The problem of finding a longest increasing subsequence (LIS) is a well known task in sequence processing. There are many variants of the basic task. We discuss a recently introduced variant of LIS, a minimal height longest increasing subsequence problem and propose a new algorithm for it, which improves its time complexity. Moreover, we define a family of similar problems and introduce algorit...

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. We discuss the “hydrodynamical approach” to the analysis of the limit behavior, which probably started with Hammersley (1972), and was subsequently further developed by several authors. We also give two proofs of an exact (non-a...

Two-dimensional Lorentzian geometry has recently found application in some models of nonrelativistic systems, most profitably for the process of boarding an aeroplane. The duration of the boarding process is assumed to be a result of passengers standing in the isle and blocking each others’ way. According to this model, the expected boarding time is equal to the length of the longest timelike g...

New algorithm for finding longest increasing subsequence is discussed. This algorithm is based on the ideas of idempotent mathematic and uses Max-Plus idempotent semiring. Problem of finding longest increasing subsequence is reformulated in a matrix form and solved with linear algebra.

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...