# 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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### How many ways to win a game between two teams with arbitrary player skills

Suppose we have $n\geq 4$ players $p_1,\cdots,p_n$ of a game between two teams: team $A$ and team $B$ (disjoint sets, each with two or more players, so that $|A|+|B|=n$). Assume that each player $p_i$ ...
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### What d.o. $\sum_i f_i(z)\partial_z^i$ correspond to subalgebras $M$ in polynoms $C[x_i]$ being Langlands dual to motive of $Spec(M) \to X$?

Briefly: The question is about presenting explicit examples of the construction discussed in the recent MO question "Relation between motives and geometric Langlands" and Will Sawin's asnwer ...
1 vote
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### Permutation graph with insert-and-shift

Motivation. I am working with a database software that allows you to sort the fields of any given table in the following peculiar way. Suppose your fields are numbered $1,\ldots, 18$. Next to every ...
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### Number of regions created by r hyper-planes in n-dimensional space [closed]

I found this formula for calculating maximum number of regions created by r hyper-planes in n-dimensional space (n<=r) ...
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### Formula for partitions of integers with no subpartition being a partition of $t$

When it comes to partitions, I know we can impose some modest restrictions (maybe even a couple) on the partitions and obtain counting formula, but I would like to impose some more serious constraints ...
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### Homology of independence complex after removing a vertex

Let $G$ be a chordal graph, $I(G)$ be its independence complex, and $v \in V(G)$ be a simplicial vertex (that is, $v$'s neighborhood is a clique). Is there a way to relate the homology of $I(G)$ and ...
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### Implementation of Friedman's algorithm of reconstructing simple polytopes

In Finding a Simple Polytope from Its Graph in Polynomial Time, Friedman gave a polynomial time algorithm on reconstructing a simple polytope from its graph. Has this algorithm been actually ...
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### Generating all possible subsets in order of sum

Given a set of positive integers, I am looking for method to algorithmically generate all possible subsets in order of their sum. Because the the count of possible subsets is exponential ($2^n$), it ...
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### Asymptotic bound on the number of simple connected graphs of bounded degree

I have posted this question on Mathematics, but unfortunately no luck so far. Let $\mathcal{G}$ be the family of simple connected graphs on $n$ vertices, where each graph has more than $m$ edges, and ...
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### Bijections on the set of integer partitions of $n$

I am looking for natural bijections from the set of integer partitions of $n$ to itself. Of course, I have no definition of natural, but for the purpose of this question it suffices that it appears ...
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### Estimating the cardinality of the set of conjugacy classes of subgroups in a finite group of given order

1. Let $G$ be a finite group of order $n$. I need an estimate for the number $c$ of conjugacy classes of subgroups $D\subseteq G$. Note that any subgroup of $G$ contains $1_G$, and so the set of all ...
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$\newcommand\calF{\mathcal{F}} \def\cupdot {\stackrel{\bullet}{\cup}} \def\minusdot {\stackrel{\bullet}{\setminus}}$This post presents a conjecture that we have with some colleagues. It is about ...
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1 vote
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### Tuples of natural numbers with no mutual divisibility and large reciprocal sums

Standard apology in case this is something simple, as I'm not a number theorist. Let $E_1, \dots, E_n$ be disjoint finite sets of natural numbers, such that for any $a_1 \in E_1, \dots, a_n \in E_n$, ...
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### If $G$ is a connected bipartite graph, then the edge ideal $I(G)$ is normally torsion free

I am studying the paper "On the Ideal Theory of Graphs" by Simis, Vasconcelos and Villarreal, Journal of Algebra 167, No. 2, 389-416 (1994), MR1283294, Zbl 0816.13003. I got stuck at theorem ...
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### Is the group of translations of an affine plane always commutative?

$\DeclareMathOperator\Dil{Dil}\DeclareMathOperator\Trans{Trans}\DeclareMathOperator\Col{Col}$An affine plane is a set of points $X$ endowed with a family $\mathcal L$ of subsets of $X$, called lines, ...
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### Given $F[N,M]=\sum_{m=0}^{N-1}(-1)^{N-1-m}(m+1)^M)/(m!(N-1-m)!)$, show $F[N,N-1]=1$ and $F[N,M]=0$ for $M<N-1$ [closed]

The function defined by $$F[N,M]=\sum_{m=0}^{N-1}\frac{(-1)^{N-1-m}(m+1)^M}{m!(N-1-m)!}$$ where $N,M$ are positive integers. I want to show $$F[N,N-1]=1,\ F[N,M]=0$$ for $N>2$ and $M<N-1$. ...
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### Do the dual graphs of hyperplane arrangements admit Hamiltonian paths?

Consider a simple hyperplane arrangement $H_1,\cdots,H_n$ in the Euclidean space $\mathbb{R}^d$. By "simple" we mean any $k$ hyperplanes in $\{H_1,\cdots,H_n\}$ intersect in codimension $k$. ...
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### Can we say a partial order set is 2-dimensional if its comparability graph does not contain an asteroidal triple?

I think it is true that if the comparability graph of a poset contains an asteroidal triple then it is at-least 3 dimensional. I want to know if the converse is true, i.e. if there exists no ...
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### How many possible Venn diagrams are there for given cardinalities of the sets?

Consider $n$ non-empty finite sets with cardinalities $c_1$, $\ldots$, $c_n$. How many possibilities are there for the Venn diagram of these sets? (I'm surprised I didn't find the answer with Google). ...
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### Is the partition tiling relation transitive?

The following is motivated by an (as of yet) unanswered question on optimal colorings of graphs. I am convinced that the question below has a positive answer in $\newcommand{\ZF}{{\sf (ZF)}}\ZF$, but ...
1 vote
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### Representing natural numbers as sums of distinct prime powers

I am investigating whether every natural number $n > 18$ can be represented as a sum $p_1^{m_1} + \dots + p_k^{m_k}$, where $p_1, \dots, p_k$ are distinct primes, and $m_1, \dots, m_k$ are distinct ...
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### Bijective proof for an identity concerning Stirling numbers of second kind

Let $\genfrac{\{}{\}}{0pt}{}{n}{k}$ the Stirling number of second kind, where $k$ is the number of parts in the partition. If we take the identity that transforms the polynomial base $x^k$ into the ...
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### Subspaces of $\mathbb{F}_2^N$ containing many pairs of far apart vectors

Let $S$ be a subset of vectors in $\mathbb{F}_2^{3n}$ having Hamming weight $n$. Suppose that $S$ contains $m$ pairs of vectors having disjoint supports (that is, they are at Hamming distance $2n$ ...
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