We investigate the poset structure on the alternating group that arises when the latter is generated by 3-cycles. We study intervals in this poset and give several enumerative results, as well as a complete description of the orbits of the Hurwitz action on maximal chains. Our motivating example is the well-studied absolute order arising when the symmetric group is generated by transpositions, i.e. 2-cycles, and we compare our results to this case along the way. In particular, noncrossing partitions arise naturally in both settings.
For an integer $n\geq 1$, we display a poset $Φ_n$ whose maximal chains are in bijection with the $n\times n$ alternating sign matrices. The Hasse diagram $\widehat Φ_n$ is obtained from the $n$-cube by adding some edges. We show that the dihedral group $D_{2n}$ acts on $\widehat Φ_n$ as a group of automorphisms.
Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1 whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We put ASMs into a larger context by studying the order ideals of subposets of a certain poset, proving that they are in bijection with a variety of interesting combinatorial objects, including ASMs, totally symmetric self--complementary plane partitions (TSSCPPs), Catalan objects, tournaments, semistandard Young tableaux, and totally symmetric plane partitions. We use this perspective to prove an expansion of the tournament generating funct…
We discuss results dealing with universal cycles (u-cycles) and $s$-overlap cycles, and contribute to the body of those results by proving existence of universal cycles of naturally labeled posets (NL posets), $s$-overlap cycles of words of weight $k$, and juggling patterns. The result on posets is, to the best of our knowledge, the first demonstration of the existence of a u-cycle whose length is unknown.
Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1 whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We present a unifying perspective on ASMs and other combinatorial objects by studying a certain tetrahedral poset and its subposets. We prove the order ideals of these subposets are in bijection with a variety of interesting combinatorial objects, including ASMs, totally symmetric self-complementary plane partitions (TSSCPPs), staircase shaped semistandard Young tableaux, Catalan objects, tournaments, and totally symmetric plane partitions. …
We show that the poset of alternating sign matrices, with Bruhat order, is isomorphic to the poset of certain submodules of the dominant Verma module for the special linear Lie algebra $\frak{sl}_n$. The latter poset consists of the intersections of Verma submodules and can also be defined in terms of a Kazhdan-Lusztig cell.
We introduce a method of decomposing the family of intervals along a cyclic permutation into chains to determine the size of the largest family of subsets of $[n]:= \{1,2,...,n\}$ not containing one or more given posets as a subposet. De Bonis, Katona and Swanepoel determined the size of the largest butterfly-free family. We strengthen this result by showing that, for certain posets containing the butterfly poset as a subposet, the same bound holds. We also obtain the corresponding LYM-type inequalities.
We study $M$-alternating Hamilton paths and $M$-alternating Hamilton cycles in a simple connected graph $G$ on $ν$ vertices with a perfect matching $M$. Let $G$ be a bipartite graph, we prove that if for any two vertices $x$ and $y$ in different parts of $G$, $d(x)+d(y)\geq ν/2+2$, then $G$ has an $M$-alternating Hamilton cycle. For general graphs, a condition for the existence of an $M$-alternating Hamilton path starting and ending with edges in $M$ is put forward. Then we prove that if $κ(G)\geqν/2$, where $κ(G)$ denotes the connectivity of $G$, then $G$ has an $M$-alternating Hamilton cycle…
We show that for two quivers without oriented cycles related by a BGP reflection, the posets of their cluster tilting objects are related by a simple combinatorial construction, which we call a flip-flop. We deduce that the posets of cluster tilting objects of derived equivalent path algebras of quivers without oriented cycles are universally derived equivalent. In particular, all Cambrian lattices corresponding to the various orientations of the same Dynkin diagram are universally derived equivalent.
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,…
A word $w=w_1w_2\cdots w_n$ is alternating if either $w_1<w_2>w_3<w_4>\cdots$ (when the word is up-down) or $w_1>w_2<w_3>w_4<\cdots$ (when the word is down-up). In this paper, we initiate the study of (pattern-avoiding) alternating words. We enumerate up-down (equivalently, down-up) words via finding a bijection with order ideals of a certain poset. Further, we show that the number of 123-avoiding up-down words of even length is given by the Narayana numbers, which is also the case, shown by us bijectively, with 132-avoiding up-down words of even length. We also give formulas for enumerating a…
In 1972, Woodall raised the following Ore type condition for directed Hamilton cycles in digraphs: Let $D$ be a digraph. If for every vertex pair $u$ and $v$, where there is no arc from $u$ to $v$, we have $d^+u)+d^-(v)\geq |D|$, then $D$ has a directed Hamilton cycle. By a correspondence between bipartite graphs and digraphs, the above result is equivalent to the following result of Las Vergnas: Let $G = (B,W)$ be a balanced bipartite graph. If for any $b \in B$ and $w \in W$, where $b$ and $w$ are nonadjacent, we have $d(w)+d(b) \geq |G|/2 + 1$, then every perfect matching of $G$ is containe…
Given two graphs, a mapping between their edge-sets is cycle-continuous, if the preimage of every cycle is a cycle. The motivation for this notion is Jaeger's conjecture that for every bridgeless graph there is a cycle-continuous mapping to the Petersen graph. Answering a question of DeVos, Nešetřil, and Raspaud, we prove that there exists an infinite set of graphs with no cycle-continuous mapping between them. Further extending this result, we show that every countable poset can be represented by graphs and existence of cycle-continuous mappings between them.
