# Tagged Questions

**17**

votes

**3**answers

2k views

### Where in mathematics do these polynomials appear?

Does anyone recognize the following sequence of polynomials?
$f_0(x) = x-1$
$f_1(x) = x^2-x$
$f_2(x) = x^4-2x^2+x$
$f_3(x) = x^8-3x^4+3x^2-x$
$f_4(x) = x^{16}-4x^8+6x^4-4x^2+x$
$\vdots$
The ...

**9**

votes

**1**answer

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 ...

**6**

votes

**1**answer

95 views

### How “accidental” are equalities between parts of Ehrhart quasi-polynomials? When do they persist to Euler-Maclaurin?

Background
What I think of Ehrhart theory (http://en.wikipedia.org/wiki/Ehrhart_polynomial) asserts that if we take a lattice polytope $P$, and count the number of lattice points in the $t$th ...

**7**

votes

**0**answers

211 views

### Do the coefficients of these irreducible polynomials always become periodic?

Fix $n\in\mathbb N$ and a starting polynomial (or seed) $p_n=a_0+a_1x+\dots+a_nx^n$ with $a_k\in\mathbb Z\ \forall k$ and $a_0a_n\ne0$.
Define $p_{n+1},p_{n+2},\dots$ recursively by $p_r = ...

**5**

votes

**1**answer

178 views

### Large gaps between consecutive irreducible polynomials with small heights

For a prime gap of length at least $n$, a trivial upper bound for its first occurrence is $N=n!$ or $N=lcm(2,\dots,n)$. A bit better is $N=p_1\cdots p_n$ where $p_k$ is the $k$th prime, as then ...

**2**

votes

**2**answers

213 views

### Irreducible Polynomials from a Reccurence

This question is inspired by a recent one : Let $c$ be a variable and define a sequence by $a_0=0$ $a_1=1$ and $a_{n+1}=a_{n}c-a_{n-1}$ . So
$$\begin{align*}
a_2 &= c
\\ a_3 &={c}^{2}-1= ...

**6**

votes

**1**answer

499 views

### “MultiCatalan numbers”

Could anyone provide a reference for the following (sort of) generalization of Catalan numbers: the multinomial coefficient
$$
\binom{2k_1+3k_2+4k_3+...}{k_1+2k_2+3k_3+...,k_1,k_2,k_3,...}
$$
is ...

**30**

votes

**2**answers

1k views

### A question on maps from $\mathbb{Z}/p\mathbb{Z}$ to itself

Let $p\geq 3$ be a prime number, and let $u:\mathbb{Z}/p\mathbb{Z}\to \mathbb{Z}/p\mathbb{Z}$ be a map such that, for all $l\in \mathbb{Z}/p\mathbb{Z}$,$l\neq 0$, the map $k\mapsto u(k+l)-u(k)$ is a ...

**3**

votes

**2**answers

318 views

### Counting polynomials with same coefficient sum and a given value at a point

Call an univariate polynomial $f(x) = \sum_{i=0}^{n}a_{i}x^{i} \in \Bbb{Z}[x]$ symmetric if $a_{i} = a_{n-i}$ and $a_{0} = a_{n} > 0$.
For a given $\sum_{i=0}^{n}a_{i}$ and $a_{i} \geq 0$, how ...

**0**

votes

**0**answers

85 views

### Enumerating certain bounded polynomials with partition type property constraints

Let $p(x) = a_{0} + a_{1}x + a_{2}x^{2} + \dots + a_{d}x^{d} + \dots + a_{2d+\alpha}x^{2d+\alpha} \in \mathbb{Z}[x]$ where $\alpha \in \{0,1\}$ with the constraints
$(1)$ $0 \leq ...

**3**

votes

**0**answers

154 views

### identity for number of monomials

Fix a positive integer $d \geq 2$, and let $n,k$ be natural numbers with $k \leq n$.
Let $b(n,k)$ denote the number of monomials of degree $kd-(n+1)$ in $n+1$ variables $x_0,\ldots,x_n$ with all ...

**0**

votes

**0**answers

63 views

### When do the even part and odd part of a hypergeometric like function have common nonnegative real root?

I encountered the following problem in my research project:
Let
$$f(x)=\sum_{k=0}^{\infty} a_{k} x^{k}$$
We can separate the even part $p(x^2)$ from the odd part $x q(x^2)$ and write
...

