All Questions
Tagged with combinatorics or co.combinatorics
11,023 questions
3
votes
0
answers
65
views
A combinatorial Dyson-Schwinger equation, tree diagrams, and compositional inversion of a Laurent series
In "Tree hook length formulae, Feynman rules and B-series", Bradley Jones and Karen Yeats state on pg. 9:
Combinatorial Dyson-Schwinger equations are functional equations with solutions in
$...
- 10.5k
1
vote
0
answers
72
views
How to understand "sparse graph limits"
For an $n$-vertex graph $G$, we say it is a sparse graph if $e(G)=o(n^2)$. Otherwise if $e(G)=\theta (n^2)$, we say it is a dense graph.
For a sequence of dense graphs $G_1,G_2,\dots,$ we know that it ...
- 349
7
votes
1
answer
445
views
What is known/expected on the co-growth series of the braid group?
The co-growth series of finitely generated group with respect to generating set $S$ is generating function for the number of words of length $n$ which are equal to 1 in the group.
Its studies ...
- 1,819
11
votes
2
answers
425
views
Maximization of a cubic form over the $14$-dimensional sphere
For any integers $i$ and $j$ such as $1\le i<j\le6$, let $x_{ij}$ be a nonnegative real number.
Is it true that, given the condition
$$\sum_{1\le i<j\le6}x_{ij}^2=1,$$
the sum
$$\sum_{1\le i<...
- 109k
15
votes
1
answer
2k
views
Sublattices of Young's Lattice
Young's Lattice is the lattice of inclusions of Young tableaux, or integer partitions.
In R. Stanley's Enumerative Combinatorics Vol. 1, Proposition 1.4.4., there is a generalization of integer ...
- 2,183
2
votes
0
answers
61
views
Algorithm for main diagonal of integer coefficients associated with Schroeder numbers
Let $T_q(n, k)$ be an integer table such that
$$T_q(n, k) = \begin{cases}
1 & \textrm{if } n = 0 \vee k = 0 \\
qT_q(n-1, n-1) + T_q(n, n-1) & \textrm{if } n = k > 0 \\
T_q(n, k-1) + T_q(n-1,...
- 7,226
4
votes
1
answer
189
views
Equation in the conjugacy class of a free group
I will pose the question in the form in which it originally appeared to me:
Let $a,b,c,d$ be different letters in a finite alphabet $\mathcal{Z}$. Let $Q$ and $R$ be finite words with letters from $\...
- 436
4
votes
1
answer
130
views
Intersecting algorithm for A065601
Let $a(n)$ be A065601 (i.e., number of Dyck paths of length $2n$ with exactly $1$ hill). Here
$$
a(n) = \frac{1}{2(n+1)}((3n-2)a(n-1) + 2(9n-19)a(n-2) + 4(2n-3)a(n-3)), \\
a(0) = a(2) = 0, a(1) = 1.
$$...
- 34.3k
7
votes
0
answers
166
views
Examples of finitary problems/theorems of high logical complexity? [duplicate]
Generally, number theoretic conjectures which are well-known and easy to explain are either obviously $\Pi^0_1$ or $\Pi^0_2$, which is to say, their truth can be decided by a single membership query ...
- 1,452
1
vote
0
answers
55
views
Combinatorial structure of the entanglement spectrum and quantum error correction in finite vector spaces
Let $V$ be a finite-dimensional vector space over $\mathbb{C}$ with dimension $d$. Consider a subspace $S \subset V^{\otimes n}$ representing the code subspace of a quantum error correcting code. We ...
- 63.9k
4
votes
1
answer
112
views
On a number of compositions of $n$ into positive triangular numbers
Let $a(n)$ be A023361 (i.e., number of compositions of $n$ into positive triangular numbers). Here
$$
a(n) = \sum\limits_{i \geqslant 1, \frac{i(i+1)}{2}\leqslant n} a(n-\frac{i(i+1)}{2}), \\
a(0) = 1....
- 34.3k
7
votes
1
answer
443
views
Road map and references for combinatorial Hodge theory
I'm a PhD student. I'm familiar with graduate level algebraic geometry and toric varieties.
