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Please show (using arithmetic) that, for $r,s,t\in\mathbb N$, $$ \prod_{i=1}^r\prod_{j=1}^s\prod_{k=1}^t\,\frac{i+j+k-1}{i+j+k-2} $$ equals $$ \prod_{i=1}^r\,\frac{\binom{s+t+i-1}{s}}{\binom{s+i-1}{s}}...
Tri's user avatar
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There are $n$ consecutive integers $m,m+1, \ldots, m+n-1$. Prove that you can choose some nonempty subset of these numbers whose sum is divisible by $1+2+\dots+n$.
Obtuse's user avatar
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For a surface $\Sigma$ with boundary and $2n$ marked points $x_1,\dots,x_{2n}\in\partial\Sigma$, we may ask: what is the number of "pairings" of the $x_i$, i.e., a partition of $\{1,\dots,2n\...
Kenta Suzuki's user avatar
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Based on some small calculations in SageMath, I conjectured that the Schur expansion of $h_n[h_k]$ contains $s_{(k,k,...,k)}$ if and only if $k$ is even. For example, this is easily seen when $n=2$. ...
Soumyadip Sarkar's user avatar
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Consider the polynomial $K(z,u)= u^e- z (p_0+p_1 u+...+p_k u^k)$, where: $k,e$ are nonnegative integers and the coefficients $p_i$ are nonnegative reals such that $0< \sum_i p_i \leq 1$ with $p_0&...
Michele's user avatar
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7 votes
3 answers
1k views

For a nonnegative integer $n$, let $N_n$ be the number of distinct values taken by the sums of $n$ 6th-degree roots of unity (with repetitions). First few counts are $N_0=1$, $N_1=6$, $N_2=19$, and ...
Max Alekseyev's user avatar
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Let $n \leq k \in \mathbb{N}$. Define unimodal $n$-tuple of weight $k$ as ordered $n$-tuple of positive integers $d_1, d_2, \dots, d_n$ such that $$ \sum_{i=1}^{n} d_i = k $$ and $\exists s \in \{1,2, ...
Oliver Bukovianský's user avatar
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Let the following data be given. Two positive integers $m$ and $n$. A family of sets $B_a \subseteq \{1, \dots, n\}$ (for $a \in \{1, \dots, m\}$). The task is to count the number $N$ of injective ...
parkingfunc's user avatar
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I'm preparing a talk on the rook polynomial, and I would like to mention some references on its variants and the reasons they were defined. I am familiar with the following two generalizations because ...
Chess's user avatar
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8 votes
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We call a family of sets $\mathcal{F}$ is weakly union-closed if for all $A,B\in\mathcal{F}$ such that $A\cap B=\varnothing$, we have $A\cup B\in\mathcal{F}$. Conjecture: For finite weakly union-...
Veronica Phan's user avatar
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This was motivated by this recent question: Expansion identity for the Eulerian polynomials of the second order Question: For each integer $m \geq 0$, is there some $2m$-dimensional lattice polytope $...
Sam Hopkins's user avatar
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Let $\mathcal{P}$ be a polyomino. Two or more rooks on $\mathcal{P}$ are called non-attacking if no path of edge-adjacent cells of $\mathcal{P}$ connects any pair of them along a row or a column. Let $...
Chess's user avatar
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57 votes
1 answer
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I'm asking this question out of curiosity, but also (and more importantly) to publicize to the research community something great that OEIS.org is doing. Recently, I put a sequence into OEIS and got ...
Nathan Reading's user avatar
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The $\mathcal{P}$-rook polynomial of a polyomino $\mathcal{P}$ is $$ r_\mathcal{P}(T) = \sum_{k=0}^{r(\mathcal{P})} r_k(\mathcal{P})\ T^k, $$ where $r_k(\mathcal{P})$ is the number of ways to place $...
Chess's user avatar
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3 votes
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148 views

Let $\mathcal{T}_n$ be the set of rooted, unlabeled trees with $n$ leaves, where each vertex either has no child or has at least two children. Let $\mathcal{T} = \bigcup_{n \ge 2} \mathcal{T}_n$. For ...
W. Wang's user avatar
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Let $I$ be a non-empty finite set, $\mathcal{F}$ be a non-empty union-closed family of subsets of $I$ except the empty set and $n$ real numbers $x_i\geq1,i\in I$. Let $d_i=\sum_{i\in J,J\in\mathcal{F}}...
Veronica Phan's user avatar
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1 vote
1 answer
225 views

