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      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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      Calculating the number of solutions of integer linear equations

      Let $N$ be a natural number. Consider the following set of matrices whose entries are non-negative integers: $$X_N:=\left\{(c_{ij})_{i,j=1}^4\in M_4(\mathbb{Z}_{\geq 0})\bigg| \sum_j c_{1j} = \sum_i ...
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      32 views

      Constructing Group Divisible Designs - Algorithms?

      I am starting my research on group divisible designs this year and I wonder if there are any algorithms/software that help with constructions. Thank you
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      60 views

      Expected number of times of choosing a word out of a given vocabulary when words are grouped into non-overlapping bins

      Two players (player C and player G) are playing a (modified) word guessing game. Both players share the same vocabulary $V$ and words in $V$ are grouped into $K$ bins, denoted as $b_1$, $b_2$, ..., $...
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      40 views

      Infinite products from the fake Laver tables-Now with no set theory

      We say that a sequence of algebras $(\{1,\dots,2^{n}\},*_{n})_{n\in\omega}$ is an inverse system of fake Laver tables if for $x\in\{1,\dots,2^{n}\}$, we have $2^{n}*_{n}x=x$, $x*_{n}1=x+1\mod 2^{n}$,...
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      89 views

      How many solutions to $p_i|a_{i,1} p_1 + \dotsc + a_{i,n} p_n$?

      Consider a system of $n$ divisibility conditions on $n$ prime variables: $$p_i|a_{i,1} p_1 + \dotsc + a_{i,n} p_n,\;\;\;\;\;1\leq i\leq n,$$ where $a_{i,j}$ are bounded integers. How many solutions ...
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      1answer
      88 views

      Sufficient conditions for the coefficients of a generating function to dominate those of its square

      Let $f(z)$ be a generating function (so in particular, its power series coefficients are nonnegative). I am interested in conditions which would ensure that for every $n$, the coefficient of $z^n$ in $...
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      combinatorics question with a deck of cards [on hold]

      I have a standard deck of 52 cards. How do I find the number of hands of 13 cards that contain 4 cards of the same rank? (A,2-10, J,Q,K) I'm starting off with $${48\choose 9}^{13}$$ but I am unsure ...
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      1answer
      133 views

      Alternating binomial-harmonic sum: evaluation request

      Let $H_k=\sum_{j=1}^k\frac1j$ be the harmonic numbers. QUESTION. Can you find an evaluation of the following sum? $$\sum_{a=1}^b(-1)^a\binom{n}{b-a}\frac{H_{b-a}}a.$$
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      148 views
      +200

      On a problem for determinants associated to Cartan matrices of certain algebras

      This is a continuation of Classification of algebras of finite global dimension via determinants of certain 0-1-matrices but this time with a concrete conjecture and using the simplification suggested ...
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      42 views

      Are contraction-sensitive graphs necessarily vertex-transitive?

      We say that a finite, simple, undirected graph $G=(V,E)$ is contraction-sensitive if collapsing any $2$ non-adjacent points increases the Hadwiger number. An example of such a graph is the icosahedron....
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      1answer
      75 views

      Algorithm to compute the convex hull of a set of $m$ possibly intersecting convex polygons in the plane

      I am trying to find an algorithm to compute the convex hull of a set of $m$ possibly intersecting convex polygons in the plane, with a total of $n$ vertices. Let $h$ denote the number of vertices on ...
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      1answer
      112 views

      Determinant of an “almost cyclic” matrix

      Let $n\geq 3$, let $Z$ be the matrix of the cyclic shift (the companion matrix of $X^n-1$), and for $\mathbf{d}\in \mathbb{C}^n$ let $\operatorname{diag}(\mathbf{d})$ be the diagonal matrix with $\...
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      2answers
      91 views

      Does any long path in a planar graph contain one of O(n) k-tuple of vertices?

      My question is a bit related to both the container method and shallow cell complexity. Let's start with that the number of length $\ell$ paths (where $\ell$ denotes the number of vertices of the path!)...
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      0answers
      28 views
      +50

      Does the bounded branching/log depth dihotomy hold for rooted trees?

      Let $T$ be a rooted tree. For any subtree $T' \subset T$ define its leaf-weight $lw$ as the number of leaves of $T'$. Further, for $T' \subset T$ define the branch-depth of a node $v \in T'$ as the ...
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      53 views

      Minimizing union of overlapping rectangles

      Believe it or not, this has something to do with making triangle-free graphs bipartite... We have a collection of $k$ axis parallel rectangles with side lengths $(a_i,b_i)$. We want to arrange them (...

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