  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.
