All Questions
11,386 questions
0
votes
0
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24
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Coxeter Matrix of Dyck Path
I am trying to understand Gjergji Zaimi's answer in What are the periodic Dyck paths?. In the third paragraph he claims that
Next, we define the matrix $X_D$
similarly to the Cartan matrix except we ...
3
votes
0
answers
54
views
While expanding Jack polynomials in monomial basis
Denote $\mathbf{z}=(z_1,\dots,z_n)$. Let $P_{\kappa}(\mathbf{z};\alpha)$ be the symmetric Jack polynomials and suppose they are expanded in terms of the monomial symmetric basis $m_{\rho}(\mathbf{z})$ ...
- 43.2k
1
vote
0
answers
28
views
Integral hull of a polyhedron Q is polyhedron
Let $Q \subseteq R^n$ be a rational polyhedron and let $Q_I=Convexhull(Q \cap Z^n)$. By finite basis theorem, we have $Q=P+C$ for some rational polytope $P$ and finitely generated cone $C$ where $C=R_+...
- 21
2
votes
1
answer
85
views
Number of disjoint set triplets in a union-closed family
Let $\mathcal{F}$ be a family of $n$ finite sets. In this case, the family can be regarded as a multiset, since it is allowed to contain multiple instances of the same set. Let $U(\mathcal{F})$ be the ...
- 988
5
votes
2
answers
720
views
Shifting an irrational binary sequence
Let $\newcommand{\tn}{\{0,1\}^\mathbb{N}}\tn$ be the collection of all infinite binary sequences. For $s\in\tn$ and $k\in\mathbb{N}$ let the left-shift of $s$ by $k$ positions, $\ell_k(s)\in \tn$, be ...
- 48.8k
6
votes
0
answers
125
views
Is there a more natural way to define the Young symmetrizer and the Specht module?
It's well known that Young symmetrizer is a fundamental tool in the representation theory of symmetric groups.
For instance, for every Young diagram $\lambda\vdash n$, we construct a Young tableau $T_\...
-1
votes
0
answers
41
views
Is it possible to backtrack an optimization solver? [closed]
I have an optimization problem and was using a linear programming optimizer to find solutions. However, I find that past a certain size, the problem becomes "infeasible" and has no solutions....
0
votes
1
answer
80
views
Median of cardinality of set union
Let $U$ be an arbitrary finite universe (you can just think of it as $[N]=\{1,2,\ldots,N\}$), and $\mathbf{S} = (S_i)_{i \in [n]}$ ($S_i \subseteq U$) be the sets that we are drawing from. Define a ...
- 29
1
vote
0
answers
107
views
Generalized identity with Stirling numbers of the second kind and falling factorials
It is known that Striling numbers of the second kind satisfy the relation
$$
\sum\limits_{k=0}^{n}{n \brace k}(x)_k = x^n.
$$
where $(x)_n$ is the falling factorials such that
$$
(x)_n = x(x-1)(x-2)\...
- 4,928
1
vote
0
answers
375
views
How to verify if sets satisfying cardinality condition exist? [migrated]
I am trying to find out if sets satisfying the following properties exist:
Call the sets $A_1, \ldots, A_{20}$ and $B_1, \ldots, B_{20}$.
For each $i \in \{1, \ldots, 20\}$, $|A_i| \in \{1,2\}, |B_i| ...
- 119
1
vote
0
answers
22
views
maximal length of saturated chains with a given terminal point in the lattice of partitions of an integer ordered by dominance
Let $Pr(n)$ be the set of partitions of the positive integer $n$. This is a lattice with respect to the dominance order: if $\lambda=(\lambda_1\geq\lambda_2\geq\cdots)$, $\mu=(\mu_1\geq\mu_2\geq\cdots)...
- 593
-1
votes
1
answer
92
views
Variance of bins for N balls into M bins [closed]
If I throw N balls independently into M bins with uniform probability, the expected mean of the M bins is N/M balls.
What is the expected variance of the M bins?
I was thinking of what bin size I ...
5
votes
3
answers
274
views
The max-clique chromatic number of a graph
Let $G = (V,E)$ be a graph. Every clique, that is, complete subgraph, is
contained in a maximal clique with respect to $\subseteq$ (this is
an easy consequence of Zorn's Lemma). Let $\newcommand{\MC}{\...
