Counting Borel Orbits in Symmetric Varieties of Types $BI$ and $CII$

TL;DR

This paper extends the combinatorial study of Borel orbits to symmetric varieties of types $BI$ and $CII$, providing generating series and exploring connections to lattice path enumeration.

Contribution

It determines the generating series for Borel orbits in specific symmetric varieties of types $BI$ and $CII$, advancing the enumeration in these classical types.

Findings

Derived explicit generating series for Borel orbits in types $BI$ and $CII$

Established connections between orbit enumeration and lattice path combinatorics

Enhanced understanding of orbit structures in classical symmetric varieties

Abstract

This is a continuation of our combinatorial program on the enumeration of Borel orbits in symmetric varieties of classical types. Here, we determine the generating series the numbers of Borel orbits in $\mathbf{SO}_{2n+1}/\mathbf{S(O}_{2p}\times \mathbf{O}_{2q+1}\mathbf{)}$ (type $BI$) and in $\mathbf{Sp}_n/\mathbf{Sp}_p\times \mathbf{Sp}_q$ (type $CII$). In addition, we explore relations to lattice path enumeration.