Title: Set partitions, tableaux, and subspace profiles under regular split semisimple matrices
Abstract: We introduce a family of univariate polynomials indexed by integer partitions. At prime powers, they count the number of subspaces in a finite vector space that transform under a regular diagonal matrix in a specified manner. At $1$, they count set partitions with specified block sizes. At $0$, they count standard tableaux of specified shape. At $-1$ they count standard shifted tableaux of a specified shape. These polynomials are generated by a new statistic on set partitions (called the interlacing number) as well as a polynomial statistic on standard tableaux. They allow us to express $q$-Stirling numbers of the second kind as sums over standard tableaux and as sums over set partitions. In a special case, these polynomials coincide with those defined by Touchard in his study of crossings of chord diagrams.