Simultaneous Avoidance of Generalized Patterns

Abstract

In [BabStein], Babson and Steingrimsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. In \([Kit1]\), Kitaev considered simultaneous avoidance (multi-avoidance) of two or more 3-patterns with no internal dashes, that is, where the patterns correspond to contiguous subwords in a permutation. There, either an explicit or a recursive formula was given for all but one case of simultaneous avoidance of more than two patterns. In this paper, we find the exponential generating function for the remaining case. Also, we consider permutations that avoid a pattern of the form \(x – yz\) or \(xy – z\) and begin with one of the patterns \(12\ldots k\), \(k(k-1)\ldots 1\), \(23\ldots 1k\), \((k-1)(k-2)\ldots 1k\), or end with one of the patterns \(12\ldots k\), \(k(k-1)\ldots 1\), \(1k(k-1)\ldots 2\), \(k12\ldots (k-1)\). For each of these cases, we find either the ordinary or exponential generating functions or a precise formula for the number of such permutations. Besides, we generalize some of the obtained results as well as some of the results given in \([Kit3]\): we consider permutations avoiding certain generalized \(3\)-patterns and beginning (ending) with an arbitrary pattern having either the greatest or the least letter as its rightmost (leftmost) letter.