Abstract
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.
Highlights
This paper is concerned with the intertwining number of a set partition, which is a combinatorial statistic introduced by Ehrenborg and Readdy in [10]
This statistic is among the combinatorial parameters on set partitions whose generating function is an important q-analog of the Stirling numbers of the second kind: Sq(n, k) =
One purpose of this article is to show that the depth-index is related in an interesting way to other set partition statistics
Summary
This paper is concerned with the intertwining number of a set partition, which is a combinatorial statistic introduced by Ehrenborg and Readdy in [10]. In the present paper we connect the intertwining number to another statistic on the set partitions, namely, the depth-index, which was recently introduced and studied in [3] by the first two authors. A purely combinatorial way to describe the above partial order on set partitions was recently introduced in [3], where the rank function of the poset was given by a certain combinatorial statistic (on arc-diagrams), called the depth-index of A and was denoted by t(A). The depth-index t(A) gives the dimension of Bnτ Bn where τ is the upper-triangular partial permutation matrix which corresponds to the set partition A It follows from Theorem 3 that the intertwining number i(A) is the rank function of the dual poset, i(A) = codim (Bnτ Bn).
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