相关论文: Noncrossing partitions, toggles, and homomesies
This paper explores the orbit structure and homomesy (constant averages over orbits) properties of certain actions of toggle groups on the collection of independent sets of a path graph. In particular we prove a generalization of a homomesy…
Homomesy is an invariance phenomenon in dynamical algebraic combinatorics which occurs when the average value of some statistic on a set of combinatorial objects is the same over each orbit generated by a map on these objects. In this paper…
This paper analyzes a certain action called "whirling" that can be defined on any family of functions between two finite sets equipped with a linear (or cyclic) ordering. Many maps of interest in dynamical algebraic combinatorics, such as…
Interval-closed sets of a poset are a natural superset of order ideals. We initiate the study of interval-closed sets of finite posets from enumerative and dynamical perspectives. In particular, we use the generalized toggle group to define…
The rowmotion operator acting on the set of order ideals of a finite poset has been the focus of a significant amount of recent research. One of the major goals has been to exhibit homomesies: statistics that have the same average along…
In this article, we consider involutions, called togglings, on the set of independent sets of the Dynkin diagram of type A, or a path graph. We are interested in the action of the subgroup of the symmetric group of the set of independent…
We generalize the notion of the toggle group, as defined in [P. Cameron-D. Fon-der-Flaass '95] and further explored in [J. Striker-N. Williams '12], from the set of order ideals of a poset to any family of subsets of a finite set. We prove…
In this paper, we analyze the toggle group on the set of antichains of a poset. Toggle groups, generated by simple involutions, were first introduced by Cameron and Fon-Der-Flaass for order ideals of posets. Recently Striker has motivated…
Much of dynamical algebraic combinatorics focuses on global dynamical systems defined via maps that are compositions of local toggle operators. The second author and Roby studied such maps that result from toggling independent sets of a…
Many invertible actions $\tau$ on a set ${\mathcal{S}}$ of combinatorial objects, along with a natural statistic $f$ on ${\mathcal{S}}$, exhibit the following property which we dub \textbf{homomesy}: the average of $f$ over each…
The dynamics of certain combinatorial actions and their liftings to actions at the piecewise-linear and birational level have been studied lately with an eye towards questions of periodicity, orbit structure, and invariants. One key…
In this paper, we study the lattice properties of posets of torsion pairs in the module category of a family of representation-finite gentle algebras called tiling algebras, introduced by Coelho Simoes and Parsons. We present a…
We consider noncrossing partitions of [n] under the action of (i) the reflection group (of order 2), (ii) the rotation group (cyclic of order n) and (iii) the rotation/reflection group (dihedral of order 2n). First, we exhibit a bijection…
The rowmotion action on order ideals or on antichains of a finite partially ordered set has been studied (under a variety of names) by many authors. Depending on the poset, one finds unexpectedly interesting orbit structures, instances of…
Let $G$ be an acylic directed graph. For each vertex $g \in G$, we define an involution on the independent sets of $G$. We call these involutions flips, and use them to define a new partial order on independent sets of $G$. Trim lattices…
In 2024, Ceballos and Chenevi{\`e}re introduced alt $\nu$-Tamari lattices, parameterized by a lattice path $\nu$ and an increment vector $\delta$, as a common generalization of $\nu$-Tamari and $\nu$-Dyck lattices. We study rowmotion on two…
Rowmotion is a simple cyclic action on the distributive lattice of order ideals of a poset: it sends the order ideal x to the order ideal generated by the minimal elements not in x. It can also be computed in "slow motion" as a sequence of…
This article explores the novel notion of gyrogroup actions, which is a natural generalization of the usual notion of group actions. As a first step toward the study of gyrogroup actions from the algebraic viewpoint, we prove three…
In \cite{striker2018rowmotion} Striker generalized Cameron and Fon-Der-Flaass's notion of a toggle group. In this paper we begin the study of transitive generalized toggle groups that contain a cycle. We first show that if such a group has…
We exhibit, for any positive integer parameter $s$, an involution on the set of integer partitions of $n$. These involutions show the joint symmetry of the distributions of the following two statistics. The first counts the number of parts…