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The 1-box pattern on pattern avoiding permutations
This paper is continuation of the study of the 1-box pattern in permutations
introduced by the authors in \cite{kitrem4}. We derive a two-variable
generating function for the distribution of this pattern on 132-avoiding
permutations, and then study some of its coefficients providing a link to the
Fibonacci numbers. We also find the number of separable permutations with two
and three occurrences of the 1-box pattern
Distributions of several infinite families of mesh patterns
Br\"and\'en and Claesson introduced mesh patterns to provide explicit
expansions for certain permutation statistics as linear combinations of
(classical) permutation patterns. The first systematic study of avoidance of
mesh patterns was conducted by Hilmarsson et al., while the first systematic
study of the distribution of mesh patterns was conducted by the first two
authors.
In this paper, we provide far-reaching generalizations for 8 known
distribution results and 5 known avoidance results related to mesh patterns by
giving distribution or avoidance formulas for certain infinite families of mesh
patterns in terms of distribution or avoidance formulas for smaller patterns.
Moreover, as a corollary to a general result, we find the distribution of one
more mesh pattern of length 2.Comment: 27 page
Harmonic numbers, Catalan's triangle and mesh patterns
The notion of a mesh pattern was introduced recently, but it has already
proved to be a useful tool for description purposes related to sets of
permutations. In this paper we study eight mesh patterns of small lengths. In
particular, we link avoidance of one of the patterns to the harmonic numbers,
while for three other patterns we show their distributions on 132-avoiding
permutations are given by the Catalan triangle. Also, we show that two specific
mesh patterns are Wilf-equivalent. As a byproduct of our studies, we define a
new set of sequences counted by the Catalan numbers and provide a relation on
the Catalan triangle that seems to be new
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