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arXiv · 1103.0173

The Möbius function of the consecutive pattern poset

Antonio Bernini·Luca Ferrari·Einar Steingrimsson
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Abstract

An occurrence of a consecutive permutation pattern $p$ in a permutation $π$ is a segment of consecutive letters of $π$ whose values appear in the same order of size as the letters in $p$. The set of all permutations forms a poset with respect to such pattern containment. We compute the Möbius function of intervals in this poset, providing what may be called a complete solution to the problem. For most intervals our results give an immediate answer to the question. In the remaining cases, we give a polynomial time algorithm to compute the Möbius function. In particular, we show that the Möbius function only takes the values -1, 0 and 1.

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Antonio Bernini, Luca Ferrari, Einar Steingrimsson. 2011-03-01. The Möbius function of the consecutive pattern poset. https://arxiv.org/abs/1103.0173

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