27,132 research outputs found
Restricted Motzkin permutations, Motzkin paths, continued fractions, and Chebyshev polynomials
We say that a permutation is a Motzkin permutation if it avoids 132 and
there do not exist such that . We study the
distribution of several statistics in Motzkin permutations, including the
length of the longest increasing and decreasing subsequences and the number of
rises and descents. We also enumerate Motzkin permutations with additional
restrictions, and study the distribution of occurrences of fairly general
patterns in this class of permutations.Comment: 18 pages, 2 figure
Generalized permutation patterns - a short survey
An occurrence of a classical pattern p in a permutation Ļ is a subsequence of Ļ whose letters are in the same relative order (of size) as those in p. In an occurrence of a generalized pattern, some letters of that subsequence may be required to be adjacent in the permutation. Subsets of permutations characterized by the avoidanceāor the prescribed number of occurrencesā of generalized patterns exhibit connections to an enormous variety of other combinatorial structures, some of them apparently deep. We give a short overview of the state of the art for generalized patterns
Restricted 132-Dumont permutations
A permutation is said to be {\em Dumont permutations of the first kind}
if each even integer in must be followed by a smaller integer, and each
odd integer is either followed by a larger integer or is the last element of
(see, for example, \cite{Z}). In \cite{D} Dumont showed that certain
classes of permutations on letters are counted by the Genocchi numbers. In
particular, Dumont showed that the st Genocchi number is the number of
Dummont permutations of the first kind on letters.
In this paper we study the number of Dumont permutations of the first kind on
letters avoiding the pattern 132 and avoiding (or containing exactly once)
an arbitrary pattern on letters. In several interesting cases the
generating function depends only on .Comment: 12 page
Some statistics on restricted 132 involutions
In [GM] Guibert and Mansour studied involutions on n letters avoiding (or
containing exactly once) 132 and avoiding (or containing exactly once) an
arbitrary pattern on k letters. They also established a bijection between
132-avoiding involutions and Dyck word prefixes of same length. Extending this
bijection to bilateral words allows to determine more parameters; in
particular, we consider the number of inversions and rises of the involutions
onto the words. This is the starting point for considering two different
directions: even/odd involutions and statistics of some generalized patterns.
Thus we first study generating functions for the number of even or odd
involutions on n letters avoiding (or containing exactly once) 132 and avoiding
(or containing exactly once) an arbitrary pattern on k letters. In
several interesting cases the generating function depends only on k and is
expressed via Chebyshev polynomials of the second kind. Next, we consider other
statistics on 132-avoiding involutions by counting an occurrences of some
generalized patterns, related to the enumeration according to the number of
rises.Comment: 22 page
Simultaneous avoidance of generalized patterns
In [BabStein] Babson and Steingr\'{\i}msson 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 or
and begin with one of the patterns , , ,
or end with one of the patterns , ,
, . 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.Comment: 18 page
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