We compute the limiting distribution, as n approaches infinity, of the number of cycles of length between gamma n and delta n in a permutation of [n] chosen uniformly at random, for constants gamma, delta such that 1/(k+1) <= gamma < delta <= 1/k for some integer k. This distribution is supported on {0, 1, ... k} and has 0th, 1st, ..., kth moments equal to those of a Poisson distribution with parameter log (delta/gamma). For more general choices of gamma, delta we show that such a limiting distribution exists, which can be given explicitly in terms of certain integrals over intersections of hypercubes with half-spaces; these integrals are analytically intractable but a recurrence specifying them can be given. The results herein provide a basis of comparison for similar statistics on restricted classes of permutations.
This paper studies mathematical properties of reaction systems that was introduced by Enrenfeucht and Rozenberg as computational models inspired by biochemical reaction in the living cells. In particular, we continue the study on the generative power of functions specified by minimal reaction systems under composition initiated by Salomaa. Allowing degenerate reaction systems, functions specified by minimal reaction systems over a quarternary alphabet that are permutations generate the alternating group on the power set of the background set.
We introduce a new natural notion of convergence for permutations at any specified scale, in terms of the density of patterns of restricted width. In this setting we prove that limits may be chosen independently at a countably infinite number of scales.
We introduce a permutation analogue of the celebrated Szemeredi Regularity Lemma, and derive a number of consequences. This tool allows us to provide a structural description of permutations which avoid a specified pattern, a result that permutations which scatter small intervals contain all possible patterns of a given size, a proof that every permutation avoiding a specified pattern has a nearly monotone linear-sized subset, and a ``thin deletion'' result. We also show how one can count sub-patterns of a permutation with an integral, and relate our results to permutation quasirandomness in a…
A geometric grid class consists of those permutations that can be drawn on a specified set of line segments of slope \pm1 arranged in a rectangular pattern governed by a matrix. Using a mixture of geometric and language theoretic methods, we prove that such classes are specified by finite sets of forbidden permutations, are partially well ordered, and have rational generating functions. Furthermore, we show that these properties are inherited by the subclasses (under permutation involvement) of such classes, and establish the basic lattice theoretic properties of the collection of all such sub…
We explore the probability that a permutation sampled from the symmetric group of order n uniformly at random has cycles of lengths not exceeding r. Asymptotic formulas valid in specified regions for the ratio n/r are obtained using the saddle point method combined with ideas originated in analytic number theory. Theorem 1 and its detailed proof are included to rectify formulas for small r which have been announced by a few other authors.
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.
We use the theory of symmetric functions to enumerate various classes of alternating permutations w of {1,2,...,n}. These classes include the following: (1) both w and w^{-1} are alternating, (2) w has certain special shapes, such as (m-1,m-2,...,1), under the RSK algorithm, (3) w has a specified cycle type, and (4) w has a specified number of fixed points. We also enumerate alternating permutations of a multiset. Most of our formulas are umbral expressions where after expanding the expression in powers of a variable E, E^k is interpreted as the Euler number E_k. As a small corollary, we obtai…
This paper considers a problem that relates to the theories of covering arrays, permutation patterns, Vapnik-Chervonenkis (VC) classes, and probability thresholds. Specifically, we want to find the number of subsets of [n]:={1,2,....,n} we need to randomly select, in a certain probability space, so as to respectively "shatter" all t-subsets of [n]. Moving from subsets to words, we ask for the number of n-letter words on a q-letter alphabet that are needed to shatter all t-subwords of the q^n words of length n. Finally, we explore the number of random permutations of [n] needed to shatter (spec…
We find an explicit combinatorial interpretation of the coefficients of Kerov character polynomials which express the value of normalized irreducible characters of the symmetric groups S(n) in terms of free cumulants R_2,R_3,... of the corresponding Young diagram. Our interpretation is based on counting certain factorizations of a given permutation.
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…
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 recurrences (alias difference equations) to prove the longstanding conjecture that the two most important permutation statistics, namely the number of inversions and the major index, are asymptotically joint-independently-normal. We even derive more-precise-than needed asymptotic formulas for the (normalized) mixed moments. This is the fully revised second edition, incorportating the many insightful comments of nine conscientious NON-anonymous referees listed under the authors' names. This article is exclusively published in the on-line journal "Personal Journal of Shalosh B. Ekhad and …
We show how the combinatorial interpretation of the normalized median Genocchi numbers in terms of multiset tuples, defined by Hetyei in his study of the alternation acyclic tournaments, is bijectively equivalent to previous models like the normalized Dumont permutations or the Dellac configurations, and we extend the interpretation to the Kreweras triangle.
