This paper studies the asymptotic distribution of descents $\des(w)$ in a permutation $w$, and its inverse, distributed according to the Mallows measure. The Mallows measure is a non-uniform probability measure on permutations introduced to study ranked data. Under this measure, permutations are weighted according to the number of inversions they contain, with the weighting controlled by a parameter $q$. The main results are a Berry-Esseen theorem for $\des(w)+\des(w^{-1})$ as well as a joint central limit theorem for $(\des(w),\des(w^{-1}))$ to a bivariate normal with a non-trivial correlation depending on $q$. The proof uses Stein's method with size-bias coupling along with a regenerative process associated to the Mallows measure.
The Mallows measure is a probability measure on $S_n$ where the probability of a permutation $π$ is proportional to $q^{l(π)}$ with $q > 0$ being a parameter and $l(π)$ the number of inversions in $π$. We prove a weak law of large numbers for the length of the longest common subsequences of two independent permutations drawn from the Mallows measure, when $q$ is a function of $n$ and $n(1-q)$ has limit in $\mathbb{R}$ as $n \to \infty$.
Introduced by Mallows in statistical ranking theory, Mallows permutation model is a class of non-uniform probability measures on the symmetric group $S_n$ that depend on a distance metric $d(σ,τ)$ on $S_n$ and a scale parameter $β$. Taking the distance metric to be the $L^1$ and $L^2$ distances--which are respectively known as Spearman's footrule and Spearman's rank correlation in the statistics literature--leads to Mallows permutation models with $L^1$ and $L^2$ distances. In this paper, we study the length of the longest increasing subsequence of random permutations drawn from Mallows permut…
We introduce a probability distribution Q on the group of permutations of the set Z of integers. Distribution Q is a natural extension of the Mallows distribution on the finite symmetric group. A one-sided infinite counterpart of Q, supported by the group of permutations of the set N of natural numbers, was studied previously in our paper [Gnedin and Olshanski, Ann. Prob. 38 (2010), 2103-2135; arXiv:0907.3275]. We analyze various features of Q such as its symmetries, the support, and the marginal distributions.
The Mallows measure is measure on permutations which was introduced by Mallows in connection with ranking problems in statistics. Under this measure, the probability of a permutation $π$ is proportional to $q^{Inv(π)}$ where $q$ is a positive parameter and $Inv(π)$ is the number of inversions in $π$. We consider the length of the longest common subsequence (LCS) of two independently permutations drawn according to $μ_{n,q}$ and $μ_{n,q'}$ for some $q,q' >0$. We show that when $0<q,q'<1$, the limiting law of the LCS is Gaussian. In the regime that $n(1-q) \to \infty$ and $n(1-q') \to \infty$ we…
Fix $q\neq 1$, and sample $w\in S_n$ from the Mallows measure. We study the distribution of $C_i(w)$, the number of $i$-cycles, as $n$ grows large. When $q<1$, they are jointly Gaussian, and this more or less follows from known ideas, but the regime $q>1$ behaves quite differently. In particular, we show that the even cycles $C_{2i}(w)$ have a mean and variance of order $n$, and jointly converge to Gaussian random variables, while the odd cycles $C_{2i+1}(w)$ have a bounded mean and variance, and converge to $C_{2i+1}(w^{even})$ or $C_{2i+1}(w^{odd})$ for some explicit random permutations $w^{…
We study the length of the longest increasing and longest decreasing subsequences of random permutations drawn from the Mallows measure. Under this measure, the probability of a permutation pi in S_n is proportional to q^{inv(pi)} where q is a real parameter and inv(pi) is the number of inversions in pi. The case q=1 corresponds to uniformly random permutations. The Mallows measure was introduced by Mallows in connection with ranking problems in statistics. We determine the typical order of magnitude of the lengths of the longest increasing and decreasing subsequences, as well as large deviati…
We study the length of cycles of random permutations drawn from the Mallows distribution. Under this distribution, the probability of a permutation $π\in \mathbb{S}_n$ is proportional to $q^{\textrm{inv}(π)}$ where $0<q\le 1$ and $\textrm{inv}(π)$ is the number of inversions in $π$. We show that the expected length of the cycle containing a given point is of order $\min\{(1-q)^{-2}, n\}$. This marks the existence of two asymptotic regimes: with high probability, when $n$ tends to infinity with $(1-q)^{-2} \ll n$ then all cycles have size $o(n)$ whereas when $n$ tends to infinity with $(1-q)^{-…
We use the Mallows permutation model to construct a new family of stationary finitely dependent proper colorings of the integers. We prove that these colorings can be expressed as finitary factors of i.i.d. processes with finite mean coding radii. They are the first colorings known to have these properties. Moreover, we prove that the coding radii have exponential tails, and that the colorings can also be expressed as functions of countable-state Markov chains. We deduce analogous existence statements concerning shifts of finite type and higher-dimensional colorings.
