Algebraic aspects of increasing subsequences

TL;DR

This paper explores algebraic structures related to increasing subsequences, connecting integrals, sums, and group invariants, and provides new formulas and proofs for distributions and sums involving permutations.

Contribution

It introduces new integral formulas, partial sums, and invariant space bases, linking increasing subsequences with classical group integrals and orthogonal polynomials.

Findings

New integral formulae for the distribution of the longest increasing subsequence in involutions

Novel partial Cauchy-Littlewood sum formulas and proofs

Explicit bases for invariant spaces of classical groups

Abstract

We present a number of results relating partial Cauchy-Littlewood sums, integrals over the compact classical groups, and increasing subsequences of permutations. These include: integral formulae for the distribution of the longest increasing subsequence of a random involution with constrained number of fixed points; new formulae for partial Cauchy-Littlewood sums, as well as new proofs of old formulae; relations of these expressions to orthogonal polynomials on the unit circle; and explicit bases for invariant spaces of the classical groups, together with appropriate generalizations of the straightening algorithm.