- 48.8k
0
votes
0
answers
112
views
Minesweeper constructions in combinatorics
In a related question I asked if constructions based on Sudoku puzzles could be used to obtain any deep results in combinatorics and noted that there were papers of Greenfeld and Tao where Sudoku ...
- 5,146
9
votes
1
answer
454
views
A conjecture related to Frankl's conjecture
Let $\mathcal{F}\subseteq2^{[n]},\emptyset\in\mathcal{F}$ be an union-closed family of sets. For $S\in\mathcal{F}$, let $w(S)$ be the number of subsets of $S$ in $\mathcal{F}$. Does there always exist ...
- 1,462
0
votes
1
answer
81
views
Image and pre-image integer choice function
Let $\newcommand{\Nplus}{\mathbb{N}^+}\Nplus$ denote the set of positive integers. Is there a function $f:\Nplus\to\Nplus$ with the following property?
For all $(a,b)\in \Nplus\times\Nplus$ there is ...
- 48.8k
1
vote
1
answer
186
views
Existence of Finite Amicable Groups
I'm interested in exploring the concept of "amicable groups" as follows:
Definition. Two finite groups $G$ and $H$ are called amicable groups if:
$G$ is the direct sum of proper subgroups ...
1
vote
0
answers
122
views
Growth polynomial of the Associahedron graph ? (Is it approximately Gaussian ?)
Consider Associahedron, consider graph build from its vertices and edges. Choose some vertex. Let us count the number of vertices on distances $k$ from the selected vertex. Write a generating ...
- 24.8k
5
votes
1
answer
259
views
Diagonal analogue of symmetric functions
Let $n$ be a positive integer and consider the ring $R$ of power series over $\mathbb{Q}$ in commuting variables $x_1,y_1,x_2,y_2,...$. Let the symmetric group $\mathfrak{S}$ of permutations of the ...
- 1,500
2
votes
0
answers
163
views
Nonabelian groups where every element has small order
Let $G$ be a finite nonabelian group with the property that if $g \in G$, then
$$\DeclareMathOperator{\ord}{ord} \ord(g) \leqslant 10 \log_2 |G|, $$
where $\ord(g)$ is the order of the element $g$, ...
- 1,354
4
votes
0
answers
106
views
Differential duality: Triangular codes vs. VT codes / Single-substitution vs. Single-deletion
Here is the introduction to my problem:
Codes correcting single-deletion. Let $q$ and $n$ be non-negative integers, and let $\mathbf{x}= \left ( x_{1}, x_{2}, \ldots, x_{n} \right )\in\mathbb{Z}_{q}^{...
- 41
1
vote
0
answers
30
views
Balanced cocircuit cover
Are there studies on matroids which can be covered by $r$ cocircuits ($r$ is the rank of the matroid), so each element is covered by a small number of times?
For example, it is known graphic matroids ...
- 613
0
votes
0
answers
34
views
separator and vertex-connectivity
A definition of "separator" is the following: Let $G$ is an $n$-vertex graph, then $S\subseteq V(G)$ is a separator if there is a partition $V=A\cup B\cup S$ such that $|A|,|B|\le 2n/3$ and ...
- 281
5
votes
0
answers
363
views
A Collatz-like map?
Consider the map $\psi$ acting on triples $(a\leq b\leq c)$ of three positive natural integers with $\mathrm{gcd}(a,b,c)=1$ as follows:
Set $$(a',b',c')=\left(\frac{a}{\mathrm{gcd}(a,bc)},\frac{b}{\...
- 17.9k
0
votes
1
answer
127
views
Closed form for $\sum\limits_{k=0}^{n} [\operatorname{wt}(k) = m]$ where $\operatorname{wt}(n)$ is the binary weight of $n$
Let $\operatorname{wt}(n)$ be A000120 (i.e., number of $1$'s in binary expansion of $n$).
Let $a(n,m)$ be the family of integer sequences such that
$$
a(n,m) = \sum\limits_{k=0}^{n} [\operatorname{wt}(...
- 4,928
0
votes
0
answers
85
views
How to prove the following equation (involving multiple binomial coefficients sum)?
I encountered the equation below, encountered a problem that has been bothering me for a long time
Does anyone have an idea how to prove it? I would be extremely grateful to you if you come up with an ...
