Skip to main content
MathOverflow

Questions tagged [enumerative-combinatorics]

Ask Question

The tag has no summary.

540 questions
Newest Active Bountied Unanswered
1 vote
1 answer
175 views

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
  • 1,917
2 votes
0 answers
395 views

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
  • 77
10 votes
2 answers
660 views

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
  • 4,732
10 votes
3 answers
448 views

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
  • 419
3 votes
0 answers
202 views

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
  • 373
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
  • 38.4k
0 votes
1 answer
192 views

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
  • 57
1 vote
2 answers
318 views

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
  • 143
0 votes
0 answers
144 views

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
  • 1,359
8 votes
2 answers
783 views

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
  • 1,558
7 votes
1 answer
371 views

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
  • 25.9k
2 votes
0 answers
152 views

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
  • 1,359
56 votes
1 answer
2k views

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
  • 2,433
2 votes
0 answers
256 views

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
  • 1,359
3 votes
0 answers
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
  • 133

15 30 50 per page
1
2 3 4 5
…
36