0
votes
0answers
101 views

Error term in formula for products of necklaces

Let us consider the $\lim_{n\to \infty}\prod_{p=1}^n N(p,a)$, where the number of fixed necklaces of length n composed of a types of beads $N(n,a)$ can be calculated via totient function: ...
9
votes
1answer
478 views

Real-rootedness, interlacing, root-bounds of a sequence of polynomials

Problem: the number $a(n,k)$ is defined by the following recurrence \begin{equation} a(n,k)=(k+1)(k+2)\, a(n-1, k)+\frac{(k+1)(k+2)(k+3)}{k} \,a(n-1, k-1), \end{equation} with $a(1,1)=1$ and ...
0
votes
1answer
251 views

Pros and cons of probability model for permutations

I am studying probability model of random permetuation Let $b(n; k)$ denote the number of permutations of {1,...,n} with precisely k inversions ($inv(\pi)$). The analytic approach was considered by ...
2
votes
0answers
78 views

A Conjecture related to minimization of product of determinants over permutations

Hi I have the following problem (and a conjecture which holds in Matlab). Given an $N\times N$ matrix $H$, represent it by its $2N\times 2N$ real-valued representation (which I will also denote by ...
1
vote
1answer
213 views

A counterexample to the Polya-Schur master theorems for half-planes

Given an integer $n\ge 1$ we say that $f\in C[z_1,\ldots,z_n]$ is stable if $f(z_1,\ldots,z_n)\neq 0$ whenever $\text{Im}\ z_i>0$ for all $1\leq i\leq n$. Stable polynomials with all real ...
1
vote
0answers
58 views

Factorization of Gegenbauer polynomials

For each natural number $n$ there is a Gegenbauer polynomial of degree n, depending on a spectral parameter $\lambda$. They fulfill many recurrent relations. The question is how to recognize their ...
5
votes
2answers
400 views

Interpolating a sum of binomial coefficients using a sin function

While studying a problem about orthogonal polynomials I encountered the following expressions \begin{equation} f(n)=\sum_{k=0}^{n}(-1)^k\binom{n+k}{2k} \frac{1}{k+1}\binom{2k}{k} \end{equation} and ...
2
votes
0answers
121 views

A subclass of log-concave functions satifying a sum inequality

Suppose $f:[0,\infty)\to[0,\infty)$ is continuous and for all positive integers $m$ and real $x\geq0$, $\alpha,\beta>0$: $$ ...
0
votes
1answer
133 views

How to Rigorize an inequalities argument

Context I'm working on a problem involving Lovasz Local Lemma, for proving that there exists a graph with a certain property. What I need to prove: There exists some constant $c$, and functions ...
11
votes
4answers
645 views

Showing that a family of polynomials has positive and real roots.

Hi everybody, for my research I am dealing with the following function: $$\alpha_n(x):=\left.\frac{\partial^{2n+1}}{\partial z^{2n+1}}\frac{\sinh(z)}{\cosh(z)-1+x}\right|_{z=0},\quad n\in ...
0
votes
2answers
681 views

Generalized multinomial coefficient

The usual binomial coefficient $\binom{n}{k} = \frac{n!}{k!(n-k)!} $ can be generalized to real upper argument, lower argument still a nonnegative integer, by the definition $\binom{\alpha}{k} = ...
18
votes
0answers
2k views

Why do polytopes pop up in Lagrange inversion?

I'd be interested in hearing people's viewpoints on this. Looking for an intuitive perspective. See Wikipedia for descriptions of polytopes and the Lagrange inversion theorem/formula (LIF) for ...
8
votes
2answers
436 views

Recognizing a measure whose moments are the motzkin numbers

The Catalan numbers are the moments of the Wigner semicircle distribution. $$ \frac{1}{2\pi} \int_{-2}^2 x^{2n} \sqrt{4 - x^2} dx = \frac{1}{n+1} \binom{2n}{n} $$ Motzkin numbers enumerate the number ...
7
votes
0answers
269 views

Reflection spectrum from a Fibonacci quasicrystal

Warning: this question may come across as too specific. Maybe it would help if I set values for the probability parameters below - they'd be based on the Fresnel equations. But anyway, I think this is ...
5
votes
1answer
233 views

Probability of false decoding with LDPC codes

It is well known that given a binary linear code $C$ under maximum likelihood decoding the probability of false decoding $P_C$ in a Binary Symmetric Channel with cross over probability $p$, in other ...
0
votes
1answer
103 views

