When the law of large numbers fails for increasing subsequences of random permutations

Abstract

Let the random variable Z_n,k denote the number of increasing subsequences of length k in a random permutation from S_n, the symmetric group of permutations of {1, …, n}. In a recent paper [Random Structures Algorithms 29 (2006) 277–295] we showed that the weak law of large numbers holds for Z_n,kn if k_n=o(n^2/5); that is, $$\lim_{n\to\infty}\frac{Z_{n,k_{n}}}{EZ_{n,k_{n}}}=1\qquad \mbox{in probability}.$$ The method of proof employed there used the second moment method and demonstrated that this method cannot work if the condition k_n=o(n^2/5) does not hold. It follows from results concerning the longest increasing subsequence of a random permutation that the law of large numbers cannot hold for Z_n,kn if k_n≥cn^1/2, with c>2. Presumably there is a critical exponent l₀ such that the law of large numbers holds if k_n=O(n^l), with l<l₀, and does not hold if $\limsup_{n\to\infty}\frac{k_{n}}{n^{l}}>0$, for some l>l₀. Several phase transitions concerning increasing subsequences occur at l=½, and these would suggest that l₀=½. However, in this paper, we show that the law of large numbers fails for Z_n,kn if $\limsup_{n\to\infty}\frac{k_{n}}{n^{4/9}}=\infty$. Thus, the critical exponent, if it exists, must satisfy $l_{0}\in[\frac{2}{5},\frac{4}{9}]$.