Intervals of permutations and the principal Möbius function

2018/06/27 by Robert Brignall, Brignall, Robert, David Marchant +1
Computer Science · Mathematics · #05A05 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1806.10362

openalex publication_date 2018/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the proportion of permutations of length n with principal Möbius function equal to zero, Z(n), is asymptotically bounded below by 0.3995. If a permutation π contains two intervals of length 2, where one interval is an ascent and the other a descent, then we show that the value of the principal Möbius function μ[1, π] is zero, and we use this result to find the lower bound for Z(n). We also show that if a permutation ϕ has certain properties, then any permutation π which contains an interval order-isomorphic to ϕ has μ[1, π] = 0.

Citations

• Simple permutations and pattern restricted permutations
• The enumeration of simple permutations

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