Constructing a monotonic function

(I'm not sure how to tag this; I'm tagging it math.CO because that's where it arose, but math.FA or something might be appropriate as well. Feel free to edit or comment on what this should be.) I ...
4
votes
2answers
268 views

Is there a closed form for $\sum_{|\gamma|=k} \gamma!$

Consider the multi-index $\gamma=(\gamma_1,\ldots, \gamma_n)$. Is there a closed form for the sum $\sum_{|\gamma|=k} \gamma!$ in terms of $n$ and $k$? Asymptotics, or good upper bounds are also very ...
2
votes
1answer
276 views

How do i maximize $\max_{\gamma}\sum_{|\alpha|=q}\binom{\alpha}{\gamma}$?

I'm trying to find the following maximum: $\max_{\gamma}\sum_{|\alpha|=q}\binom{\alpha}{\gamma}$. Here $\alpha=(\alpha_1,\ldots, \alpha_n),\gamma=(\gamma_1,\ldots, \gamma_n)$ are multi-indices. The ...
9
votes
0answers
226 views

Positivity of polynomial sequences via generating series

In this question I address the problem of proving the nonnegativity of a numerical sequence $a_0,a_1,a_2,\dots$ via generating series technique. In the notation $A(x)=\sum_{n=0}^\infty a_nx^n\ge0$ ...
12
votes
11answers
2k views

Applications of Measure, Integration and Banach Spaces to Combinatorics

I'm going to be teaching a Master's level analysis course(measure theory, Lebesgue integration, Banach and Hilbert spaces, and if there's time, some spectral or PDE stuff) in the fall. My problem is ...
26
votes
0answers
1k views

Curious $q$-analogues

Consider the Fibonacci polynomials $$F_n (x) = \sum_{j = 0}^{\left\lfloor {n/2} \right\rfloor }\binom{n-j}{j} x^{n - 2j} $$ and the Lucas polynomials $$L_n (x) = \sum_{j = 0}^{\left\lfloor {n/2} ...
11
votes
2answers
899 views

Evaluation of a combinatorial sum (that comes from random matrices)

I'm looking for an elementary combinatorial/generating function/etc proof of the following result: For nonnegative integers $r$, $$\frac{1}{r!} = \sum_{p_0+p_1+\cdots = r} ...
18
votes
0answers
1k views

Trigonometry related to Rogers--Ramanujan identities

For integers $n\ge2$ and $k\ge2$, fix the notation $$ [m]=\sin\frac{\pi m}{nk+1} \quad\text{and}\quad [m]!=[1][2]\dots[m], \qquad m\in\mathbb Z_{>0}. $$ Consider the following trigonometric ...
12
votes
0answers
603 views

Polynomials with presumably positive coefficients

After seeing that some positivity problems get their solutions on MO, I am quite enthusiastic of posing my (and not only) problem of positive flavour. In order to state it, I have to introduce the ...
3
votes
1answer
433 views

What is exactly the (singularity) confinement property ?

This property seems to be used both in the context of differential equations and several kinds of discrete equation systems or automata. It seems to be related in certain case to the Painlevé ...
2
votes
2answers
312 views

A bound on linear functionals over cotype 2 spaces

This is a modification of the somewhat naive question that I asked below. Suppose $X$ is a real Banach space of cotype-2, and $u_1, u_2, ... u_n$ are unit vectors in this space. For $\gamma = ...
16
votes
3answers
1k views

Recursions which define polynomials

There are many examples (Somos sequences, special polynomials related to rational solutions of the Painleve equations) when a recurrence relation, which a priori produces a sequence of rational ...
14
votes
5answers
2k views

Discrete harmonic function on a planar graph

Given a graph $G$ we will call a function $f:V(G)\to \mathbb{R}$ discrete harmonic if for all $v\in V(G)$ , the value of $f(v)$ is equal to the average of the values of $f$ at all the neighbors of ...
4
votes
3answers
976 views

Asymptotics of a hypergeometric series/Taylor series coefficient.

I was planning on figuring this problem out for myself, but I also wanted to try out mathoverflow. Here goes: I wanted to know the asymptotics of the sum of the absolute values of the Fourier-Walsh ...
14
votes
5answers
903 views

Is there a topological description of combinatorial Euler characteristic?

There are a collection of definitions of "combinatorial Euler characteristic", which is different from the "homotopy Euler characteristic". I will describe a few of them and give some references, and ...