The maximal-inversion statistic and pattern-avoiding permutations

We define a new combinatorial statistic, maximal-inversion, on a permutation. We remark that the number M(n,k) of permutations in S_n with k maximal-inversions is the signless Stirling number c(n,n−k) of the first kind. A permutation π in S_n is uniquely determined by its maximal-inversion set . We prove it by making an algorithm for retrieving the permutation from its maximal-inversion set. Also, we remark on how the algorithm can be used directly to determine whether a given set is the maximal-inversion set of a permutation. As an application of the algorithm, we characterize the maximal-inversion set for pattern-avoiding permutations. Then we give some enumerative results concerning permutations with forbidden patterns.     

Reconstruction of permutations distorted by reversal errors

The problem of reconstructing permutations on n elements from their erroneous patterns which are distorted by reversal errors is considered in this paper. Reversals are the operations reversing the order of a substring of a permutation. To solve this problem, it is essential to investigate structural and combinatorial properties of a corresponding Cayley graph on the symmetric group Sym_n generated by reversals. It is shown that for any n?3 an arbitrary permutation π is uniquely reconstructible from four distinct permutations at reversal distance at most one from π where the reversal distance is defined as the least number of reversals needed to transform one permutation into the other. It is also proved that an arbitrary permutation is reconstructible from three permutations with a probability p₃→1 and from two permutations with a probability as n→∞. A reconstruction algorithm is presented. In the case of at most two reversal errors it is shown that at least erroneous patterns are required in order to reconstruct an arbitrary permutation.     

A bijective proof of Jackson's formula for the number of factorizations of a cycle

Factorizations of the cyclic permutation into two permutations with respectively n and m cycles, or, equivalently, unicellular bicolored maps with N edges and n white and m black vertices, have been enumerated independantly by Jackson and Adrianov using evaluations of characters of the symmetric group. In this paper we present a bijection between unicellular partitioned bicolored maps and couples made of an ordered bicolored tree and a partial permutation, that allows for a combinatorial derivation of these results.Our work is closely related to a recent construction of Goulden and Nica for the celebrated Harer-Zagier formula, and indeed we provide a unified presentation of both bijections in terms of Eulerian tours in graphs.     

Short proofs for cut-and-paste sorting of permutations

We consider the problem of determining the maximum number of moves required to sort a permutation of [n] using cut-and-paste operations, in which a segment is cut out and then pasted into the remaining string, possibly reversed. We give short proofs that every permutation of [n] can be transformed to the identity in at most ⌊2n/3⌋ such moves and that some permutations require at least ⌊n/2⌋ moves.     

Sorting with networks of data structures

We consider the problem of sorting a permutation using a network of data structures as introduced by Knuth and Tarjan. In general the model as considered previously was restricted to networks that are directed acyclic graphs (DAGs) of stacks and/or queues. In this paper we study the question of which are the smallest general graphs that can sort an arbitrary permutation and what is their efficiency. We show that certain two-node graphs can sort in time Θ(n²) and no simpler graph can sort all permutations. We then show that certain three-node graphs sort in time Ω(n^3/2), and that there exist graphs of k nodes which can sort in time Θ(nlog_kn), which is optimal.     

Connected permutation graphs

A permutation graph is a simple graph associated with a permutation. Let c_n be the number of connected permutation graphs on n vertices. Then the sequence {c_n} satisfies an interesting recurrence relation such that it provides partitions of n! as . We also see that, if uniformly chosen at random, asymptotically almost all permutation graphs are connected.     

Random induced subgraphs of Cayley graphs induced by transpositions

In this paper we study random induced subgraphs of Cayley graphs of the symmetric group induced by an arbitrary minimal generating set of transpositions. A random induced subgraph of this Cayley graph is obtained by selecting permutations with independent probability, λ_n. Our main result is that for any minimal generating set of transpositions, for probabilities where and δ>0, a random induced subgraph has a.s. a unique largest component of size . Here x(?_n) is the survival probability of a Poisson branching process with parameter λ=1+?_n.     

