Permutations In Sn Research Articles

Pinnacle sets revisited

In 2017, Davis, Nelson, Petersen, and Tenner (Davis et al. (2018) [1]) initiated the combinatorics of pinnacles in permutations. We provide a simple and efficient recursion to compute pn(S), the number of permutations in Sn with pinnacle set S, and a conjectural closed formula for the related numbers qn(S). This conjecture was proved between the first draft of this paper and its final version by Fang (2022) [4] and later by Minnich [7]. We also determine the lexicographically minimal elements of the orbits of the modified Foata-Strehl action, prove that these elements form a lower ideal of the left weak order and characterize and count the maximal elements of this ideal.

Pinnacle set properties

Let π=π1π2…πn be a permutation in the symmetric group Sn written in one-line notation. The pinnacle set of π, denoted Pin π, is the set of all πi such that πi−1<πi>πi+1. This is an analogue of the well-studied peak set of π where one considers values rather than positions. The pinnacle set was so named by Davis, Nelson, Petersen, and Tenner who showed that it has many interesting properties. In particular, they proved that the number of subsets of [n]={1,2,…,n} which can be the pinnacle set of some permutation is a binomial coefficient. Their proof involved a bijection with lattice paths and was somewhat involved. We give a simpler demonstration of this result which does not need lattice paths. Moreover, we show that our map and theirs are different descriptions of the same function. Davis et al. also studied the number of pinnacle sets with maximum m and cardinality d which they denoted by p(m,d). We show that these integers are ballot numbers and give two proofs of this fact: one using finite differences and one bijective. Diaz-Lopez, Harris, Huang, Insko, and Nilsen found a summation formula for calculating the number of permutations in Sn having a given pinnacle set. We derive a new expression for this number which is faster to calculate in many cases. We also show how this method can be adapted to find the number of orderings of a pinnacle set which can be realized by some π∈Sn.

The infinite limit of separable permutations

AbstractLet denote the uniform probability measure on the set of separable permutations in Sn. Let with an appropriate metric and denote by the compact metric space consisting of functions from to which are injections when restricted to ; that is, if , i ≠ j, then . Extending permutations by defining , for j &gt; n, we have . We show that converges weakly on to a limiting distribution of regenerative type, which we calculate explicitly.

A formula for enumerating permutations with a fixed pinnacle set

In 2017 Davis, Nelson, Petersen, and Tenner pioneered the study of pinnacle sets of permutations and asked whether there exists a class of operations, which applied to a permutation in Sn, can produce any other permutation with the same pinnacle set and no others. In this paper, we adapt a group action defined by Foata and Strehl to provide a way to generate all permutations with a given pinnacle set. From this we give an answer to a second question asked by Davis, Nelsen, Peterson, and Tenner, which asks for a closed non-recursive formula enumerating permutations with a given pinnacle set.

Stack-sorting with consecutive-pattern-avoiding stacks

We introduce consecutive-pattern-avoiding stack-sorting maps SCσ, which are natural generalizations of West's stack-sorting map s and natural analogues of the classical-pattern-avoiding stack-sorting maps sσ recently introduced by Cerbai, Claesson, and Ferrari. We characterize the patterns σ such that Sort(SCσ), the set of permutations that are sortable via the map s∘SCσ, is a permutation class, and we enumerate the sets Sort(SCσ) for σ∈{123,132,321}. We also study the maps SCσ from a dynamical point of view, characterizing the periodic points of SCσ for all σ∈S3 and computing maxπ∈Sn⁡|SCσ−1(π)| for all σ∈{132,213,231,312}. In addition, we characterize the periodic points of the classical-pattern-avoiding stack-sorting map s132, and we show that the maximum number of iterations of s132 needed to send a permutation in Sn to a periodic point is n−1. The paper ends with numerous open problems and conjectures.

Quantum query complexity of symmetric oracle problems

We study the query complexity of quantum learning problems in which the oracles form a group G of unitary matrices. In the simplest case, one wishes to identify the oracle, and we find a description of the optimal success probability of a t-query quantum algorithm in terms of group characters. As an application, we show that Ω(n) queries are required to identify a random permutation in Sn. More generally, suppose H is a fixed subgroup of the group G of oracles, and given access to an oracle sampled uniformly from G, we want to learn which coset of H the oracle belongs to. We call this problem coset identification and it generalizes a number of well-known quantum algorithms including the Bernstein-Vazirani problem, the van Dam problem and finite field polynomial interpolation. We provide character-theoretic formulas for the optimal success probability achieved by a t-query algorithm for this problem. One application involves the Heisenberg group and provides a family of problems depending on n which require n+1 queries classically and only 1 query quantumly.

