To each permutation $σ$ in $S_{n}$ we associate a triangulation of a fixed $(n+2)$-gon. We then determine the fibers of this association and show that they coincide with the sylvester classes depicted By Novelli, Hivert and Thibon. A signed version of this construction allows us to reformulate the four color theorem in terms of the existence of a signable path between any two permutations in the Cayley graph of the symmetric group $S_{n}.
Starting from the two-species partially asymmetric simple exclusion process, we study a subclass of signed permutations, the partially signed permutations, using the combinatorics of Laguerre histories. From this physical and bijective point of view, we obtain a natural descent statistic on partially signed permutations; as well as partially signed permutations patterns.
We show that the number of signed permutations avoiding 1234 equals the number of signed permutations avoiding 2143 (also called vexillary signed permutations), resolving a conjecture by Anderson and Fulton. The main tool that we use is the generating tree developed by West. Many further directions are mentioned in the end.
In this paper, we start by giving the definitions and basic facts about hyperoctahedral number system. There is a natural correspondence between the integers expressed in the latter and the elements of the hyperoctahedral group when we use the inversion statistic on this group to code the signed permutations. We use this correspondence to define a way with which the signed permutations group can be ordered. With this classification scheme, we can find the r-th signed permutation from a given number r and vice versa without consulting the list in lexicographical order of the elements of the sig…
In order to study signed Eulerian numbers, we introduce permutations of a particular type, called parity-alternate permutations, because they take even and odd entries alternately. The objective of this paper is twofold. The first is to derive several properties of those permutations, by subdividing them into even and odd ones. The second is to discuss relationships between those and signed Eulerian numbers. Divisibility properties by prime powers are also deduced for signed Eulerian numbers and several related numbers.
Theta-vexillary signed permutations are elements in the hyperoctahedral group that index certain classes of degeneracy loci of type B and C. These permutations are described using triples of $s$-tuples of integers subject to specific conditions. The objective of this work is to present different characterizations of theta-vexillary signed permutations, describing them in terms of corners in the Rothe diagram and pattern avoidance.
In this paper we study the generating polynomials obtained by enumerating signed simsun permutations by number of the descents. Properties of the polynomials, including the recurrence relations and generating functions are studied.
We give some interpretations to certain integer sequences in terms of parameters on Grand-Dyck paths and coloured noncrossing partitions, and we find some new bijections relating Grand-Dyck paths and signed pattern avoiding permutations. Next we transfer a natural distributive lattice structure on Grand-Dyck paths to coloured noncrossing partitions and signed pattern avoiding permutations, thus showing, in particular, that it is isomorphic to the structure induced by the (strong) Bruhat order on a certain set of signed pattern avoiding permutations.
MacMahon's classic theorem states that the 'length' and 'major index' statistics are equidistributed on the symmetric group S_n. By defining natural analogues or generalizations of those statistics, similar equidistribution results have been obtained for the alternating group A_n by Regev and Roichman, for the hyperoctahedral group B_n by Adin, Brenti and Roichman, and for the group of even-signed permutations D_n by Biagioli. We prove analogues of MacMahon's equidistribution theorem for the group of signed even permutations and for its subgroup of even-signed even permutations.
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.
In this paper, we construct bijections between Dyck paths, noncrossing partitions, and 231-avoiding permutations, which send the area statistic on Dyck paths to the inversion number on noncrossing partitions and on 231-avoiding permutations. This bijection has the additional property that it simultaneously sends the major index on Dyck paths to the sum of the major index and the inverse major index on noncrossing partitions and on 231-avoiding permutations, respectively. Moreover, we provide generalizations of these constructions to the group of signed permutations.