**6**

votes

**3**answers

397 views

### What is the correspondence between combinatorial problems and the location of the zeroes of polynomials called?

(From MSE)
In the wikipedia article on the Italian-born American mathematician and philosopher Gian-Carlo Rota, it is stated that the one combinatorial idea he would like to be remembered for
...

**26**

votes

**1**answer

700 views

### How many polynomial Morse functions on the sphere?

Let $f$ be a homogeneous polynomial of degree $d$ in $n$ variables. Restricted to the unit sphere $S^{n-1}$, it might or might not be a Morse function.
If $f$ is a Morse function of degree $1$, you ...

**1**

vote

**1**answer

279 views

### Combinatorics: Product Rules.

I couldn't find a way to figure this out, though it is a somewhat basic question that came up when studying the stationary phase expansion of an integral. The abstract version is the following:
I ...

**5**

votes

**1**answer

825 views

### Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants

Zeta functions abound in mathematics. Audrey Terras describes in Zeta Functions and Chaos three zeta functions--the zeta fct. of a projective non-singular algebraic variety; the Artin-Mazur zeta ...

**6**

votes

**2**answers

1k views

### Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus

I was exploring some raising and lowering operators related to an infinitesimal generator for fractional integro-derivatives and found an Appell sequence of polynomials, i.e., an infinite sequence of ...

**12**

votes

**0**answers

232 views

### Are the zeros of Tutte polynomials dense in $\mathbb C^2$?

For the chromatic polynomials of graphs we have two nice theorems which describe the behavior of their zeros: Thomassen proved that the set of real zeros of all chromatic polynomials is the union of ...

**14**

votes

**1**answer

654 views

### Reconstructing a word

Let $w(a,b)$ be a word in two letter alphabet. Let $$A=\left(\begin{array}{lll}x_1 & x_2 & x_3\\\ x_4 &x_5 & x_6\\\ x_7 & x_8 & x_9\end{array}\right), ...

**16**

votes

**4**answers

901 views

### A mixing property for finite fields of characteristic $2$

In connection with this MO post, here is a question somewhat implicitly contained in a joint paper of S. Kopparty, S. Saraf, M. Sudan, and myself.
Let ${\mathbb F}$ be a finite field, and suppose ...

**11**

votes

**4**answers

641 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 ...

**4**

votes

**2**answers

285 views

### A combinatorial formula involving the necklace polynomial

This question is motivated by the answers given to my previous one. In combinatorics, the necklace polynomials are given by
$$M(X,n)=\frac1n\sum_{d|n}\mu\left(\frac{n}{d}\right)X^d,$$
where $\mu$ is ...

**8**

votes

**2**answers

754 views

### The number of irreducible polynomials over ${\mathbb F}_p$

Let $p$ be a prime number. The number of monic irreducible polynomial $P\in{\mathbb F}_p[X]$, in terms of the degree $d$, begins with
$${\rm irr}(1)=p,\qquad{\rm irr}(2)=\frac{p(p-1)}2,\qquad{\rm ...

**10**

votes

**1**answer

550 views

### Irreducibility of Schur polynomials

A natural question covering both this and this question would be
Let $n>2$. Describe Young diagrams $\lambda$ with at most $n$ nonempty rows (or equivalently non-increasing sequences ...

**12**

votes

**0**answers

360 views

### Catalan objects associated to a univariate polynomial

Given a monic degree $n$ polynomial $f(z)$ with no double roots, and a phase $0\leq \theta < \pi$, there are natural constructions which associate to this data:
a noncrossing matching on $2n$ ...

**3**

votes

**0**answers

97 views

### Are there existing resources on modular-esque recurrence relations?

Does anyone know where I would be able to get information on analyzing a class of polynomial recurrence relations of a form like this?
$\begin{align*}
f_{n,k}(x) & ...

**2**

votes

**0**answers

266 views

### Characteriszation of certain kinds of polynomials

My question:
Is there a known characterization for polynomials $P\in \mathbb{R}[x,y]$
with the property that
$$P(x,y) = 0 \wedge P(y,x) =0 \Rightarrow x = \overline{y}$$
and the number of the ...

**0**

votes

**1**answer

379 views

### Find recurrence in Pascal-like triangle of polynomials

Consider an infinite, upper-triangular Toepliz-matrix, with first row $x_1,x_2,\dots,x_n.$
Then there is a sequence of determinants obtained from the $m \times m-$ sub-matrices
with upper left corner ...

**8**

votes

**2**answers

653 views

### What is known about zero-sets of Schur polynomials?

Consider a set S of partitions not containing the empty partition (I would be happy with, say, all the partitions of size less than k, except for the empty one).
Let $U_\lambda^{(r)}$ be the ...

