# 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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### Length of walking on a graph

Given a finite directed connected graph $G$, let $P_{circle}$ be the set of finitely long circle paths on $G$ (a circle path is a path with identical starting and ending vertex). It is well known that ...
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### How many non-isomorphic abelian subgroups of the permutation group $S_n$?

I am interested in how many (pairwise non-isomorphic) subgroups of the permutation group $S_n$ are abelian. ($n \in \mathbb{N}$ arbitrary and possibly big) Are you aware of any references which treat ...
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### Limit of the Schröder numbers ratio

I have been playing around with interesting integer sequences and came across Schröder number which defines the number of lattice paths of n x n grid. The recurrence formula to calculate these numbers ...
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### Another generalization of parity of Catalan numbers

Recently, a question by T. Amdeberhan gathered up many enjoyable proofs that a Catalan number $C_n$ is odd if and only if $n=2^r-1$. Noam D. Elkies' answer considered $F=\sum_{n=0}^\infty C_n x^{n+1}$....
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### A variant of node-disjoint path problem

Given a graph $G$, I want to find $2$ (or $k$) node-disjoint paths with minimum total cost (or minimum maximum cost). The problem is a classical problem, but I have the following non-trivial setting. ...
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### Does every $4$-connected nonplanar graph contain a $K_5$-minor?

By Kuratowski's theorem, every nonplanar graph contains a (topological) minor of $K_5$ or $K_{3,3}$. But I observed that every time I construct a $4$-connected nonplanar graph, it always contains not ...
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### On the sum $\sum_{i=1}^{m}\binom{m}{i}\frac{(-1)^{i+1}}{2^i-1}$

I want to find out the result of the following summation if some (maybe big) positive integer $m$ is given. $$\sum_{i=1}^{m}\binom{m}{i}\frac{(-1)^{i+1}}{2^i-1}$$ It doesn't seem much possible to ...
Let $p, q \in \mathbb{Z}$. Let $\operatorname{wt}(n)$ is A000120, number of $1$'s in binary expansion of $n$ (or the binary weight of $n$) and n=2^{t_1}(1+2^{t_2+1}(1+\dots(1+2^{t_{wt(n)}+1}))\dots)$... 1answer 158 views ### Controlling iterated sum sets of "most" of$A+B$I am reading Tao-Vu book on Additive combinatorics and came across the following lemma. I know that it is better to ask this question on MathStack but I asked few questions before and no one answered ... 0answers 137 views ### Combinatorial interpretation of a determinant This is a continuation of Worpitzky-like identities?. Let$ r_k(x)=\prod_{j=1}^k {(\frac{x+j}{j}})^{\min(j,k-j)}.$As Sam Hopkins has remarked$r_k(x)$is the number of plane partitions in a$ \...
Let $G$ be a subcubic graph with only vertices of degree 1 or degree 3. Suppose that $G$ has an edge coloring $\varphi$ using colors from $\{1,2,3,4,5\}$ such that each edge is colored with a set of ...