Mallows permutation model, introduced by Mallows in statistical ranking theory, is a class of non-uniform probability measures on the symmetric group $S_n$. The model depends on a distance metric $d(σ,τ)$ on $S_n$, which can be chosen from a host of metrics on permutations. In this paper, we focus on Mallows permutation models with $L^1$ and $L^2$ distances, respectively known in the statistics literature as Spearman's footrule and Spearman's rank correlation. Unlike most of the random permutation models that have been analyzed in the literature, Mallows permutation models with $L^1$ and $L^2$…
We consider permutations avoiding a pattern of length three under the family of Mallows distributions. In particular, for any pattern $τ\in S_3-\{321\}$, we obtain rather precise results on the asymptotic probability as $n\to\infty$ that a permutation $σ\in S_n$ under the Mallows distribution with parameter $q\in(0,1)$ avoids the pattern. By a duality between the parameters $q$ and $\frac1q$, we also obtain rather precise results on the above probability for $q>1$ and any pattern $τ\in S_3-\{123\}$.
The Mallows measure is a probability measure on $S_n$ where the probability of a permutation $π$ is proportional to $q^{l(π)}$ with $q > 0$ being a parameter and $l(π)$ the number of inversions in $π$. We show the convergence of the random empirical measure of the product of two independent permutations drawn from the Mallows measure, when $q$ is a function of $n$ and $n(1-q)$ has limit in $\mathbb{R}$ as $n \to \infty$.
We use various combinatorial and probabilistic techniques to study growth rates for the probability that a random permutation from the Mallows distribution avoids consecutive patterns. The Mallows distribution behaves like a $q$-analogue of the uniform distribution by weighting each permutation $π$ by $q^{inv(π)}$, where $inv(π)$ is the number of inversions in $π$ and $q$ is a positive, real-valued parameter. We prove that the growth rate exists for all patterns and all $q>0$, and we generalize Goulden and Jackson's cluster method to keep track of the number of inversions in permutations avoid…
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 $π$ is proportional to $q^{{\rm inv}(π)}$ where $q$ is a positive parameter and ${\rm inv}(π)$ is the number of inversions in $π$. In our main result we show that when $0<q<1$, 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 cas…
Random binary search trees are obtained by recursively inserting the elements $σ(1),σ(2),\ldots,σ(n)$ of a uniformly random permutation $σ$ of $[n]=\{1,\dots,n\}$ into a binary search tree data structure. Devroye (1986) proved that the height of such trees is asymptotically of order $c^*\log n$, where $c^*=4.311\ldots$ is the unique solution of $c \log((2e)/c)=1$ with $c \geq 2$. In this paper, we study the structure of binary search trees $T_{n,q}$ built from Mallows permutations. A $\textrm{Mallows}(q)$ permutation is a random permutation of $[n]=\{1,\ldots,n\}$ whose probability is proporti…
For a positive number $q$ the Mallows measure on the symmetric group is the probability measure on $S_n$ such that $P_{n,q}(π)$ is proportional to $q$-to-the-power-$\mathrm{inv}(π)$ where $\mathrm{inv}(π)$ equals the number of inversions: $\mathrm{inv}(π)$ equals the number of pairs $i<j$ such that $π_i>π_j$. One may consider this as a mean-field model from statistical mechanics. The weak large deviation principle may replace the Gibbs variational principle for characterizing equilibrium measures. In this sense, we prove absence of phase transition, i.e., phase uniqueness.
