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arXiv · 1401.2691

The Location of the First Ascent in a 123-Avoiding Permutation

Samuel Connolly·Zachary Gabor·Anant Godbole
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Abstract

It is natural to ask, given a permutation with no three-term ascending subsequence, at what index the first ascent occurs. We shall show, using both a recursion and a bijection, that the number of 123-avoiding permutations at which the first ascent occurs at positions $k,k+1$ is given by the $k$-fold Catalan convolution $C_{n,k}$. For $1\le k\le n$, $C_{n,k}$ is also seen to enumerate the number of 123-avoiding permutations with $n$ being in the $k$th position. Two interesting discrete probability distributions, related obliquely to the Poisson and geometric random variables, are derived as a result.

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Samuel Connolly, Zachary Gabor, Anant Godbole. 2014-01-13. The Location of the First Ascent in a 123-Avoiding Permutation. https://arxiv.org/abs/1401.2691

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