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Enumerating Permutations Avoiding Split Patterns 3|12 and 23|1

Travis Grigsby, Edward Richmond

Abstract

In this paper, we give a formula for the number of permutations that avoid the split patterns $3|12$ and $23|1$ with respect to a position $r$. Such permutations count the number of Schubert varieties for which the projection map from the flag variety to a Grassmannian induces a fiber bundle structure. We also study the corresponding bivariate generating function and show how it is related to modified Bessel functions.

Enumerating Permutations Avoiding Split Patterns 3|12 and 23|1

Abstract

In this paper, we give a formula for the number of permutations that avoid the split patterns and with respect to a position . Such permutations count the number of Schubert varieties for which the projection map from the flag variety to a Grassmannian induces a fiber bundle structure. We also study the corresponding bivariate generating function and show how it is related to modified Bessel functions.
Paper Structure (4 sections, 8 theorems, 47 equations, 1 table)

This paper contains 4 sections, 8 theorems, 47 equations, 1 table.

Table of Contents

  1. Introduction
  2. Connections with Schubert varieties
  3. Enumerating split patterns
  4. Generating functions

Key Result

Theorem 1.2

For any, $0\leq r\leq n$, we have

Theorems & Definitions (16)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 6 more