Motivation coming from the study of affine Weyl groups, a structure of ranked poset is defined on the set of circular permutations in $S_n$ (that is, $n$-cycles). It is isomorphic to the poset of so-called admitted vectors, and to an interval in the affine symmetric group $\tilde S_n$ with the weak order. The poset is a semidistributive lattice, and the rank function, whose range is cubic in $n$, is computed by some special formula involving inversions. We prove also some links with Eulerian numbers, triangulations of an $n$-gon, and Young's lattice.
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.
Cubic factor-invariant graphs of cycle quotient type -- the alternating case
University of Newcastle（纽卡斯尔大学）；Institute of Mathematics, Physics and Mechanics（数学物理力学研究所）；University of Ljubljana, Faculty of Education（卢布尔雅那大学教育学院）；University of Primorska, Institute Andrej Marušič（滨海大学安德烈亚·马鲁西奇研究所）
An alternating cycle in a 2-two-edge-colored graph is a cycle such that any two consecutive edges have different colors. Let $G_1, \ldots, G_k$ be a collection of pairwise vertex disjoint 2-edge-colored graphs. The colored generalized sum of $G_1, \ldots, G_k$, denoted by $ \oplus_{i=1}^k G_i$, is the set of all 2-edge-colored graphs $G$ such that: (i) $V(G)=\bigcup_{i=1}^k V(G_i)$, (ii) $G\langle V(G_i)\rangle\cong G_i$ for $i=1,\ldots, k$ as edge-colored graphs where $G\langle V(G_i)\rangle$ has the same coloring as $G_i$ and (iii) between each pair of vertices in different summands of $G$ t…
The main objects of the present paper are (i) Hibi rings (toric rings arising from order polytopes of posets), (ii) stable set rings (toric rings arising from stable set polytopes of perfect graphs), and (iii) edge rings (toric rings arising from edge polytopes of graphs satisfying the odd cycle condition). The goal of the present paper is to analyze those three toric rings and to discuss their structures in the case where their class groups have small rank. We prove that the class groups of (i), (ii) and (iii) are torsionfree. More precisely, we give descriptions of their class groups. Moreov…
Generalized permutahedra are a family of polytopes with a rich combinatorial structure and strong connections to optimization. We prove that they are the universal family of polyhedra with a certain Hopf algebraic structure. Their antipode is remarkably simple: the antipode of a polytope is the alternating sum of its faces. Our construction provides a unifying framework to organize numerous combinatorial structures, including graphs, matroids, posets, set partitions, linear graphs, hypergraphs, simplicial complexes, building sets, and simple graphs. We highlight three applications: 1. We obtai…
Let $\mathcal{G}_α$ be a hereditary graph class (i.e, every subgraph of $G_α\in \mathcal{G}_α$ belongs to $\mathcal{G}_α$) such that every graph $G_α$ in $\mathcal{G}_α$ has minimum degree at most 1, or contains either an edge $uv$ such that $d_{G_α}(u)+d_{G_α}(v)\leq α$ or a 2-alternating cycle. It is proved that every graph in $\mathcal{G}_α$ ($α\geq 5$) with maximum degree $Δ$ can be edge-partitioned into two forests $F_1$, $F_2$ and a subgraph $H$ such that $Δ(F_i)\leq \max\{2,\lceil\frac{Δ-α+6}{2}\rceil\}$ for $i=1,2$ and $Δ(H)\leq α-5$.
