All Questions
Tagged with co.combinatorics polynomials
69 questions
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 ...
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 ...
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 ...
3
votes
0
answers
151
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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 ...
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 ...
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 ...
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}...
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 ...
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
$$\begin{align*}
a_2 &= c
\\ a_3 &={c}^{2}-1= \...
3
votes
0
answers
144
views
Noncrossing partitions in Hopf algebras/monoids via compositional inversion
Partition polynomials constructed from the face structures of the associahedra (OEIS A133437) and permutahedra (A133314) comprise the antipodes/compositional inverses in a Faa-di-Bruno-type Hopf ...
2
votes
0
answers
337
views
Enumerating certain types of permutation polynomials
Given a prime power $q$, I would like to enumerate (preferably up to isomorphism*) all the permutation polynomials $f(x)$ on $K = GF(q^3)$ satisfying the following conditions:
$f(ax) = af(x)$ for all ...
2
votes
1
answer
431
views
Lagrange interpolation vs homogeneous symmetric polynomials?
This question is a follow-up on another MO query here.
Question. For $r\geq$ an integer, is it true that there exists homogeneous symmetric polynomial $P_r(x_1,\dots,x_n)$ with positive ...
2
votes
0
answers
345
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Combinatorics of iterated composition of noncrossing partition polynomials
A combinatorial problem arises in relating connected and disconnected Green functions associated to a "zero-dimensional" quantum field theory presented by Brezin, Itzykson, Parisi, and Zuber ...
2
votes
1
answer
385
views
Determinants of striped Hankel matrices
This question is related to the matrices described in Deyi Chen's recent MO post (look at some examples there). The main difference: we are asking for a determinant evaluation instead of a permanent, ...
2
votes
2
answers
452
views
These polynomials are always either even or odd [duplicate]
The falling factorials are $(x)_n=x(x-1)\cdots(x-n+1)$ with $(x)_0:=1$. Define the forward shift $E$ and the discrete derivative $\delta=E-1$, respectively, by
$$Ef(x)=f(x+1) \qquad \text{and} \qquad \...
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 $...
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}[...
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 ...
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 ...