On two conjectures about pattern avoidance of cyclic permutations

2025/05/04 by Junyao Pan, Pan, Junyao
Engineering · Mathematics · Computer Science · #graph theory and CDMA systems #Advanced Combinatorial Mathematics #Coding theory and cryptography

paper · pdf · doi:10.48550/arxiv.2505.02045

Abstract

Let π be a cyclic permutation that can be expressed in its one-line form as π= π₁π₂ ⋅⋅⋅ π_n and in its standard cycle form as π= (c₁,c₂, ..., c_n) where c₁=1. Archer et al. introduced the notion of pattern avoidance of one-line and the standard cycle form for a cyclic permutation π, defined as both π₁π₂ ⋅⋅⋅ π_n and its standard cycle form c₁c₂⋅⋅⋅ c_n avoiding a given pattern. Let A_n(σ₁,...,σ_k; τ) denote the set of cyclic permutations in the symmetric group S_n that avoid each pattern of \σ₁,...,σ_k\ in their one-line forms and avoid τ in their standard cycle forms. In this paper, we obtain some results about the cyclic permutations avoiding patterns in both one-line and cycle forms. In particular, we resolve two conjectures of Archer et al.

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