A geometric interpretation of the intertwining number

We exhibit a connection between two statistics on set partitions, the intertwining number and the depth-index. In particular, results link the intertwining number to the algebraic geometry of Borel orbits. Furthermore, by studying the generating polynomials of our statistics, we determine the q = −1 specialization of a q-analogue of the Bell numbers. Finally, by using Renner’s H-polynomial of an algebraic monoid, we introduce and study a t-analog of q-Stirling numbers.