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5 votes
1 answer
211 views

degree of a polynomial over set-partitions

Denote $(x)_t = x(x-1)(x-2)\cdots(x-t+1)$ and fix some $t_1,\dots,t_n\in\mathbb{N}$. Now consider the polynomials $$f_n(x)=\sum_{\pi\in L[n]}(-1)^{\vert\pi\vert-1}(\vert\pi\vert-1)!\prod_{A\in\pi}(x)_{...
T. Amdeberhan's user avatar
4 votes
1 answer
239 views

A discrete operator begets even/odd polynomials

Given a function $f(x)$ define the forward shift operator by $Ef(x)=f(x+1)$ and the discrete derivative $\delta f(x)=(E-1)f(x)=f(x+1)-f(x)$. Given a partition $\lambda=(\lambda_1,\lambda_2,\dots,\...
T. Amdeberhan's user avatar
4 votes
0 answers
145 views

Validating a result on evaluating Jack polynomials

I am currently working through the following paper: Lapointe L., Lascoux A., Morse J. Determinantal Expression and Recursion for Jack Polynomials Electron. J. Combin. 7 (2000), Notes 1. DOI: 10.37236/...
J. M. isn't a mathematician's user avatar
3 votes
1 answer
372 views

How many ways can $N$ be written as a sum of terms in the form $2^i3^j$?

Given a positive integer $N$, let $f(N)$ be the number of ways $N$ can be decomposed as a sum of terms of the form $2^i3^j$, where each such term appears at most once in the sum. For example, $f(10) = ...
Gautam's user avatar
  • 1,703
3 votes
1 answer
272 views

Generating function for parity in hooks

Let $\lambda\vdash n$ denote an integer partition of $n$ and $\frak{H}_{\lambda}$ be the multiset of hook lengths of $\lambda$. Further, let $o(\lambda)=\#$ of odd entries and $e(\lambda)=\#$ of even ...
T. Amdeberhan's user avatar
3 votes
1 answer
159 views

Proving that two sequences of polynomials defined over partitions are inverse to each other

For any fixed $c>0$ consider the polynomials \begin{align*} & p_n(X_1,X_2,\ldots) := \frac{n!}{c} \sum\limits_{b=1}^n \frac{c^b}{b!(n+1-b)!} \sum\limits_{\substack{l_1,\ldots,l_b \geq 1 \\ ...
Ben Deitmar's user avatar
  • 1,295
3 votes
0 answers
214 views

A family of polynomials related to integer partitions

For a positive integer $n$, let $p(n)$ be the number of partitions of $n$. For $1\le k\le n$, let $p(n,k)$ denote the number of partitions of $n$ having exactly $k$ terms; in other words, $p(n,k)$ is ...
Zhi-Wei Sun's user avatar
  • 15.6k
3 votes
0 answers
312 views

Enumerating a class of polynomials

How many equivalence classes of $\Bbb F_2[x,y]$ polynomials with $x$ degree $n_x$ and $y$ degree $n_y$ are there such that each $y^i$ coefficient (polynomial in $\Bbb Z[x]$) is distinct and $x^i$ ...
user avatar
2 votes
1 answer
142 views

Reading off top hook-lengths in partitions

Given an integer partition $\lambda$ and its Young diagram $Y_{\lambda}$, let $h_{\lambda}(i,j)$ stand for the corresponding hook length of the cell $(i,j)\in Y_{\lambda}$. Write $\lambda\vdash n$ for ...
T. Amdeberhan's user avatar
1 vote
1 answer
196 views

Largest part and length of a partition in play

If $\lambda=(\lambda_1\geq\lambda_2\geq\dots\geq\lambda_k)\vdash n$ is an integer partition of $n$ then $\lambda_1$ is its largest part and $k$ is its length, $\ell(\lambda)$. Define the statistic $...
T. Amdeberhan's user avatar