| Mon, Nov 04, 2019
|
Graduate Student Seminar
5:00 PM
MSCS 514 | | On the rational generating function $Q_k$ for Young's Lattice
Faqruddin Ali Azam, Oklahoma State University
Snacks
| | | Abstract: A partition $\lambda=(\lambda_1,\lambda_2,\lambda_3,\cdots)$ is a sequence of weekly decreasing non-negative integers such that all but finitely many $\lambda_i$ are zero. For a positive integer $k,$ we define $$\displaystyle{Q_{k}(x_1,x_2,\cdots,x_k,y):=\sum_{\substack{\lambda\text{, partition} \\ l(\lambda)=k}}^{}g_{\lambda}(y)x^{\lambda},}$$ where $g_{\lambda}(y)$ is the rank generating polynomial of the Young's lattice $[0, \lambda]$. Here $l(\lambda)$ denotes the number of nonzero parts in $\lambda$ and $x^\lambda:= x_1^{\lambda_1}x_2^{\lambda_2}\cdots x_{k}^{\lambda_k},$ provided that $\lambda=({\lambda_1},{\lambda_2},\cdots,{\lambda_k}).$ Thus $Q_{k}(x_1,x_2,\cdots,x_k,y)$ is a series of polynomials in $x_1, x_2, \cdots, x_k\text{ and } y$. We will show that for each positive integer $k,$ $Q_{k}(x_1,x_2,\cdots,x_k,y)$ can be expressed as a rational function in $x_1, x_2, \cdots, x_k\text{ and } y.$ |
|
|
|