第二届艾青诗歌节将举办
百度 而每次不多的几句话，更语出惊人，一副小大人儿的成熟，跟他的年龄相比，更彰显无厘头的童趣。
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531 questions
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Counterexample to a generalization of Frankl's union-closed sets conjecture
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-...
- 1,478
6
votes
1
answer
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Stirling number lattice polytope
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 $...
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Is there a canonical name for this variant of the rook polynomial?
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 $...
- 703
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When did the OEIS get even better?
I'm asking this question out of curiosity, but also (and more importantly) to publicize to the research community something great that OEIS.org is doing.
Recently, I put a sequence into OEIS and got ...
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0
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$\mathcal{P}$-rook polynomial of a grid
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 $...
- 703
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Does a random rooted tree with sufficiently many leaves almost surely contain a specific rooted tree as a subtree?
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 ...
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1
vote
1
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An approach to a generalization of Frankl's union-closed sets conjecture
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}}...
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1
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1
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Is the generating function for triple-turn-avoiding grid Hamiltonian paths D-finite?
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 ...
1
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0
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Counting edge-simple walks of length $k$ in the complete graph $K_n$ that cover the whole graph
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 ...
- 675
3
votes
1
answer
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An analogue of the sum of binomial coefficients
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 $p=2$, ...
1
vote
1
answer
92
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Counting multidimensional arrays up to reindexing and relabeling
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 ...
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An identity involving binomial coefficients
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\...
- 139k
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1
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Using Pascal's identity and summation of binomial coefficients: proof of a weighted sum formula
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\...
- 179
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Number of squares in a sub-grid
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 ...
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Is the rook polynomial of a convex polyomino determined?
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 $...
- 703