Let the random variable $Z_{n,k}$ denote the number of increasing subsequences of length $k$ in a random permutation from $S_n$, the symmetric group of permutations of $\{1,...,n\}$. We show that $Var(Z_{n,k_n})=o((EZ_{n,k_n})^2)$ as $ n\to\infty$ if and only if $k_n=o(n^\frac25)$. In particular then, the weak law of large numbers holds for $Z_{n,k_n}$ if $k_n=o(n^\frac25)$. We also show the following approximation result for the uniform measure $U_n$ on $S_n$. Define the probability measure $μ_{n;k_n}$ on $S_n$ as follows: Consider $n$ cards, numbered from 1 to $n$, and laid out on a table from left to right in increasing order. Place a mark on $k_n$ of the cards, chosen at random. Then pick up all the unmarked cards and randomly insert them between the $k_n$ marked cards that remained on the table. Denote the resulting distribution on $S_n$ by $μ_{n;k_n}$. The weak law of large numbers holds for $Z_{n,k_n}$ if and only if the total variation distance between $μ_{n;k_n}$ and $U_n$ converges to 0 as $n\to\infty$.
In order to evaluate the asymptotic behavior of the second moment, we need to analyze certain occupation times of certain conditioned two-dimensional random walks.
We find a two term asymptotic expansion for the optimal expected value of a sequentially selected monotone subsequence from a random permutation of length n. A striking feature of this expansion is that tells us that the expected value of optimal selection from a random permutation is quantifiably larger than optimal sequential selection from an independent sequences of uniformly distributed random variables; specifically, it is larger by at least (1/6)log n +O(1).
We address a question and a conjecture on the expected length of the longest common subsequences of two i.i.d.$\ $random permutations of $[n]:=\{1,2,...,n\}$. The question is resolved by showing that the minimal expectation is not attained in the uniform case. The conjecture asserts that $\sqrt{n}$ is a lower bound on this expectation, but we only obtain $\sqrt[3]{n}$ for it.
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…
Connections between longest increasing subsequences in random permutations and eigenvalues of random matrices with complex entries have been intensely studied. This note applies properties of random elements of the finite general linear group to obtain results about the longest increasing subsequence in non- uniform random permutations.
We compute the limit distribution for (centered and scaled) length of the longest increasing subsequence of random colored permutations. The limit distribution function is a power of that for usual random permutations computed recently by Baik, Deift, and Johansson (math.CO/9810105). In two--colored case our method provides a different proof of a similar result by Tracy and Widom about longest increasing subsequences of signed permutations (math.CO/9811154). Our main idea is to reduce the `colored' problem to the case of usual random permutations using certain combinatorial results and element…
A random $n$-permutation may be generated by sequentially removing random cards $C_1,...,C_n$ from an $n$-card deck $D = \{1,...,n\}$. The permutation $σ$ is simply the sequence of cards in the order they are removed. This permutation is itself uniformly random, as long as each random card $C_t$ is drawn uniformly from the remaining set at time $t$. We consider, here, a variant of this simple procedure in which one is given a choice between $k$ random cards from the remaining set at each step, and selects the lowest numbered of these for removal. This induces a bias towards selecting lower num…
We survey the theory of increasing and decreasing subsequences of permutations. Enumeration problems in this area are closely related to the RSK algorithm. The asymptotic behavior of the expected value of the length is(w) of the longest increasing subsequence of a permutation w of 1,2,...,n was obtained by Vershik-Kerov and (almost) by Logan-Shepp. The entire limiting distribution of is(w) was then determined by Baik, Deift, and Johansson. These techniques can be applied to other classes of permutations, such as involutions, and are related to the distribution of eigenvalues of elements of the…
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 $n \to \infty$.
We study the shape of the Young diagram λassociated via the Robinson-Schensted-Knuth algorithm to a random permutation in S_n such that the length of the longest decreasing subsequence is not bigger than a fixed number d; in other words we study the restriction of the Plancherel measure to Young diagrams with at most d rows. We prove that in the limit n\to\infty the rows of λbehave like the eigenvalues of a certain random matrix (traceless Gaussian Unitary Ensemble) with d rows and columns. In particular, the length of the longest increasing subsequence of such a random permutation behaves asy…
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…
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…
We study the fluctuations, 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.
