# The Möbius function of separable and decomposable permutations

Alexander Burstein, Vít Jelínek, Eva Jelínková, Einar Steingrimsson

Research output: Contribution to journalArticle

11 Citations (Scopus)

### 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 language English 2346–2364 19 Journal of Combinatorial Theory Series A 118 8 https://doi.org/10.1016/j.jcta.2011.06.002 Published - Nov 2011

### Fingerprint

Decomposable
Pi
Permutation
Interval
Recursive Formula
Poset
Skew
Explicit Formula

### Keywords

• Möbius function
• poset
• permutations
• pattern containment

### Cite this

Burstein, Alexander ; Jelínek, Vít ; Jelínková, Eva ; Steingrimsson, Einar. / The Möbius function of separable and decomposable permutations. In: Journal of Combinatorial Theory Series A . 2011 ; Vol. 118, No. 8. pp. 2346–2364.
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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.",
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The Möbius function of separable and decomposable permutations. / Burstein, Alexander; Jelínek, Vít; Jelínková, Eva; Steingrimsson, Einar.

In: Journal of Combinatorial Theory Series A , Vol. 118, No. 8, 11.2011, p. 2346–2364.

Research output: Contribution to journalArticle

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T1 - The Möbius function of separable and decomposable permutations

AU - Burstein, Alexander

AU - Jelínek, Vít

AU - Jelínková, Eva

AU - Steingrimsson, Einar

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N2 - 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.

AB - 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.

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