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Vertex coloring of the Rado graph

Is there a reference for the following fact about the Rado graph (the random countable graph) which came up in an answer to this question? If the vertices of the Rado graph $G=(V,E)$ are colored with ...
bof's user avatar
  • 13.4k
3 votes
0 answers
73 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 $m_{\rho}(\mathbf{z})$ ...
T. Amdeberhan's user avatar
8 votes
1 answer
671 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 ...
T. Amdeberhan's user avatar
3 votes
1 answer
405 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 ...
Pluviophile's user avatar
  • 1,608
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)...
T. Amdeberhan's user avatar
0 votes
0 answers
138 views

State of the art on attempts to solve the elliptic curve discrete logarithm problem through transfering the problem to a weaker curve

Let an elliptic curve $E$, and 2 points on such curve $P$ and $O$ the methods I’m talking about consist in creating a weaker elliptic curve $F$ and mapping $P$ and $O$ to $F$ while successfully ...
user2284570's user avatar
0 votes
0 answers
52 views

Reference request for the determinant of a matrix constructed from Pascal's triangle

One can prove by induction that the matrix $M^{(n)}$ given by $$ \begin{pmatrix} 1 & 1 & 1 & 1 & \dots & \binom{n}{0} \\ 1 & 2 & 3 & 4 & \dots & \binom{n+1}{1} \...
Jeff Harvey's user avatar
  • 5,546
1 vote
1 answer
76 views

Determinant formula for a certain parametrized M-matrix

Let $P_{ij}$ be variables, and let $A \in \mathbb{R}^{n\times n}$ be the matrix defined by $$ A_{ij} = \begin{cases} -P_{ij} & i \neq j,\\ P_{i1} + P_{i2} + \dots + P_{in} & i=j. \end{cases} $$...
Federico Poloni's user avatar
8 votes
0 answers
244 views

Strengthening of Frankl's union-closed sets conjecture: An algebraic approach

Let $\mathcal F$ be a union-closed family of subsets of $[n]=\{1,2,...n\}$ and $n$ real numbers $x_1,x_2,...,x_n\geq 1$. Conjecture: There exists $k\in [n]$ such that: $$\sum_{k\in A,A\in \mathcal F}\...
Veronica Phan's user avatar
3 votes
0 answers
101 views

Tuple rearrangement: a combinatoric problem emerging from the Hurwitz action on Coxeter groups

I am working on Artin Groups, so called Dual Artin groups and the conjecture that they are isomorphic. Tuples of $n$ group elements can be acted on by the braid group $B_n$ in a particular way called ...
Sean O'Brien's user avatar
10 votes
0 answers
287 views

Coefficients of polynomials vs trigonometric product

Let's consider the family of sequences of coefficients in the expansion $$\prod_{i=0}^{n-1}(1+x^{3^i}+x^{3^{i+1}})=\sum_{k\geq0}a_n(k)\, x^k.$$ Remark. Evidently, the RHS is a finite sum. Here is a ...
T. Amdeberhan's user avatar
2 votes
1 answer
431 views

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,\...
T. Amdeberhan's user avatar
2 votes
1 answer
276 views

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? ...
T. Amdeberhan's user avatar
2 votes
0 answers
30 views

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 ...
Martin Rubey's user avatar
  • 5,822
4 votes
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 ...
Zach H's user avatar
  • 1,989
8 votes
4 answers
1k views

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 ...
T. Amdeberhan's user avatar
23 votes
4 answers
2k views

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} - \...
T. Amdeberhan's user avatar
3 votes
0 answers
61 views

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 ...
M. Winter's user avatar
  • 13.6k
11 votes
1 answer
340 views

Number of odd-dimensional irreducible representations of $S_n$

In this paper the structure of odd-dimensional irreducible representations of the symmetric group is described, but what is the asymptotic behaviour of the number of such representations? (Or, if it ...
Fedor Petrov's user avatar
2 votes
1 answer
215 views

Number of binary matroids of rank $r$ on a ground set with $n$ elements

How many simple binary matroids are there, up to isomorphism, of rank $r$ on an $n$-element ground set, where $r \le n < 2^r$? Write this number as $a_r(n)$. Is there somewhere where I can get this ...
Colin Tan's user avatar
  • 331
7 votes
1 answer
440 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 ...
It'sMe's user avatar
  • 839
4 votes
1 answer
170 views

About $CW(512,16^2)$

Definitions: A weighing matrix $W = W(n,k)$ with weight $k$ is a square matrix of order $n$ and entries $w_{ij}$ in $\{0, \pm 1\}$ such that $WW^T=kI$, where $I$ is the identity matrix. A circulant ...
user369335's user avatar
2 votes
1 answer
217 views