Let $$ \mathcal{G}_{k,n}:=\{1,\dots ,k\}\times\{1,\dots ,n\}\subset\mathbb{Z}^{2}, \qquad k,n\ge1, $$ be a $k\times n$ rectangular lattice graph. A Hamiltonian path (a walk that visits every vertex ...
Alex Cooper's user avatar
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1 vote
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Let $K_n$ be the complete graph on $n$ vertices. I am interested in counting the number of walks of length $k$ in $K_n$ with the following constraint: Edges may not be repeated (each edge is used at ...
Victor's user avatar
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Let $a$ and $p$ be positive integers, and consider the polynomial $$(1+x+\cdots+x^{p-1})^a = \sum_{i=0}^{a(p-1)} a_ix^i.$$ I'm looking for an asymptotic estimate of $\sum_{i=0}^{b} a_i$. If $p=2$, ...
FABIO MASTROGIACOMO's user avatar
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1 answer
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Define a $d_0\times d_1\times\cdots\times d_{k-1}$-grid to be a surjective function $G$ with domain $\prod_{i=0}^{k-1}\{0,1,\dots,d_i-1\}$. A $d_0\times d_1\times\cdots\times d_{k-1}$-grid can also be ...
Brett L's user avatar
  • 11
16 votes
2 answers
970 views

How to prove that $$\sum _{j=0}^{n-i} \frac{(-1)^j \binom{n-i}{j}} {(i+j) (i+j+1) (n+i+j+1)\binom{n+i+j}{n-i}} =\frac{4 (2 i-1)!\, (2 n-2 i+1)!}{(2n+2)!},$$ where $n$ and $i$ are integers such $1\le i\...
Iosif Pinelis's user avatar
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0 votes
1 answer
181 views

I am trying to prove the following identity involving binomial coefficients. It is known that: -For any $\ell \in \mathbb{N}$ and any $r \geq \ell$, we have the summation formula: $$ \sum_{j=\ell}^r\...
Mourad Khattari's user avatar
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0 votes
0 answers
188 views

Let $ G $ be an $ m \times n $ grid in the plane, and consider an embedded diagram $P$, which is either a Ferrers diagram or, more generally, a convex polyomino. For a fixed positive integer $ h $, I ...
Chess's user avatar
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6 votes
0 answers
236 views

Consider a convex polyomino, which can be uniquely defined by its horizontal and vertical projection vectors. More precisely, the horizontal projection vector is $h = (h_1, h_2, \dots, h_m)$, where $...
Chess's user avatar
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3 votes
1 answer
152 views

Fixing a number of vertices $N$, how many distinct partial cubes (isometric subgraphs of hypercubes) with $N$ vertices exist up to isomorphism? Is there a method for enumeration? Note that the problem ...
Ning Bao's user avatar
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1 vote
0 answers
382 views

Continue my previous question, consider the first conjecture: Let $\mathcal{F}$ be a union-closed family of subsets of $[n]=\{1,2,...n\}$ and $n$ real numbers $x_1,x_2,...,x_n\geq 1$. Conjecture: ...
Veronica Phan's user avatar
  • 1,558
9 votes
1 answer
1k views

This question is rooted in combinatorics and concerns a possible relationship between the so-called $\mathcal{P}$-rook polynomial and a reformulation of it. Background and definitions on the $\mathcal{...
Chess's user avatar
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5 votes
0 answers
253 views

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}\DeclareMathOperator\PSO{PSO}\DeclareMathOperator\HSpin{HSpin}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\PSp{...
Andrea Aveni's user avatar
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4 votes
2 answers
947 views

This question relates to https://oeis.org/A051708 The question is whether priorly submitted and accepted into A051708 formula by Vladimir Kruchinin: ...
Alex's user avatar
  • 181
4 votes
0 answers
217 views

Let $A_n$ be the number of permutations $\pi$ of $[n]=\{1,2,\ldots,n\}$ such that $\pi(i)\neq i$ for all $i \in [n]$ and $|\pi(i+1) - \pi(i)| > 1$ for all $i \in [n-1]$. So $A_n$ counts the ...
user967210's user avatar
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1 vote
0 answers
279 views

Let $C_n$ be a chain (totally ordered set) of $n$ elements. For each $x \in C_n$, denote by $\iota(x)$ the number of elements in $C_n$ that are less than $x$. Consider the set of order-preserving ...
Eremphōs Chieh's user avatar
  • 11
0 votes
0 answers
112 views

Let $a(n)$ be the number of equivalence classes of $3$-dimensional central arrangements of $n$ hyperplanes in general position. I believe that $a(n)= 1$ for $n \leq 5$ and $a(6)=3$. Is this correct? ...
Bipolar Minds's user avatar
  • 1,912
4 votes
2 answers
200 views

I am interested in estimating the number of non-isomorphic simple graphs on $n$ vertices with $O(n)$ edges. Specifically, I am wondering whether it is correct that the number of such graphs is at ...
sisylana's user avatar
  • 43
16 votes
0 answers
374 views

Let $w=a_1 a_2\cdots a_n\in S_n$, the symmetric group of all permutations of $1,2,\dots,n$. The descent set $D(w)$ is defined by $D(w)=\{1\leq i\leq n-1\,:\, a_i>a_{i+1}\}$. Let $f(n)$ be the ...
Richard Stanley's user avatar
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2 votes
0 answers
90 views