We prove two estimates for the expectation of the exponential of a complex function of a random permutation or subset. Using this theory, we find asymptotic expressions for the expected number of copies and induced copies of a given graph in a uniformly random graph with degree sequence $(d_1,\ldots,d_n)$ as $n \rightarrow \infty$. We also determine the expected number of spanning trees in this model. The range of degrees covered includes $d_j = λn + O(n^{1/2+\varepsilon})$ for some $λ$ bounded away from $0$ and $1$.
We present a bijective algorithm with which an arbitrary permutation decomposes canonically into elementary blocks which we call families, which are sets with a specified number of ascents and descents. We show that families, arranged in an arbitrary order in a sequence, are in bijection with permutations. The permutation decomposes canonically, by inserting parentheses, into a tree having as nodes a class of permutations which we call primitive. Primitive permutations can be assembled from very simple data. The data for the trees into which a permutation decomposes can be written in a form si…
Richard Stanley proved that the centralized/normalized version of the random variable "length of largest up-down subsequence" in a random permutation of length n is asymptotically normal. We go beyond and present a more refined asymptotic formula for the moments.
We give a new expression for the number of factorizations of a full cycle into an ordered product of permutations of specified cycle types. This is done through purely algebraic means, extending work of Biane. We deduce from our result a formula of Poulalhon and Schaeffer that was previously derived through an intricate combinatorial argument.
We derive a closed-form expression for all genus 1 Hurwitz numbers, and give a simple new graph-theoretic interpretation of Hurwitz numbers in genus 0 and 1. (Hurwitz numbers essentially count irreducible genus g covers of the sphere, with arbitrary specified branching over one point, simple branching over other specified points, and no other branching. The problem is equivalent to counting transitive factorisations of permutations into transpositions.) These results prove a conjecture of Goulden and Jackson, and extend results of Hurwitz and many others.
Hurwitz numbers count branched covers of the Riemann sphere with specified ramification, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers, related to the expansion of complete symmetric functions in the Jucys-Murphy elements, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra-Itzykson-Zuber integral. In previous work we gave an explicit formula for monotone Hurwitz numbers in genus zero. In this paper we consider monotone Hurwitz…
In this paper, we look at the number of factorizations of a given permutation into star transpositions. In particular, we give a natural explanation of a hidden symmetry, answering a question of I.P. Goulden and D.M. Jackson. We also have a new proof of their explicit formula. Another result is the normalized class expansion of some central elements of the symmetric group algebra introduced by P. Biane. To obtain this results, we use natural analogs of Jucys-Murphy elements in the algebra of partial permutations of V. Ivanov and S. Kerov. We investigate their properties and use a formula of A.…
The $(p,q,n)$-dipole problem is a map enumeration problem, arising in perturbative Yang-Mills theory, in which the parameters $p$ and $q$, at each vertex, specify the number of edges separating of two distinguished edges. Combinatorially, it is notable for being a permutation factorization problem which does not lie in the centre of $\mathbb{C}[\mathfrak{S}_n]$, rendering the problem inaccessible through the character theoretic methods often employed to study such problems. This paper gives a solution to this problem on all orientable surfaces when $q=n-1$, which is a combinatorially significa…
We study the $\bG_a^M$ degenerations $\Fl^a_\la$ of the type $A$ flag varieties $\Fl_\la$. We describe these degenerations explicitly as subvarieties in the products of Grassmanians. We construct cell decompositions of $\Fl^a_\la$ and show that for complete flags the number of cells is equal to the normalized median Genocchi numbers $h_n$. This leads to a new combinatorial definition of the numbers $h_n$. We also compute the Poincar\' e polynomials of the complete degenerate flag varieties via a natural statistics on the set of Dellac's configurations, similar to the length statistics on the s…
Hurwitz numbers count branched covers of the Riemann sphere with specified ramification data, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of the branched covers counted by the Hurwitz numbers, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra-Itzykson-Zuber integral. In this paper we begin a detailed study of monotone Hurwitz numbers. We prove two results that are reminiscent of those for classical Hurwitz numbers. The first is the monotone join-cu…
Two well-known polytopes whose vertices are indexed by permutations in the symmetric group $\mathfrak{S}_n$ are the permutohedron $P_n$ and the Birkhoff polytope $B_n$. We consider polytopes $P_n(Π)$ and $B_n(Π)$, whose vertices correspond to the permutations in $\mathfrak{S}_n$ avoiding a set of patterns $Π$. For various choices of $Π$, we explore the Ehrhart polynomials and $h^*$-vectors of these polytopes as well as other aspects of their combinatorial structure. For $P_n(Π)$, we consider all subsets $Π\subseteq \mathfrak{S}_3$ and are able to provide results in most cases. To illustrate, $…
