We prove a formula for the number of permutations in Sn such that their first n− k entries are increasing and their longest increasing subsequence has length n− k. This formula first appeared as a consequence of character polynomial calculations in recent work of Adriano Garsia and Alain Goupil. We give two ‘elementary’ bijective proofs of this result and of its q-analogue, one proof using the ...

Let Ln be the length of the longest increasing subsequence of a random permutation of the numbers 1, . . . , n, for the uniform distribution on the set of permutations. Hammersley’s interacting particle process, implicit in Hammersley (1972), has been used in Aldous and Diaconis (1995) to provide a “soft” hydrodynamical argument for proving that limn→∞ ELn/ √ n = 2. We show in this note that th...

It is proved in [BOO], [J2] and [Ok1] that the joint distribution of suitably scaled rows of a partition with respect to the Plancherel measure of the symmetric group converges to the corresponding distribution of eigenvalues of a Hermitian matrix from the Gaussian Unitary Ensemble. We introduce a new measure on strict partitions, which is analogous to the Plancherel measure, and prove that the...

This note gives a simple proof of the existence and monotonicity of optimal debt contracts in simple models of borrowing and lending with ex−post asymmetric information, risk−averse agents and heterogeneous beliefs. Our argument is based on the concept of nondecreasing rearrangement and on a supermodular version of Hardy−Littlewood inequality. Citation: Renou, Ludovic and Guillaume Carlier, (20...

1.1. Plancherel measures. Given a finite group G, by the corresponding Plancherel measure we mean the probability measure on the set G∧ of irreducible representations of G which assigns to a representation π ∈ G∧ the weight (dim π)/|G|. For the symmetric group S(n), the set S(n)∧ is the set of partitions λ of the number n, which we shall identify with Young diagrams with n squares throughout th...