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

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

2004-03-29

arxiv.org/abs/math/0403502v2

Mathematics

Combinatorics

10 pages, 1 figure

Scientific paper

We show the first known example for a pattern $q$ for which $\lim_{n\to \infty} \sqrt[n]{S_n(q)}$ is not an integer. We find the exact value of the limit and show that it is irrational. Then we generalize our results to an infinite sequence of patterns. Finally, 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 $L(q)\geq (k-1)^2$ holds.

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