I wanted to know a road map for getting into combinatorial Hodge theory and other prerequisites that I'll ...
- 1,046
3
votes
1
answer
178
views
Algorithm for the sum with binomial coefficients and Bell numbers
Let $a(n)$ be A000110 (i.e., Bell or exponential numbers: number of ways to partition a set of $n$ labeled elements).
Let $b(n)$ be A355247 (i.e., expansion of exponential generating function $\exp(2(\...
- 34.3k
0
votes
1
answer
81
views
If the matroids associated to two finite subsets of the same vector space are isomorphic, are these two finite subsets linearly equivalent?
Let $E$ be a finite subset of ${\mathbb{F}_2}^n$, the $n$-dimensional vector space over the finite field $\mathbb{F}_2$ of $2$ elements. Let $M_E$ denote the associated matroid on $E$ where the ...
3
votes
1
answer
159
views
Proving that two sequences of polynomials defined over partitions are inverse to each other
For any fixed $c>0$ consider the polynomials
\begin{align*}
& p_n(X_1,X_2,\ldots) := \frac{n!}{c} \sum\limits_{b=1}^n \frac{c^b}{b!(n+1-b)!} \sum\limits_{\substack{l_1,\ldots,l_b \geq 1 \\ ...
- 10.5k
1
vote
0
answers
56
views
Find a set satisfying a specific condition
Let $S$ be a set of numbers of size $n$.
Let $M_S$ be the set of all multisets of size $n$ made of elements from $S$, with at least one element repeated. For example, if $S=\{1, 2, 3\}$, then $M_S = \{...
2
votes
0
answers
46
views
On A088352 as an antidiagonal sums of A129179
Let $a(n)$ be A088352. Here $a(n)$ is an integer sequence with generating function $A(x)$ such that
$$
A(x) = \cfrac{1}{1-x-\cfrac{x^2}{1-x^3-\cfrac{x^4}{1-x^5-\cfrac{x^6}{1-x^7-\cfrac{x^8}{\ddots}}}}}...
- 12.9k
0
votes
1
answer
82
views
Is there a stiff graph that is not a core?
By a graph, I mean a simple, undirected graph with no loops. A graph homomorphism $f : G \to H$ is a function from the vertexset of $G$ to the vertexset of $H$ such that if $u$ and $v$ are adjacent ...
- 331
4
votes
1
answer
190
views
Is the transpose of an infinite Hadamard matrix also Hadamard?
Let $\omega$ be the set of non-negative integers. If $f,g:\omega\to\{-1,1\}$ are maps, then we say $f,g$ are almost orthogonal if there is a positive integer $C_0\in \omega$ such that for all $n\in\...
- 9,720
0
votes
2
answers
251
views
Compute the average path weights of paths with the same path length in a directed acyclic graph (DAG)
Given a weighted directed acyclic graph (DAG) $G=(V,E)$ with each edge $e\in E$ has a non-negative weight $w(e)$. For a path $p=(e_1,e_2,\dotsc,e_n)$ in $G$, define the path weight as : $w(p)=\sum_{i=...
CommunityBot
- 1
0
votes
0
answers
54
views
Functional equations with coupled arguments and additive sructure
Let $G$ be a locally compact abelian group and let $f: G \to \mathbb{R}^+$ be a continuous function satisfying the functional equation
$$f(x + \phi(y)) + f(y + \phi(x)) = 1 + f(x+y)$$
for all $x, y \...
59
votes
2
answers
4k
views
For a finite set A of positive reals, prove that the set A + A - A contains at least as many positive as negative elements
I am currently working on a proof that would need to use the following theorem that I cannot prove:
"Let $A$ be a finite set of positive real numbers. Then, the set $A + A - A$ contains at least ...
- 3,499
4
votes
0
answers
66
views
Convergence of graph geodesics to geodesics on metric spaces
Let $(X,d)$ be a compact length space metric space $\mathbb{X}_{\delta}$ be a $\delta$-packing on $X$ and, for every $k\in \mathbb{N}_+$, let $G_{k,\delta}=(\mathbb{X}_{\delta},\mathcal{E}_k,W_k)$ ...
