Dynamical algebraic combinatorics and the homomesy phenomenon

Roby, Tom

doi:10.1007/978-3-319-24298-9_25

Cited by 40 publications

(35 citation statements)

References 35 publications

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“…Many of the algebraic properties that hold in the combinatorial setting have also been proven for the piecewise-linear setting, and furthermore generalized to the birational setting [EP18, GR14,Rob16]. We will show that almost all that we proved for the relationship between toggles in Tog A (P ) and Tog J (P ) also extends to the piecewise-linear setting.…”

Section: Piecewise-linear Generalizationmentioning

confidence: 63%

Antichain Toggling and Rowmotion

Joseph

2019

Electron. J. Combin.

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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 the study of toggle groups on general families of subsets, including antichains. This paper expands on this work by examining the relationship between the toggle groups of antichains and order ideals, constructing an explicit isomorphism between the two groups (for a finite poset). We also focus on the rowmotion action on antichains of a poset that has been well-studied in dynamical algebraic combinatorics, describing it as the composition of antichain toggles. We also describe a piecewise-linear analogue of toggling to Stanley's chain polytope. We examine the connections with the piecewise-linear toggling Einstein and Propp introduced for order polytopes and prove that almost all of our results for antichain toggles extend to the piecewise-linear setting.2010 Mathematics Subject Classification. 05E18.

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Section: Piecewise-linear Generalizationmentioning

confidence: 63%

Antichain Toggling and Rowmotion

Joseph

2019

Electron. J. Combin.

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“…In recent years, especially in the context of "dynamical algebraic combinatorics" [68,86], other aspects of rowmotion beyond its orbit structure have been investigated. There has been a particular focus on exhibiting homomesies for rowmotion.…”

Section: Rowmotionmentioning

confidence: 99%

Minuscule Doppelgängers, The Coincidental down-Degree Expectations Property, and Rowmotion

Hopkins

2020

Experimental Mathematics

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We relate Reiner, Tenner, and Yong's coincidental down-degree expectations (CDE) property of posets to the minuscule doppelgänger pairs studied by Hamaker, Patrias, Pechenik, and Williams. Via this relation, we put forward a series of conjectures which suggest that the minuscule doppelgänger pairs behave "as if" they had isomorphic comparability graphs, even though they do not. We further explore the idea of minuscule doppelgänger pairs pretending to have isomorphic comparability graphs by considering the rowmotion operator on order ideals. We conjecture that the members of a minuscule doppelgänger pair behave the same way under rowmotion, as they would if they had isomorphic comparability graphs. Moreover, we conjecture that these pairs continue to behave the same way under the piecewise-linear and birational liftings of rowmotion introduced by Einstein and Propp. This conjecture motivates us to study the homomesies (in the sense of Propp and Roby) exhibited by birational rowmotion. We establish the birational analog of the antichain cardinality homomesy for the major examples of posets known or conjectured to have finite birational rowmotion order (namely: minuscule posets and root posets of coincidental type).

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“…Roughly speaking, a toggle is an involution on a set of combinatorial objects that only makes a small local change. See [36] for more information about toggles.…”

Section: Toggles Trees and Permutationsmentioning

confidence: 99%

Polyurethane Toggles

Defant

2020

Electron. J. Combin.

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We consider the involutions known as toggles, which have been used to give simplified proofs of the fundamental properties of the promotion and evacuation maps. We transfer these involutions so that they generate a group $\mathscr P_n$ that acts on the set $S_n$ of permutations of $\{1,\ldots,n\}$. After characterizing its orbits in terms of permutation skeletons, we apply the action in order to understand West's stack-sorting map. We obtain a very simple proof of a result that clarifies and extensively generalizes a theorem of Bouvel and Guibert and also generalizes a theorem of Bousquet-M\'elou. We also settle a conjecture of Bouvel and Guibert. We prove a result related to the recently-introduced notion of postorder Wilf equivalence. Finally, we investigate an interesting connection among the action of $\mathscr P_n$ on $S_n$, the group structure of $S_n$, and the stack-sorting map.

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Dynamical algebraic combinatorics and the homomesy phenomenon

Cited by 40 publications

References 35 publications

Antichain Toggling and Rowmotion

Antichain Toggling and Rowmotion

Minuscule Doppelgängers, The Coincidental down-Degree Expectations Property, and Rowmotion

Polyurethane Toggles

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