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Enumerating pattern-avoiding permutations by leading terms

Ömer Eğecioğlu, Collier Gaiser, Mei Yin

Abstract

The number of 123-avoiding permutation on $\{1,2,\ldots,n\}$ with a fixed leading terms is counted by the ballot numbers. The same holds for $132$-avoiding permutations. These results were proved by Miner and Pak using the Robinson-Schensted-Knuth (RSK) correspondence to connect permutations with Dyck paths. In this paper, we first provide an alternate proof of these enumeration results via a direct counting argument. We then study the number of pattern-avoiding permutations with a fixed prefix of length $t\geq1$, generalizing the $t=1$ case. We find exact expressions for single and pairs of patterns of length three as well as the pair $3412$ and $3421$. These expressions depend on $t$, the extrema, and the order statistics. We also define $r$-Wilf equivalence for permutations with a single fixed leading term $r$, and classify the $r$-Wilf-equivalence classes for both classical and vincular patterns of length three.

Enumerating pattern-avoiding permutations by leading terms

Abstract

The number of 123-avoiding permutation on with a fixed leading terms is counted by the ballot numbers. The same holds for -avoiding permutations. These results were proved by Miner and Pak using the Robinson-Schensted-Knuth (RSK) correspondence to connect permutations with Dyck paths. In this paper, we first provide an alternate proof of these enumeration results via a direct counting argument. We then study the number of pattern-avoiding permutations with a fixed prefix of length , generalizing the case. We find exact expressions for single and pairs of patterns of length three as well as the pair and . These expressions depend on , the extrema, and the order statistics. We also define -Wilf equivalence for permutations with a single fixed leading term , and classify the -Wilf-equivalence classes for both classical and vincular patterns of length three.
Paper Structure (8 sections, 32 theorems, 59 equations, 3 tables)

This paper contains 8 sections, 32 theorems, 59 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A new proof of \ref{['Theorem:ClassicalLeading123AND132']}
  4. Single patterns of length three
  5. Pairs of patterns of length three
  6. The pair 3412 and 3421
  7. $r$-Wilf-equivalence classes
  8. Concluding remarks

Key Result

Theorem 1.1

Knuth1973 For all $n\geq1$ and $\sigma\in S_3$, we have

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 58 more