The limit of a Stanley-Wilf sequence is not always rational, and layered patterns beat monotone patterns

Authors:
Miklós Bóna
Affiliations:
University of Florida, P.O. Box 118105, 358 Little Hall, Gainesville FL
Venue:
Journal of Combinatorial Theory Series A
Year:
2005

Citing 3
Cited 2

Enumerative combinatorics

Enumerative combinatorics
Permutations avoiding certain patterns: the case of length 4 and some generalizations

Discrete Mathematics
Exact enumeration of 1342-avoiding permutations: a close link with labeled trees and planar maps

Journal of Combinatorial Theory Series A

New records in Stanley-Wilf limits

European Journal of Combinatorics
Upper bounds for the Stanley-Wilf limit of 1324 and other layered patterns

Journal of Combinatorial Theory Series A

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Abstract

We show the first known example for a pattern q for which L(q)=limn-∞ √Sn(q) is not an integer. where Sn(q) denotes the number of permutations of length n avoiding the pattern q. We find the exact value of the limit and show that it is irrational, but algebraic. Then we generalize our results to an infinite sequence of patterns. We provide further generalizations that start explaining why certain patterns are easier to avoid than others. Finally, we show that if q is a layered pattern of length k. Then L(q)≥(k-1)2 holds.