1,557 research outputs found
Restricted 132-avoiding permutations
We study generating functions for the number of permutations on n letters
avoiding 132 and an arbitrary permutation on k letters, or containing
exactly once. In several interesting cases the generating function
depends only on k and is expressed via Chebyshev polynomials of the second
kind.Comment: 10 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
On the Asymptotic Statistics of the Number of Occurrences of Multiple Permutation Patterns
We study statistical properties of the random variables ,
the number of occurrences of the pattern in the permutation . We
present two contrasting approaches to this problem: traditional probability
theory and the ``less traditional'' computational approach. Through the
perspective of the first one, we prove that for any pair of patterns
and , the random variables and are jointly
asymptotically normal (when the permutation is chosen from ). From the
other perspective, we develop algorithms that can show asymptotic normality and
joint asymptotic normality (up to a point) and derive explicit formulas for
quite a few moments and mixed moments empirically, yet rigorously. The
computational approach can also be extended to the case where permutations are
drawn from a set of pattern avoiders to produce many empirical moments and
mixed moments. This data suggests that some random variables are not
asymptotically normal in this setting.Comment: 18 page
Enumerating pattern avoidance for affine permutations
In this paper we study pattern avoidance for affine permutations. In
particular, we show that for a given pattern p, there are only finitely many
affine permutations in that avoid p if and only if p avoids
the pattern 321. We then count the number of affine permutations that avoid a
given pattern p for each p in S_3, as well as give some conjectures for the
patterns in S_4.Comment: 11 pages, 3 figures; fixed typos and proof of Proposition
- β¦