We study the lengths of monotone subsequences for permutations drawn from the Mallows measure. The Mallows measure was introduced by Mallows in connection with ranking problems in statistics. Under this measure, the probability of a permutation @math is proportional to @math where @math is a positive parameter and @math is the number of inversions in @math . In our main result we show that when @math , the limiting distribution of the longest increasing subsequence (LIS) is Gaussian, answering an open question in [Bhatnagar and Peled, PTRF, 2015]. This is in contrast to the case when @math where the limiting distribution of the LIS when scaled appropriately is the GUE Tracy-Widom distribution. We also obtain a law of large numbers for the length of the longest decreasing subsequence (LDS) and identify the precise constant in the order of the expectation, answering a further open question in [Bhatnagar and Peled, PTRF, 2015].
 In a famous paper [8] Hammersley investigated the lengthL n of the longest increasing subsequence of a randomn-permutation. Implicit in that paper is a certain one-dimensional continuous-space interacting particle process. By studying a hydrodynamical limit for Hammersley's process we show by fairly “soft” arguments that limn ′1 2 EL n =2. This is a known result, but previous proofs [14, 11] relied on hard analysis of combinatorial asymptotics. We consider the concentration of measure for n i.i.d., two-dimensional random variables under the conditioning that they form a record. Under mild conditions, we show that all random variables tend to concentrate, as n → ∞, around limiting curves, which are the solutions of an appropriate variational problem. We also show that the same phenomenon occurs, without the records conditioning, for the longest increasing subsequence in the sample. Adding a column of numbers produces "carries" along the way. We show that random digits produce a pattern of carries with a neat probabilistic description: the carries form a one-dependent determinantal point process. This makes it easy to answer natural questions: How many carries are typical? Where are they located? We show that many further examples, from combinatorics, algebra and group theory, have essentially the same neat formulae, and that any one-dependent point process on the integers is determinantal. The examples give a gentle introduction to the emerging fields of one-dependent and determinantal point processes.  The Mallows measure on the symmetric group S n is the probability measure such that each permutation has probability proportional to q raised to the power of the number of inversions, where q is a positive parameter and the number of inversions of π is equal to the number of pairs i π j . We prove a weak law of large numbers for the length of the longest increasing subsequence for Mallows distributed random permutations, in the limit that n→∞ and q→1 in such a way that n(1−q) has a limit in R. Abstract Stanley posed the problem of minimizing the functional H(f)= ∫ 0 ∞ dx ∫ 0 f(x) dy log (f (x) −y+f −1 (y) − x) over nonincreasing nonnegative f on (0, ∞) of integral unity. We show that the minimum is unique and has the value − 1 2 , as was conjectured by Stanley. The minimizing function f 0 , H(f 0 ) = − 1 2 , is given parametrically by f 0 (x)=( 2 π )(sin θ −θ cos θ), x =f o (x) + 2 cos θ, 0 ⩽ θ ⩽ π (2) for 0 ⩽ x ⩽ 2; and f 0 ( x ) = 0 for x ⩾ 2. Closely related unpublished results have been obtained by Hammersley. We also find the minimum of H ( f ) subject to the constraints f (0) ⩽ a and f −1 (0) = inf ( x : f ( x ) = 0) ⩽ b where a and b are given. Proofs of the results for the case of constraints are complicated and will be given elsewhere. Let λ n be the shape of the random Young tableau with n unit squares obtained from sampling from the Schensted distribution where P(λ n = n! π 2 (λ n ) (3) where π ( λ n ) is the product of the n hook lengths of λ n . Consider the stochastic processes λ n (t)=( 1 n 1 2 )λ(t n 1 2 , n ⩾ 1, t ⩾ 0 (4) where λ n ( t ) is the height of the tableau λ n at a horizontal distance t from the corner. We show λ n → f 0 in the sense of weak convergence in a certain metric, where f 0 is the deterministic function in (2). Let l ( σ n ) denote the length of the longest increasing subsequence of a random permutation σ n of 1,2,…, n . Hammersley showed that l(ω n ) n 1 2 →c in probability, n→∞ (6) Schensted showed that l ( σ n ) has the same distribution as λ n (0) under the distribution (3) on λ n . It has long been conjectured (apparently first by Baer and Brock) that c = 2. We show here that c ⩾ 2 as a by-product of (5).