- 63.9k
6
votes
0
answers
155
views
How to characterize this condition for commutative squares in $\Delta$
In the simplex category $\Delta$ we have the situation, that
pullbacks exist for cospans $[a] \xrightarrow{\alpha} [n] \xleftarrow{\beta} [b]$ in $\Delta_\text{mono}$ and
pushouts exist spans $[a] \...
10
votes
1
answer
207
views
Generating function for A225114
Let $a(n)$ be A225114 (i.e., number of skew partitions of $n$ whose diagrams have no empty rows and columns).
Let $b(n)$ be an integer sequence with generating function $B(x)$ such that
$$
B(x) = \...
- 14k
0
votes
1
answer
71
views
Forced monochromatic pairs in graphs
Starting point. Consider the "$V$-graph" on the vertex set $\{1,2,3\}$ and let the edges be $\{1,2\}$ and $\{2,3\}$. This graph is clearly bipartite. It is a trivial observation that ...
- 7,226
18
votes
6
answers
9k
views
calculating Möbius function
I wonder if there is any efficient way to calculate Möbius function for a array of number 1:1000000
http://en.wikipedia.org/wiki/M%C3%B6bius_function
2
votes
0
answers
71
views
Are the ranks of the following matrices given by these simple expressions?
The question itself is formulated in the title, so below I specify the matrices and expressions mentioned there. In case if this is something known or can be easily deduced from something known, this ...
- 321
2
votes
1
answer
310
views
Generating function for A300483 (related to Chebyshev polynomial of first kind)
Let $a(n)$ be A300483. Here
$$
a(n) = 2\int\limits_{t \geqslant 0}T_n\left(\frac{t+1}{2}\right)\exp(-t)\,dt.
$$
where $T_n(x)$ is $n$-th Chebyshev polynomial of first kind.
Let $b(n)$ be an integer ...
- 34.3k
10
votes
1
answer
625
views
Generating function for A261041
Let $a(n)$ be A261041 (i.e., number of partitions of subsets of $\{1,2,\dotsc,n\}$, where consecutive integers are required to be in different parts).
Let $b(n)$ be an integer sequence with generating ...
- 34.3k
5
votes
0
answers
190
views
Number of discrete Lipschitz functions with given Lipschitz constant
Fix $T, K, N \in \mathbb Z_+$. How many distinct Lipschitz functions $f: \{0, \dots, T\} \to \mathbb Z$ are there with Lipschitz constant $K$, and supremum norm at most $N$ satisfying $f(0) = 0$?
In ...
- 6,205
10
votes
2
answers
398
views
Length of optimal play in Hex as a function of size
Consider Hex on an $n \times n$ board without a swap rule, so that the first player wins. Assume the first player tries to minimize the length of the game, and the second player tries to maximize the ...
- 631
5
votes
0
answers
137
views
Looking for a certain finite lattice
I don't think it actually exists, and it should be difficult proving that it doesn't (some background here), but is it possible to build a finite lattice $L$ where the only meet-irreducible elements ...
- 1,000
3
votes
1
answer
460
views
Generating uniquely $k$-optimal point sets
This question is motivated by the observation that finding an optimal tour through a set of points in the Euclidean plane is especially simple, if the points are in convex configuration and, that the ...
CommunityBot
- 1
3
votes
1
answer
168
views
Maximal zero-sum free sequences of $C_3^n$
I am working on the Davenport constant for groups, $D(G)$, which is the minimal number $d$ such that every sequence or multiset of $d$ elements of the group $G$ always contains some non-empty zero-sum ...
3
votes
0
answers
81
views
Can we remove the restriction on a parameter in Talagrand concentration inequality?
Recently I am trying to use Talagrand concentration inequality to do something on graphs. I find a version from the book of Molloy and Reed ''Graph Colouring and Probabilistics Method''. I attached a ...
- 4,219
2
votes
0
answers
95
views
Finding a well-spaced interval of natural numbers
For $b>0$, let us say that a subset $I\subseteq \{1,\ldots,n\}$ is $b$-good if
(1) $\# I\geq b n$,
(2) $\ell_{i+1}-\ell_i\leq \frac{1}{b}$ for all $i=1,\ldots,\#I-1$, where we write $I=\{\ell_i\}_{...