An approximation algorithm for sorting by reversals and transpositions

Genome rearrangement algorithms are powerful tools to analyze gene orders in molecular evolution. Analysis of genomes evolving by reversals and transpositions leads to a combinatorial problem of sorting by reversals and transpositions, the problem of finding a shortest sequence of reversals and transpositions that sorts one genome into the other. In this paper we present a 2k-approximation algorithm for sorting by reversals and transpositions for unsigned permutations where k is the approximation ratio of the algorithm used for cycle decomposition. For the best known value of k our approximation ratio becomes 2.8386+δ for any δ>0. We also derive a lower bound on reversal and transposition distance of an unsigned permutation.     

A Note on Average-Case Sorting

This note studies the average-case comparison-complexity of sorting n elements when there is a known distribution on inputs and the goal is to minimize the expected number of comparisons. We generalize Fredman’s algorithm which is a variant of insertion sort and provide a basically tight upper bound: If μ is a distribution on permutations on n elements, then one may sort inputs from μ with expected number of comparisons that is at most H(μ) + 2n, where H is the entropy function. The algorithm uses less comparisons for more probable inputs: For every permutation π, the algorithm sorts π by using at most \(\log _{2}(\frac {1}{\Pr _{\mu }(\pi )})+2n\) comparisons. A lower bound on the expected number of comparisons of H(μ) always holds, and a linear dependence on n is also required.     

Asymptotic bounds for permutations containing many different patterns

We say that a permutation σ∈Sn contains a permutation π∈Sk as a pattern if some subsequence of σ has the same order relations among its entries as π. We improve on results of Wilf, Coleman, and Eriksson et al. that bound the asymptotic behavior of pat(n), the maximum number of distinct patterns of any length contained in a single permutation of length n. We prove that by estimating the amount of redundancy due to patterns that are contained multiple times in a given permutation. We also consider the question of k-superpatterns, which are permutations that contain all patterns of a given length k. We give a simple construction that shows that Lk, the length of the shortest k-superpattern, is at most . This may lend evidence to a conjecture of Eriksson et al. that .     

On the corners of certain determinantal ranges

Let A be a complex n×n matrix and let SO(n) be the group of real orthogonal matrices of determinant one. Define Δ(A)={det(A°Q):Q∈SO(n)}, where ° denotes the Hadamard product of matrices. For a permutation σ on {1,…,n}, define It is shown that if the equation z_σ=det(A°Q) has in SO(n) only the obvious solutions (Q=(ε_iδ_σi,j),ε_i=±1 such that ε₁…ε_n=sgnσ), then the local shape of Δ(A) in a vicinity of z_σ resembles a truncated cone whose opening angle equals , where σ₁, σ₂ differ from σ by transpositions. This lends further credibility to the well known de Oliveira Marcus Conjecture (OMC) concerning the determinant of the sum of normal n×n matrices. We deduce the mentioned fact from a general result concerning multivariate power series and also use some elementary algebraic topology.     

On generation of permutations through decomposition of symmetric groups into cosets

A hardware-oriented algorithm for generating permutations is presented that takes as a theoretic base an iterative decomposition of the symmetric groupS _n into cosets. It generates permutations in a new order. Simple ranking and unranking algorithms are given. The construction of a permutation generator is proposed which contains a cellular permutation network as a main component. The application of the permutation generator for solving a class of combinatorial problems on parallel computers is suggested.     

