All Questions
13 questions
13
votes
2
answers
2k
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 ...
- 10.5k
8
votes
1
answer
498
views
Loday's characterization and enumeration of faces of associahedra (Stasheff polytopes)
From "The multiple facets of the associahedra" by Loday:
Let us consider the formal power series
$$f(x) = x+a_1 x^2 +a_2 x^3 + \cdots+ a_n x^{n+1} + \cdots$$
and let
$$ g(x) = x+b_1 x^2 + ...
- 10.5k
7
votes
2
answers
789
views
"MultiCatalan numbers"
Could anyone provide a reference for the following (sort of) generalization of Catalan numbers: the multinomial coefficient
$$
\binom{2k_1+3k_2+4k_3+...}{k_1+2k_2+3k_3+...,k_1,k_2,k_3,...}
$$
is ...
- 18.4k
11
votes
4
answers
1k
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
".....
- 4,829
7
votes
0
answers
229
views
Has anyone seen these binary trees (Catalan-type related to the Gegenbauer polynomials and Motzkin paths)?
The OEIS entry A121448 enumerates binary trees with $n$ edges and $k$ vertices with outdegree 1.
Has anyone seen these trees?
The o.g.f. for this entry, $G(x,t)$, is essentially a discriminant ...
- 10.5k
6
votes
1
answer
588
views
A numerical matrix of power sum polynomials
Let $p_i=x_1^i+x_2^i+\cdots+x_m^i=\sum_{k=1}^mx_k^i$ be the power sum polynomials. Then, the determinant of the $m\times m$ Hankel matrix $M_m=(p_{i+j-2})$, for $1\leq i,j\leq m$, has a neat ...
- 43.2k
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 ...
- 5,622
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 ...
- 10.5k
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}...
- 43.2k
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 ...
- 43.2k
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 ...
- 10.5k
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 ...
- 43.2k
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 ...
- 10.5k