Let $A_n\subseteq S_n$ denote the alternating and the symmetric groups on $1,...,n$. MacMahaon's theorem, about the equi-distribution of the length and the major indices in $S_n$, has received far reaching refinements and generalizations, by Foata, Carlitz, Foata-Schutzenberger, Garsia-Gessel and followers. Our main goal is to find analogous statistics and identities for the alternating group $A_{n}$. A new statistic for $S_n$, {\it the delent number}, is introduced. This new statistic is involved with new $S_n$ equi-distribution identities, refining some of the results of Foata-Schutzenberger and Garsia-Gessel. By a certain covering map $f:A_{n+1}\to S_n$, such $S_n$ identities are `lifted' to $A_{n+1}$, yielding the corresponding $A_{n+1}$ equi-distribution identities.
This survey of alternating permutations and Euler numbers includes refinements of Euler numbers, other occurrences of Euler numbers, longest alternating subsequences, umbral enumeration of classes of alternating permutations, and the cd-index of the symmetric group.
We prove generalized versions of some conjectures of Joel Lewis on the number of alternating permutations avoiding certain patterns. Our main tool is the perhaps surprising observation that a classic bijection on pattern avoiding permutations often preserves the alternating property.
We construct a refined bijection $ϕ$ between alternating permutations and 0-1-2 increasing trees with degree at most 2. It satisfies that the first element of alternating permutation $π$ is equal to the first vertex in $ϕ(π)$ in the postorder.
A conjecture by R. Stanley on a class of alternating permutations, which is proved by R. Chapman and L. Williams states that alternating permutations with the maximal number of fixed points is equidistributed with derangements. We extend this (type $A$) result to type $B$: We prove that various classes of alternating signed permutations with the maximal number of fixed points is equidistributed with certain types of derangements (of type $B$), respectively.
Using the correspondence between a cycle up-down permutation and a pair of matchings, we give a combinatorial proof of the enumeration of alternating permutations according to the given peak set.
Guibert and Linusson introduced the family of doubly alternating Baxter permutations, i.e. Baxter permutations $σ\in S_n$, such that $σ$ and $σ^{-1}$ are alternating. They proved that the number of such permutations in $S_{2n}$ and $S_{2n+1}$ is the Catalan number $C_n$. In this paper we explore the expected limit shape of such permutations, following the approach by Miner and Pak.
This paper is continuation of the systematic study of distribution of quadrant marked mesh patterns. We study quadrant marked mesh patterns on up-down and down-up permutations, also known as alternating and reverse alternating permutations, respectively. In particular, we refine classic enumeration results of André on alternating permutations by showing that the distribution of the quadrant marked mesh pattern of interest is given by $(\sec(xt))^{1/x}$ on up-down permutations of even length and by $\int_0^t (\sec(xz))^{1+\frac{1}{x}}dz$ on down-up permutations of odd length.
In this paper, we give bijections between the set of 4123-avoiding down-up alternating permutations of length $2n$ and the set of standard Young tableaux of shape $(n,n,n)$, and between the set of 4123-avoiding down-up alternating permutations of length $2n-1$ and the set of shifted standard Young tableaux of shape $(n+1, n, n-1)$ via an intermediate structure of Yamanouchi words. Moreover, we get the enumeration of 4123-avoiding up-down alternating permutations of even and odd length by presenting bijections between 4123-avoiding up-down alternating permutations and standard Young tableaux.
Let R(n,k) denote the number of permutations of {1,2,...,n} with k alternating runs. We find a grammatical description of the numbers R(n,k) and then present several convolution formulas involving the generating function for the numbers R(n,k). Moreover, we establish a connection between alternating runs and Andre permutations.
The finite difference equation system introduced by Christiane Poupard in the study of tangent trees is reinterpreted in the alternating permutation environment. It makes it possible to make a joint study of both tangent and secant trees and calculate the generating polynomial for alternating permutations by a new statistic, referred to as being the greater neighbor of the maximum.
