Counting Occurrences of a Pattern of Length Three with at most two Distinct Letters in a k-ary Word.

Abstract

Define $\tau(\pi)$ to be the number of subsequences of $\pi$ that are order-isomorphic to $\tau$. Let~$\tau$ be a pattern of length three with at most two distinct letters, namely, \[\tau\in\{111,112,121,122,211,212,221\}.\] In this paper, we give an algorithm for finding the generating function \[w_{\tau;r}(n;y)=\sum_{k\geq1}\,\,\sum_{\pi\in[k]^n,\tau(\pi)=r}y^k\] for the number of $k$-ary words of length $n$ that contain exactly $r$ occurrences of the pattern $\tau$, for given $r\geq0$. In particular, we obtain explicit formulas for the generating functions $w_{\tau;r}(n;y)$, where $r=0,1$.