Generalized Pattern Avoidance

Abstract

Recently, Babson and Steingrımsson have introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We will consider pattern avoidance for such patterns, and give a complete solution for the number of permutations avoiding any single pattern of length three with exactly one adjacent pair of letters. For eight of these 12 patterns the answer is given by the Bell numbers. For the remaining four the answer is given by the Catalan numbers. We also give some results for the number of permutations avoiding two different patterns. These results relate the permutations in question to Motzkin paths, involutions and non-overlapping partitions. Furthermore, we define a new class of set partitions, called monotone partitions, and show that these partitions are in one-to-one correspondence with non-overlapping partitions

Increasing and decreasing subsequences and their variants

“…In particular, we can look at patterns where some of the terms must appear consecutively. This concept was introduced by Babson and Steingrímsson [10] and further investigated by Claesson [39] and others. For instance, the generalized pattern 1-32 indicates a subsequence a i a j a j +1 of a permutation w = a 1 a 2 .…”

Continued Fractions and Generalized Patterns

“…Reference [3] presented a complete solution for the number of permutations avoiding any pattern of length three with exactly one adjacent pair of letters. Reference [4] presented a complete solution for the number of permutations avoiding any two patterns of length three with exactly one adjacent pair of letters.…”

Enumerating Permutations Avoiding Three Babson-Steingrímsson Patterns

“…They contain two (or more) elements which are adjacent (see below). Claesson [3] solved the problem of the enumeration of permutations avoiding any single generalized pattern of length 3 of type (1,2) or (2, 1), called Babson-Steingrímsson patterns. Subsequently, Claesson and Mansour [4] gave the solution for any pair of Babson-Steingrímsson patterns and made conjectures on the number of permutations avoiding any set of three or more such patterns.…”

“…In [3] and [4] we can find the results concerning the enumeration of permutations avoiding one and two Babson-Steingrímsson patterns. In [4] the authors also make conjectures on the cardinality of S n (P) for sets P, when P contains three or more such patterns.…”