A locally uniform random permutation is generated by sampling $n$ points independently from some absolutely continuous distribution $ρ$ on the plane and interpreting them as a permutation by the rule that $i$ maps to $j$ if the $i$th point from the left is the $j$th point from below. As $n$ tends to infinity, decreasing subsequences in the permutation will appear as curves in the plane, and by interpreting these as level curves, a union of decreasing subsequences give rise to a surface. We show that, under the correct scaling, for any $r\ge0$, the largest union of $\lfloor r\sqrt{n}\rfloor$ de…
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…
This paper is interested in independent sets (or equivalently, cliques) in uniform random cographs. We also study their permutation analogs, namely, increasing subsequences in uniform random separable permutations. First, we prove that, with high probability as $n$ gets large, the largest independent set in a uniform random cograph with $n$ vertices has size $o(n)$. This answers a question of Kang, McDiarmid, Reed and Scott. Using the connection between graphs and permutations via inversion graphs, we also give a similar result for the longest increasing subsequence in separable permutations. …
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…
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 $\R$.
For a word $π$ and integer $i$, we define $L^i(π)$ to be the length of the longest subsequence of the form $i(i+1)\cdots j$, and we let $L(π):=\max_i L^i(π)$. In this paper we estimate the expected values of $L^1(π)$ and $L(π)$ when $π$ is chosen uniformly at random from all words which use each of the first $n$ integers exactly $m$ times. We show that $\mathbb{E}[L^1(π)]\sim m$ if $n$ is sufficiently larger in terms of $m$ as $m$ tends towards infinity, confirming a conjecture of Diaconis, Graham, He, and Spiro. We also show that $\mathbb{E}[L(π)]$ is asymptotic to the inverse gamma function …
By a classical principle of probability theory, sufficiently thin subsequences of general sequences of random variables behave like i.i.d.\ sequences. This observation not only explains the remarkable properties of lacunary trigonometric series, but also provides a powerful tool in many areas of analysis, such the theory of orthogonal series and Banach space theory. In contrast to i.i.d.\ sequences, however, the probabilistic structure of lacunary sequences is not permutation-invariant and the analytic properties of such sequences can change after rearrangement. In a previous paper we showed t…
Let the random variable $Z_{n,k}$ denote the number of increasing subsequences of length $k$ in a random permutation from $S_n$, the symmetric group of permutations of $\{1,...,n\}$. In a recent paper [Random Structures Algorithms 29 (2006) 277--295] we showed that the weak law of large numbers holds for $Z_{n,k_n}$ if $k_n=o(n^{2/5})$; that is, \[\lim_{n\to\infty}\frac{Z_{n,k_n}}{EZ_{n,k_n}}=1\qquad in probability.\] The method of proof employed there used the second moment method and demonstrated that this method cannot work if the condition $k_n=o(n^{2/5})$ does not hold. It follows from re…
Let $X=(X_1,\ldots,X_n)$ be a vector of i.i.d. random variables where $X_i$'s take values over $\mathbb{N}$. The purpose of this paper is to study the number of weakly increasing subsequences of $X$ of a given length $k$, and the number of all weakly increasing subsequences of $X$. For the former, it is shown that a central limit theorem holds. Also, the first two moments of each of those two random variables are analyzed, their asymptotics are investigated, and results are related to the case of similar statistics in uniformly random permutations. We conclude the paper with applications on a …
Let $LA_{n}(τ)$ be the length of the longest alternating subsequence of a uniform random permutation $τ\in[n]$. Classical probabilistic arguments are used to rederive the asymptotic mean, variance and limiting law of $LA_{n}(τ)$. Our methodology is robust enough to tackle similar problems for finite alphabet random words or even Markovian sequences in which case our results are mainly original. A sketch of how some cases of pattern restricted permutations can also be tackled with probabilistic methods is finally presented.
