96 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
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
Continued fractions for permutation statistics
We explore a bijection between permutations and colored Motzkin paths that
has been used in different forms by Foata and Zeilberger, Biane, and Corteel.
By giving a visual representation of this bijection in terms of so-called cycle
diagrams, we find simple translations of some statistics on permutations (and
subsets of permutations) into statistics on colored Motzkin paths, which are
amenable to the use of continued fractions. We obtain new enumeration formulas
for subsets of permutations with respect to fixed points, excedances, double
excedances, cycles, and inversions. In particular, we prove that cyclic
permutations whose excedances are increasing are counted by the Bell numbers.Comment: final version formatted for DMTC
Permutations Restricted by Two Distinct Patterns of Length Three
Define to be the number of permutations on letters which avoid
all patterns in the set and contain each pattern in the multiset
exactly once. In this paper we enumerate and
for all . The
results for follow from two papers by Mansour and
Vainshtein.Comment: 15 pages, some relevant reference brought to my attention (see
section 4
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
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