Number of distinct higher dimensional integer partitions

By a distinct partition, I mean a partition into distinct parts, i.e., $10 = 5+4+1$ is one, but $10=6+2+2$ is not. The number of distinct partitions of $k$ all whose parts are at most $n$ is given by ...
Bubaya's user avatar
  • 281
12 votes
1 answer
238 views

Number of planes generated by integer vectors

For fixed dimension $d$ and large $R$ consider all non-zero integer vectors in the ball $B(0,R)\subset \mathbb{R} ^d$ of radius $R$ centered at the origin. The number of such vectors grows as $c_d\...
Fedor Petrov's user avatar
7 votes
1 answer
224 views

Generating set of permutation group such that generators do not "contain" other group elements

Let $(G, X)$ be a permutation group with domain $X$. Let $O=\{o_1,\dots,o_m\}$ be the set of orbits of $G$. I am interested in generating sets $S$ with the following property: Let $g\in S$ be a ...
Martin Rubey's user avatar
  • 5,822
1 vote
0 answers
76 views

Shellable non-pseudomanifolds with dimension greater than 2

Shellability of simplicial balls and spheres (simplicial complexes whose geometric realizations are homeomorphic to balls and spheres) has been studied quite extensively. There are many explicit ...
mashedcarrots's user avatar
3 votes
1 answer
162 views

Counting equal covering sets

Definition. We call a set $C$ of sets to be an equal covering set of $S$ if the elements of $C$ are all the same size and each element of $S$ is contained an equal number of times throughout the sets ...
chirpyboat73's user avatar
1 vote
1 answer
232 views

Looking for q-analog of derangement anagrams for a word

I have already known QPermutationDerangement: It describes the distribution $$ d_n(q)=\sum_{\sigma \in D_n} q^{\operatorname{maj}(\sigma)} $$ Where we sum over all derangements of an $n$ element set. ...
138 Aspen's user avatar
  • 175
15 votes
3 answers
1k views

Does anyone remember what happened to the experimental search for polynomial identities for $\pi$?

So a while back I was on the internet and had encountered a website containing an experimental search for identities for $\pi$. My memory was that the page belonged to either Jonathan Sondow or ...
Sidharth Ghoshal's user avatar
7 votes
1 answer
295 views

Is this known? As $p,q\to\infty$, most elements of the power set of $\{1,\dots,p\}\times\{1,\dots,q\}$ are in free $\Sigma_p\times\Sigma_q$-orbits

Let $p,q$ be nonnegative integers. The product of symmetric groups $\Sigma_p\times\Sigma_q$ acts on the power set $P(\{1, \dots ,p\}\times\{1, \dots ,q\})$ in the evident way. You can ask what ...
user509184's user avatar
  • 1,335
2 votes
0 answers
54 views

Transform connecting powers of integration and differentiation operators

Just by a chance, I found the following power series identity, which holds for any analytic function $F(\cdot)$, nonnegative integer $m$, and constants $u,v$ not depending on indeterminates $z,t$: $$\...
Max Alekseyev's user avatar
6 votes
1 answer
244 views

Linear independence over field of rational functions

To prove that functions $f_1(x), \dots, f_n(x)$ with $x \in \mathbb R$ are linearly independent, we only need to show that the Wronskian of these functions is non-zero at a certain value of $x$. Now ...
Pluviophile's user avatar
  • 1,608
3 votes
1 answer
90 views

Asymptotic growth rate for primitve S-adic systems

It is known that for a primitive substitution $S:\mathcal{A}\to \mathcal{A}^+$, there exists constants $c,C>0$ such that $$ c\theta_S^n \leq \vert S^n(a)\vert \leq C \theta_S^n \quad \text{for all} ...
Keen-ameteur's user avatar
3 votes
1 answer
140 views

Generalizations of a theorem of Edmonds/Tutte on existence of a perfect matching in a graphs

It is well known that for a bipartite graph $G$ with bi-adjacency matrix $A$, then $\det A \neq 0$ (as a polynomial) iff $G$ has a perfect matching (there is a similar result for general graphs with ...
Agile_Eagle's user avatar
3 votes
1 answer
343 views

Reference request: about “SNF” (Smith Normal Form)

I've read about some studies on the Paley I Construction. Among them I found the following notations ( See this page: https://documents.uow.edu.au/~jennie/matrices/32P02.html ). $$SNF:1,2^a,4^{b},8^{b}...
Matthers's user avatar
3 votes
0 answers
125 views

Short path problem on Cayley graphs as language translation task (from "Permutlandski" to "Cayleylandski"(s) :). Reference/suggestion request

Context: Algorithms to find short paths on Cayley graphs of (finite) groups are of some interest - see below. There can be several approaches to that task. One of ideas coming to my mind - in some ...
Alexander Chervov's user avatar
0 votes
0 answers
117 views

