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...

We consider scenarios in which long sequences of data are analyzed and subsequences must be traced that are monotone and maximum, according to some measure. A classical example is the online Longest Increasing Subsequence Problem for numeric and alphanumeric data. We extend the problem in two ways: (a) we allow data from any partially ordered set, and (b) we maximize subsequences using much mor...

We study the uctuations, in the large deviations regime, of the longest increasing subsequence of a random i.i.d. sample on the unit square. In particular, our results yield the precise upper and lower exponential tails for the length of the longest increasing subsequence of a random permutation. i=1 denote a sequence of i.i.d. random variables with marginal law on the unit square Q = 0; 1] 2. ...

A number of learning models are commonly employed in the simulation of social behaviour. These include population learning, lifetime learning and cultural learning. Population learning allows populations as a whole to evolve over time, typically through a Darwinian model of natural selection. Lifetime learning allows individuals to acquire knowledge during their lifetimes and cultural learning ...

Given a sequence of integers, we want to find a longest increasing subsequence of the sequence. It is known that this problem can be solved in O(n log n) time and space. Our goal in this paper is to reduce the space consumption while keeping the time complexity small. For √ n ≤ s ≤ n, we present algorithms that use O(s log n) bits and O( 1 s · n · log n) time for computing the length of a longe...

The deterministic space complexity of approximating the length of the longest increasing subsequence of a stream of N integers is known to be Θ̃( √ N). However, the randomized complexity is wide open. We show that the technique used in earlier work to establish the Ω( √ N) deterministic lower bound fails strongly under randomization: specifically, we show that the communication problems on which...

By reviewing Longest Increasing Subsequence (LIS) and Longest Common Subsequence (LCS), the Longest Common Increasing Subsequence (LCIS) problem is explored for two non-random input cases in details. Specifically, we designed two algorithms, one solving the input sequence scenario with the case that one sequence is ordered and duplicate elements are allowed in each of sequences, and the second ...