We study the generating polynomial of the flag major index with each one-dimensional character, called signed Mahonian polynomial, over the colored permutation group, the wreath product of a cyclic group with the symmetric group. Using the insertion lemma of Han and Haglund-Loehr-Remmel and a signed extension established by Eu et al., we derive the signed Mahonian polynomial over the quotients of parabolic subgroups of the colored permutation group, for a variety of systems of coset representatives in terms of subsequence restrictions. This generalizes the related work over parabolic quotients…
In this paper we present a topological framework for studying signed permutations and their reversal distance. As a result we can give an alternative approach and interpretation of the Hannenhalli-Pevzner formula for the reversal distance of signed permutations. Our approach utlizes the Poincaré dual, upon which reversals act in a particular way and obsoletes the notion of "padding" of the signed permutations. To this end we construct a bijection between signed permutations and an equivalence class of particular fatgraphs, called $π$-maps, and analyze the action of reversals on the latter. We …
Arc permutations, which were originally introduced in the study of triangulations and characters, have recently been shown to have interesting combinatorial properties. The first part of this paper continues their study by providing signed enumeration formulas with respect to their descent set and major index. Next, we generalize the notion of arc permutations to the hyperoctahedral group in two different directions. We show that these extensions to type $B$ carry interesting analogues of the properties of type $A$ arc permutations, such as characterizations by pattern avoidance, and elegant u…
The well-known Worpitzky identity provides a connection between two bases of $\mathbb{Q}[x]$: The standard basis $(x+1)^n$ and the binomial basis ${{x+n-i} \choose {n}}$, where the Eulerian numbers for the Coxeter group of type $A$ (the symmetric group) serve as the entries of the transformation matrix. Brenti has generalized this identity to the Coxeter groups of types $B$ and $D$ (signed and even-signed permutations groups, respectively) using generating function techniques. Motivated by Foata-Schützenberger and Rawlings' proof for the Worpitzky identity in the symmetric group, we provide co…
Many important statistics of signed permutations are realized in the corresponding permutation tableaux or bare tableaux of type $B$: Alignments, crossings and inversions of signed permutations are realized in the corresponding permutation tableaux of type $B$, and the cycles of signed permutations are understood in the corresponding bare tableaux of type $B$. This leads us to relate the number of alignments and crossings with other statistics of signed permutations and also to characterize the covering relation in weak Bruhat order on Coxeter system of type $B$ in terms of permutation tableau…
The derangement polynomial for the symmetric group enumerates derangements by the number of excedances. It can be interpreted as the local $h$-polynomial, in the sense of Stanley, of the barycentric subdivision of the simplex. Motivated by this interpretation, we define a derangement polynomial for the even-signed permutation group. The coefficients of this polynomial are nonnegative, symmetric and unimodal. We show that they enumerate derangements in the even-signed permutation group according to a notion of excedance, which is analogous to the one introduced by Brenti for signed permutations…
The flag-major index "fmaj" and the classical length function "$\ell$" are used to construct two $q$-analogs of the generating polynomial for the hyperoctahedral group~$B_n$ by number of positive and negative fixed points (resp. pixed points). Specializations of those $q$-analogs are also derived dealing with signed derangements and desarrangements, as well as several classical results that were previously proved for the symmetric group.
Pinnacle sets record the values of the local maxima for a given family of permutations. They were introduced by Davis-Nelson-Petersen-Tenner as a dual concept to that of peaks, previously defined by Billey-Burdzy-Sagan. In recent years pinnacles and admissible pinnacles sets for the type $A$ symmetric group have been widely studied. In this article we define the pinnacle set of signed permutations of types $B$ and $D$. We give a closed formula for the number of type $B$/$D$ admissible pinnacle sets and answer several other related enumerative questions.
Computing the reversal distances of signed permutations is an important topic in Bioinformatics. Recently, a new lower bound for the reversal distance was obtained via the plane permutation framework. This lower bound appears different from the existing lower bound obtained by Bafna and Pevzner through breakpoint graphs. In this paper, we prove that the two lower bounds are equal. Moreover, we confirm a related conjecture on skew-symmetric plane permutations, which can be restated as follows: let $p=(0,-1,-2,\ldots -n,n,n-1,\ldots 1)$ and let $$ \tilde{s}=(0,a_1,a_2,\ldots a_n,-a_n,-a_{n-1},\l…
We study the combinatorial properties of vexillary signed permutations, which are signed analogues of the vexillary permutations first considered by Lascoux and Schützenberger. We give several equivalent characterizations of vexillary signed permutations, including descriptions in terms of essential sets and pattern avoidance, and we relate them to the vexillary elements introduced by Billey and Lam.
