15 research outputs found
Continued fractions and Catalan problems
We find a generating function expressed as a continued fraction that
enumerates ordered trees by the number of vertices at different levels. Several
Catalan problems are mapped to an ordered-tree problem and their generating
functions also expressed as a continued fraction. Among these problems is the
enumeration of (132)-pattern avoiding permutations that have a given number of
increasing patterns of length k. This extends and illuminates a result of
Robertson, Wilf and Zeilberger for the case k=3.Comment: 9 pages, 1 figur
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
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
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
Counting Permutations by Their Rigid Patterns
AbstractIn how many permutations does the patternτ occur exactly m times? In most cases, the answer is unknown. When we search for rigid patterns, on the other hand, we obtain exact formulas for the solution, in all cases considered
Patterns in random permutations avoiding the pattern 132
We consider a random permutation drawn from the set of 132-avoiding
permutations of length and show that the number of occurrences of another
pattern has a limit distribution, after scaling by
where is the length of plus
the number of descents. The limit is not normal, and can be expressed as a
functional of a Brownian excursion. Moments can be found by recursion.Comment: 32 page
On multi-avoidance of generalized patterns
In [Kit1] Kitaev discussed simultaneous avoidance of two 3-patterns with no
internal dashes, that is, where the patterns correspond to contiguous subwords
in a permutation. In three essentially different cases, the numbers of such
-permutations are , the number of involutions in ,
and , where is the -th Euler number. In this paper we give
recurrence relations for the remaining three essentially different cases.
To complete the descriptions in [Kit3] and [KitMans], we consider avoidance
of a pattern of the form (a classical 3-pattern) and beginning or
ending with an increasing or decreasing pattern. Moreover, we generalize this
problem: we demand that a permutation must avoid a 3-pattern, begin with a
certain pattern and end with a certain pattern simultaneously. We find the
number of such permutations in case of avoiding an arbitrary generalized
3-pattern and beginning and ending with increasing or decreasing patterns.Comment: 26 page