**2**

votes

**0**answers

203 views

### Matrices preserving interlacing/stable polynomials

Let $v_1 = (p_1,\dots,p_k)$ be a vector of interlacing polynomials, and non-negative coefficients,
i.e. $p_1,\dots,p_k$ are real-rooted, and the roots of $p_i$, $p_{i+1}$ interlace.
Let $M$ be a $k ...

**21**

votes

**6**answers

1k views

### Relations between sums of powers

This question is so naive that it could have been asked before on this site. If so, I'll delete it.
Among beautiful formula, I like a lot this one:
$$\left(\sum_{n=1}^Nn\right)^2=\sum_{n=1}^Nn^3.$$
...

**6**

votes

**5**answers

2k views

### Number of spanning forests in a graph

Hello,
I have two questions that have been bugging me recently. The first is about the number of spanning forests in a graph and the second is about enumerating these with edge labels.
Q1: I am ...

**3**

votes

**1**answer

260 views

### Counting some polynomials that have a zero in $\mathbb{Z}_n[X]$

This is a question I asked on Math.SE and got only a partial answer. I hope I will have better chances here.
Given the ring of polynomials $\mathbb{Z}_n[X]$, consider $$\mathbb{P}_n = \lbrace a_0 ...

**13**

votes

**1**answer

569 views

### Symmetric polynomial from graphs

Let $g$ be a directed, connected multigraph, on $n$ vertices, without loops.
Define
$$P_g(x_1,\dots,x_n) := Sym\left[ \prod_{(i,j) \in g} (x_i-x_j) \right]$$
where $(i,j)$ is the directed edge ...

**8**

votes

**1**answer

538 views

### When does $P(a−b)=0$ for $a≠b$ ensure $P(0)=0$? (Continued.)

As a natural (and expectable) extension of my earlier question:
How large must be a set $A\subset F_2^n$ to ensure that if $P$ is a cubic polynomial in $n$ variables over the field $F_2$, ...

**9**

votes

**0**answers

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
$A(x)=\sum_{n=0}^\infty a_nx^n\ge0$ ...

**6**

votes

**4**answers

573 views

### Can this nested sum be expressed in terms of generalized harmonic numbers and the cycle index polynomials of the symmetric groups?

For a paper I was working on recently I needed to find the value of the following sum:
$$S(n,k) = \sum_{i_1 = 1}^n \sum_{i_2 = i_1+1}^n \cdots \sum_{i_k=i_{k-1}+1}^n \frac{1}{i_1 i_2 \cdots i_k}.$$
...

**1**

vote

**3**answers

427 views

### Characteristic polynomials for $K$-Bonacci numbers: what's their name?

Fibonacci numbers are defined by the recurrence relation
$f_{n+2}=f_{n+1}+f_{n}$ and
Tribonacci numbers by
$f_{n+3}=f_{n+2}+f_{n+1}+f_{n}$
One can define, in general, K-Bonacci numbers as
...

**8**

votes

**1**answer

769 views

### Is this sequence of polynomials well-known?

While working on a problem in p-adic Hodge theory, and needing to write down a solution to a certain equation involving p-adic power series, I stumbled across a certain sequence of polynomials. Define ...

**7**

votes

**4**answers

492 views

### If the series Σ pᵃ⁽ʷ⁾·xᴵʷᴵ is rational, is Σ a(w)·xᴵʷᴵ also rational (summation over words w in a regular language)?

Let $p$ be a prime number and let $a_i$ be a sequence of natural numbers such that the series $\sum_{i=1}^\infty p^{a_i} x^i$ is rational. A warm-up question:
Question 1. Does it follow that the ...

**12**

votes

**0**answers

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 ...

**16**

votes

**3**answers

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 ...

**10**

votes

**1**answer

624 views

### Counting colored rook configurations in the cube - when is it even?

Informal Statement
In the $n\times n \times n$ grid, we can places rooks (those from chess) such that no two rooks can attack each other. One way to achieve this is to place a rook in position ...

**2**

votes

**3**answers

497 views

### Variant of binomial coefficients

I've recently come across a variant of the binomial polynomials, and I'm curious if anyone has seen these before. If so, I'd love a reference, a name, etc.
First recall the following. If z is a ...

**5**

votes

**1**answer

627 views

### Specializations of Schur functions at consecutive integers

Given a partition λ = (λ1, λ2, ..., λn) denote with sλ the associated Schur function.
There exists a nice product formula for the principal specializations:
...