A signed permutation π= π_1π_2 \ldots π_n in the hyperoctahedral group B_n is a word such that each π_i \in {-n, \ldots, -1, 1, \ldots, n} and {|π_1|, |π_2|, \ldots, |π_n|} = {1,2,\ldots,n}. An index i is a peak of πif π_{i-1}<π_i>π_{i+1} and P_B(π) denotes the set of all peaks of π. Given any set S, we define P_B(S,n) to be the set of signed permutations π\in B_n with P_B(π) = S. In this paper we are interested in the cardinality of the set P_B(S,n). In 2012, Billey, Burdzy and Sagan investigated the analogous problem for permutations in the symmetric group, S_n. In this paper we extend their…
We introduce diagrams and essential sets for signed permutations, extending the analogous notions for ordinary permutations. In particular, we show that the essential set provides a minimal list of rank conditions defining the Schubert variety or degeneracy locus corresponding to a signed permutation. Our essential set is in bijection with the poset-theoretic version defined by Reiner, Woo, and Yong, and thus gives an explicit, diagrammatic method for computing the latter.
In this paper, we study Eulerian polynomials for permutations and signed permutations of the multiset $\{1,1,2,2,\ldots,n,n\}$. Properties of these polynomials, including recurrence relations and unimodality are discussed. In particular, we give a unified proof of the fact that these polynomials are unimodal with modes in the middle.
Permutation-partition pairs were introduced by Stahl in 1980. These pairs are generalizations of graphs and graphs on surfaces. They were used to solve some problems for orientable embeddings of graphs. In this paper, we introduce a particular type of permutation-partition pair, called permutation-bipartition pair, which can be seen as generalizations of signed graphs and signed graph embeddings. Some applications are given.
There are several ways to generalize graph coloring to signed graphs. Máčajová, Raspaud and Škoviera introduced one of them and conjectured that in this setting, for signed planar graphs four colors are always enough, generalising thereby The Four Color Theorem. We disprove the conjecture.
Let B_n be the hyperoctahedral group; that is, the set of all signed permutations on n letters, and let B_n(T) be the set of all signed permutations in B_n which avoids a set T of signed patterns. In this paper, we find all the cardinalities of the sets B_n(T) where $T \subseteq B_2$. This allow us to express these cardinalities via inverse of binomial coefficients, binomial coefficients, Catalan numbers, and Fibonacci numbers.
In this paper, we extend the notion of labeled partitions with ordinary permutations to colored permutations in the sense that the colors are endowed with a cyclic structure. We use labeled partitions with colored permutations to derive the generating function of the $\mathrm{fmaj}_k$ indices of colored permutations. The second result is a combinatorial treatment of a relation on the q-derangement numbers with respect to colored permutations which leads to the formula of Chow for signed permutations and the formula of Faliharimalala and Zeng [10] on colored permutations. The third result is an…
We define a method for edge coloring signed graphs and what it means for such a coloring to be proper. Our method has many desirable properties: it specializes to the usual notion of edge coloring when the signed graph is all-negative, it has a natural definition in terms of vertex coloring of a line graph, and the minimum number of colors required for a proper coloring of a signed simple graph is bounded above by Δ + 1 in parallel with Vizing's Theorem. In fact, Vizing's Theorem is a special case of the more difficult theorem concerning signed graphs.
We define a new statistic $\mathsf{sor}$ on the set of colored permutations $\mathsf{G}_{r,n}$ and prove that it has the same distribution as the length function. For the set of restricted colored permutations corresponding to the arrangements of $n$ non-attacking rooks on a fixed Ferrers shape we show that the following two sequences of set-valued statistics are joint equidistributed: $(\ell,\mathsf{Rmil}^0,\mathsf{Rmil}^1,...,\mathsf{Rmil}^{r-1}$, $\mathsf{Lmil}^0,\mathsf{Lmil}^1,...,\mathsf{Lmil}^{r-1}$, $\mathsf{Lmal}^0,\mathsf{Lmal}^1,...,\mathsf{Lmal}^{r-1}$, $\mathsf{Lmap}^0,\mathsf{Lma…
This paper builds on two covering Hopf algebras of the Hopf algebra QSym of quasi-symmetric functions, with linear bases parameterized by compositions. One is the Malvenuto-Reutenauer Hopf algebra SSym of permutations, mapped onto QSym by taking descents of permutations. The other one is the recently introduced Hopf algebra RQSym of weak quasi-symmetric functions, mapped onto QSym by extracting compositions from weak compositions. We extend these two surjective Hopf algebra homomorphisms into a commutative diagram by introducing a Hopf algebra HSym, linearly spanned by signed permutations from…
A signed labeled forest is defined as a (plane) forest labeled by {1,2,..., n} along with minus signs associated to some vertices. Signed labeled forests can be viewed as an extension of signed permutations. We define the inversion number, the flag major index and the R-major index on signed labeled forests. They can be considered as type B analogues of the indices for labeled forests introduced by Bjorner and Wachs. The flag major index for signed labeled forests is based on the flag major index on signed permutations introduced by Adin and Roichman, whereas the R-major index for signed label…
Motivated by the work of Chung, Claesson, Dukes, and Graham, we define a natural type B analog of the classic bubble sort, and use it to define a type B analog of the maximum drop statistic. We enumerate (by explicit, recursive, and generating function formulas) signed permutations with r type B descents and type B maximum drop at most k. We also find a connection between these signed permutations and certain 2-colored juggling sequences.