- 41
8
votes
1
answer
666
views
Infinite series and sum of two squares
Consider the following infinite sequence $a(n)$ generated by
$$\sum_{n\geq0} a(n)q^n
=\frac{\sum_{k\geq0}F(2k+1)q^{\binom{k+1}2}}{\sum_{k\geq0} q^{\binom{k+1}2}}$$
where the $F(2k+1)$ are the odd ...
- 43.2k
3
votes
0
answers
80
views
Combinatorial/probabilistic interpretation of a quantity of union closed family
Let $\mathcal{F}\subseteq2^{[n]}$ be a union-closed family of sets. For a set $S\in[n]$ (not necessary belong to $\mathcal{F}$), define $w_{\mathcal{F}}(S)$ to be the number of subset of $S$ which ...
- 1,462
3
votes
1
answer
404
views
Moments of a random variable related to uniform distribution on sphere
Let $u$ be taken uniformly from the unit sphere $\mathbb S^{n-1}$ and $D$ be a diagonal matrix. I'd like to find a general formula for
$$
\mathbb E[(u^\top D u)^m]
$$
for $m=1,2,3, \dots$, in terms of ...
- 1,608
0
votes
0
answers
21
views
Easy instance of set cover
I am trying to prove that a natural greedy algorithm solves the following instance of the set cover problem: for a set of elements $e\in U$ with a set of weights $w_e$, we define the cost of a subset ...
- 4,049
4
votes
0
answers
80
views
Classification of nilpotent orbits over local fields (for type ABCD via partitions )
Let $\mathfrak g$ be a simple Lie algebra over a char $0$ local field $F$ (e.g. $F=\mathbb R$ or $F=\mathbb Q_p$) with its adjoint group $G$. Let $\mathcal N \subseteq \mathfrak g$ be its nilpotent ...
- 6,612
0
votes
0
answers
78
views
Solution modulo $9$ of certain linear equation implies triviality modulo $3$
Question: Let $k \geq 2$ and $r \geq 4$ be two natural numbers. We are given eight integers $\nu_{ij} \geq 0$ for every $1 \leq i \leq k$ and $1 \leq j \leq r$ such that the following two conditions ...
- 119
1
vote
1
answer
80
views
What are the efficient algorithms to compute Hamiltonian paths on Cayley graphs of finite groups ? Can GAP do it?
The famous Lovasz conjecture predicts existence of the Hamiltonian path on Cayley graphs. In general finding such a path is NP-complete problem, but there are many heuristic algorithms.
Question 1: ...
- 24.8k
0
votes
1
answer
168
views
Partial sums of binomial coefficients and related family of polynomials
Let $a(n)$ be A302117. Here
$$
a(n) = 4(n-1)a(n-1) - \frac{1}{3}\prod\limits_{k=0}^{n-1}(2k-3), \\
a(0) = 0.
$$
Let
$$
T(n,k) = \sum\limits_{i=0}^{k} \binom{n}{i}.
$$
Let $P_n(z)$ be the family of ...
- 4,928
2
votes
0
answers
83
views
Formula for sum involving products of (symplectic) Schur functions
This question is a continuation of a question asked yesterday which had a very nice answer.
Consider the summation
$$\sum_{\lambda \subset (k)^n} \dim S_{\lambda^t} (\mathbb{C}^k) \cdot \dim S_{[\...
- 553
5
votes
0
answers
215
views
Are there Sudoku variants which are useful or mathematically deep?
I was recently watching a Sudoku Youtube channel which shows a large number of variants on the traditional Sudoku puzzle, some of them non-trivial to solve. I think there was some mention of a Sudoku ...
- 5,146
6
votes
1
answer
131
views
Number of semistandard tableaux of all possible shapes fitting within some rectangle
Suppose $n$ and $k$ are two integers. Then I am interested in having a closed form for the sum
$$\sum_{\lambda \subset k \times n} S_\lambda (\mathbb{C}^n),$$
where $S_\lambda$ denotes the Schur ...
- 553
7
votes
4
answers
497
views
Distinguishing finite families of sets by algebras of bounded size
Say that an algebra of sets $K$ distinguishes set $B$ from set $C$ provided that for some $A\in K$, we have exactly one of $A\cap B$ and $A\cap C$ non-empty. Given families $F$ and $G$ of sets, say ...
