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5 votes
1 answer
213 views

Matrix-valued periodic Fibonacci polynomials

Consider the Fibonacci polynomials $f_n(x)$, defined by the recursion $f_n(x)=xf_{n-1}(x)-f_{n-2}(x)$ with initial values $f_0(x)=0$ and $f_1(x)=1$. It is well known that the values of these ...
Johann Cigler's user avatar
4 votes
1 answer
295 views

A Conjecture about the integral related to Chebyshev polynomial

I am interested in the following integral related to the Chebyshev polynomials $$I_{n,m}:= \int_0^\pi \left(\frac {\sin nx}{\sin x}\right)^{m} dx,$$ where $n,m\in \mathbb{Z}^+.$ It is easy to see ...
Jacob.Z.Lee's user avatar
3 votes
0 answers
144 views

Flat polynomials with factors of big height

Let $p(x)$ be a polynomial of degree $n$ with all coefficients in $\{-1,0,1\}$ (such polynomials are sometimes called flat). I am wondering how big the coefficients of a factor of $p$ can be. Call ...
Wolfgang's user avatar
  • 13.4k
3 votes
0 answers
151 views

Extension of work by Novelli and Thibon on noncommutative symmetric functions and Lagrange inversion

(Edit May 12, 2023: I just put up a brief summary of some of my notes on the partition polynomials described below in my WordPress mini-arXiv at "As Above, So Below". It contains multinomial ...
Tom Copeland's user avatar
  • 10.5k
3 votes
1 answer
447 views

A number array related to colored necklaces and the primes

I stumbled upon entry OEIS-A208535 on the enumeration of certain kinds of colored necklaces and noticed that the integers for the odd prime rows of the table there seem to be given by the Moreau ...
Tom Copeland's user avatar
  • 10.5k
3 votes
0 answers
203 views

Combinatorial characterizations of complex weight supports

This question is related to my last question and is originally motivated by recent advances in quantum physics. I am looking for combinatorial characterizations of some algebraically defined families ...
Alex Ravsky's user avatar
  • 5,409
3 votes
1 answer
165 views

The inverse of a symbolic matrix (with reciprocal binomials) has Laurent entries

Recalling the $q$-binomials (Gaussian polynomials). Let $[n]_q!=\prod_{j=1}^n\frac{1-q^j}{1-q}$ and $\binom{n}k_q=\frac{[n]_q!}{[k]_q!\cdot[n-k]_q!}$. Now, consider the $n\times n$ matrix $\mathbf{M}...
T. Amdeberhan's user avatar
3 votes
0 answers
207 views

On a variation of the Vandermonde matrix

The ubiquitous Vandermonde matrix, of entries $(x_i^{j-1})_{i,j}^{1,n}$, and its determinant $$\prod_{i<j}^{1,n}(x_j-x_i)$$ have found many utilities in Combinatorics and Physics, among other ...
T. Amdeberhan's user avatar
3 votes
2 answers
405 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= \...
Aaron Meyerowitz's user avatar
3 votes
0 answers
144 views

Noncrossing partitions in Hopf algebras/monoids via compositional inversion

Partition polynomials constructed from the face structures of the associahedra (OEIS A133437) and permutahedra (A133314) comprise the antipodes/compositional inverses in a Faa-di-Bruno-type Hopf ...
Tom Copeland's user avatar
  • 10.5k
2 votes
0 answers
337 views

Enumerating certain types of permutation polynomials

Given a prime power $q$, I would like to enumerate (preferably up to isomorphism*) all the permutation polynomials $f(x)$ on $K = GF(q^3)$ satisfying the following conditions: $f(ax) = af(x)$ for all ...
Anurag's user avatar
  • 1,197
2 votes
1 answer
431 views

Lagrange interpolation vs homogeneous symmetric polynomials?

This question is a follow-up on another MO query here. Question. For $r\geq$ an integer, is it true that there exists homogeneous symmetric polynomial $P_r(x_1,\dots,x_n)$ with positive ...
T. Amdeberhan's user avatar
2 votes
0 answers
345 views

Combinatorics of iterated composition of noncrossing partition polynomials

A combinatorial problem arises in relating connected and disconnected Green functions associated to a "zero-dimensional" quantum field theory presented by Brezin, Itzykson, Parisi, and Zuber ...
Tom Copeland's user avatar
  • 10.5k
2 votes
1 answer
385 views

Determinants of striped Hankel matrices

This question is related to the matrices described in Deyi Chen's recent MO post (look at some examples there). The main difference: we are asking for a determinant evaluation instead of a permanent, ...
T. Amdeberhan's user avatar
2 votes
2 answers
452 views

These polynomials are always either even or odd [duplicate]

The falling factorials are $(x)_n=x(x-1)\cdots(x-n+1)$ with $(x)_0:=1$. Define the forward shift $E$ and the discrete derivative $\delta=E-1$, respectively, by $$Ef(x)=f(x+1) \qquad \text{and} \qquad \...
T. Amdeberhan's user avatar
1 vote
0 answers
143 views

Polynomial interpolation of binary vectors

Let $\mathbb{F}$ be a finite field and let $\boldsymbol{x} = (x_1, x_2, \dots, x_n)$ be $n$ pairwise distinct points in $\mathbb{F}$. Given the vector $\boldsymbol{y} = (y_1, y_2, \dots, y_n)$, with $...
Bean Guy's user avatar
1 vote
0 answers
147 views

Counting Hamiltonian cycles in graph and finding a coefficient of polynomial

Exact result is #P-Hard, so we are looking for bounds. Let $G$ be simple graph or simple digraph and $A$ its adjacency matrix. $A$ is $n \times n$ with entries only zeros or ones. Let $K=\mathbb{Z}[...
joro's user avatar
  • 25.4k
1 vote
0 answers
89 views

Combinatorial models of the refined inverse Eulerian numbers

If I evaluate substitution of an infinite set of indeterminates $(c_1,c_2,c_3,\cdots)$ into the infinite set of refined Eulerian polynomials $[E]$ of OEIS A145271, I obtain the Taylor series ...
Tom Copeland's user avatar
  • 10.5k
0 votes
1 answer
349 views

Log associahedra and log noncrossing partitions--raising ops and symmetric function theory for $A_n$ (references)

Where do the following three sets $[LA]$, $[ILA]$, and $[LN]$ of partition polynomials appear in the literature? There are two sets of partition polynomials, not in the OEIS, that serve as the ...
Tom Copeland's user avatar
  • 10.5k

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