3,482 research outputs found
Generating trees and pattern avoidance in alternating permutations
We extend earlier work of the same author to enumerate alternating
permutations avoiding the permutation pattern 2143. We use a generating tree
approach to construct a recursive bijection between the set A_{2n}(2143) of
alternating permutations of length 2n avoiding 2143 and standard Young tableaux
of shape (n, n, n) and between the set A_{2n + 1}(2143) of alternating
permutations of length 2n + 1 avoiding 2143 and shifted standard Young tableaux
of shape (n + 2, n + 1, n). We also give a number of conjectures and open
questions on pattern avoidance in alternating permutations and generalizations
thereof.Comment: 21 pages. To be presented at FPSAC 2010. Comments welcome
Asymptotic distribution of fixed points of pattern-avoiding involutions
For a variety of pattern-avoiding classes, we describe the limiting
distribution for the number of fixed points for involutions chosen uniformly at
random from that class. In particular we consider monotone patterns of
arbitrary length as well as all patterns of length 3. For monotone patterns we
utilize the connection with standard Young tableaux with at most rows and
involutions avoiding a monotone pattern of length . For every pattern of
length 3 we give the bivariate generating function with respect to fixed points
for the involutions that avoid that pattern, and where applicable apply tools
from analytic combinatorics to extract information about the limiting
distribution from the generating function. Many well-known distributions
appear.Comment: 16 page
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