Continued Fractions, Statistics, and Generalized Patterns

Abstract

Recently, Babson and Steingrimsson (see \([BS]\)) introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation.

Following \([BCS]\), let \(e_k,m\) (respectively, \(f_k\pi\)) be the number of occurrences of the generalized pattern \(12-3-\ldots-k\) (respectively, \(21-3-\ldots-k\)) in a permutation \(\pi\). In the present note, we study the distribution of the statistics \(e_k,f_k\) and \(f_k\pi\) in a permutation avoiding the classical pattern \(1-3-2\).

We also present some applications of our results, which relate the enumeration of permutations avoiding the classical pattern \(1-3-2\) according to the statistics \(e_k\) and \(f_k\) to Narayana numbers and Catalan numbers.