Fertility monotonicity and average complexity of the stack-sorting map

Let Dn denote the average number of iterations of West’s stack-sorting map s that are needed to sort a permutation in Sn into the identity permutation 123⋯n. We prove that 0.62433≈λ≤lim infn→∞Dnn≤lim supn→∞Dnn≤35(7−8log2)≈0.87289, where λ is the Golomb–Dickman constant. Our lower bound improves upon West’s lower bound of 0.23, and our upper bound is the first improvement upon the trivial upper bound of 1. We then show that fertilities of permutations increase monotonically upon iterations of s. More precisely, we prove that |s−1(σ)|≤|s−1(s(σ))| for all σ∈Sn, where equality holds if and only if σ=123⋯n. This is the first theorem that manifests a law-of-diminishing-returns philosophy for the stack-sorting map that Bóna has proposed. Along the way, we note some connections between the stack-sorting map and the right and left weak orders on Sn.

Counting 3-stack-sortable permutations

We prove a “decomposition lemma” that allows us to count preimages of certain sets of permutations under West's stack-sorting map s. As a first application, we give a new proof of Zeilberger's formula for the number W2(n) of 2-stack-sortable permutations in Sn. Our proof generalizes, allowing us to find an algebraic equation satisfied by the generating function that counts 2-stack-sortable permutations according to length, number of descents, and number of peaks. This is also the first proof of this formula that generalizes to the setting of 3-stack-sortable permutations. Indeed, the same method allows us to obtain a recurrence relation for W3(n), the number of 3-stack-sortable permutations in Sn. Hence, we obtain the first polynomial-time algorithm for computing these numbers. We compute W3(n) for n≤174, vastly extending the 13 terms of this sequence that were known before. We also prove the first nontrivial lower bound for limn→∞⁡W3(n)1/n, showing that it is at least 8.659702. Invoking a result of Kremer, we also prove that limn→∞⁡Wt(n)1/n≥(t+1)2 for all t≥1, which we use to improve a result of Smith concerning a variant of the stack-sorting procedure. Our computations allow us to disprove a conjecture of Bóna, although we do not yet know for sure which one.In fact, we can refine our methods to obtain a recurrence for W3(n,k,p), the number of 3-stack-sortable permutations in Sn with k descents and p peaks. This allows us to gain a large amount of evidence supporting a real-rootedness conjecture of Bóna. Using part of the theory of valid hook configurations, we give a new proof of a γ-nonnegativity result of Brändén, which in turn implies an older result of Bóna. We then answer a question of the current author by producing a set A⊆S11 such that ∑σ∈s−1(A)xdes(σ) has nonreal roots. We interpret this as partial evidence against the same real-rootedness conjecture of Bóna that we found evidence supporting. Examining the parities of the numbers W3(n), we obtain strong evidence against yet another conjecture of Bóna. We end with some conjectures of our own.

Signatures of representations of Hecke algebras and rational Cherednik algebras

Determining whether an irreducible representation of a group (or ⁎-algebra) admits a non-degenerate invariant, positive-definite Hermitian form is an important problem in representation theory. In this paper, we study the related notion of signatures. Let q∈C with |q|=1, κ=−1/c with c∈Q<0 with certain additional (mild) restrictions, and n∈Z≥2. We study representations Sλ(q) of Hn(q), the Hecke algebra of type An−1, and representations Mκ(λ) of Hκ, the rational Cherednik algebra of type An−1, which have unique (up to scaling) invariant Hermitian forms (here λ is a partition of n). The signature is the number of elements with positive norm minus the number of elements with negative norm, and we analogously define the signature character in the case that there is a natural grading on the module. We provide formulas for (1) signatures of modules over Hn(q) and (2) signature characters of modules over Hκ. We study the limit c→−∞, in which case the signature character has a simpler form in terms of inversions and descents of permutations in Sn. We provide examples corresponding to some special shapes, and small values of n. Finally, when q=e2πic, we show that the asymptotic signature character of the Hκ-module Mκ(τ) is the signature of the Hn(q)-module Sτ(q).

Constraining strong c-Wilf equivalence using cluster poset asymptotics

Let π∈Sm and σ∈Sn be permutations. An occurrence of π in σ as a consecutive pattern is a subsequence σiσi+1⋯σi+m−1 of σ with the same order relations as π. We say that patterns π,τ∈Sm are strongly c-Wilf equivalent if for all n and k, the number of permutations in Sn with exactly k occurrences of π as a consecutive pattern is the same as for τ. In 2018, Dwyer and Elizalde [6] conjectured (generalizing a conjecture of Elizalde [8] from 2012) that if π,τ∈Sm are strongly c-Wilf equivalent, then (τ1,τm) is equal to one of (π1,πm), (πm,π1), (m+1−π1,m+1−πm), or (m+1−πm,m+1−π1). We prove this conjecture using the cluster method introduced by Goulden and Jackson in 1979 [12], which Dwyer and Elizalde used to prove that |π1−πm|=|τ1−τm|. A consequence of our result is the full classification of c-Wilf equivalence for a special class of permutations, the non-overlapping permutations. Our approach uses analytic methods to approximate the number of linear extensions of the “cluster posets” defined by Elizalde and Noy in 2012 [11].

