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.
11,024 questions
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Another version of Sidorenko's conjecture(?)
I would like to ask a question about Sidorenko's conjecture. Here is the background of my question:
Quasi-random graphs
A sequence of graphs $(G_n)$ is called quasi-random if it satisfies certain ...
- 349
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36
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Construct a maximum matching from a minimum vertex cover in bipartite graph?
Konig's theorem in graph theory says that for a bipartite graph $G$, the size of maximum matching in $G$ is equal to the size of minimum vertex cover of $G$.
Typically, one of the proofs is to ...
- 281
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27
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Projection onto polytopes as tropical polynomial
Let $C$ be a convex polytope in $\mathbb{R}^n$ with $m$ extremal points. Let $p\in \{1,2\}$.
Can the $\ell^p$-projection $\Pi_C:\mathbb{R}^n\to C$
$$
\Pi_C(x) \in \operatorname{argmin}_{z\in C}\, \|x-...
- 364
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1
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Equivalence of sequences related to A033264
Let $a(n)$ be A033264 (i.e., number of blocks of $\{1,0\}$ in the binary expansion of $n$). Here
$$
a(4n) = a(4n+1) = a(2n), \\
a(4n+2) = a(n)+1, \\
a(4n+3) = a(n), \\
a(0) = 0.
$$
Let
$$
\ell(n) = \...
- 4,938
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votes
4
answers
1k
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Generalization of a mind-boggling box-opening puzzle
Motivation. Suppose we are given $6$ boxes, arranged in the following manner:
$$\left[\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right]$$
Two of these boxes contain a ...
- 48.9k
2
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1
answer
431
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Shadows of partitions of lcm
$\DeclareMathOperator\lcm{lcm}$Fix an integer $n\geq1$. Denote the least common multiple $L_n=\lcm(1,2,\dots,n)$.
QUESTION. Is the following true? For each integer partition $\lambda=(\lambda_1,\...
- 43.2k
7
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1
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195
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The tilde species
Consider a combinatorial species $F$, that is, an action of the symmetric group $\mathfrak S_n$ on a finite set $F[n]$. Recall that the elements of $F[n]$ are called structures. Furthermore, recall ...
- 5,822
2
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1
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276
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Estimating a sum over set partitions
Let $[n]:=\{1,\dots,n\}$. Fix a set partition $\rho$ of $[n]$, with an abuse of notation we shall use $\rho\vdash [n]$.
I would like to estimate the following alternating sum.
QUESTION. Is this true?
...
- 43.2k
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1
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90
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Characterizing the family of maximal cliques of a cograph
Preamble #1
There are two common equivalent definitions of cographs:
the smallest class that includes $K_1$ and is closed under disjoint union and complementation (or join);
the finite $P_4$-free ...
- 113
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89
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Test for odd prime triples in a $2p-1$ progression
Let $a(n)$ be A057326 (i.e., first member of a prime triple in a $2p-1$ progression).
Let $b(n) = B$ after $n-1$ iterations where we start with $A=n, B=1$ and for $i$ from $1$ to $n-1$ simultaneously ...
- 4,938
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172
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How many maximal length snakes are there?
This problem was motivated by the classic phone game Snake.
Consider the square grid graph with vertex set $V := \{1, \dots, N\}^2$, for fixed odd positive integer $N$, and an edge between $(x, y)$ ...
- 6,155
4
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228
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A definite integral of a hypergeometric series related to the enumeration of fusenes
If my calculations are correct, the number of (not necessarily planar) fusenes of perimeter $2n$ grows asymptotically like $\mu^n$ where
\begin{equation}
\mu = \frac{6}{\int_0^{1/6} H(t)\mathrm{d}t} = ...
- 3,927
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90
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Definition of Loop in an Oriented Matroid
I had posted this on Stackexchange because I don't believe this is a particlarly difficult question, but there were no answers, so I'm posting it on here now.
I just had a quick question about the ...