- 2,555
0
votes
2
answers
99
views
Is there an uncountable extension of the Ramsey set $[\omega]^2$?
We say that a family ${\cal A}\subseteq {\cal P}(\omega)$ is Ramsey
if for every map $c:{\cal A}\to\{0,1\}$ there is an infinite set $X\subseteq \omega$
with the following properties:
${\cal A}\cap {\...
- 48.8k
3
votes
1
answer
192
views
Density of Pisot polynomials
Recall that a Pisot polynomial $P=x^n+ a_{n-1}x^{n-1}\ldots a_1$ has integer coefficients, a real root $x_1>1$ and all other roots $|x_i|<1$ for $1\leq i \leq n$. One key result is that $\{(...
- 680
5
votes
2
answers
240
views
Expansion of key polynomials in terms of non-symmetric Hall-Littlewood polynomials and charge-like statistics
Edit: The problem I pose here is impossible to solve with the basis $H$, in the answer I made to this post I explain why. The only way I can think it to amend the situation would be to try with ...
- 161
0
votes
0
answers
93
views
Class multiplication coefficients of symmetric groups
My question is that I was working with some counting problems, and finally the answer should be
$$
\nu_{\mu_1,\mu_2,\mu_3}=\#\{(\sigma_1,\sigma_2,\sigma_3): \sigma_1\sigma_2\sigma_3=1, \sigma_1\in C_{\...
8
votes
1
answer
1k
views
GAP cannot solve Rubik's cube 4x4x4 and higher ? (Practical limits of Schreier–Sims algorithm)
According to our practical experiments and literature search - computer algebra system GAP cannot "solve" Rubik's cube 4x4x4 and higher. That means cannot decompose given random element of ...
- 24.8k
4
votes
1
answer
197
views
Solving a three-parameter recursive sequence
Consider the triple-indexed sequence of integers defined by
\begin{align} \label{coefficientsV} \nonumber
f(\alpha,\beta,\gamma)
&:=(2\alpha+8\beta+12\gamma-1)\cdot f(\alpha-1,\beta,\gamma)...
- 43.2k
9
votes
0
answers
143
views
Which polytopes have compact realization spaces?
Let $P\subset\Bbb R^d$ be a convex polytope.
Its reduced realization space is the space of all combinatorially equivalent polytopes modulo projective transformations.
I am interested in polytopes for ...
- 13.6k
2
votes
0
answers
89
views
Conjugacy between piecewise linear circle maps
Let $\mathcal{M}$ the Mandelbrot set,
$\mathcal{M}=\{c \in \mathbb{C}: \{Q_c^n(0) \}_{n \in \mathbb{N}} \text{ is bounded, where } Q_c(z)=z^2+c \}$
And let the hyperbolic or stable component, $H_n=\{ ...
- 271
8
votes
1
answer
534
views
The cars problem, again
Consider the following simple problem: We are given $2n$ parking spots, labelled from 0 to $2n-1$. There are $n$ cars on the first $n$ spots, and the remaining $n$ spots are free. At every step, every ...
6
votes
1
answer
243
views
Is it possible that the number of $\mathcal{T}$-algebras is an arithmetic progression?
For a single-sorted algebraic theory $\mathcal{T}$ denote by $t_n$ the number of $\mathcal{T}$-algebras with $n$ elements (up to isomorphism). Is there an example for $\mathcal{T}$ such that ...
- 63.1k
1
vote
0
answers
57
views
Step back step forward algorithm for A108442
Let $a(n)$ be A108442. Here generating function is $\frac{1}{1-zA(z)}$ where
$$
A(z) = 1 + z(A(z))^2 + z(A(z))^3.
$$
Also
$$
a(n) = \sum\limits_{k=1}^{n}\frac{k}{2n-k}\sum\limits_{i=0}^{n-k} \binom{2n-...
- 4,928
2
votes
1
answer
111
views
Is there a ternary Cayley graph on 27 vertices that is a non-complete core?
Is there a non-complete ternary Cayley graph that is a core with $3^3 = 27$ vertices?
By a ternary Cayley graph, I mean a (simple, undirected) graph whose vertex set is $\mathbb{Z}_3^n := \bigoplus_{i ...
- 331