-element monoids, $n$-element commutative monoids, etcetera - always up to isomorphism. The algebraic structures I've listed, and many more, are studied ...
3
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Divide Euclidean space by surfaces
It is well known that $n$ hyperplanes in $\mathbb{R}^k$ can divide $\mathbb{R}^k$ into at most $p$ regions where $p$ is
\begin{equation}
1 + n + C^2_n + \cdots + C^k_n
\end{equation}
Is there similar ...
0
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163
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Sum of square of parts, and sum of binomials over integer partition
Let $n$ be positive integer. Consider its integer partitions denoting as $(m_1,\cdots,m_k)$, where $m_1+\cdots+m_k=n$ and the order does not matter. We ignore the case of $(m_1,\cdots,m_k)=n$.
I am ...
6
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2
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285
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The number of boolean function with given Fourier degree
How many boolean functions $\{-1,1\}^n \to \{-1,1\}$ with Fourier degree at most $d$?
By Fourier degree I mean the maximal cardinality of $S$ such that the Fourier coefficient $\hat{f}(S)$ is not ...
1
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0
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104
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The number of boolean functions with given decision tree complexity
How many boolean function with $n$ variables with decision tree complexity $k$?
By decision tree complexity of a function $f$ I mean the smallest depth among all deterministic decision trees that ...
3
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1
answer
260
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Enumerating possible number of satisfied linear equations
Consider a system of linear equations of variable $x=(x_1,\cdots,x_n)$ where each $x_i\in\{ 0,1,\cdots,L-1 \}$. Clearly, there are $\frac{n(n-1)}{2}$ number of equations in the system.
$$x_i-x_j=0, \ \...
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1
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343
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A combinatorial proof: where art thou?
Start by introducing the finite sums
$$A_n:=\sum_{m=1}^nq^m\prod_{j=1}^{m-1}(1-q^j) \qquad \text{and} \qquad
B_n:=\sum_{m=1}^nq^m\prod_{j=m+1}^n(1-q^j).$$
An algebraic proof is facile: Clearly, $A_1=...
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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 ...
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3
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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 ...
4
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2
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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 ...
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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
$$\binom{n}...
4
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1
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On a double sum involving binomial coefficients
For natural $n$, let
\begin{equation}
p_n:=2^{1-n}\sum_{v=1}^l \binom l{(v+l)/2}1(v\equiv l)
\sum_{u=1-v}^{v-1}\binom k{(u+k)/2}1(u\equiv k), \tag{1}\label{1}
\end{equation}
where $k:=\lfloor(n+1)/...
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Counting $n$-edge directed graphs
I would like to count the $n$-edge directed graphs. The graphs might contain self-loops (edges connecting a vertex to itself) and multiple edges (multiple edges connecting the same pair of vertices). ...
2
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0
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Skewed plane partition with only row fillings reversed
The number of plane partitions in a bounded box is well-studied and dates back to MacMahon, at the start of this paper by Sam Hopkins and Tri Lai, p9, they summarized current results on the ...
4
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"Convolving" a general Catalan with classical Catalan
Consider what is sometimes known as generalized Catalan sequence
$$\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 ...
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74
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Distribution of colour pairs from a random matching
Given $a$ many red balls and $b$ many blue balls, with $a+b$ even, suppose we chose a random matching between the $a+b$ balls. What is the distribution of the number of red-red and red-blue (and blue-...
3
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0
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Number of planar bipartite graphs
How many planar bipartite graphs are there with $m$ vertices of one color and $n$ vertices of the other color?
How many non-isomorphic classes exist?
6
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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-...
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Lattice paths avoiding holes
Consider lattice paths from $(0,0)$ to $(2n,2n)$ with steps $N=(0,1)$ and $E=(1,0)$ avoiding the points $(2i-1,2i-1)$ for all $1\leq i\leq n$. There are Catalan many $C_{2n}=\frac1{2n+1}\binom{4n}{2n}$...
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What is the Möbius function for the lattice of partial partitions?
Let $n$ be a positive integer. Let $P$ be the set of partitions of subsets of
$\{ 1, 2, \dotsc, n \}$ (so, for example, when $n=2$, the set $P$ contains $\emptyset$, $\{ \{1 \} \}$, $\{ \{2 \} \}$, $\{...
3
votes
1
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Seeking for a combinatorial argument for partition identities
Given an integer partition $\lambda$, introduce the following quantities:
\begin{align*}
c(\lambda)&=\sum_{i\geq1}\left\lceil\frac{\lambda_i}2\right\rceil, \qquad c_o(\lambda)=\sum_{i\geq1}\left\...
4
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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{...
2
votes
1
answer
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Catalan and path pairs in polynomials
Define $\mathbf{K}_n$ to be the set of all $(2n+1)$-tuple sequences $\mathbf{a}=(a_0,a_1,\dots,a_{2n})\in\{-1,1\}^{n+1}$ satisfying: (a) there are $n$ occurrences of $-1$ and $n+1$ of $+1$; (b) all ...
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Super Catalan (super ballot) numbers
We refer to this article by Ira Gessel. In section 6, page 10, equation (28), the Super Catalan numbers are defined as
$$S(m,n)=\frac{(2m)!(2n)!}{m!n!(m+n)!}.$$
On page 12, equation (31), there goes ...
3
votes
1
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382
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A combinatorial identity involving binomial coefficients
When I was reading an article by CHUN-GANG JI (A SIMPLE PROOF OF A CURIOUS CONGRUENCE BY ZHAO), he mentioned in the acknowledgement
the following identity
$$\sum_{i+j+k=p,\text{ } i,j,k\gt 0}{p\choose ...
6
votes
2
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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 (...
2
votes
1
answer
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Proof of a binomial identity
Computations with Maple suggest the following binomial identity
\begin{equation*}
\forall{p,j}: \sum_{k=j+1}^{p+1} (-1)^j \dfrac{1}{k}\binom{k-1}{j} =
\sum_{k=j+1}^{p+1} (-1)^{k-1} \dfrac{1}{...
1
vote
1
answer
91
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Existence of some lattice path connecting all given lattice paths
My daily work concerns analysis on metric spaces and some time ago it turned out that the problem I am dealing with boils down to a certain combinatorial problem. I've checked a lot of examples and it ...
4
votes
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216
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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 $\...
5
votes
1
answer
389
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Catalan sequences vs composition sequences
In the paper, A polytope related to empirical distributions, plane trees, parking functions, and the associahedron, Pitman and Stanley studied the $n$-dimensional polytope
$$\Pi_n(\mathbf x)=\{y\in\...
0
votes
2
answers
174
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Asymptotic approximation of a convolution of binomial coefficients
I would like to find the following limit which is somewhat similar to the usual Vandermonde's convolution for binomial coefficients. Define $L$ as follows.
$$ L \triangleq \lim_{N\to\infty} \frac{1}{2^...
0
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0
answers
41
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Enumerating directed cacti by the number of vertices and edges
Let's say that a directed cactus is a labeled directed graph, such that each vertex belongs to at most one simple cycle. In other words, it is a directed graph such that all its strongly connected ...