A Geometric Interpretation of the Intertwining Number

TL;DR

This paper explores the relationship between combinatorial statistics on set partitions and algebraic geometry, linking intertwining numbers to Borel orbits and introducing new polynomial analogs of classical numbers.

Contribution

It establishes a geometric interpretation of the intertwining number, connects it to algebraic geometry, and introduces new polynomial analogs of Stirling and Bell numbers.

Findings

Connected intertwining number to Borel orbit geometry.

Determined the $q=-1$ specialization of a $q$-Bell number analogue.

Introduced a $t$-analog of $q$-Stirling numbers using Renner's $H$-polynomial.

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.