Using symmetric function theory, we study the cycle structure and increasing subsequence structure of permutations after various shuffling methods, emphasizing the role of Cauchy type identities and the Robinson-Schensted-Knuth correspondence. One consequence is a cycle index for riffle shuffles when one deals from the bottom of the deck; conclusions are drawn concerning the distribution of fixed points and the asymptotic distribution of cycle structure. In studying cycle structure of unimodal permutations and their generalizations we are led to an interesting variation of the RSK corresponden…
The Preisach graph is a directed graph associated with a permutation $ρ\in{\cal S}_N$. We give an explicit bijection between its vertices and increasing subsequences of $ρ$ with the property that the length of a subsequence equals to the degree of nesting of the corresponding vertex inside a hierarchy of cycles and sub-cycles of the graph. As a consequence, the nesting degree of the Preisach graph equals the length of the longest increasing subsequence.
The longest increasing subsequence problem for permutations has been studied extensively in the last fifty years. The interpretation of the longest increasing subsequence as the longest 21-avoiding subsequence in the context of permutation patterns leads to many interesting research directions. We introduce and study the statistical properties of Hammersleytype interacting particle processes related to these generalizations and explore the finer structures of their distributions. We also propose three different interacting particle systems in the plane analogous to the Hammersley process in on…
We present a number of results relating partial Cauchy-Littlewood sums, integrals over the compact classical groups, and increasing subsequences of permutations. These include: integral formulae for the distribution of the longest increasing subsequence of a random involution with constrained number of fixed points; new formulae for partial Cauchy-Littlewood sums, as well as new proofs of old formulae; relations of these expressions to orthogonal polynomials on the unit circle; and explicit bases for invariant spaces of the classical groups, together with appropriate generalizations of the str…
For large $q$, does the (discrete) uniform distribution on the set of $q!$ permutations of the vector $(1,2,\dots,q)$ closely approximate the (continuous) uniform distribution on the $(q-2)$-sphere that contains them? These permutations comprise the vertices of the regular permutohedron, a $(q-1)$-dimensional convex polyhedron. Surprisingly to me, the answer is emphatically no: these permutations are confined to a negligible portion of the sphere, and the regular permutohedron occupies a negligible portion of the ball. However, $(1,2,\dots,q)$ is not the most favorable configuration for approx…
We investigate the variance of the length of the longest common subsequences of two independent random words of size $n$, where the letters of one word are i.i.d. uniformly drawn from $\{α_1, α_2, \cdots, α_m\}$, while the letters of the other word are i.i.d. drawn from $\{α_1, α_2, \cdots, α_m, α_{m+1}\}$, with probability $p > 0$ to be $α_{m+1}$, and $(1-p)/m > 0$ for all the other letters. The order of the variance of this length is shown to be linear in $n$.
This paper is concerned with certain connections between the ensemble of n x n unitary matrices -- specifically the characteristic function of the random variable tr(U) -- and combinatorics -- specifically Ulam's problem concerning the distribution of the length of the longest increasing subsequence in permutation groups -- and the appearance of Painleve functions in the answers to apparently unrelated questions. Among the results is a representation in terms of a Painleve V function for the characteristic function of tr(U) and (using recent results of Baik, Deift and Johansson) an expression …
Let $X=(X_i)_{i\ge 1}$ and $Y=(Y_i)_{i\ge 1}$ be two sequences of independent and identically distributed (iid) random variables taking their values, uniformly, in a common totally ordered finite alphabet. Let LCI$_n$ be the length of the longest common and (weakly) increasing subsequence of $X_1\cdots X_n$ and $Y_1\cdots Y_n$. As $n$ grows without bound, and when properly centered and normalized, LCI$_n$ is shown to converge, in distribution, towards a Brownian functional that we identify.
Let $(X_i)_{i \geq 1}$ and $(Y_i)_{i\geq1}$ be two independent sequences of independent identically distributed random variables taking their values in a common finite alphabet and having the same law. Let $LC_n$ be the length of the longest common subsequences of the two random words $X_1\cdots X_n$ and $Y_1\cdots Y_n$. Under a lower bound assumption on the order of its variance, $LC_n$ is shown to satisfy a central limit theorem. This is in contrast to the limiting distribution of the length of the longest common subsequences in two independent uniform random permutations of $\{1, \dots, n\}…
We present some Markovian approaches to prove universality results for some functions on the symmetric group. Some of those statistics are already studied in [Kammoun, 2018, 2020] but not the general case. We prove, in particular, that the number of occurrences of a vincular patterns satisfies a CLT for conjugation invariant random permutations with few cycles and we improve the results already known for the longest increasing subsequence. The second approach is a suggestion of a generalization to other random permutations and other sets having a similar structure than the symmetric group.