- 61
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91
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Finite projective geometry and the Krasner hyperfield
The Krasner hyperfield is an algebraic structure of two operations on $K=\{0,1\}$ called $+\colon K\times K\to \mathcal{P}(K)$ and $\cdot\colon K\times K\to K$ with
$0+0=0$
$0+1=1+0=1$
$1+1=\{0,1\}$
...
- 10.4k
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111
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Map between Weyl modules in terms of Young tableaux
The irreducible algebraic representations of $\text{GL}_n$ over the complex numbers are given by highest weight representations of dominant weights $\lambda=(k_1,k_2,\ldots,k_n): k_1 \ge k_2 \ge \...
- 461
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145
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Is it easier to exit a box to the right of a box in $\mathbb{Z}^2$ if I remove some edges to the left?
Suppose that I am given the graph $G = (V,E)$ where
$V = \{ 1, 2, \dots 2N+1 \} \times \{ 1, 2, \dots 2N+1 \} $
and there is an edge between two vertices $(n,m)$ and $(n',m')$ if and only if
$\vert n-...
- 1,054
4
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answers
124
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LIS-based permutation property
Let $S_n$ be the set of all permutations of $\{1, \ldots, n\}$, thereafter treated as integer sequences. Let $A_n$ be the set of all such permutations $\sigma \in S_n$ that we can choose two ...
- 5,590
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113
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Numbers of positive terms in polynomials equal A069999
Let $a(n)$ be A069999 (i.e., number of possible dimensions for commutators of $n \times n$ matrices; it is independent of the field). OEIS states that no generating function is known.
Let $P(n,k)$ be ...
- 4,938
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128
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The smallest dihedral angle of convex polyhedrons
Given a set of points $\{x_{k}\}_{k=0}^{m} \subset \mathbb{R}^n$, is it always possible to find a constant $c=c(m,n)>0$, depending only on the dimension $n$ and the number $m$, such that, after ...
- 705
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30
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An algorithm to decompose a directly indecomposable permutation group into a wreath product
I am considering the following two binary operations on permutation groups:
the direct product, and
the wreath product.
It turns out that there is an efficient algorithm to factor a given ...
- 5,822
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0
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47
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Harmonic numbers multifold convolution
I have a question. If I define the multifold convolution of Harmonic numbers as
$\sum_{n_1=1}^{\infty} \cdots \sum_{n_k=1}^{\infty} H_{n_1} \cdots H_{n_k} \mathbf{1}_{\{n\}}(n_1+\dots+n_k)$
for the $k$...
- 471
1
vote
1
answer
60
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Optimal transport for sum of two costs
Let $X$ be a finite set and $\sigma_0$, $\sigma_1$ two fixed measures on $X$ with $\sigma_0(X)=\sigma_1(X)$. A transportation plan is a measure $\mu$ on $X\times X$ whose projections on the first and ...
- 1,074
4
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0
answers
91
views
Reference for fact about flags of vexillary permutations
Vexillary permutations are an important family of permutations in Schubert calculus. There are several definitions, for example that they avoid the pattern 2143.
Recall the Lehmer code of a ...
- 1,989
2
votes
2
answers
210
views
Rank of adjacency matrix of a graph on a sphere all of whose faces have four vertices
Let $G$ be a graph drawn on the sphere such that every face of $G$ has exactly four vertices. Question: can anything be said about the rank of the adjacency matrix of $G$ in terms of other (preferably ...
- 2,964
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4
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1k
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Counting with trees
Let $\mathcal{U}_n$ denote the set of unrooted unlabelled trees with $n$ edges. For $T\in\mathcal{U}_n$, let $1^{u_1}2^{u_2}\cdots n^{u_n}$ be its degree distribution, that is, $u_i=\#$ of vertices ...
- 43.2k
6
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164
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Can one naturally transform Tamari lattices into distributive lattices with the same number of elements?
