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Large Deviations and Ratio Limit Theorems for Pattern-Avoiding Permutations

Published online by Cambridge University Press:  13 December 2013

MAHSHID ATAPOUR  and
NEAL MADRAS
MAHSHID ATAPOUR
Affiliation:
Department of Finance and Management Science, University of Saskatchewan, 25 Campus Drive, Saskatoon, Saskatchewan S7N 5A7, Canada (e-mail: atapour@math.usask.ca)
NEAL MADRAS
Affiliation:
Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario M3J 1P3, Canada (e-mail: madras@mathstat.yorku.ca)
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Abstract

For a fixed permutation τ, let $\mathcal{S}_N(\tau)$ be the set of permutations on N elements that avoid the pattern τ. Madras and Liu (2010) conjectured that $\lim_{N\rightarrow\infty}\frac{|\mathcal{S}_{N+1}(\tau)|}{ |\mathcal{S}_N(\tau)|}$ exists; if it does, it must equal the Stanley–Wilf limit. We prove the conjecture for every permutation τ of length 5 or less, as well as for some longer cases (including 704 of the 720 permutations of length 6). We also consider permutations drawn at random from $\mathcal{S}_N(\tau)$, and we investigate properties of their graphs (viewing permutations as functions on {1,. . .,N}) scaled down to the unit square [0,1]2. We prove exact large deviation results for these graphs when τ has length 3; it follows, for example, that it is exponentially unlikely for a random 312-avoiding permutation to have points above the diagonal strip |y−x| < ε, but not unlikely to have points below the strip. For general τ, we show that some neighbourhood of the upper left corner of [0,1]2 is exponentially unlikely to contain a point of the graph if and only if τ starts with its largest element. For patterns such as τ=4231 we establish that this neighbourhood can be extended along the sides of [0,1]2 to come arbitrarily close to the corner points (0,0) and (1,1), as simulations had suggested.

Keywords

Primary 60C05Secondary 05A0505A1660F10

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 2 , March 2014 , pp. 161 - 200
Copyright
Copyright © Cambridge University Press 2013 

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