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 $i<j$ such that $π_i > π_j$. 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 $n$ tends to infinity and $q$ tends to 1 in such a way that $n(1-q)$ has a limit in $\R$.
We study cycle counts in permutations of $1,\dots,n$ drawn at random according to the Mallows distribution. Under this distribution, each permutation $π\in S_n$ is selected with probability proportional to $q^{\text{inv}(π)}$, where $q>0$ is a parameter and $\text{inv}(π)$ denotes the number of inversions of $π$. For $\ell$ fixed, we study the vector $(C_1(Π_n),\dots,C_\ell(Π_n))$ where $C_i(π)$ denotes the number of cycles of length $i$ in $π$ and $Π_n$ is sampled according to the Mallows distribution. Here we show that if $0<q<1$ is fixed and $n\to\infty$ then there are positive constants $m…
Motivated by recent studies of large Mallows$(q)$ permutations, we propose a class of random permutations of $\mathbb{N}_{+}$ and of $\mathbb{Z}$, called regenerative permutations. Many previous results of the limiting Mallows$(q)$ permutations are recovered and extended. Three special examples: blocked permutations, p-shifted permutations and p-biased permutations are studied.
Let $A^{(n)}_{l;k}\subset S_n$ denote the set of permutations of $[n]$ for which the set of $l$ consecutive numbers $\{k, k+1,\cdots, k+l-1\}$ appears in a set of consecutive positions. Under the uniformly probability measure $P_n$ on $S_n$, one has $P_n(A^{(n)}_{l;k})\sim\frac{l!}{n^{l-1}}$ as $n\to\infty$. In one part of this paper we consider the probability of clustering of consecutive numbers under Mallows distributions $P_n^q$, $q>0$. Because of a duality, it suffices to consider $q\in(0,1)$. We show that for $q_n=1-\frac c{n^α}$, with $c>0$ and $α\in(0,1)$, $P_n^q(A^{(n)}_{l;k_n})$ is o…
Asymptotics of the normalizing constant is computed for a class of one parameter exponential families on permutations which includes Mallows model with Spearmans's Footrule and Spearman's Rank Correlation Statistic. The MLE, and a computable approximation of the MLE are shown to be consistent. The pseudo-likelihood estimator of Besag is shown to be $\sqrt{n}$-consistent. An iterative algorithm (IPFP) is proved to converge to the limiting normalizing constant. The Mallows model with Kendall's Tau is also analyzed to demonstrate flexibility of the tools of this paper.
In 2018, Kahle and Stump raised the following problem: identify sequences of finite Coxeter groups $W_n$ for which the two-sided descent statistics on a uniform random element of $W_n$ is asymptotically normal. Recently, Brück and Röttger provided an almost-complete answer, assuming some regularity condition on the sequence $W_n$. In this note, we provide a shorter proof of their result, which does not require any regularity condition. The main new proof ingredient is the use of the second Wasserstein distance on probability distributions, based on the work of Mallows (1972).
Using the recently developed notion of permutation limits this paper derives the limiting distribution of the number of fixed points and cycle structure for any convergent sequence of random permutations, under mild regularity conditions. In particular this covers random permutations generated from Mallows Model with Kendall's Tau, $μ$ random permutations introduced in [11], as well as a class of exponential families introduced in [15].
The exponential functional of simple, symmetric random walks with negative drift is an infinite polynomial $Y = 1 + ξ_1 + ξ_1 ξ_2 + ξ_1 ξ_2 ξ_3 + ...$ of independent and identically distributed non-negative random variables. It has moments that are rational functions of the variables $μ_k = \ev(ξ^k) < 1$ with universal coefficients. It turns out that such a coefficient is equal to the number of permutations with descent set defined by the multiindex of the coefficient. A recursion enumerates all numbers of permutations with given descent sets in the form of a Pascal-type triangle.
