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 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…
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 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 $n \to \infty$.
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.
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 correlatio…
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…
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 $\R$.
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 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 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 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…
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.
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…
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…
We evaluate the probabilities of various events under the uniform distribution on the set of 312-avoiding permutations of 1,...,N. We derive exact formulas for the probability that the ith element of a random permutation is a specific value less than i, and for joint probabilities of two such events. In addition, we obtain asymptotic approximations to these probabilities for large N when the elements are not close to the boundaries or to each other. We also evaluate the probability that the graph of a random 312-avoiding permutation has k specified decreasing points, and we show that for large…
A classical result for the simple symmetric random walk with $2n$ steps is that the number of steps above the origin, the time of the last visit to the origin, and the time of the maximum height all have exactly the same distribution and converge when scaled to the arcsine law. Motivated by applications in genomics, we study the distributions of these statistics for the non-Markovian random walk generated from the ascents and descents of a uniform random permutation and a Mallows($q$) permutation and show that they have the same asymptotic distributions as for the simple random walk. We also g…
We initiate the study of limit shapes for random permutations avoiding a given pattern. Specifically, for patterns of length 3, we obtain delicate results on the asymptotics of distributions of positions of numbers in the permutations. We view the permutations as 0-1 matrices to describe the resulting asymptotics geometrically. We then apply our results to obtain a number of results on distributions of permutation statistics.
We consider a random permutation drawn from the set of permutations of length $n$ that avoid some given set of patterns of length 3. We show that the number of occurrences of another pattern $σ$ has a limit distribution, after suitable scaling. In several cases, the number is asymptotically normal; this contrasts to the cases of permutations avoiding a single pattern of length 3 studied in earlier papers.
We set up a new notion of local convergence for permutations and we prove a characterization in terms of proportions of \emph{consecutive} pattern occurrences. We also characterize random limiting objects for this new topology introducing a notion of "shift-invariant" property (corresponding to the notion of unimodularity for random graphs). We then study two models in the framework of random pattern-avoiding permutations. We compute the local limits of uniform $ρ$-avoiding permutations, for $|ρ|=3,$ when the size of the permutations tends to infinity. The core part of the argument is the desc…
We consider a random permutation drawn from the set of 321-avoiding permutations of length $n$ and show that the number of occurrences of another pattern $σ$ has a limit distribution, after scaling by $n^{m+\ell}$ where $m$ is the length of $σ$ and $\ell$ is the number of blocks in it. The limit is not normal, and can be expressed as a functional of a Brownian excursion.
We consider a random permutation drawn from the set of 132-avoiding permutations of length $n$ and show that the number of occurrences of another pattern $σ$ has a limit distribution, after scaling by $n^{λ(σ)/2}$ where $λ(σ)$ is the length of $σ$ plus the number of descents. The limit is not normal, and can be expressed as a functional of a Brownian excursion. Moments can be found by recursion.
Permutations that avoid given patterns are among the most classical objects in combinatorics and have strong connections to many fields of mathematics, computer science and biology. In this paper we study fixed points of both 123- and 231-avoiding permutations. We find an exact description for a scaling limit of the empirical distribution of fixed points in term of Brownian excursion. This builds on the connections between pattern-avoiding permutations and Brownian excursion developed in Part I of this series and strengthens the recent results of Elizalde (2012) and Miner and Pak (2014) on fix…
We describe the distribution of the number and location of the fixed points of permu- tations that avoid the pattern 321 via a bijection with rooted plane trees on n + 1 vertices. Using the local limit theorem for Galton-Watson trees, we are able to give an explicit description of the limit of this distribution.
In this paper we study the degree sequence of the permutation graph $G_{π_n}$ associated with a sequence $π_n\in S_n$ of random permutations. Joint limiting distributions of the degrees are established using results from graph and permutation limit theories. In particular, for the uniform random permutation, the joint distribution of the degrees of the vertices labelled $\lceil nr_1 \rceil, \lceil nr_2 \rceil, \ldots, \lceil nr_s \rceil$ converges (after scaling by $n$) to independent random variables $D_1, D_2, \ldots, D_s$, where $D_i\sim \text{Unif}(r_i, 1-r_i)$, for $r_i\in [0,1]$ and $i\i…
