Related papers: Rowmotion on hook and two-row alt $\nu$-Tamari lat…
Thomas and Williams conjectured that rowmotion acting on the rational $(a,b)$-Tamari lattice has order $a+b-1$. We construct an equivariant bijection that proves this conjecture when $b\equiv 1\pmod a$; in fact, we determine the entire…
This article illustrates the dynamical concept of $homomesy$ in three kinds of dynamical systems -- combinatorial, piecewise-linear, and birational -- and shows the relationship between these three settings. In particular, we show how the…
We introduce $n(n-1)/2$ natural involutions ("toggles") on the set $S$ of noncrossing partitions $\pi$ of size $n$, along with certain composite operations obtained by composing these involutions. We show that for many operations $T$ of…
Birational rowmotion is an action on the space of assignments of rational functions to the elements of a finite partially-ordered set (poset). It is lifted from the well-studied rowmotion map on order ideals (equivariantly on antichains) of…
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…
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…
We study rowmotion dynamics on interval-closed sets. Our first main result proves a simplification of the global definition of interval-closed set rowmotion from (Elder, Lafreni\`ere, McNicholas, Striker, and Welch 2024). We then completely…
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…
We introduce new partial order structures on the underlying sets of free nonsymmetric operads. These posets involve decorated ordered rooted trees, and their terminal intervals are lattices. These lattices are not graded, not self-dual, and…
Rowmotion is an invertible operator on the order ideals of a poset which has been extensively studied and is well understood for the rectangle poset. In this paper, we show that rowmotion is equivariant with respect to a bijection of…
We count the number of linear intervals in the Tamari and the Dyck lattices according to their height, using generating series and Lagrange inversion. Surprisingly, these numbers are the same in both lattices. We define a new family of…
Non-Hermitian effects could create rich dynamical and topological phase structures. In this work, we show that the collaboration between lattice dimerization and non-Hermiticity could generally bring about mobility edges and multiple…
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…
We study various aspects of the dynamics induced by integer matrices on the invariant rational lattices of the torus in dimension 2 and greater. Firstly, we investigate the orbit structure when the toral endomorphism is not invertible on…
We first find an explicit formula for the square root of positive $2 \times 2$ operator matrices with commuting entries, and then use it to define and study semi-hyponormality for commuting pairs of Hilbert space operators. \ For the…
We introduce a new group action on set systems, constructed as a semidirect product of a permutation group and a group generated by twist and loop complementation operations on a single element. This action extends the ribbon group…
Given a lattice path $\nu$, the $\nu$-Tamari lattice and the $\nu$-Dyck lattice are two natural examples of partial order structures on the set of lattice paths that lie weakly above $\nu$. In this paper, we introduce a more general family…
It is well known in general relativity that trajectories of Hamiltonian systems lift to geodesics of pp-wave spacetimes, an example of a more general phenomenon known as the "Eisenhart lift." We review and expand upon the benefits of this…
We study the dynamics of countable groups on their respective spaces of quasimorphisms. For cohomologically non-trivial quasimorphisms we show that there are no invariant measures and classify stationary measures. Within the equivalence…
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…