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We introduce a family of univariate polynomials indexed by integerpartitions. At prime powers, they count the number of subspaces in a finitevector space that transform under a regular diagonal matrix in a specifiedmanner. At 1, they count set partitions with specified block sizes. At 0, theycount standard tableaux of specified shape. At -1, they count standard shiftedtableaux of a specified shape. These polynomials are generated by a newstatistic on set partitions (called the interlacing number) as well as apolynomial statistic on standard tableaux. They allow us to express q-Stirlingnumbers of the second kind as sums over standard tableaux and as sums over setpartitions.  For partitions whose parts are at most two, these polynomials are thenon-zero entries of the Catalan triangle associated to the q-Hermite orthogonalpolynomial sequence. In particular, when all parts are equal to two, theycoincide with the polynomials defined by Touchard that enumerate chord diagramsby number of crossings.

Set partitions, tableaux, and subspace profiles under regular split semisimple matrices