The Möbius function of separable and decomposable permutations

Alexander Burstein, Vít Jelínek, Eva Jelínková, Einar Steingrimsson

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Abstract

We give a recursive formula for the Moebius function of an interval $[\sigma,\pi]$ in the poset of permutations ordered by pattern containment in the case where $\pi$ is a decomposable permutation, that is, consists of two blocks where the first one contains all the letters 1, 2, ..., k for some k. This leads to many special cases of more explicit formulas. It also gives rise to a computationally efficient formula for the Moebius function in the case where $\sigma$ and $\pi$ are separable permutations. A permutation is separable if it can be generated from the permutation 1 by successive sums and skew sums or, equivalently, if it avoids the patterns 2413 and 3142. A consequence of the formula is that the Moebius function of such an interval $[\sigma,\pi]$ is bounded by the number of occurrences of $\sigma$ as a pattern in $\pi$. We also show that for any separable permutation $\pi$ the Moebius function of $(1,\pi)$ is either 0, 1 or -1.
Original languageEnglish
Pages (from-to)2346–2364
Number of pages19
JournalJournal of Combinatorial Theory Series A
Volume118
Issue number8
DOIs
Publication statusPublished - Nov 2011

Keywords

  • Möbius function
  • poset
  • permutations
  • pattern containment

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