Recently Richard Stanley initiated a study of the distribution of the length as(w) of the longest alternating subsequence in a random permutation w from the symmetric group $S_n$. Among other things he found an explicit formula for the generating function (on n and k) for the probability that as(w) is at most k and conjectured that the distribution, suitably centered and normalized, tended to a Gaussian with variance 8/45. In this note we present a proof of the conjecture based on the generating function.
In testing the independence of two Gaussian populations, one computes the distribution of the sample canonical correlation coefficients, given that the actual correlation is zero. The "Laplace transform" of this distribution is not only an integral over the Grassmannian of p-dimensional planes in complex n-space, but is also related to a generalized hypergeometric function. Such integrals are solutions of Painlevé-like equations. They also have expansions, related to random words of length l formed with an alphabet of p letters. Given that each letter appears in the word, the maximal length of…
We provide a framework for proving convergence to the directed landscape, the central object in the Kardar-Parisi-Zhang universality class. For last passage models, we show that compact convergence to the Airy line ensemble implies convergence to the Airy sheet. In i.i.d. environments, we show that Airy sheet convergence implies convergence of distances and geodesics to their counterparts in the directed landscape. Our results imply convergence of classical last passage models and interacting particle systems. Our framework is built on the notion of a directed metric, a generalization of metri…
The rate of convergence of the distribution of the length of the longest increasing subsequence, toward the maximal eigenvalue of certain matrix ensembles, is investigated. For finite-alphabet uniform and nonuniform i.i.d. sources, a rate of $\log n/\sqrt{n}$ is obtained. The uniform binary case is further explored, and an improved $1/\sqrt{n}$ rate obtained.
We consider the problem of estimating the average tour length of the asymmetric TSP arising from the disk scheduling problem with a linear seek function and a probability distribution on the location of I/O requests. The optimal disk scheduling algorithm of Andrews, Bender and Zhang is interpreted as a simple peeling process on points in a 2 dimensional space-time w.r.t the causal structure. The patience sorting algorithm for finding the longest increasing subsequence in a permutation can be given a similar interpretation. Using this interpretation we show that the optimal tour length is the l…
Pop-Stack Sorting is an algorithm that takes a permutation as an input and sorts its elements. It consists of several steps. At one step, the algorithm reads the permutation it has to process from left to right and reverses each of its maximal decreasing subsequences of consecutive elements. It terminates at the first step that outputs the identity permutation. In this note, we answer a question of Defant on the running time of Pop-Stack Sorting on the uniform random permutation $σ_n$. More precisely, we show that there is a constant $c > 0.5$ such that asymptotically almost surely, the algori…
We define and study the Plancherel-Hecke probability measure on Young diagrams; the Hecke algorithm of [Buch-Kresch-Shimozono-Tamvakis-Yong '06] is interpreted as a polynomial-time exact sampling algorithm for this measure. Using the results of [Thomas-Yong '07] on jeu de taquin for increasing tableaux, a symmetry property of the Hecke algorithm is proved, in terms of longest strictly increasing/decreasing subsequences of words. This parallels classical theorems of [Schensted '61] and of [Knuth '70], respectively, on the Schensted and Robinson-Schensted-Knuth algorithms. We investigate, and co…
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…
We present a recursive construction of a (2t + 1)-wise uniform set of permutations on 2n objects using a (2t + 1) - (2n, n, \cdot) combinatorial design, a t-wise uniform set of permutations on n objects and a (2t+1)-wise uniform set of permutations on n objects. Using the complete design in this procedure gives a t-wise uniform set of permutations on n objects whose size is at most t^2n, the first non-trivial construction of an infinite family of t-wise uniform sets for t \geq 4. If a non-trivial design with suitable parameters is found, it will imply a corresponding improvement in the constru…
The nature of the alignment with gaps corresponding to a longest common subsequence (LCS) of two independent iid random sequences drawn from a finite alphabet is investigated. It is shown that such an optimal alignment typically matches pieces of similar short-length. This is of importance in understanding the structure of optimal alignments of two sequences. Moreover, it is also shown that any property, common to two subsequences, typically holds in most parts of the optimal alignment whenever this same property holds, with high probability, for strings of similar short-length. Our results sh…
