Catalan numbers and Schubert polynomials for $w=1(n+1)... 2$

TL;DR

This paper demonstrates that the Schubert polynomial for a specific permutation specializes to the Catalan number, providing multiple proofs, a q-analog, and an application to Schubert variety singularities.

Contribution

It establishes a new connection between Schubert polynomials and Catalan numbers, including proofs, a q-analog, and an application to singularity analysis.

Findings

Schubert polynomial for w=1(n+1)...2 equals Catalan number C_n

Multiple proofs of this specialization are provided

A q-analog of the result is introduced

Abstract

We show that the Schubert polynomial S_w specializes to the Catalan number C_n when $w=1(n+1)...2$. Several proofs of this result as well as a q-analog are given. An application to the singularities of Schubert varieties is given.