35 research outputs found
The Eulerian numbers on restricted centrosymmetric permutations
We study the descent distribution over the set of centrosymmetric
permutations that avoid the pattern of length 3. Our main tool in the most
puzzling case, namely, and even, is a bijection that associates
a Dyck prefix of length to every centrosymmetric permutation in
that avoids 123.Comment: 17 pages, 6 figure
Combinatorial properties of the numbers of tableaux of bounded height
We introduce an infinite family of lower triangular matrices ¡(s), where
°s
n;i counts the standard Young tableaux on n cells and with at most
s columns on a suitable subset of shapes. We show that the entries
of these matrices satisfy a three-term row recurrence and we deduce
recursive and asymptotic properties for the total number Âżs(n) of
tableaux on n cells and with at most s columns
Two permutation classes enumerated by the central binomial coefficients
We define a map between the set of permutations that avoid either the four
patterns or , and the set of Dyck
prefixes. This map, when restricted to either of the two classes, turns out to
be a bijection that allows us to determine some notable features of these
permutations, such as the distribution of the statistics "number of ascents",
"number of left-to-right maxima", "first element", and "position of the maximum
element"Comment: 26 pages, 3 figure
Descent sets on 321-avoiding involutions and hook decompositions of partitions
We show that the distribution of the major index over the set of involutions
in S_n that avoid the pattern 321 is given by the q-analogue of the n-th
central binomial coefficient. The proof consists of a composition of three
non-trivial bijections, one being the Robinson-Schensted correspondence,
ultimately mapping those involutions with major index m into partitions of m
whose Young diagram fits inside an n/2 by n/2 box. We also obtain a refinement
that keeps track of the descent set, and we deduce an analogous result for the
comajor index of 123-avoiding involutions
Restricted involutions and Motzkin paths
AbstractWe show how a bijection due to Biane between involutions and labelled Motzkin paths yields bijections between Motzkin paths and two families of restricted involutions that are counted by Motzkin numbers, namely, involutions avoiding 4321 and 3412. As a consequence, we derive characterizations of Motzkin paths corresponding to involutions avoiding either 4321 or 3412 together with any pattern of length 3. Furthermore, we exploit the described bijection to study some notable subsets of the set of restricted involutions, namely, fixed point free and centrosymmetric restricted involutions
and
Abstract. We study the descent distribution over the set of centrosymmetric permutations that avoid a pattern of length 3. In the most puzzling case, namely, Ď„ = 123 and n even, our main tool is a bijection that associates a Dyck pre x of length 2n to every centrosymmetric permutation in S2n that avoids 123. Mathematics Subject Classi cation(2000). 05A05, 05A15, 05A19
Permutations Avoiding a Simsun Pattern
A permutation \u3c0 avoids the simsun pattern \u3c4 if \u3c0 avoids the consecutive pattern \u3c4 and the same condition applies to the restriction of \u3c0 to any interval [k]. Permutations avoiding the simsun pattern 321 are the usual simsun permutation introduced by Simion and Sundaram. Deutsch and Elizalde enumerated the set of simsun permutations that avoid in addition any set of patterns of length 3 in the classical sense. In this paper we enumerate the set of permutations avoiding any other simsun pattern of length 3 together with any set of classical patterns of length 3.
The main tool in the proofs is a massive use of a bijection between permutations and increasing binary trees
Pattern avoiding meandric permutations
We study and characterize meandric permutations avoiding one or more
patterns of length three, and nd explicit formulae for the cardinality of
each of these sets. We determine the distribution of the descent statistic
for the set of meandric permutations avoiding the pattern 231. The
sets of meandric permutations avoiding any other pattern of length three
can be either trivially determined, or deduced from the 231 case via the
symmetries of the square. In the 231 case we provide a bijection with
a set of Motzkin paths that maps the statistic \number of descents of a
permutation" to the statistic \number of non-horizontal steps of a path"