Proofs and generalizations of a homomesy conjecture of Propp and Roby

Bloom, Jonathan; Pechenik, Oliver; Saracino, Dan

doi:10.1016/j.disc.2015.08.011

Cited by 25 publications

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References 17 publications

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“…On the other hand, e is contained in a maximum chain of length three. We obtain δ(e) = 2 and ν(e) = 0, making the restriction for e the interval [3,5]. Applying this definition to every element in the poset results in a consistent labeling.…”

Section: 2mentioning

confidence: 99%

Rowmotion and increasing labeling promotion

Dilks¹,

Striker²,

Vorland³

2019

Journal of Combinatorial Theory, Series A

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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 ...

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Section: 2mentioning

confidence: 99%

Rowmotion and increasing labeling promotion

Dilks¹,

Striker²,

Vorland³

2019

Journal of Combinatorial Theory, Series A

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“…The columns L j,k 1,2 are defined to correspond to boxes in an increasing tableau, as in As a result, we can say the global average is f L = nb 2 , and hence c must also be nb 2 . Pechenik further generalized the results of [1] and the result stated in Theorem 4.15. From this, we get a more general analogue of Corollary 5.2.…”

Section: Tableaux and Refined Resultsmentioning

confidence: 70%

“…, (j n , k n ) are rotationally symmetric about the point ( 3 2 , a+1 2 ). If f L (I) denotes the cardinality of I on L j 1 ,k 1 1,2 , L j 2 ,k 2 1,2 , . .…”

Section: Tableaux and Refined Resultsmentioning

confidence: 99%

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Homomesy in Products of Three Chains and Multidimensional Recombination

Vorland¹

2019

Electron. J. Combin.

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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.

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“…In the cases = 2, 3, there are proofs that proceed by giving a bijection to other combinatorial objects (noncrossing matchings and webs, respectively) that sends promotion to rotation [PePyRh09]; see Figure 7. There are also homomesy results in this and more general settings; see [BlPeSa16].…”

Section: Theorem 15 ([Rh10]) Promotion On ( × ) Exhibits the Cyclic mentioning

confidence: 99%

Dynamical Algebraic Combinatorics: Promotion, Rowmotion, and Resonance

Striker¹

2017

Notices Amer. Math. Soc.

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Mathematical inquiry often begins with the study of objects (numbers, shapes, variables, matrices, ideals, metric spaces, …) and the question, "What are the objects like?" It then moves to the study of actions (functions, rotations, reflections, multiplication, derivatives, shifts, …) and the question, "How do the objects behave?" The study of actions in various mathematical contexts has been extremely fruitful; consider the study of metric spaces through the lens of dynamical systems or the study of symmetries arising from group actions. For objects and actions arising from algebraic combinatorics, we call this study dynamical algebraic combinatorics.Let be a bijective action on a finite set . Such an action breaks the space into orbits. Often, the study of the orbits of provides insight into the structure of the objects in , revealing hidden symmetries and connections. One typically first seeks to understand the order of the action and then finds interesting properties the action exhibits. One surprisingly ubiquitous property is the cyclic sieving phenomenon [ReStWh04], which occurs when the evaluation of a generating function for at the primitive th root of unity (where = 2 / ) counts the number of elements of fixed under .For example, let be the set of binary words composed of two zeros and two ones. Let be the cyclic shift that acts by moving the first digit to the end of the word. Each word has four digits, so is of order four. Consider the inversion number statistic inv( ) of ∈ , which equals the number of pairs ( , ) with < such that the th digit of is 1 and the th digit is 0. See Figure 1 for the orbits of under and the inversion numbers of each binary word.

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Proofs and generalizations of a homomesy conjecture of Propp and Roby

Cited by 25 publications

References 17 publications

Rowmotion and increasing labeling promotion

Rowmotion and increasing labeling promotion

Homomesy in Products of Three Chains and Multidimensional Recombination

Dynamical Algebraic Combinatorics: Promotion, Rowmotion, and Resonance

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