300 research outputs found
Frame patterns in n-cycles
In this paper, we study the distribution of the number of occurrences of the
simplest frame pattern, called the pattern, in -cycles. Given an
-cycle , we say that a pair matches the
pattern if and as we traverse around in a clockwise direction
starting at and ending at , we never encounter a with .
We say that is a nontrivial -match if .
Also, an -cycle is incontractible if there is no such that
immediately follows in .
We show that the number of incontractible -cycles in the symmetric group
is , where is the number of derangements in .
Further, we prove that the number of -cycles in with exactly
-matches can be expressed as a linear combination of binomial coefficients
of the form where . We also show that the
generating function of raised to the number of nontrivial
-matches in over all incontractible -cycles in is a new
-analogue of , which is different from the -analogues of the
derangement numbers that have been studied by Garsia and Remmel and by Wachs.
We show that there is a rather surprising connection between the charge
statistic on permutations due to Lascoux and Sch\"uzenberger and our
polynomials in that the coefficient of the smallest power of in
is the number of permutations in whose charge
path is a Dyck path. Finally, we show that and are the number of partitions
of for sufficiently large
A self-dual poset on objects counted by the Catalan numbers and a type-B analogue
We introduce two partially ordered sets, and , of the same
cardinalities as the type-A and type-B noncrossing partition lattices. The
ground sets of and are subsets of the symmetric and the
hyperoctahedral groups, consisting of permutations which avoid certain
patterns. The order relation is given by (strict) containment of the descent
sets. In each case, by means of an explicit order-preserving bijection, we show
that the poset of restricted permutations is an extension of the refinement
order on noncrossing partitions. Several structural properties of these
permutation posets follow, including self-duality and the strong Sperner
property. We also discuss posets and similarly associated with
noncrossing partitions, defined by means of the excedence sets of suitable
pattern-avoiding subsets of the symmetric and hyperoctahedral groups.Comment: 15 pages, 2 figure
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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