37 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
Permutations with restricted patterns and Dyck paths
We exhibit a bijection between 132-avoiding permutations and Dyck paths.
Using this bijection, it is shown that all the recently discovered results on
generating functions for 132-avoiding permutations with a given number of
occurrences of the pattern follow directly from old results on the
enumeration of Motzkin paths, among which is a continued fraction result due to
Flajolet. As a bonus, we use these observations to derive further results and a
precise asymptotic estimate for the number of 132-avoiding permutations of
with exactly occurrences of the pattern . Second,
we exhibit a bijection between 123-avoiding permutations and Dyck paths. When
combined with a result of Roblet and Viennot, this bijection allows us to
express the generating function for 123-avoiding permutations with a given
number of occurrences of the pattern in form of a continued
fraction and to derive further results for these permutations.Comment: 17 pages, AmS-Te
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