We define a notion of vexillary signed permutation in types B, C, and D, corresponding to natural degeneracy loci for vector bundles with symmetries of those types. We show that the classes of these loci are given by explicit Pfaffian formulas. The Grassmannian formulas of Kazarian are important special cases, and the corresponding double Schubert polynomials of Ikeda, Mihalcea, and Naruse are shown to be equal to these Pfaffians.
Motivated by a conjecture of Savage and Visontai about the equidistribution of the descent statistic on signed permutations of the multiset $\{1,1,2,2,\ldots,n,n\}$ and the ascent statistic on $(1,4,3,8,\ldots,2n-1,4n)$-inversion sequences, we investigate the descent polynomial of the signed permutations of a general multiset. We obtain a factorial generating function formula for a $q$-analog of these descent polynomials and apply it to show that they have only real roots. Two different proofs of the conjecture of Savage and Visontai are provided.
The signed permutation modules are a simultaneous generalization of the ordinary permutation modules and the twisted permutation modules of the symmetric group. In a recent paper Dave Benson and Peter Symonds defined a new invariant $γ_G(M)$ for a finite dimensional module $M$ of a finite group $G$ which attempts to quantify how close a module is to being projective. In this paper, we determine this invariant for all the signed permutation modules of the symmetric group using tools from representation theory and combinatorics.
Let $E_n^r=\{[τ]_a=(τ_1^{(a_1)},...,τ_n^{(a_n)})| τ\in S_n,\ 1\leq a_i\leq r\}$ be the set of all signed permutations on the symbols 1,2,...,n with signs 1,2,...,r. We prove, for every 2-letter signed pattern $[τ]_a$, that the number of $[τ]_a$-avoiding signed permutations in $E_n^r$ is given by the formula $\sum\limits_{j=0}^n j!(r-1)^j{n\choose j}^2$. Also we prove that there are only one Wilf class for r=1, four Wilf classes for r=2, and six Wilf classes for $r\geq 3$.
It is a classical result that the parity-balance of the number of weak excedances of all permutations (derangements, respectively) of length $n$ is the Euler number $E_n$, alternating in sign, if $n$ is odd (even, respectively). Josuat-Vergès obtained a $q$-analog of the results respecting the number of crossings of a permutation. One of the goals in this paper is to extend the results to the permutations (derangements, respectively) of types B and D, on the basis of the joint distribution in statistics excedances, crossings and the number of negative entries obtained by Corteel, Josuat-Vergès…
As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, signed bigrassmannian polynomials and bigrassmannian determinant. First, a signed bigrassmannian polynomial is a variant of the statistic given by the number of bigrassmannian permutations below a permutation in Bruhat order as Reading suggested (2002) and afterward the author developed (2011). Second, bigrassmannian determinant is a $q$-analog of the determinant with respect to our statistic. It plays a key role for a determinantal expression of those polynomials. We further show that bigrassmannian…
A classical result states that the parity balance of the number of excedances of all permutations (derangements, respectively) of length $n$ is the Euler number. In 2010, Josuat-Vergès gives a $q$-analogue with $q$ representing the number of crossings. We extend this result to the permutations (derangements, respectively) of type B and D. It turns out that the signed countings are related to the derivative polynomials of $\tan$ and $\sec$. Springer numbers defined by Springer can be regarded as an analogue of Euler numbers defined on every Coxeter group. In 1992 Arnol'd showed that the Springe…
The periodic (ordinal) patterns of a map are the permutations realized by the relative order of the points in its periodic orbits. We give a combinatorial characterization of the periodic patterns of an arbitrary signed shift, in terms of the structure of the descent set of a certain cyclic permutation associated to the pattern. Signed shifts are an important family of one-dimensional dynamical systems that includes shift maps and the tent map as particular cases. Defined as a function on the set of infinite words on a finite alphabet, a signed shift deletes the first letter and, depending on …
We give some new Wilf equivalences for signed patterns which allow the complete classification of signed patterns of lengths three and four. The problem is considered for pattern avoidance by general as well as involutive signed permutations.