In this paper, we first consider a generalization of the David-Barton identity which relate the alternating run polynomials to Eulerian polynomials. By using context-free grammars, we then present a combinatorial interpretation of a family of q-alternating run polynomials. Furthermore, we introduce the definition of semi-gamma-positive polynomial and we show the semi-gamma-positivity of the alternating run polynomials of dual Stirling permutations. A connection between the up-down run polynomials of permutations and the alternating run polynomials of dual Stirling permutations is established.
We extend earlier work of the same author to enumerate alternating permutations avoiding the permutation pattern 2143. We use a generating tree approach to construct a recursive bijection between the set A_{2n}(2143) of alternating permutations of length 2n avoiding 2143 and standard Young tableaux of shape (n, n, n) and between the set A_{2n + 1}(2143) of alternating permutations of length 2n + 1 avoiding 2143 and shifted standard Young tableaux of shape (n + 2, n + 1, n). We also give a number of conjectures and open questions on pattern avoidance in alternating permutations and generalizati…
A Parity Alternating Permutation of the set $[n] = \{1, 2,\ldots, n\}$ is a permutation with even and odd entries alternatively. We deal with parity alternating permutations having an odd entry in the first position, PAPs. We study the numbers that count the PAPs with even as well as odd parity. We also study a subclass of PAPs being derangements as well, Parity Alternating Derangements (PADs). Moreover, by considering the parity of these PADs we look into their statistical property of excedance.
We give two simple proofs of a conjecture of Richard Stanley concerning the equidistribution of derangements and alternating permutations with the maximal number of fixed points.
Let $R(n,k)$ denote the number of permutations of ${1,2,...,n}$ with $k$ alternating runs. In this note we present an explicit formula for the numbers $R(n,k)$.
We study the problem of counting alternating permutations avoiding collections of permutation patterns including 132. We construct a bijection between the set S_n(132) of 132-avoiding permutations and the set A_{2n + 1}(132) of alternating, 132-avoiding permutations. For every set p_1, ..., p_k of patterns and certain related patterns q_1, ..., q_k, our bijection restricts to a bijection between S_n(132, p_1, ..., p_k), the set of permutations avoiding 132 and the p_i, and A_{2n + 1}(132, q_1, ..., q_k), the set of alternating permutations avoiding 132 and the q_i. This reduces the enumeration…
In this paper, we take interest in finding applications for a hook-length formula recently proved in (Morales Pak Panova 2016). This formula can be applied to give a non trivial relation between alternating permutations and weighted Dyck paths. First, we give an alternative proof for this result using continued fractions, and then we apply a similar reasoning to the more general case of $k$-alternating permutations.
The combinatorial theory for the set of parity alternating permutations is expounded. In view of the numbers of ascents and inversions, several enumerative aspects of the set are investigated. In particular, it is shown that signed Eulerian numbers have intimate relationships to the set.
In this paper we present an explicit formula for the number of permutations with a given number of alternating descents. Moreover, we study the interlacing property of the real parts of the zeros of the generating polynomials of these numbers.
An alternating permutation of length $n$ is a permutation $π=π_1 π_2 ... π_n$ such that $π_1 < π_2 > π_3 < π_4 > ...$. Let $A_n$ denote set of alternating permutations of ${1,2,..., n}$, and let $A_n(σ)$ be set of alternating permutations in $A_n$ that avoid a pattern $σ$. Recently, Lewis used generating trees to enumerate $A_{2n}(1234)$, $A_{2n}(2143)$ and $A_{2n+1}(2143)$, and he posed several conjectures on the Wilf-equivalence of alternating permutations avoiding certain patterns. Some of these conjectures have been proved by Bóna, Xu and Yan. In this paper, we prove the two relations $|A_…
In this paper, we count a dual set of Stirling permutations by the number of alternating runs. Properties of the generating functions, including recurrence relations, grammatical interpretations and convolution formulas are studied.
A permutation is said to be \emph{alternating} if it starts with rise and then descents and rises come in turn. In this paper we study the generating function for the number of alternating permutations on $n$ letters that avoid or contain exactly once 132 and also avoid or contain exactly once an arbitrary pattern on $k$ letters. In several interesting cases the generating function depends only on $k$ and is expressed via Chebyshev polynomials of the second kind.