Many of the zillions of combinatorial objects counted by Catalan numbers come with various lattice structures.
The $n$th Tamari lattice $T_n$, as originally defined, is the lattice of all those maps $...
- 18.4k
5
votes
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185
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Gaps in sumsets and difference sets
a) Let $S\subset \{1,2,\dotsc,N\}$ be a fairly thick set (with at least $N^{1-\epsilon}$ elements, say). Suppose that the intersection of, say,
$$3 S - 3 S = \{a_1+a_2+a_3-(a_4+a_5+a_6): a_1,\dotsc,...
- 20.2k
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1
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123
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Petersen graph does not have a nowhere-zero 4-flow
I try to prove that the Petersen graph does not have a nowhere-zero 4-flow (i.e., over $\mathbb{Z}_4$), but I don't know how a proof could work...
I'm happy about every hint, thank you in advance!
6
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3
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550
views
Conjecture about commutative semigroups
Conjecture: given any commutative semigroup $S$ of order $n \ge 4$, there exist $a, b \in S$ with $a \ne b$, an integer $m \ge \lfloor (n-1)/2 \rfloor$, and two $m$-element subsets $X = \{x_1, \ldots, ...
- 1,000
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115
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A slight strengthening of the union-closed sets conjecture
Consider a union-closed family $\mathcal{F}=\{A_1,…,A_n\}$ of $n \gt 1$ finite sets.
I was not able to find a counterexample to the following conjecture:
there exist two sets $A,B \in \mathcal{F}$ ...
- 1,000
1
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1
answer
106
views
Iterated optimal transport
Suppose we are interested in two consecutive transport plans (in the Kantorovich formulation). That is, we are given finite sets $X$, $Y$ and $Z$, endowed with probability measures $\mu_X$, $\mu_Y$ ...
- 13
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0
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82
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Generating functions related to generating function of Catalan numbers
Let $C_n$ be A000108 (i.e., Catalan numbers). Here generating function is $C(x)$ such that
$$
C(x) = \frac{1-\sqrt{1-4x}}{2x}.
$$
Let $a(n)$ be an integer sequence with generating function $A(x)$ such ...
- 4,938
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84
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Generate two bijectively mapped sets subject to certain conditions on choice of elements
$\DeclareMathOperator\setsum{setsum}$Let there be two sets of numbers of size $n$ each given by $S_1, S_2$.
Let there be a one-to-one onto mapping $f: S_1 \rightarrow S_2$.
Let us denote the sum of ...
1
vote
1
answer
197
views
Probability distribution on Python-dictionary-like objects?
I would like to examine information-theoretical properties of random variables that take as values objects which are akin to dictionaries in the Python programing language.
That is, each sample of the ...
- 11
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"Center" of a set of binary strings
For a finite set $A$ in a metric space define its diameter ${\rm diam} (A)$ as the maximal distance between two points in $A$ and radius $r(A)$ as the minimal radius of a ball containing $A$. ...
2
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1
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100
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Clique number and a special partition
Let $G=(V,E)$ be a finite, simple, undirected, connected graph, and let $\omega(G)$ denote its clique number. Assume that $G$ has a partition into $m$ independent subsets $U_1,\dots, U_m$ such that ...
- 21
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226
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Expanders except for commutativity?
What would you call a graph that is an expander except for commutativity, in the following sense?
Say that, from every vertex, you have $d$ edges ($d$ large) labelled $x_1,\dotsc, x_d$. Say that your ...
- 20.2k
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65
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Algorithm and equivalent recursion for A258173 (related to Dyck paths)
Let $a(n)$ be A258173 i.e. sum over all Dyck paths of semilength $n$ of products over all peaks $p$ of $y_p$, where $y_p$ is the $y$-coordinate of peak $p$.
A Dyck path of semilength $n$ is a $(x,y)$-...
