All Questions
19 questions with no upvoted or accepted answers
7
votes
0
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229
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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
0
answers
207
views
Parameter independence of Stanley's "content formula". Why?
For a cell $\square$ in the Young diagram of a partition $\lambda$, let $h_{\square}$ and $c_{\square}$ denote the hook length and content of $\square$, respectively.
R. Stanley remarked following ...
- 43.2k
6
votes
0
answers
115
views
Recursions which define polynomials?
Let $k$ be a positive integer and let
$$h(n,k,q)=\frac{1-(1+q^{k})q^{2k(n-1)+1}+q^{2}}{1-q^{2n-1}}h(n-1,k,q)-\frac{(1-q^{k(2n-3)})(1-q^{2k(n-1)})q^2}{(1-q^{2n-1})(1-q^{2n-3})}h(n-2,k,q)$$
with ...
- 5,622
4
votes
0
answers
128
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Inequality for $q$-binomials
Recall the constructions $[n]_q=\frac{1-q^n}{1-q}, [n]!_q=[1]_q[2]_q\cdots[n]_q$ with $[0]!_q:=1$ and the $q$-binomials (Gaussian polynomials)
$$\binom{n}k_q=\frac{[n]!_q}{[k]!_q[n-k]!_q}.$$
Given two ...
- 43.2k
4
votes
0
answers
186
views
A problem in the spirit of P. Borwein's polynomials
A well-known conjecture (now a theorem) of P. Borwein (see Wang and Krattenthaler - An asymptotic approach to Borwein-type sign pattern theorems) states:
For all positive integers $n$, the sign ...
- 43.2k
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
0
answers
116
views
A theory of refined h- and f-polynomials for the permutahedra, associahedra, noncrossing partitions, and tropical Grassmannians (references)
Looking for references (insights) on a theory encompassing a notion of refined face polynomials and their associated refined h-polynomials that are generalizations of the relation between ordinary f-...
- 10.5k
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
0
answers
195
views
Congruence for the polynomials $(t+1)^n$
An interesting polynomial congruence is given by
$$A_n(t^m)\equiv \left(\frac{1+t+\cdots+t^{m-1}}m\right)^{n+1}A_n(t) \qquad \mod (t-1)^{n+1}, \tag1$$
where $A_n(t)$ are the Eulerian polynomials with ...
- 43.2k
3
votes
0
answers
111
views
When does the constant term in the following expansion is nonzero?
Dyson's Theorem
The constant term in the expansion of
$$\prod_{1\leq i\neq j\leq n}\left(1-\frac{x_i}{x_j}\right)^{a_i}$$
is the multinomial coefficient
$$\frac{(a_1+\cdots+a_n)!}{a_1!\cdots a_n!},$$
...
- 1,997
2
votes
0
answers
73
views
An iterative formula for the Kreweras-Voiculescu polynomials (reference request)
Let
$$N(x) = 1 + \sum_{k \ge 1} N_k(h_1,h_2,...,h_k) \;x^k$$
$$ = 1 + h_1 x + (h_1^2 + h_2) x^2 + (h_1^3 + 3h_1h_2 + h_3)x^3 + (h_1^4 + 6 h_2 h_1^2 + 4 h_3 h_1 + 2 h_2^2 + h_4) x^4 + \cdots$$
be the ...
- 10.5k
2
votes
1
answer
358
views
q-polynomials in terms of a basis
Consider the polynomials
$$f_n(q)=\prod_{j=1}^n(1+q^j) \qquad \text{and} \qquad g_m(q)=1+q+q^2+\cdots+q^m.$$
I'll list a few examples to motivate my question. Direct calculations show that
$$f_1=g_1, \...
- 43.2k
2
votes
0
answers
77
views
Flexagons and noncrossing partitions
Turns out a couple of series related to the faces of flexagons
popped up in my explorations of combinatorial reciprocities in a group algebra for sets of partition polynomial (ParPs) related to the ...
- 10.5k
2
votes
0
answers
80
views
Inequality on polynomials
Recall $[n]_q=\frac{1-q^n}{1-q}, [n]!_q=[1]_q[2]_q\cdots[n]_q$ and the Gaussian polynomial $\binom{a}{b}_q=\frac{[a]!_q}{[b]!_q[a-b]!_q}$ with $[0]!_q:=1$.
Given two polynomials $U(q)=\sum_k\alpha_kq^...
- 43.2k
2
votes
0
answers
117
views
A multi-variable "Fibonacci polynomial"?
There is a tremendous literature on the Fibonacci sequence, including its polynomial analogue $F_{-1}=0, F_0=1$ and
$$F_n(x)=xF_{n-1}(x)+F_{n-2} \qquad \text{for $n\geq1$}.$$
What I have found is the ...
- 43.2k
1
vote
0
answers
63
views
Factorization of the symmetric function identity $E(t)=1/H(t)$ with the refined Euler characteristic polynomials of associahedra / Lagrange inversion
I've come across two matrix identities, flagged with daggers below, relating the two sets of elementary and complete homogeneous symmetric polynomials/functions via the two sets of refined Lah and ...
- 10.5k
1
vote
0
answers
159
views
A follow up on Bergeron's conjecture and a question
We say two polynomials satisfy $P(x)\geq Q(x)$ iff $P(x)-Q(x)$ has non-negative coefficients. Recall $(n)_q!=\prod_{j=1}^n(1-q^j)$ and the Gaussian polynomials $\binom{n}k_q=\frac{(n)_q!}{(k)_q!(n-k)...
- 43.2k
0
votes
0
answers
109
views
Applications of Jack polynomials
I developed four libraries (Julia, R, Python, Haskell) for the computation of Jack polynomials. I developed them for fun because I found this was programmatically interesting. But now I'd like them to ...
- 2,560
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