Permutation tableaux are combinatorial objects related with permutations and various statistics on them. They appeared in connection with total positivity in Grassmannians, and stationary probabilities in a PASEP model. In particular they gave rise to an interesting q-analog of Eulerian numbers. The purpose of this article is to study some combinatorial properties of type B permutation tableaux, defined by Lam and Williams, and links with signed permutation statistics. We show that many of the tools used for permutation tableaux generalize in this case, including: the Matrix Ansatz (a method o…
We complete the Wilf classification of signed patterns of length 5 for both signed permutations and signed involutions. New general equivalences of patterns are given which prove Jaggard's conjectures concerning involutions in the symmetric group avoiding certain patterns of length 5 and 6. In this way, we also complete the Wilf classification of S_5, S_6, and S_7 for both permutations and involutions.
The allowed patterns of a map are those permutations in the same relative order as the initial segments of orbits realized by the map. In this paper, we characterize and provide enumerative bounds for the allowed patterns of signed shifts, a family of maps on infinite words.
We give a combinatorial description for the weak order on the hyperoctahedral group. This characterization is then used to analyze the order-theoretic properties of the shifted products of hyperoctahedral groups. It is shown that each shifted product is a disjoint union of some intervals, which can be convex embedded into a hyperoctahedral group. As an application, we investigate the monomial basis for the Hopf algebra $\mathfrak{H}Sym$ of signed permutations, related to the fundamental basis via Möbius inversion on the weak order on hyperoctahedral groups. It turns out that the image of a mon…
A signed graph is a graph in which each edge is labeled with $+1$ or $-1$. A (proper) vertex coloring of a signed graph is a mapping $\f$ that assigns to each vertex $v\in V(G)$ a color $\f(v)\in \mz$ such that every edge $vw$ of $G$ satisfies $\f(v)\not= \sg(vw)\f(w)$, where $\sg(vw)$ is the sign of the edge $vw$. For an integer $h\geq 0$, let $\Ga_{2h}=\{\pm1,\pm2, \ldots, \pm h\}$ and $\Ga_{2h+1}=\Ga_{2h} \cup \{0\}$. Following \cite{MaRS2015}, the signed chromatic number $\scn(G)$ of $G$ is the least integer $k$ such that $G$ admits a vertex coloring $\f$ with ${\rm im}(\f)\subseteq \Ga_k$…
The numbers of even and odd permutations with a given ascent number are investigated using an operator that was previously introduced by the author. Their difference is called a signed Eulerian number. By means of the operator the recurrence relation for signed Eulerian numbers is deduced, which was obtained by an analytic method. Our approach is straightforward and enables us to deduce other properties including divisibility by prime powers.
The notion of the circular coloring of signed graphs is a recent one that simultaneously extends both notions of the circular coloring of graphs and $0$-free coloring of signed graphs. A circular $r$-coloring of a signed graph $(G, σ)$ is to assign points of a circle of circumference $r$, $r\geq 2$, to the vertices of $G$ such that vertices connected by a positive edge are at circular distance at least $1$ and vertices connected by a negative edge are at circular distance at most $\frac{r}{2}-1$. The infimum of all $r$ for which $(G, σ)$ admits a circular $r$-coloring is said to be the circula…
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 $\…
Let G be a finite group. Let pi be a permutation from S{n}. We study the distribution of probabilities of equality a{1} a{2} ...a{n-1}a{n}=a{pi{1}}^{epsilon{1}} a{pi_{2}}^{epsilon{2}}...a{pi{n-1}}^{epsilon_{n-1}} a_{pi_{n}}^{epsilon{n}}, when pi varies over all the permutations in S{n}, and epsilon{i} varies over the set {+1, -1}. By the paper "Hultman Numbers and Generalized Commuting Probability in Finite Groups" (2017), The case where all epsilon{i} are +1 led to a close connection to Hultman numbers. In this paper we generalize the results, permitting epsilon{i} to be -1. We describe the s…
