Samrith Ram: Set Partitions, Tableaux, and Subspace Profiles under Regular Split Semisimple Matrices
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Abstract: In this talk we will introduce a family of polynomials $b_\lambda(q)$ indexed by integer partitions $\lambda$. These polynomials arise from an intriguing connection between two classical combinatorial classes, namely set partitions and standard tableaux. The polynomials $b_\lambda(q)$ can be derived from a new statistic on set partitions called the interlacing number which is a variant of the well-known crossing number of a set partition. These polynomials also have several interesting specializations: $b_\lambda(1)$ enumerates the number of set partitions of shape $\lambda$ and $b_\lambda(0)$ counts the number of standard tableaux of shape $\lambda$ while $b_\lambda(-1)$ equals the number of standard shifted tableaux of shape $\lambda$ respectively. When $q$ is a prime power $b_\lambda(q)$ counts (up to factors of $q$ and $q-1$) the number of subspaces in a finite vector space that transform under a regular diagonal matrix in a specified manner.
This is joint work with Amritanshu Prasad.