# On generating bijections for permutations and inversion sequences

## Author ORCID Identifier

https://orcid.org/0000-0002-9099-1578

Spring 2024

Thesis (Ph.D.)

Mathematics

Peter Doyle

Sergi Elizalde

## Abstract

Given an algebraic proof of a combinatorial identity, we use recursive methods to construct a bijection demonstrating the identity.

Our first application centers around derangements and nonderangements. A derangement is a permutation with no fixed point, and a nonderangement is a permutation with at least one fixed point. There is a one-term recurrence for the number of derangements of n elements, and we describe a bijective proof of this recurrence which can be found using a recursive map. We then show the combinatorial interpretation of this bijection and how it compares with other known bijections, and show how this extends to an involution. Nonderangements satisfy a similar recurrence. We convert the bijective proof of the one-term identity for derangements into a bijective proof of the one-term identity for nonderangements.

We then consider identities proven via inclusion-exclusion. An inversion sequence is an integer sequence e = e_1 e_2 ... e_n such that 0 ≤ e_i < i for all 1 ≤ i ≤ n. Inversion sequences of length n are in bijection with permutations in S_n. A pattern is a sequence p = p_1 p_2 … p_r with p_i in {0, 1, …, r - 1} for all 1 ≤ i ≤ r such that j can only appear in p if j - 1 also appears, and an inversion sequence e is said to contain the consecutive pattern p if there is a consecutive subsequence of e whose reduction is p; otherwise, e is said to avoid p. We say two patterns p and q are Wilf equivalent if the number of inversion sequences of length n which avoid p is equal to the number of inversion sequences of length n which avoid q. In previous work, Auli and Elizalde characterized generalized Wilf equivalences among all 75 consecutive patterns of length 4, proving some with direct bijections and some via inclusion-exclusion arguments. For those proved by inclusion-exclusion, we derive bijective proofs of these equivalences. We also give new bijective proofs of a stronger relation among some consecutive patterns.

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