- 24.2k
6
votes
1
answer
120
views
Amalgamation problem for the 11-cell and 57-cell
Are there any finite regular abstract 5-polytopes whose facets are 11-cells and whose vertex figures are 57-cells?
2
votes
2
answers
315
views
5 different ways to define the same family of integer sequences
Let ${n \brace k}$ be a Stirling number of the second kind.
Let $A_n(x)$ be an Eulerian polynomial. Here
$$
A_n(x) = \sum_{i=0}^{n}i!{n \brace i}(x-1)^{n-i}.
$$
Let $a_1(n,p,q)$ be the family of ...
- 17k
2
votes
2
answers
196
views
Estimates on the number of vertices of reflexive polytopes
Suppose $M \cong \mathbb{Z}^n$ is a rank $n$ lattice, with dual lattice $N$. Suppose $\Delta$ is a full dimensional lattice polytope (i.e. convex hull of finite lattice points) in $M$. Then $\Delta$ ...
- 511
5
votes
2
answers
308
views
Majority voting on $\{0,1\}^\mathbb{Z}$
Motivation. Sometimes in life, people seem to do what the majority of their friends are doing. Do we all become more similar over time? Do we split up into pockets of similarity? This post aims to ...
- 577
1
vote
1
answer
207
views
A candidate for one-way functions
For every $n \geq 3$ consider a bipartite random $3$-regular graph $G_n$ with two parts $X=\{x_1, \dotsc, x_n\}$ and $Y=\{y_1, \dotsc, y_n\}$. For any $i \leq n$ assign either 0 or 1 to each vertex $...
- 3,319
20
votes
1
answer
557
views
Almost orthogonal maps $f:\omega \to \{-1,1\}$
Let $\omega$ denote the set of non-negative integers. For sets $A,B$, let $B^A$ denote the set of maps $f:A\to B$. For $f,g\in\{-1,1\}^\omega$ we say that $f,g$ are almost orthogonal if there is $C_0\...
- 6,353
9
votes
1
answer
216
views
Distinct closed walks with $2n$ steps in the $n$-dimensional hypercube
I am interested in finding out what are the shortest closed walks that touch all $n$ dimensions in an $n$-dimensional hypercube. Because they must be closed and they must touch all $n$ dimensions, the ...
- 156k
1
vote
0
answers
87
views
All matroid polytope is a generalized permutohedron
In many texts, the authors say something like "a matroid polytope lives in the family of generalized permutohedra". We can quickly check the veracity of this claim by describing the matroid ...
- 251
7
votes
2
answers
406
views
Proving an identity for flagged Schur without use of determinants?
In proposition 3 of Determinantal transition kernels for some interacting particles on the line, Dieker and Warren prove the following identity: consider vector $a:=(a_1,\dotsc,a_N)$ and kernels
$$\...
- 1,989
9
votes
1
answer
790
views
Properties a triangulation must have in order to describe a manifold
I am mainly interested in the $3$-dimensional case. It is a well-known fact, following from the work of E. E. Moise and R. H. Bing in the 1950s, that every $3$-dimensional topological manifold (with ...
- 1,429
3
votes
1
answer
199
views
Mutually equal Hamming distance of members of ${\cal P}(\mathbb{N})$
This is inspired by an older, as of yet unanswered question.
If $X$ is a set and $A,B\subseteq X$, we let the Hamming distance of $A, B$ be defined as $d_H:=\text{card}\big((A\setminus B)\cup (B\...
- 328
0
votes
0
answers
81
views
A generalized permutohedron as the sum of the dilatations of the faces of the standard simplex
I am trying to understand the proof of the statement, specifically it refers to a theorem stated by Postnikov in his text on permutohedra. So, this sentence claims the following:
If $\{Y_I \}$ is a ...
- 2,858
1
vote
0
answers
121
views
Simple algorithm for A107670
Let $T(n, k)$ be A107670 (i.e., matrix square of triangle A107667). Here we define the triangular matrix $P$ by $P(n, k) = \frac{(n+1)^{2(n-k)}}{(n-k)!}$ for $0 \leqslant k \leqslant n$ and the ...
- 175