# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $2n$ ...
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### 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: $$\left(\sum_{n=1}^Nn\right)^2=\sum_{n=1}^Nn^3.$$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...