In 2012, N. Williams and the second author showed that on order ideals of ranked partially ordered sets (posets), rowmotion is conjugate to (and thus has the same orbit structure as) a different toggle group action, which in special cases is equivalent to promotion on linear extensions of posets constructed from two chains. In 2015, O. Pechenik and the first and second authors extended these results to show that increasing tableaux under K-promotion naturally corresponds to order ideals in a product of three chains under a toggle group action conjugate to rowmotion they called hyperplane promotion.In this paper, we generalize these results to the setting of arbitrary increasing labelings of any finite poset with given restrictions on the labels. We define a generalization of K-promotion in this setting and show it corresponds to a toggle group action we call toggle-promotion on order ideals of an associated poset. When the restrictions on labels are particularly nice (for example, specifying a global bound on all labels used), we show that toggle-promotion is conjugate to rowmotion. Additionally, we show that any poset that can be nicely embedded into a Cartesian product has a natural toggle-promotion action conjugate to rowmotion. This paper builds upon work of N. Williams and the second author [38] regarding promotion and rowmotion on posets with a two-dimensional lattice projection and subsequent work of O. Pechenik with the first and second authors [12] which n-dimensionalized this result, relating K-promotion on increasing tableaux to rowmotion on the product of three chains poset.We describe our main results in Subsection 1.1 and then give a brief history of promotion and rowmotion in Subsection 1.2, with a focus on the motivating results from these two papers.1.1. Main Results. Throughout this paper, let P be a finite partially ordered set (poset).Definition 1.1. We say that a function f : P → Z is an increasing labeling if p 1 < p 2 in P implies that f (p 1 ) < f (p 2 ) (with the usual total ordering on the integers). We will be interested in sets of increasing labelings on P given a restriction function R : P → P(Z) indicating which labels each poset element is allowed to attain (where P(Z) is the power set of Z). We require R(p) to be nonempty and finite for each p ∈ P . Call the set of such increasing labelings Inc R (P ).See Figures 1 and 2 for examples.Remark 1.2. Up to conventions of increasing versus decreasing, increasing labelings can also be thought of as strict P -partitions with restricted parts. We use the terminology of increasing labelings rather than P -partitions since we are generalizing from increasing tableaux rather than from integer partitions. See Remarks 1.4, 2.30, and 2.31 for more on the connection to P -partitions.We consider a natural partial order on Inc R (P ), where f ≤ g if and only if f (p) ≤ g(p) for all p ∈ P . This is a distributive lattice, so we may apply Birkhoff's Representation Theorem to obtain a representation in terms of order ideals in a poset. A subset I of P ...
J. Propp and T. Roby isolated a phenomenon in which a statistic on a set has the same average value over any orbit as its global average, naming it homomesy. They proved that the cardinality statistic on order ideals of the product of two chains poset under rowmotion exhibits homomesy. In this paper, we prove an analogous result in the case of the product of three chains where one chain is of length two. In order to prove this result, we generalize from two to n dimensions the recombination technique that D. Einstein and J. Propp developed to study homomesy. We also prove a number of corollaries, including a partial generalization of this homomesy result to an arbitrary product of three chains and a new result on increasing tableaux. We conclude with a generalization of recombination to any ranked poset and a homomesy result for the Type B minuscule poset cross a two element chain.
Abstract. We define and enumerate two new two-parameter permutation families, namely, placements of a maximum number of non-attacking rooks on k chained-together n × n chessboards, in either a circular or linear configuration. The linear case with k = 1 corresponds to standard permutations of n, and the circular case with n = 4 and k = 6 corresponds to a three-person chessboard. We give bijections of these rook placements to matrix form, one-line notation, and matchings on certain graphs. Finally, we define chained linear and circular alternating sign matrices, enumerate them for certain values of n and k, and give bijections to analogues of monotone triangles, square ice configurations, and fully-packed loop configurations.
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