The birational Lalanne-Kreweras involution

Hopkins, Sam; Joseph, Michael

doi:10.48550/arxiv.2012.15795

Cited by 3 publications

(8 citation statements)

References 30 publications

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“…That the antichain cardinality statistic is homomesic for rowmotion for the Type A root poset was first proved by Armstrong, Stump, and Thomas [2, Theorem 1.2(iii)] (and in fact, they showed this for all root posets, as we will discuss in a moment). The homomesy of the R i statistics under rowmotion was observed recently by Hopkins and Joseph [18,Corollary 4.15]. The stronger result that these statistics are ≡ const follows from the work of Chan, Haddadan, Hopkins, and Moci [6], in exactly the way we have presented here.…”

Section: 4supporting

confidence: 84%

Homomesy via Toggleability Statistics

Defant¹,

Hopkins²,

Poznanović³

et al. 2021

Preprint

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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 every orbit of the action. We systematize a technique for proving that various statistics of interest are homomesic by writing these statistics as linear combinations of "toggleability statistics" (originally introduced by Striker) plus a constant. We show that this technique recaptures most of the known homomesies for the posets on which rowmotion has been most studied. We also show that the technique continues to work in modified contexts. For instance, this technique also yields homomesies for the piecewise-linear and birational extensions of rowmotion; furthermore, we introduce a q-analogue of rowmotion and show that the technique yields homomesies for "q-rowmotion" as well.

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Section: 4supporting

confidence: 84%

Homomesy via Toggleability Statistics

Defant¹,

Hopkins²,

Poznanović³

et al. 2021

Preprint

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“…It was recently shown by the second author and Joseph [15] that the Lalanne-Kreweras involution is rowvacuation for each root poset of Type A. Together with Panyushev's prior work from [18], this proves Theorem 1.4 for Types A, B, and C. So the only case we have to address here is Type D. 1 However, along the way, we prove results concerning rowvacuation of NN(W ) for arbitrary Φ.…”

Section: Introduction and Statement Of Resultssupporting

confidence: 54%

“…Panyushev was unable to define P in general, but in [18], he was able to come up with a definition in Type A. In fact, Panyushev's involution P in Type A is equivalent to the Lalanne-Kreweras involution on Dyck paths (see [15]). A simple "folding" argument allows one to obtain the appropriate involution P in Types B/C from the one in Type A, so Panyushev was also able to treat Types B/C.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…Observe that #A + # Rvac(A) = 2 + 2 = 4. ♦ Rowvacuation was first formally defined, briefly, in [14, §5.1] and was further studied in [15]. As mentioned in Section 1, rowmotion and rowvacuation are "partner" operators in exactly the same way that Schützenberger's promotion and evacuation operators are.…”

Section: 2mentioning

confidence: 99%

“…In general, it seems hard to "describe" the rowvacuation of an antichain. But in [15], the second author and Joseph showed that rowvacuation acting on a root poset of Type A can be computed as follows.…”

Section: 2mentioning

confidence: 99%

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Symmetry of Narayana numbers and rowvacuation of root posets

Defant,

Hopkins

2021

Preprint

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For a Weyl group W of rank r, the W -Catalan number is the number of antichains of the poset of positive roots, and the W -Narayana numbers refine the W -Catalan number by keeping track of the cardinalities of these antichains. The W -Narayana numbers are symmetric, i.e., the number of antichains of cardinality k is the same as the number of cardinality r − k. However, this symmetry is far from obvious. Panyushev posed the problem of defining an involution on root poset antichains that exhibits the symmetry of the W -Narayana numbers.Rowmotion and rowvacuation are two related operators, defined as compositions of "toggles," that give a dihedral action on the set of antichains of any ranked poset. Rowmotion acting on root posets has been the subject of a significant amount of research in the recent past. We prove that for the root posets of classical types, rowvacuation is Panyushev's desired involution.

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Rowmotion on 321-Avoiding Permutations

Ben

Elizalde

2023

Electron. J. Combin.

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We give a natural definition of rowmotion for $321$-avoiding permutations, by translating, through bijections involving Dyck paths and the Lalanne--Kreweras involution, the analogous notion for antichains of the positive root poset of type $A$. We prove that some permutation statistics, such as the number of fixed points, are homomesic under rowmotion, meaning that they have a constant average over its orbits. Our setting also provides a more natural description of the celebrated Armstrong--Stump--Thomas equivariant bijection between antichains and non-crossing matchings in types $A$ and $B$, by showing that it is equivalent to the Robinson--Schensted--Knuth correspondence on $321$-avoiding permutations permutations.

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The birational Lalanne-Kreweras involution

Cited by 3 publications

References 30 publications

Homomesy via Toggleability Statistics

Homomesy via Toggleability Statistics

Symmetry of Narayana numbers and rowvacuation of root posets

Rowmotion on 321-Avoiding Permutations

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