In the study of permutations, generalized patterns extend classical patterns by adding the requirement that certain adjacent integers in a pattern must be adjacent in the permutation.For any generalized pattern $\pi_0^*$ of length $k$ with $1 \leq b \leq k$ blocks, we prove that for all $\mu > 0$, there exists $0 < c =c(k, \mu) < 1$ so that whenever $n \geq n_0(k, \mu, c)$, all but $c^n n!$ many $\pi \in S_n$ admit $(1 \pm \mu) \tfrac{1}{k!}\tbinom{n}{b}$ occurrences of $\pi_0^*$. Up to the choice of $c$, this result is best possible for all $\pi_0^*$ with $k \geq 2$.We also give a lower bound on avoidance of the generalized pattern $12$-$34$, which answers a question of S. Elizalde (2006).
@article{10_37236_2692,
author = {Joshua Cooper and Erik Lundberg and Brendan Nagle},
title = {Generalized pattern frequency in large permutations},
journal = {The electronic journal of combinatorics},
year = {2013},
volume = {20},
number = {1},
doi = {10.37236/2692},
zbl = {1267.05029},
url = {http://geodesic.mathdoc.fr/articles/10.37236/2692/}
}
TY - JOUR
AU - Joshua Cooper
AU - Erik Lundberg
AU - Brendan Nagle
TI - Generalized pattern frequency in large permutations
JO - The electronic journal of combinatorics
PY - 2013
VL - 20
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/2692/
DO - 10.37236/2692
ID - 10_37236_2692
ER -
%0 Journal Article
%A Joshua Cooper
%A Erik Lundberg
%A Brendan Nagle
%T Generalized pattern frequency in large permutations
%J The electronic journal of combinatorics
%D 2013
%V 20
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/2692/
%R 10.37236/2692
%F 10_37236_2692
Joshua Cooper; Erik Lundberg; Brendan Nagle. Generalized pattern frequency in large permutations. The electronic journal of combinatorics, Tome 20 (2013) no. 1. doi: 10.37236/2692