Sorting permutations with transpositions in O(n3) amortized time

The problem of sorting a permutation π over alphabet ∑={1,2,3…n} with the minimum number of transpositions is known to be intractable. That is, the computation of transposition distance i.e. dt(π), is hard. Such sorting is equivalent to determining the distance between any α, β of the symmetric group Sn over ∑ under the transposition operation. It has applications in computer interconnection networks where a node is denoted by a permutation, and genetics where a genome is modeled by a permutation. The maximum distance between any pair of permutations in Sn is the diameter i.e. diam(Γ(Sn)), of the corresponding Cayley graph Γ(Sn). It corresponds to the maximum latency in the computer interconnection network modeled by Γ(Sn) and the maximum genetic dissimilarity between any pair α, β in Sn. We call transpositions, prefix transpositions, suffix transpositions and prefix/suffix transpositions as block-moves. We design an algorithm to compute dt(π) ∀π in Sn and compute diam(Γ(Sn)) with transpositions in O(n!n3) time and O(n!n2) space; at an amortized time of O(n3). The same can be computed for the remaining block moves in O(n2log⁡n) amortized time and O(n!n) space.

Pattern-avoiding polytopes

Two well-known polytopes whose vertices are indexed by permutations in the symmetric group Sn are the permutohedron Pn and the Birkhoff polytope Bn. We consider polytopes Pn(Π) and Bn(Π), whose vertices correspond to the permutations in Sn 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 Pn(Π), we consider all subsets Π⊆S3 and are able to provide results in most cases. To illustrate, Pn(123,132) is a Pitman–Stanley polytope, the number of interior lattice points in Pn(132,312) is a derangement number, and the normalized volume of Pn(123,231,312) is the number of trees on n vertices.The polytopes Bn(Π) seem much more difficult to analyze, so we focus on four particular choices of Π. First we show that the Bn(231,321) is exactly the Chan–Robbins–Yuen polytope. Next we prove that for any Π containing {123,312} we have h∗(Bn(Π))=1. Finally, we study Bn(132,312) and B˜n(123), where the tilde indicates that we choose vertices corresponding to alternating permutations avoiding the pattern 123. In both cases we use order complexes of posets and techniques from toric algebra to construct regular, unimodular triangulations of the polytopes. The posets involved turn out to be isomorphic to the lattices of Young diagrams contained in a certain shape, and this permits us to give an exact expression for the normalized volumes of the corresponding polytopes via the hook formula. Finally, Stanley’s theory of (P,ω)-partitions allows us to show that their h∗-vectors are symmetric and unimodal.Various questions and conjectures are presented throughout.

Wilf equivalence relations for consecutive patterns

Two permutations π and τ are c-Wilf equivalent if, for each n, the number of permutations in Sn avoiding π as a consecutive pattern (i.e., in adjacent positions) is the same as the number of those avoiding τ. In addition, π and τ are strongly c-Wilf equivalent if, for each n and k, the number of permutations in Sn containing k occurrences of π as a consecutive pattern is the same as for τ. In this paper we introduce a third, more restrictive equivalence relation, defining π and τ to be super-strongly c-Wilf equivalent if the above condition holds for any set of prescribed positions for the k occurrences. We show that, when restricted to non-overlapping permutations, these three equivalence relations coincide.We also give a necessary condition for two permutations to be strongly c-Wilf equivalent. Specifically, we show that if π,τ∈Sm are strongly c-Wilf equivalent, then |πm−π1|=|τm−τ1|. In the special case of non-overlapping permutations π and τ, this proves a weaker version of a conjecture of the second author stating that π and τ are c-Wilf equivalent if and only if π1=τ1 and πm=τm, up to trivial symmetries. Finally, we strengthen a recent result of Nakamura and Khoroshkin–Shapiro giving sufficient conditions for strong c-Wilf equivalence.

