Let X = (X1, . . . , Xn) be a vector of i.i.d. random variables where Xi's take values over N. The purpose of this paper is to study the number of weakly increasing subsequences of X of a given length k, and the number of all weakly increasing subsequences of X. For the former, it is shown that a central limit theorem holds. Also, the first two moments of each of those two random variables are analyzed, their asymptotics are investigated, and results are related to the case of similar statistics in uniformly random permutations. We conclude the paper with applications on a similarity measure of Steele, and on increasing subsequences of riffle shuffles.

We provide a martingale proof of the well-known fact that the number of descents in random permutations is asmyptotically normal. The same technique is shown to be applicable to other descent and descent-related statistics as they satisy certain recurrence relation conditions. These statistics include inversions, descents in signed permutations, descents in Stirling permutations, the length of the longest alternating subsequences, descents in matchings and two-sided Eulerian numbers.

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This paper studies Markov chains on the symmetric group Sn where the transition probabilities are given by Ewens distribution with parameter θ > 0. The eigenvalues are identified to be content polynomials of partitions divided by nth rising factorial of θ. The mixing time analysis is carried out for two cases: If θ is a constant integer, the chain is fast enough that the mixing time does not depend on n. If θ = n, the chain exhibits a total variation cutoff at log n log 2 steps.