Rearrangement of conditionally convergent series on a small set

We consider ideals I of subsets of the set of natural numbers such that for every conditionally convergent series _∑n∈ωan and every there is a permutation such that _∑n∈ωaπr(n)=r and

     

Coding for locality in reconstructing permutations

The problem of storing permutations in a distributed manner arises in several common scenarios, such as efficient updates of a large, encrypted, or compressed data set. This problem may be addressed in either a combinatorial or a coding approach. The former approach boils down to presenting large sets of permutations with locality, that is, any symbol of the permutation can be computed from a small set of other symbols. In the latter approach, a permutation may be coded in order to achieve locality. Both approaches must present low query complexity to allow the user to find an element efficiently. We discuss both approaches, and give a particular focus to the combinatorial one. In the combinatorial approach, we provide upper and lower bounds for the maximal size of a set of permutations with locality, and provide several simple constructions which attain the upper bound. In cases where the upper bound is not attained, we provide alternative constructions using a variety of tools, such as Reed-Solomon codes, permutation polynomials, and multi-permutations. In addition, several low-rate constructions of particular interest are discussed. In the coding approach we discuss an alternative representation of permutations, present a paradigm for supporting arbitrary powers of the stored permutation, and conclude with a proof of concept that permutations may be stored more efficiently than ordinary strings over the same alphabet.     

A note on the total number of cycles of even and odd permutations

We prove bijectively that the total number of cycles of all even permutations of [n]={1,2,…,n} and the total number of cycles of all odd permutations of [n] differ by (−1)ⁿ(n−2)!, which was stated as an open problem by Miklós Bóna. We also prove bijectively the following more general identity:

     

Random and algebraic permutations’ statistics

An algebraic permutation $\hat{A}\in S(N=n^{m})$ is the permutation of the N points of the finite torus ? _n ^m , realized by a linear operator A∈SL(m,? _n ). The statistical properties of algebraic permutations are quite different from those of random permutations of N points. For instance, the period length T(A) grows superexponentially with N for some (random) permutations A of N elements, whereas $T(\hat{A})$ is bounded by a power of N for algebraic permutations  $\hat{A}$ . The paper also contains a strange mean asymptotics formula for the number of points of the finite projective line P¹(? _n ) in terms of the zeta function.     

Combinatorial families enumerated by quasi-polynomials

We say that the sequence (an) is quasi-polynomial in n if there exist polynomials P₀,…,Ps−1 and an integer n₀ such that, for all n?n₀, an=Pi(n) where . We present several families of combinatorial objects with the following properties: Each family of objects depends on two or more parameters, and the number of isomorphism types of objects is quasi-polynomial in one of the parameters whenever the values of the remaining parameters are fixed to arbitrary constants. For each family we are able to translate the problem of counting isomorphism types of objects into the problem of counting integer points in a union of parametrized rational polytopes. The families of objects to which this approach is applicable include combinatorial designs, linear and unrestricted codes, and dissections of regular polygons.     

On the dual canonical and Kazhdan-Lusztig bases and 3412-, 4231-avoiding permutations

Using Du’s characterization of the dual canonical basis of the coordinate ring O(GL(n,C)), we express all elements of this basis in terms of immanants. We then give a new factorization of permutations w avoiding the patterns 3412 and 4231, which in turn yields a factorization of the corresponding Kazhdan-Lusztig basis elements of the Hecke algebra H_n(q). Using the immanant and factorization results, we show that for every totally nonnegative immanant and its expansion with respect to the basis of Kazhdan-Lusztig immanants, the coefficient d_w must be nonnegative when w avoids the patterns 3412 and 4231.     

Long zero-free sequences in finite cyclic groups

A sequence in an additively written abelian group is called zero-free if each of its nonempty subsequences has sum different from the zero element of the group. The article determines the structure of the zero-free sequences with lengths greater than n/2 in the additive group Z_n of integers modulo n. The main result states that for each zero-free sequence  of length ?>n/2 in Z_n there is an integer g coprime to n such that if denotes the least positive integer in the congruence class ga_i (modulo n), then . The answers to a number of frequently asked zero-sum questions for cyclic groups follow as immediate consequences. Among other applications, best possible lower bounds are established for the maximum multiplicity of a term in a zero-free sequence with length greater than n/2, as well as for the maximum multiplicity of a generator. The approach is combinatorial and does not appeal to previously known nontrivial facts.