# Questions tagged [co.combinatorics]

Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.

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### Problem concerning cutting of 2n*2n square into 2 equal area connected figures using various cuts without self crossings

We have a square 2n*2n, where n belongs to N. The main problem is to find how many different equal area connected figures could be produced by cuttings without self-crossings. The orientability of the ...
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### Alternating sum of hook lengths: Part I

Given $\lambda$ an integer partition of $n$, let $h_{ij}(\lambda)$ denote the hook length of cell $(i,j)$ in the Young diagram of $\lambda$. Is there a closed formula or a generating function for the ...
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### Higher order Leibniz rule and ordered multiindex notation

Although I think this is probably known, I am making here a short exposition on the multiindex notations I am using to make this question self-contained. I note that there is at least two different ...
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### Directed version of this lemma

On a paper by Shoham Letzter, available Here, there's a lemma that says as follows: Lemma 0: For every graph $G$, there exist two disjoint sets $U,W\subseteq V(G)$ of equal size, such that there are ...
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### Number of linear inequalities describing a polyhedron with prescribed number of vertices

If a polytope has $d$ vertices in $k$ dimensions how many linear inequalities is required to describe it?
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### Packing equal-size disks in a unit disk

Inspired by the delicious buns and Siu Mai in bamboo steamers I saw tonight in a food show about Cantonese Dim Sum, here is a natural question. It probably has been well studied in the literature, but ...
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Let $n>1$ be an integer, and let $S_n$ be the symmetric group of all the permutatins of $\{1,\ldots,n\}$. I'm curious whether there is a permutation $\tau\in S_n$ such that $$\tau(1)^{\tau(2)}+\... 1answer 56 views ### Low-Hamming weight vectors in low-dimensional subspaces of \mathbb{F}_p^n What is the maximum number vectors of Hamming weight at most w in a d-dimensional subspace of \mathbb{F}_p^n, where w,d,p are constant and p is odd. (The Hamming weight is the number of ... 0answers 122 views ### Finding a tree with adjacency matrix near a given matrix For defining a distance between trees, one can code them into \mathbb{R}^n and use norms in \mathbb{R}^n as distance. (For example we can use adjacency matrices as a tool for this coding) After ... 1answer 82 views ### Almost-parallel corners of the hypercube in high dimensions Say I would like a collection of k "almost-parallel" boolean vectors X_1,...,X_k \in \{\pm 1\}^n, such that (X_i,X_j)/n \approx 1-\epsilon for some small \epsilon. How many ways are ... 0answers 81 views ### A connection between the Bell numbers and Bell polynomial Let B(n,x) = \sum_{k=0}^n {n\brace k}x^k be the Bell polynomials and B_n = B(n,1) be the Bell numbers. I recently proved a nice relation between the two:$$ B(n,x)^{1/n}/x \ge B_{n/x}^{x/n}, $$... 0answers 78 views ### When is it possible to extend several linear orders defined “locally” into a single linear order defined “globally”? This is a somewhat fuzzy question, so I will try my best to give a formulation which includes everything relevant while excluding everything else. I would like to find out if anyone else has studied ... 1answer 54 views ### “Circuit rank” but for vertices A graph's circuit rank is the minimum number of edges that have to be removed for the graph to become a tree or forest. Is there a term that represents the minimum number of vertices that we must ... 1answer 138 views ### Expansion in hypergraphs Is there a useful concept of expansion in hypergraphs, generalizing the concept for graphs (see: expander graphs)? Of course, expander graphs can be characterized in several qualitatively equivalent ... 2answers 124 views ### growth of the permanent of some band matrix Consider such special band matrix of dimension n. It is a 0-1 matrix, and only the first few diagonals are nonzero. Specifically,$$ H_{ij} = 1 $$if and only if |i-j| \leq 2. How does the ... 1answer 409 views ### Hurwitz numbers and t-cores For integers k \geq 0 and d \geq 1 let H(k,d) be the Hurwitz number which, for the purposes of this posting, will be defined by: H(k,d) \, := \ d! \, \sum_{\lambda \, \vdash d}... 0answers 118 views ### Square root of a function on a finite set Let S be a finite set and f \colon S \to S be an arbitrary function. How can we find all functions g \colon S \to S with f = g \circ g? If f and g are both required to be invertible, the ... 0answers 97 views ### Asymptotics of a combinatorial series I am interested in the exact asymptotics of the following combinatorial series (which arises from the study of a Markov chain):$$F(q):=\sum_{k \ge 1} \frac{q^{k^2}}{(q;q)_k^2}\quad \mbox{as } q \to 1^...
Given an undirected weighted graph, we seek a pair of node-disjoint path between $2$ nodes $s$ and $t$: if the objective is to minimize the total path cost, the Suurballe algorithm can be applied; now ...
In number theory, an odious number is a positive integer that has an odd number of $1$s in its binary expansion. The first odious numbers are: \$1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 21, 22, 25, 26, 28, ...