Set partitions, tableaux, and subspace profiles under regular diagonal matrices

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. This enumeration formula is a combinatorial solution to a problem introduced by Bender, Coley, Robbins and Rumsey. 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.

For partitions whose parts are at most two, these polynomials are the non-zero entries of the Catalan triangle associated to the $q$-Hermite orthogonal polynomial sequence. In particular, when all parts are equal to two, they coincide with the polynomials defined by Touchard that enumerate chord diagrams by the number of crossings.