We show that the moments of the trace of a random unitary matrix have combinatorial interpretations in terms of longest increasing subsequences of permutations. To be precise, we show that the 2n-th moment of the trace of a random k-dimensional unitary matrix is equal to the number of permutations of length n with no increasing subsequence of length greater than k. We then generalize this to ot...

Let 〈a1, a2, . . . , an〉 be a sequence of comparable elements. In this paper, we study two constrained versions of the longest increasing subsequence (LIS) problem. The first problem is the range-constrained longest increasing subsequence (RLIS) problem. Given 0 < LI ≤ UI < n and 0 ≤ LV ≤ UV , the objective of the RLIS problem is to deliver a maximum-length increasing subsequence 〈ai1 , ai2 , ....

The number of even 321-avoiding permutations of length n is equal to the number of odd ones if n is even, and exceeds it by the n−1 2 th Catalan number otherwise. We present an involution that proves a refinement of this sign-balance property respecting the length of the longest increasing subsequence of the permutation. In addition, this yields a combinatorial proof of a recent analogous resul...

where Rk,r is the number of permutations of the integers {1, . . . , k} that contain an increasing subsequence of length at least r. Let Xy denote a positive integer valued random variable such that Prob (Xy ≥ r) = F (r; y). If R k,r denotes the complement of Rk,r, i.e. the number of permutations σ ∈ Sk all of whose increasing subsequences have length strictly less than r, then clearly R k,r = ...