We present a short, direct proof of the fact that the generating function of all permutations of a fixed length $n\geq 4$ is divisible by $(1+z)^m$, where $m=\lfloor (n-2)/2 \rfloor$.
Let R(n,k) be the number of permutations of $\{1,2,\ldots,n\}$ with k alternating runs. In this paper, we establish the relationships between R(n,k) and the central factorial numbers of even indices as well as the number of signed permutations with a given number of alternating runs and the central factorial numbers of odd indices. The explicit formulas of the peak and left peak polynomials for permutations and the derivative polynomials of the tangent and secant functions are also established.
We investigate pattern avoidance in alternating permutations and generalizations thereof. First, we study pattern avoidance in an alternating analogue of Young diagrams. In particular, we extend Babson-West's notion of shape-Wilf equivalence to apply to alternating permutations and so generalize results of Backelin-West-Xin and Ouchterlony to alternating permutations. Second, we study pattern avoidance in the more general context of permutations with restricted ascents and descents. We consider a question of Lewis regarding permutations that are the reading words of thickened staircase Young t…
At a crossroads of calculus and combinatorics, the generating function of secant and tangent numbers (Euler numbers) provides enumeration of alternating permutations. In this article, we present a new refinement of Euler numbers to answer the combinatorial question on some particular relation of Euler numbers proved by Heneghan-Petersen, Power series for up-down min-max permutations, College Math. Journal, Vol. 45, No. 2 (2014), 83-91.
The length is(w) of the longest increasing subsequence of a permutation w in the symmetric group S_n has been the object of much investigation. We develop comparable results for the length as(w) of the longest alternating subsequence of w, where a sequence a,b,c,d,... is alternating if a>b<c>d<.... For instance, the expected value (mean) of as(w) for w in S_n is exactly (4n+1)/6 if n>1.
We find the exponential generating function for permutations with all valleys even and all peaks odd, and use it to determine the asymptotics for its coefficients, answering a question posed by Liviu Nicolaescu. The generating function can be expressed as the reciprocal of a sum involving Euler numbers. We give two proofs of the formula. The first uses a system of differential equations. The second proof derives the generating function directly from general permutation enumeration techniques, using noncommutative symmetric functions. The generating function is an "alternating" analogue of Davi…
This paper is continuation of the systematic study of distribution of quadrant marked mesh patterns initiated in "S. Kitaev and J. Remmel, Quadrant marked mesh patterns, J. Integer Sequences 12, Issue 4 (2012), Article 12.4.7.". We study quadrant marked mesh patterns on up-down and down-up permutations.
Inspired by a recent note of Zeilberger (arXiv:1110.4379), Alejandro Morales asked whether one can count alternating (i.e., up-down) permutations that contain the pattern 123 or 321 exactly once. In this note we answer the question in the affirmative; in particular, we show that for m > 1, a_(2m)(123) = 10 (2m)!/((m - 2)! (m + 3)!), a_(2m)(321) = 4(m - 2) (2m + 3)!/((m + 1)! (m + 4)!), and a_(2m + 1)(123) = a_(2m + 1)(321) = 3(3m + 4)(m - 1) (2m + 2)!/((m + 1)! (m + 4)!) where a_n(p) is the number of alternating permutations of length n containing the pattern p exactly once.
A classical result of Euler states that the tangent numbers are an alternating sum of Eulerian numbers. A dual result of Roselle states that the secant numbers can be obtained by a signed enumeration of derangements. We show that both identities can be refined with the following statistics: the number of crossings in permutations and derangements, and the number of patterns 31-2 in alternating permutations. Using previous results of Corteel, Rubey, Prellberg, and the author, we derive closed formulas for both q-tangent and q-secant numbers. There are two different methods to obtain these formu…
Alternating sign triangles were introduced by Carroll and Speyer in relation to cube recurrence, by analogy to alternating sign matrices for octahedron recurrence. Permutation triangles are the alternating sign triangles whose entries are either 0 or 1, by analogy with permutation matrices. In this paper, we prove a simple characterization of permutation triangles, originally conjectured by Glick. We will also prove some properties of alternating sign triangles.
