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Bijections for generalized Wilf equivalences

Melanie Ferreri

Abstract

Starting with an inclusion-exclusion proof of a combinatorial identity, a direct bijection can be produced using recursive subtraction (sometimes with a direct combinatorial description). We apply this method to identities for generalized Wilf equivalences among consecutive patterns in inversion sequences, giving direct bijective proofs of some generalized Wilf equivalences shown by Auli and Elizalde. We also give new bijective proofs of a stronger relation among some consecutive patterns.

Bijections for generalized Wilf equivalences

Abstract

Starting with an inclusion-exclusion proof of a combinatorial identity, a direct bijection can be produced using recursive subtraction (sometimes with a direct combinatorial description). We apply this method to identities for generalized Wilf equivalences among consecutive patterns in inversion sequences, giving direct bijective proofs of some generalized Wilf equivalences shown by Auli and Elizalde. We also give new bijective proofs of a stronger relation among some consecutive patterns.

Paper Structure

This paper contains 8 sections, 14 theorems, 18 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Equivalence relations for patterns in inversion sequences
  3. Bijections defined recursively
  4. Iterative interpretation
  5. Patterns that are not mutually non-overlapping
  6. Reciprocal relations
  7. The remaining equivalences
  8. Extended example

Key Result

Theorem 1

A complete list of the generalized Wilf equivalences between consecutive patterns of length 4 is as follows: All of the other (super strong) Wilf equivalence classes contain just one pattern.

Figures (3)

  • Figure 5: Underdiagonal lattice path diagrams for 01022304, 01122304, 01120334, 01122324, 01120324.
  • Figure 6: Diagrams of replacements made by Algorithm \ref{['alg:sd']} in each case of an occurrence at $i$.
  • Figure 7: Diagram of replacements made when applying Algorithm \ref{['alg:sd']} twice to the sequence $\epsilon = 011223042526$.

Theorems & Definitions (37)

  • Example 1
  • Example 2
  • Example 3
  • Theorem 1: Extension of Theorem 2.2 of aulielizalde
  • Example 4
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • ...and 27 more