The depth of a permutation

For the elements of a Coxeter group, we present a statistic called depth, defined in terms of factorizations of the elements into products of reflections. Depth is bounded above by length and below by the average of length and reflection length. In this article, we focus on the case of the symmetric group, where we show that depth is equal to $\sum_{i} \max {\lbrace w(i) - 1, 0 \rbrace}$. We characterize those permutations for which depth equals length: these are the 321-avoiding permutations (and hence are enumerated by the Catalan numbers). We also characterize those permutations for which depth equals reflection length: these are permutations avoiding both 321 and 3412 (also known as boolean permutations, which we can hence also enumerate). In this case, it also happens that length equals reflection length, leading to a new perspective on a result of Edelman.

Keywords

Coxeter group, permutation, reflection, depth

2010 Mathematics Subject Classification

Primary 20F55. Secondary 05A05, 05A15.

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Published 20 March 2015