Springer numbers are an analog of Euler numbers for the group of signed permutations. Arnol'd showed that they count some objects called snakes, that generalize alternating permutations. Hoffman established a link between Springer numbers, snakes, and some polynomials related with the successive derivatives of trigonometric functions. The goal of this article is to give further combinatorial properties of derivative polynomials, in terms of snakes and other objects: cycle-alternating permutations, weighted Dyck or Motzkin paths, increasing trees and forests. We obtain the generating functions,…
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…
We define a class L_{n, k} of permutations that generalizes alternating (up-down) permutations and give bijective proofs of certain pattern-avoidance results for this class. As a special case of our results, we give two bijections between the set A_{2n}(1234) of alternating permutations of length 2n with no four-term increasing subsequence and standard Young tableaux of shape (3^n), and between the set A_{2n + 1}(1234) and standard Young tableaux of shape (3^{n - 1}, 2, 1). This represents the first enumeration of alternating permutations avoiding a pattern of length four. We also extend previ…
We consider the structure of roller coaster permutations as introduced by Ahmed & Snevily[1]. A roller coaster permutation is described as a permuta- tion that maximizes the total switches from ascending to descending or visa versa for the permutation and all of its subpermutations simultaneously. This paper looks at the alternating structure of these permutations and then we introduce a notion of a condition stronger than alternating for a permutation that we shall refer to as recursively alternating. We also examine the behav- ior of what entries can show up in even, odd, and end positions w…
We define and study a new family of polytopes which are formed as convex hulls of partial alternating sign matrices. We determine the inequality descriptions, number of facets, and face lattices of these polytopes. We also study partial permutohedra that we show arise naturally as projections of these polytopes. We enumerate facets and also characterize the face lattices of partial permutohedra in terms of chains in the Boolean lattice. Finally, we have a result and a conjecture on the volume of partial permutohedra when one parameter is fixed to be two.
We investigate alternating sign matrices that are not permutation matrices, but have finite order in a general linear group. We classify all such examples of the form $P+T$, where $P$ is a permutation matrix and $T$ has four non-zero entries, forming a square with entries $1$ and $-1$ in each row and column. We show that the multiplicative orders of these matrices do not always coincide with those of permutation matrices of the same size. We pose the problem of identifying finite subgroups of general linear groups that are generated by alternating sign matrices.
We study new statistics on permutations that are variations on the descent and the inversion statistics. In particular, we consider the alternating descent set of a permutation sigma = sigma_1sigma_2...sigma_n defined as the set of indices i such that either i is odd and sigma_i > sigma_{i+1}, or i is even and sigma_i < sigma_{i+1}. We show that this statistic is equidistributed with the 3-descent set statistic on permutations sigma = sigma_1sigma_2...sigma_{n+1} with sigma_1 = 1, defined to be the set of indices i such that the triple sigma_i sigma_{i+1} sigma_{i+2} forms an odd permutation o…
We extend Stanley's work on alternating permutations with extremal number of fixed points in two directions: first, alternating permutations are replaced by permutations with a prescribed descent set; second, instead of simply counting permutations we study their generating polynomials by number of excedances. Several techniques are used: Desarmenien's desarrangement combinatorics, Gessel's hook-factorization and the analytical properties of two new permutation statistics "DEZ" and "lec". Explicit formulas for the maximal case are derived by using symmetric function tools.
The group of alternating colored permutations is the natural analogue of the classical alternating group, inside the wreath product $\mathbb{Z}_r \wr S_n$. We present a 'Coxeter-like' presentation for this group and compute the length function with respect to that presentation. Then, we present this group as a covering of $\mathbb{Z}_{\frac{r}{2}} \wr S_n$ and use this point of view to give another expression for the length function. We also use this covering to lift several known parameters of $\mathbb{Z}_{\frac{r}{2}} \wr S_n$ to the group of alternating colored permutations.