We have a function $\eta(s,n,m)$ that counts the number of nondecreasing (or nonincreasing) partitions of $s$ with at most $n$ parts, each of size at most $m$. The function can be defined by a ...
xiver77's user avatar
  • 121
0 votes
0 answers
57 views

Let $G = (A, B, E)$ be a bipartite graph representing a matching market with $|A| = n+1$ and $|B| = n$. Each vertex $a \in A$ has a preference list that is a uniformly random permutation of $B$, and ...
user avatar
3 votes
1 answer
236 views

Suppose $\lambda = (1^{m_1},\ldots,n^{m_n})$ is a partition of $n$ with length $\ell$. I am seeking a bijection between the following two sets. The first is the set of all permutations of the multiset ...
ArB's user avatar
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1 vote
0 answers
106 views

Let $A, B, C \subseteq \mathbb{Z}_2^n$. Define $r(A, B, C) := |\{(a, b, c) \in A \times B \times C : a + b = c\}|$. For $1 \leq a \leq 2^n - 1$, let $A_a \subseteq \mathbb{Z}_2^n$ denote the set ...
RFZ's user avatar
  • 448
23 votes
3 answers
3k views

I was writing a research paper in Computer Science. I had to provide an upper bound for the number of steps of the algorithm I had found with my colleagues; the nature of the algorithm is totally ...
Melanzio's user avatar
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2 votes
0 answers
150 views

This question is a continuation of a question asked yesterday which had a very nice answer. Consider the summation $$\sum_{\lambda \subset (k)^n} \dim S_{\lambda^t} (\mathbb{C}^k) \cdot \dim S_{[\...
Rellek's user avatar
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6 votes
1 answer
275 views

Suppose $n$ and $k$ are two integers. Then I am interested in having a closed form for the sum $$\sum_{\lambda \subset k \times n} S_\lambda (\mathbb{C}^n),$$ where $S_\lambda$ denotes the Schur ...
Rellek's user avatar
  • 563
4 votes
0 answers
155 views

Denote $\Phi(n)$ as the root system of Lie algebra $\mathfrak{g}$ of type $A_{n}$. Call a disjoint union $\Phi(n) = \Phi_{1}\sqcup\dotsb\sqcup\Phi_{s}$ a decomposition of $\Phi$ if each $\Phi_{k}$ is ...
Yuanjiu Lyu's user avatar
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1 vote
0 answers
148 views

Assume that $A$ and $B$ are subsets of $\mathbb N$, with counting functions verifying $A(x)\gg x^\alpha$ and $B(x)\gg x^\beta$, with $\alpha+\beta<1$. Let $C=A+B$ and $C(x)$ its counting function. ...
G. Melfi's user avatar
  • 640
4 votes
0 answers
201 views

Let $S_n$ be the set of all permutations of $\{1, \ldots, n\}$, thereafter treated as integer sequences. Let $A_n$ be the set of all such permutations $\sigma \in S_n$ that we can choose two ...
Mikhail Tikhomirov's user avatar
  • 5,812
3 votes
2 answers
358 views

The general linear group $GL_n(\mathbb{F}_2)$ acts on the powerset $2^{{\mathbb{F}_2}^n \setminus \{0\}}$ by multiplication: $A \cdot S := \{Ax \in {\mathbb{F}_2}^n : \, x \in S\}$, for an invertible ...
Colin Tan's user avatar
  • 49
3 votes
1 answer
207 views

For a prime $p$, fix two bases $U=\{v_1,\dots,v_n\}$ and $W=\{w_1,\dots,w_n\}$ of the vector space $V=\mathbf{F}_p^n$. We may assume $U$ is the standard basis without loss of generality. For $s_1,\...
Connor's user avatar
  • 551
2 votes
1 answer
291 views

How many simple binary matroids are there, up to isomorphism, of rank $r$ on an $n$-element ground set, where $r \le n < 2^r$? Write this number as $a_r(n)$. Is there somewhere where I can get this ...
Colin Tan's user avatar
  • 49
4 votes
1 answer
469 views

QUESTION. Let $p$ be an odd prime and let $a,b,c\in\mathbb Z$. How to determine the parity of the sum $$S_p(a,b,c)=\sum_{1\le j<k\le p-1\atop p\nmid aj^2+bjk+ck^2}(aj^2+bjk+ck^2)$$ in terms of $a,b,...
Zhi-Wei Sun's user avatar
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8 votes
0 answers
335 views

Famously, the alternating sign matrix theorem gives a product formula for the number $a(n)$ of ASMs of size $n$. There are multiple proofs of this formula, all somewhat involved. My question is ...
Igor Pak's user avatar
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-3 votes
1 answer
151 views

Given $a,b$ positive numbers such that $gcd(a,b)=1$.Prove that there are infinitely many $n$ positive integers such that $x_n=a+nb$ sequence has many terms such that it is not divisible by any prime'...
Nuran Nurməmmədov's user avatar
  • 1

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