The genesis of involutions (polarizations and lattice paths)

TL;DR

This paper investigates the enumeration of Borel orbits in polarizations of symmetric varieties, deriving generating functions and exploring their connections to lattice path combinatorics.

Contribution

It introduces new generating functions for Borel orbits in polarizations and establishes links to lattice path combinatorics, advancing understanding of symmetric variety structures.

Findings

Derived bivariate generating functions for Borel orbits

Established relations between orbit counts and lattice path combinatorics

Analyzed the structure of polarizations in symmetric varieties

Abstract

The number of Borel orbits in polarizations (the symmetric variety $SL(n)/S(GL(p)\times GL(q))$) is analyzed, various (bivariate) generating functions are found. Relations to lattice path combinatorics are explored.