We introduce a new directed graph structure into the set of alternating sign matrices. This includes Bruhat graph (Bruhat order) of the symmetric groups as a subgraph (subposet). Drake-Gerrish-Skandera (2004, 2006) gave characterizations of Bruhat order in terms of total nonnegativity (TNN) and subtraction-free Laurent (SFL) expressions for permutation monomials. With our directed graph, we extend their idea in two ways: first, from permutations to alternating sign matrices; second, $q$-analogs (which we name $q$TNN and $q$SFL properties). %In our discussion, essential sets, introduced by Fult…
The alternating-runs polynomial enumerates alternating runs in the symmetric group. There are three formulae for the number of permutations, $R_{n,k}$ in $\mathfrak{S}_n$ with $k$ alternating runs, but all of them are complicated. We show that when enumerated with sign taken into account, one gets a {\it neat formula}. As a consequence, we get a near refinement of a result of Wilf on the exponent of $(1+t)$ when it divides the alternating-runs polynomial in the alternating group $\mathcal{A}_n$. Other applications include a moment-type identity and enumeration of alternating permutations in $\…
Two permutations $(x_1,\dots,x_w)$ and $(y_1,\dots,y_w)$ are weakly similar if $x_i<x_{i+1}$ if and only if $y_i<y_{i+1}$ for all $1\leqslant i \leqslant w$. Let $π$ be a permutation of the set $[n]=\{1,2,\dots, n\}$ and let $wt(π)$ denote the largest integer $w$ such that $π$ contains a pair of disjoint weakly similar sub-permutations (called weak twins) of length $w$. Finally, let $wt(n)$ denote the minimum of $wt(π)$ over all permutations $π$ of $[n]$. Clearly, $wt(n)\le n/2$. In this paper we show that $\tfrac n{12}\le wt(n)\le\tfrac n2-Ω(n^{1/3})$. We also study a variant of this problem.…
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.
André proved that the number of alternating permutations on $\{1, 2, \dots, n\}$ is equal to the Euler number $E_n$. A refinement of André's result was given by Entringer, who proved that counting alternating permutations according to the first element gives rise to Seidel's triangle $(E_{n,k})$ for computing the Euler numbers. In a series of papers, using generating function method and induction, Poupard gave several further combinatorial interpretations for $E_{n,k}$ both in alternating permutations and increasing trees. Kuznetsov, Pak, and Postnikov have given more combinatorial interpretat…
We study the symmetric function and polynomial combinatorial invariants of Hopf algebras of permutations, posets and graphs. We investigate their properties and the relations among them. In particular, we show that the chromatic symmetric function and many other invariants have a property we call positively $h$-alternating. This property of positively $h$-alternating leads to Schur positivity and $e$-positivity when applying the operator $\nabla$ at $q=1$. We conclude by showing that the invariants we consider can be expressed as scheduling problems.
The Euler number $E_n$ (resp. Entringer number $E_{n,k}$) enumerates the alternating (down-up) permutations of $\{1,\dots,n\}$ (resp. starting with $k$). The Springer number $S_n$ (resp. Arnold number $S_{n,k}$) enumerates the type $B$ alternating permutations (resp. starting with $k$). In this paper, using bijections we first derive the counterparts in {\em André permutations} and {\em Simsun permutations} for the Entringer numbers $(E_{n,k})$, and then the counterparts in {\em signed André permutations} and {\em type $B$ increasing 1-2 trees} for the Arnold numbers $(S_{n,k})$.
In 2005 J.L. Waldspurger proved the following theorem: given a finite real reflection group $W$, the closed positive root cone is tiled by the images of the open weight cone under the action of the linear transformations $id-w$. Shortly thereafter E. Meinrencken extended the result to affine Weyl groups. P.V. Bibikov and V.S. Zhgoon then gave a uniform proof for a discrete reflection group acting on a simply-connected space of constant curvature. In this paper we show that the Waldspurger and Meinrenken theorems of type A give a new perspective on the combinatorics of the symmetric group. In p…