Given integers $k,l\geq 2$, where either $l$ is odd or $k$ is even, let $n(k,l)$ denote the largest integer $n$ such that each element of $A_n$ is a product of $k$ many $l$-cycles. In 2008, M. Herzog, G. Kaplan and A. Lev conjectured that $\lfloor \frac{2kl}{3} \rfloor \leq n(k,l)\leq \lfloor \frac{2kl}{3}\rfloor+1$. It is known that the conjecture holds when $k=2,3,4$. Moreover, it is also true when $3\mid l$. In this article, we determine the exact value of $n(k,l)$ when $3\nmid l$ and $k\geq 5$. As an immediate consequence, we get that $n(k,l)<\lfloor \frac{2kl}{3}\rfloor$ when $k\geq 5$, w…
A bipartite graph B is called a brace if it is connected and every matching of size at most two in B is contained in some perfect matching of B and a cycle C in B is called conformal if B-V(C) has a perfect matching. We show that there do not exist two disjoint alternating paths that form a cross over a conformal cycle C in a brace B if and only if one can reduce B, by an application of a matching theoretic analogue of small clique sums, to a planar brace H in which C bounds a face. We then utilise this result and provide a polynomial time algorithm which solves the 2-linkage problem for alter…
The Razumov-Stroganov correspondence, an important link between statistical physics and combinatorics proved in 2011 by L. Cantini and A. Sportiello, relates the ground state eigenvector of the O(1) dense loop model on a semi-infinite cylinder to a refined enumeration of fully-packed loops, which are in bijection with alternating sign matrices. This paper reformulates a key component of this proof in terms of posets, the toggle group, and homomesy, and proves two new homomesy results on general posets which we hope will have broader implications.
Let $S_n$ and $A_n$ denote the symmetric group and alternating group of degree $n$ with $n\geq 3$, respectively. Let $S$ be the set of all $3$-cycles in $S_n$. The \emph{complete alternating group graph}, denoted by $CAG_n$, is defined as the Cayley graph $\mathrm{Cay}(A_n,S)$ on $A_n$ with respect to $S$. In this paper, we show that $CAG_n$ ($n\geq 4$) is not a normal Cayley graph. Furthermore, the automorphism group of $CAG_n$ for $n\geq 5$ is obtained, which equals to $\mathrm{Aut}(CAG_n)=(R(A_n)\rtimes \mathrm{Inn}(S_n))\rtimes \mathbb{Z}_2\cong (A_n\rtimes S_n)\rtimes \mathbb{Z}_2$, where…
We explore and relate two notions of monotonicity, stochastic and realizable, for a system of probability measures on a common finite partially ordered set (poset) S when the measures are indexed by another poset A. We give counterexamples to show that the two notions are not always equivalent, but for various large classes of S we also present conditions on the poset A that are necessary and sufficient for equivalence. When A = S, the condition that the cover graph of S have no cycles is necessary and sufficient for equivalence. This case arises in comparing applicability of the perfect sampl…
A fascinating problem on digraphs is the existence problem of the finiteupper bound on s for all vertex-primitive s-arc-transitive digraphs except directed cycles (which is known to be reduced to the almost simple groups case). In this paper,we prove that s 2 for all G-vertex-primitive s-arc-transitive digraphs with G an (insoluble) alternating or symmetric group, which makes an important progress towards a solution of the problem. The proofs involves some methods that may be used to investigate other almost simple groups cases.
This note shows there are infinitely many finite groups G, such that every connected Cayley graph on G has a hamiltonian cycle, and G is not solvable. Specifically, for every prime p that is congruent to 1, modulo 30, we show there is a hamiltonian cycle in every connected Cayley graph on the direct product of the cyclic group of order p with the alternating group A_5 on five letters.
A general explicit upper bound is obtained for the proportion $P(n,m)$ of elements of order dividing $m$, where $n-1 \le m \le cn$ for some constant $c$, in the finite symmetric group $S_n$. This is used to find lower bounds for the conditional probabilities that an element of $S_n$ or $A_n$ contains an $r$-cycle, given that it satisfies an equation of the form $x^{rs}=1$ where $s\leq3$. For example, the conditional probability that an element $x$ is an $n$-cycle, given that $x^n=1$, is always greater than 2/7, and is greater than 1/2 if $n$ does not divide 24. Our results improve estimates of…
The second author had previously obtained explicit generating functions for moments of characteristic polynomials of permutation matrices (n points). In this paper, we generalize many aspects of this situation. We introduce random shifts of the eigenvalues of the permutation matrices, in two different ways: independently or not for each subset of eigenvalues associated to the same cycle. We also consider vastly more general functions than the characteristic polynomial of a permutation matrix, by first finding an equivalent definition in terms of cycle-type of the permutation. We consider other…
There are left and right actions of the 0-Hecke monoid of the affine symmetric group $\tilde{S}_n$ on involutions whose cycles are labeled periodically by nonnegative integers. Using these actions we construct two bijections, which are length-preserving in an appropriate sense, from the set of involutions in $\tilde{S}_n$ to the set of $\mathbb{N}$-weighted matchings in the $n$-element cycle graph. As an application, we compute a formula for the bivariate generating function counting the involutions in $\tilde S_n$ by length and absolute length. The 0-Hecke monoid of $\tilde{S}_n$ also acts on…
Cycle-Structure Generating Functions for Special Breakpoint Graphs
The George Washington University（乔治华盛顿大学）；Joint Center for Quantum Information and Computer Science and Joint Quantum Institute, NIST/University of Maryland（量子信息与计算联合中心及联合量子研究所，美国国家标准与技术研究院/马里兰大学）；Microsoft Quantum（微软量子）
