Burstein, Alexander and Jelínek, Vít and Jelínková, Eva and Steingrimsson, Einar (2011) The Möbius function of separable and decomposable permutations. Journal of Combinatorial Theory Series A, 118 (8). 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.
Item type:  Article 

ID code:  33802 
Keywords:  Möbius function , poset , permutations, pattern containment, Electronic computers. Computer science, Discrete Mathematics and Combinatorics, Computational Theory and Mathematics, Theoretical Computer Science 
Subjects:  Science > Mathematics > Electronic computers. Computer science 
Department:  Faculty of Science > Computer and Information Sciences 
Depositing user:  Pure Administrator 
Date Deposited:  19 Oct 2011 15:40 
Last modified:  18 Jun 2015 08:29 
URI:  http://strathprints.strath.ac.uk/id/eprint/33802 
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