- 4,938
2
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1
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210
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Maximum number of ones in a full rank matrix with a restriction
Consider $n \times n$ binary matrices. I am interested in the largest number of ones possible in an $n \times n$ binary matrix with full rank over the field of integers mod 2 with the following ...
- 3,377
2
votes
1
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173
views
What do we know about the action of the symmetric group by conjugation on the set of permutation groups?
Motivation:
I have co-authored a package for sagemath to compute with combinatorial species, also known as sequences of group actions of the symmetric groups. In an effort to find good tests for that ...
- 5,822
5
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1
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345
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Are parabolic Kazhdan-Lusztig polynomials truncations of the usual Kazhdan-Lusztig polynomials?
Let $(W,S)$ be the affine Weyl group associated to a simple root system.
For $x,y \in W$ we have the usual Kazhdan--Lustig polynomials $h_{y,x} \in \mathbb{Z}[v]$ in Soergel's normalisation, and if ...
- 10.5k
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0
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60
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Algorithm for $q$-Bell numbers
Let $T(n,k)$ be A126347 (i.e., triangle, read by rows, with row polynomials $B(n, q)$). Here
$$
B(n, q) = \sum\limits_{k=0}^{n-1}\binom{n-1}{k}B(k, q)q^k, \\
B(0, q) = 1.
$$
Start with vector $\nu$ of ...
- 4,938
7
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208
views
How biased is $(x_i x_j)_{i,j}$, $x_i\in \mathbb{F}_2$?
Let $N = \frac{n (n-1)}{2}$. Let $V$ be the $N$-dimensional vector space over $\mathbb{F}_2$ consisting of tuples $(x_{(i,j)})_{1\leq i <j \leq n}$, $x_{(i,j)}\in \mathbb{F}_2$. Let $S$ be the set ...
- 20.2k
-2
votes
1
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298
views
Is polynomial not bijective, on this finited field?
Let $(a,b,c) \in \mathbb F_p,p=2^{127}-1$ and $P(x)=x^{16}+ax^{11}+bx^{5}+c$.
Is it true that $P(x)$ not bijective on $\mathbb F_p$?
I have asked this question here (*), but no answer.
(*) : https://...
- 4,074
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0
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98
views
Number of tetrahedra inside a sphere with boundary A
I understand, that there are some combinatorial problems which are not yet solved regarding gluing triangulations in 3D. At least last time I checked, it was not yet known exactly how many ...
- 183
1
vote
1
answer
178
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Algorithm for A127782
Let $a(n)$ be A127782 (i.e., an integer sequence with generating function $A(x)$ such that $A(x)=1+xA(x+x^2)$). Here
$$
a(n) = \sum\limits_{k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor} \binom{n-k-1}{...
- 4,938
23
votes
4
answers
2k
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Identity for an infinite product
Here is an experimental "result" exhibiting the difference of two (formal) infinite products that "almost factorizes".
QUESTION. Is this true?
$$\prod_{n\geq1}(1+x^{2n-1})^{24} - \...
- 43.2k
4
votes
1
answer
195
views
Optimal partition of $n$ points
Given an integer $n$, and 3 real sequences $\{x_1, \dots, x_n\}, \{y_1, \dots, y_n\}$ and $\{w_1, \dots, w_n\}$ with $x_k, y_k, w_k > 0$, for all $k \in \{1, \dots, n\}$. For a fixed $p < n$ ...
- 43
1
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0
answers
151
views
Decide if a group is abelian
Let $G = \langle X: R\rangle$ be a finitely presented group. The following problem seems very natural to me, yet I cannot find any reference for it: Decide if $G$ is abelian or not.
With a reduction ...
- 11
3
votes
0
answers
61
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Is this bipartite equivalent of 1-walk-regular graphs known?
A graph $G$ is 1-walk-regular if
for each vertex $v$ the number of closed walks of length $\ell$ starting (and ending) at $v$ depends only on $\ell$ but not on $v$.
for each edge $vw$ the number of ...
- 13.6k