It is known from the work of Baik, Deift, and Johansson [1999] that we have Tracy-Widom fluctuations for the longest increasing subsequence of uniform permutations. In this paper, we prove that this result holds also in the case of the Ewens distribution and more generally for a class of random permutation with distribution invariant under conjugation. Moreover, we obtain the convergence of the first components of the associated Young tableaux to the Airy Ensemble as well as the global convergence to the Vershik-Kerov-Logan-Shepp shape. Using similar techniques, we also prove that the limiting…
In this note we investigate correlation inequalities for `up-sets' of permutations, in the spirit of the Harris--Kleitman inequality. We focus on two well-studied partial orders on $S_n$, giving rise to differing notions of up-sets. Our first result shows that, under the strong Bruhat order on $S_n$, up-sets are positively correlated (in the Harris--Kleitman sense). Thus, for example, for a (uniformly) random permutation $π$, the event that no point is displaced by more than a fixed distance $d$ and the event that $π$ is the product of at most $k$ adjacent transpositions are positively correla…
This paper develops techniques to study the number of descents in random permutations via martingales. We relax an assumption in the Berry-Esseen theorem of Bolthausen (1982) to extend the theorem's scope to martingale differences of time-dependent variances. This extension leads to a new proof of the fact that the number of descents in random permutations is asymptotically normal with an error bound of order $1/\sqrt{n}.$ The same techniques are shown to be applicable to other descent and descent-related statistics as they satisfy certain recurrence relation conditions. These statistics inclu…
Normal approximations for descents and inversions of permutations of the set $\{1,2,...,n\}$ are well known. A number of sequences that occur in practice, such as the human genome and other genomes, contain many repeated elements. Motivated by such examples, we consider the number of inversions of a permutation $π(1), π(2),...,π(n)$ of a multiset with $n$ elements, which is the number of pairs $(i,j)$ with $1\leq i < j \leq n$ and $π(i)>π(j)$. The number of descents is the number of $i$ in the range $1\leq i < n$ such that $π(i) > π(i+1)$. We prove that, appropriately normalized, the distribut…
The graph of zigzag diagrams is a close relative of Young's lattice. The boundary problem for this graph amounts to describing coherent random permutations with descent-set statistic, and is also related to certain positive characters on the algebra of quasi-symmetric functions. We establish connections to some further relatives of Young's lattice and solve the boundary problem by reducing it to the classification of spreadable total orders on integers, as recently obtained by Jacka and Warren.
We study the joint distribution of descents and sign for elements of the symmetric group and the hyperoctahedral group (Coxeter groups of types $A$ and $B$). For both groups, this has an application to riffle shuffling: for large decks of cards the sign is close to random after a single shuffle. In both groups, we derive generating functions for the Eulerian distribution refined according to sign, and use them to give two proofs of central limit theorems for positive and negative Eulerian numbers.
We study Stirling permutations defined by Gessel and Stanley. We prove that their generating function according to the number of descents has real roots only. We use that fact to prove that the distribution of these descents, and other, equidistributed statistics on these objects converge to a normal distribution.
The distribution of descents in fixed conjugacy classes of $S_n$ has been studied, and it is shown that its moments have interesting properties. Fulman proved that the descent numbers of permutations in conjugacy classes with large cycles are asymptotically normal, and Kim proved that the descent numbers of fixed point free involutions are also asymptotically normal. In this paper, we generalize these results to prove a central limit theorem for descent numbers of permutations in any conjugacy class of $S_n$.
Concentration of measure is a phenomenon in which a random variable that depends in a smooth way on a large number of independent random variables is essentially constant. The random variable will "concentrate" around its median or expectation. In this work, we explore several theories and applications of concentration of measure. The results of the thesis are divided into three main parts. In the first part, we explore concentration of measure for several random operator compressions and for the length of the longest increasing subsequence of a random walk evolving under the asymmetric exclus…
We prove a quantitative local limit theorem for the number of descents in a random permutation. Our proof uses a conditioning argument and is based on bounding the characteristic function $ϕ(t)$ of the number of descents. We also establish a central limit theorem for the number of 3-term arithmetic progressions (3-APs) in a random subset of $\mathbb{Z}/p\mathbb{Z}$. We conjecture that there is no local limit theorem for 3-APs, but a proof of this remains elusive. A promising avenue of proof is to condition on the size of the subset and show that the resulting distributions are too far apart fo…
This paper does three things: It proves a central limit theorem for novel permutation statistics (for example, the number of descents plus the number of descents in the inverse). It provides a clear illustration of a new approach to proving central limit theorems more generally. It gives us an opportunity to acknowledge the work of our teacher and friend B. V. Rao.