An interesting identity involving skew-Schur functions

Denote $\rho=(-\frac12,-\frac32,-\frac52,\dots)$. I was reading this interesting paper, where in particular, the authors claim that one can get the expression in (2.9) \begin{align*} \prod_{k\geq1}(1+...
T. Amdeberhan's user avatar
0 votes
0 answers
57 views

Reference for packing property and König property

Can someone please suggest reference material to study about the packing property and König property of ideals and some examples?
Sowbarnika R's user avatar
4 votes
0 answers
90 views

Symmetric functions and pattern avoidance

It is known that the number of $k$-regular simple graphs with vertices labeled by $1,2,\dots,n$ can be expressed as the coefficient of $x_1^k \dots x_n^k$ in a symmetric function, which is $$ \prod_{1\...
Pluviophile's user avatar
  • 1,608
4 votes
1 answer
261 views

What is the convergence rate of this "infinite monkey"-type probability?

Cross-posted from Math Stack Exchange, where it hasn’t received an answer yet: Let $S$ be a finite set and $n,m\in\mathbb N$. Consider the process $R=(R_i)_{i\in\mathbb N}$ where all $R_i$ are iid ...
Joseph Expo's user avatar
4 votes
0 answers
82 views

Expansion of Schubert polynomials into standard elementary monomials

I have an explicit formula for expressing any Schubert polynomial in terms of standard elementary monomials that may or may not be cancelation-free. I haven't determined this yet, but it seems likely ...
Matt Samuel's user avatar
  • 2,168
2 votes
0 answers
278 views

On $(k,\ell)$-sumfree sets

Call a set $\mathcal S \subset \mathbb N$ to be $(k,\ell)$-sumfree if there are no non-trivial solutions to the equation $$x_1+\dots +x_k = y_1+\dots +y_\ell$$ in the set (for distinct $x_i$'s and $...
Sayan Dutta's user avatar
0 votes
2 answers
96 views

Isometric path cover number of the 2 dimensional grid graph

I am looking for a proof of the fact that at least $2n/3$ isometric paths (i.e. shortest paths between the end points) are required to cover the vertices of the $n\times n$ grid graph (i.e. Cartesian ...
Pritam Majumder's user avatar
5 votes
0 answers
141 views

If chromatic polynomials for two graphs agree, can I always find an edge such that the two deletion-contraction minors have same chromatic polynomial?

Suppose I have non-isomorphic graphs $G$ and $H$ (which have at least one edge), but such that their chromatic polynomials are the same. Can I then always find an edge $e$ in $G$ and $f$ in $H$ such ...
Per Alexandersson's user avatar
1 vote
0 answers
59 views

A question on generalized bases

I just came to know that it is possible to define a generalized base as an infinite sequence of natural numbers $\mathbf b=(b_1,b_2,\dots)$ where $b_i\ge 2$ for all $i$. With this definition, any $m\...
Dumbest person on earth's user avatar
6 votes
0 answers
102 views

The meet of two dominant permutations in weak order of $S_n$

A permutation is called dominant if its Lehmer code is a partition, or equivalently if it avoids the pattern $132$. I can prove that given a permutation $v\in S_n$, there is a unique dominant ...
Matt Samuel's user avatar
  • 2,168
4 votes
1 answer
222 views

Double cover the edges of a complete graph by smaller complete graphs

Suppose we have a complete graph $K_n$ on $n$ vertices. Are there any results on the ways to cover $K_n$ with $k$ copies of $K_m$, for $m<n$, such that each edge of $K_n$ is contained in exactly ...
Wallace Rin's user avatar
2 votes
2 answers
77 views

Reference request for a subfamily of regular graphs

[Repost of same question math stack exchange which got no answers] I'm looking for literature on the following family of graphs: Call a regular graph $G=(V,E)$ (of regularity degree $d$) nice if there ...
jojo's user avatar
  • 21
1 vote
0 answers
148 views

Exploring the Intersection of Expander Graphs, Number Theory, Representation Theory and Recent Computer Science Developments [closed]

I have a solid understanding of the basics of expander graphs and their properties and the recent development of High-Dimensional Expanders and their application to Random Walks, along with other ...
total dependent random choice's user avatar
5 votes
0 answers
107 views

Generalized Puiseux series for diagonal reflections of the curves $y = \frac{x}{(1-ax)(1-bx)^m}$

Reflection of the curve $y = f_m(x) = \frac{x}{(1-ax)(1-bx)^m}$ through the diagonal line $y=x$ in the $xy$-plane can be regarded as local compositional inversion of the curve $y=f_m(x)$. ($x,y,a,b$ ...
Tom Copeland's user avatar
  • 10.5k

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