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Mon, Nov 11, 2019
5:00 PM
MSCS 514
On the rank generating function for an interval of partitions
Faqruddin Ali Azam, Oklahoma State University
Snaaaaacks
 [Abstract] [PDF]
 Abstract: A partition $\lambda=(\lambda_1,\lambda_2,\lambda_3,\cdots)$ is a finite sequence of weekly decreasing non-negative integers. If $\lambda=(\lambda_n)$ and $\delta=(\delta_n)$ are two partitions such that $\delta_i\leq \lambda_i$ for all $i,$ then we say that $\delta\leq \lambda.$ For two partitions $\delta, \lambda,$ we define $\displaystyle{[\delta, \lambda]:=\{ \alpha: \alpha \text{ is a partition and }\delta\leq \alpha\leq \lambda \} }.$ An interval of two partitions is a graded partially ordered set. The rank generating polynomial of a graded partially ordered set P is defined as $\displaystyle{ g_{P}(y)=\sum_{\alpha\in P} y^{\rho(\alpha)},}$ where $\rho(\alpha)$ is the rank of $\alpha.$ We will discuss some formulae to find the rank generating polynomial for any arbitrary interval of partitions.