Ordered set partitions, generalized coinvariant algebras, and the Delta Conjecture

The symmetric group Sn acts on the polynomial ring Q[xn]=Q[x1,…,xn] by variable permutation. The invariant ideal In is the ideal generated by all Sn-invariant polynomials with vanishing constant term. The quotient Rn=Q[xn]In is called the coinvariant algebra. The coinvariant algebra Rn has received a great deal of study in algebraic and geometric combinatorics. We introduce a generalization In,k⊆Q[xn] of the ideal In indexed by two positive integers k≤n. The corresponding quotient Rn,k:=Q[xn]In,k carries a graded action of Sn and specializes to Rn when k=n. We generalize many of the nice properties of Rn to Rn,k. In particular, we describe the Hilbert series of Rn,k, give extensions of the Artin and Garsia–Stanton monomial bases of Rn to Rn,k, determine the reduced Gröbner basis for In,k with respect to the lexicographic monomial order, and describe the graded Frobenius series of Rn,k. Just as the combinatorics of Rn are controlled by permutations in Sn, we will show that the combinatorics of Rn,k are controlled by ordered set partitions of {1,2,…,n} with k blocks. The Delta Conjecture of Haglund, Remmel, and Wilson is a generalization of the Shuffle Conjecture in the theory of diagonal coinvariants. We will show that the graded Frobenius series of Rn,k is (up to a minor twist) the t=0 specialization of the combinatorial side of the Delta Conjecture. It remains an open problem to give a bigraded Sn-module Vn,k whose Frobenius image is even conjecturally equal to any of the expressions in the Delta Conjecture; our module Rn,k solves this problem in the specialization t=0.

Number of Right Ideals and a q-analogue of Indecomposable Permutations

AbstractWe prove that the number of right ideals of codimension n in the algebra of noncommutative Laurent polynomials in two variables over the finite field 𝔽q is equal to where the sum is over all indecomposable permutations in Sn+1 and where inv(θ) stands for the number of inversions of θ

On Sillke's bijection

Indecomposable permutations in Sn+1, subgroups of index n of the free group on two generators and doubly pointed hypermaps of cardinality n are equinumerous. We give here a proof of a bijection, due to Sillke, between these three sets.

A non-trivial intersection theorem for permutations with fixed number of cycles

Let Sn denote the set of permutations of [n]={1,2,…,n}. For a positive integer k, define Sn,k to be the set of all permutations of [n] with exactly k disjoint cycles, i.e., Sn,k={π∈Sn:π=c1c2⋯ck}, where c1,c2,…,ck are disjoint cycles. The size of Sn,k is given by [nk]=(−1)n−ks(n,k), where s(n,k) is the Stirling number of the first kind. A family A⊆Sn,k is said to be t-cycle-intersecting if any two elements of A have at least t common cycles. A family A⊆Sn,k is said to be trivially t-cycle-intersecting if A is the stabiliser of t fixed points, i.e., A consists of all permutations in Sn,k with some t fixed cycles of length one. For 1≤j≤t, let Qj={σ∈Sn,k:σ(i)=ifor alli∈[k]∖{j}}. For t+1≤s≤k, let Bs={σ∈Sn,k:σ(i)=ifor alli∈[t]∪{s}}. In this paper, we show that, given any positive integers k,t with k≥2t+3, there exists an n0=n0(k,t), such that for all n≥n0, if A⊆Sn,k is non-trivially t-cycle-intersecting, then |A|≤|B|, where B=⋃s=t+1kBs∪⋃j=1tQj. Furthermore, equality holds if and only if A is a conjugate of B, i.e., A=β−1Bβ for some β∈Sn.

Permutations weakly avoiding barred patterns and combinatorial bijections to generalized Dyck and Motzkin paths

We explore the weak avoidance of barred patterns in permutations introduced recently by J.-L. Baril. We give its general properties in comparison with the barred pattern avoidance. By showing explicit bijections, we revisit Dyck paths with no peak at height p, Dyck paths with no ud…du and Motzkin paths via weakly avoiding permutations in Sn(132).

A Refinement of Weak Order Intervals into Distributive Lattices

In this paper we consider arbitrary intervals in the left weak order on the symmetric group Sn. We show that the Lehmer codes of permutations in an interval form a distributive lattice under the product order. Furthermore, the rank-generating function of this distributive lattice matches that of the weak order interval. We construct a poset such that its lattice of order ideals is isomorphic to the lattice of Lehmer codes of permutations in the given interval. We show that there are at least \({\left(\lfloor {\frac{n}{2}} \rfloor \right)!}\) permutations in Sn that form a rank-symmetric interval in the weak order.

Enumeration of permutations by number of alternating runs

Let Sn denote the symmetric group of all permutations of {1,2,…,n}. In this paper, we consider the number R(n,k) of permutations in Sn with k alternating runs, and the number ak(n) of permutations in Sn with the longest alternating subsequence of length k. By using the grammatical method due to Chen, we obtain grammatical interpretations of the generating functions of these numbers, as well as convolution identities involving these generating functions. Moreover, we establish a connection between alternating runs and André permutations.