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arXiv · 1605.05490

On pattern avoiding indecomposable permutations

Alice L. L. Gao·Sergey Kitaev·Philip B. Zhang
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Abstract

Comtet introduced the notion of indecomposable permutations in 1972. A permutation is indecomposable if and only if it has no proper prefix which is itself a permutation. Indecomposable permutations were studied in the literature in various contexts. In particular, this notion has been proven to be useful in obtaining non-trivial enumeration and equidistribution results on permutations. In this paper, we give a complete classification of indecomposable permutations avoiding a classical pattern of length 3 or 4, and of indecomposable permutations avoiding a non-consecutive vincular pattern of length 3. Further, we provide a recursive formula for enumerating $12\cdots k$-avoiding indecomposable permutations for $k\geq 3$. Several of our results involve the descent statistic. We also provide a bijective proof of a fact relevant to our studies.

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Alice L. L. Gao, Sergey Kitaev, Philip B. Zhang. 2016-05-23. On pattern avoiding indecomposable permutations. https://arxiv.org/abs/1605.05490

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