Wilf-Classification of Mesh Patterns of Short Length

Abstract

This paper starts the Wilf-classification of mesh patterns of length 2. Although there are initially 1024 patterns to consider we introduce automatic methods to reduce the number of potentially different Wilf-classes to at most 65. By enumerating some of the remaining classes we bring that upper-bound further down to 56. Finally, we conjecture that the actual number of Wilf-classes of mesh patterns of length 2 is 46.

Distributions of several infinite families of mesh patterns

“…34 (in Theorems 5.7 and 5.8). In this way, one can obtain any previous results in [4,10] related to the patterns appearing in this paper. In this paper, we need the following result.…”

“…Note that even though the subsequences 251 and 451 are order isomorphic to 231 (the τ in the drawn pattern), they are not occurrences of the pattern because of the elements 4 and 3, respectively, be in the shaded squares. See [4] for more examples of occurrences of mesh patterns in permutations.…”

“…12, 13, and 17; see Section 2) but also to derive the previously unknown distribution of the pattern Nr. 66 = (pattern's number is introduced by us in this paper) which is not equivalent to any of the patterns in [4].…”

Distributions of mesh patterns of short lengths

“…see [1,3,7,9,10,11,15,16]. However, the first systematic study of mesh patterns was not done until [6], where 25 out of 65 non-equivalent avoidance cases of patterns of length 2 were solved. That is, in the 25 cases, the number of permutations avoiding the respective mesh patterns was found.…”

“…In this paper, we initiate a systematic study of distributions of mesh patterns by giving 27 distribution results for the patterns considered in [6], including 14 distributions for which avoidance was not known. Moreover, for the unsolved cases, we prove an equidistribution result (out of 6 equidistribution results we prove in total), and conjecture 6 more equidistributions (see Table 2).…”