ALGEBRAIC COMBINATORICS

no. 2
The birational Lalanne–Kreweras involution
Hopkins, Sam1; Joseph, Michael2
1 Department of Mathematics Howard University Washington DC USA
2 Department of Technology and Mathematics Dalton State College Dalton GA USA
Algebraic Combinatorics, Volume 5 (2022) no. 2, pp. 227-265.

The Lalanne–Kreweras involution is an involution on the set of Dyck paths which combinatorially exhibits the symmetry of the number of valleys and major index statistics. We define piecewise-linear and birational extensions of the Lalanne–Kreweras involution. Actually, we show that the Lalanne–Kreweras involution is a special case of a more general operator, called rowvacuation, which acts on the antichains of any graded poset. Rowvacuation, like the closely related and more studied rowmotion operator, is a composition of toggles. We obtain the piecewise-linear and birational lifts of the Lalanne–Kreweras involution by using the piecewise-linear and birational toggles of Einstein and Propp. We show that the symmetry properties of the Lalanne–Kreweras involution extend to these piecewise-linear and birational lifts.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.201
Classification: 05E18, 06A07, 05A19
Keywords: Lalanne–Kreweras involution, rowvacuation, rowmotion, toggles, piecewise-linear and birational lifts, homomesy
Hopkins, Sam 1; Joseph, Michael 2

1 Department of Mathematics Howard University Washington DC USA
2 Department of Technology and Mathematics Dalton State College Dalton GA USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Hopkins, Sam; Joseph, Michael. The birational Lalanne–Kreweras involution. Algebraic Combinatorics, Volume 5 (2022) no. 2, pp. 227-265. doi : 10.5802/alco.201. https://alco.centre-mersenne.org/articles/10.5802/alco.201/

[1] Adenbaum, Ben; Elizalde, Sergi Rowmotion on 321-avoiding permutations, in preparation, 2021

[2] Armstrong, Drew; Stump, Christian; Thomas, Hugh A uniform bijection between nonnesting and noncrossing partitions, Trans. Amer. Math. Soc., Volume 365 (2013) no. 8, pp. 4121-4151 | DOI | MR | Zbl

[3] Bessis, David; Reiner, Victor Cyclic sieving of noncrossing partitions for complex reflection groups, Ann. Comb., Volume 15 (2011) no. 2, pp. 197-222 | DOI | MR | Zbl

[4] Billey, Sara C.; Jockusch, William; Stanley, Richard P. Some combinatorial properties of Schubert polynomials, J. Algebraic Combin., Volume 2 (1993) no. 4, pp. 345-374 | DOI | MR | Zbl

[5] Brouwer, Andries; Schrijver, Alexander On the period of an operator, defined on antichains, ZW 24/74, Stichting Mathematisch Centrum, 1974, pp. 1-13

[6] Callan, David Bijections from Dyck paths to 321-avoiding permutations revisited (2007) (https://arxiv.org/abs/0711.2684)

[7] Cameron, Peter J.; Fon-Der-Flaass, Dmitri G. Orbits of antichains revisited, European J. Combin., Volume 16 (1995) no. 6, pp. 545-554 | DOI | MR | Zbl

[8] Chan, Melody; Haddadan, Shahrzad; Hopkins, Sam; Moci, Luca The expected jaggedness of order ideals, Forum Math. Sigma, Volume 5 (2017), Paper no. e9, 27 pages | DOI | MR | Zbl

[9] Defant, Colin; Hopkins, Sam Symmetry of Narayana numbers and rowvacuation of root posets, Forum Math. Sigma, Volume 9 (2021), Paper no. e53, 24 pages | DOI | MR | Zbl

[10] Defant, Colin; Hopkins, Sam; Poznanović, Svetlana; Propp, James Homomesy via toggleability statistics (2021) (https://arxiv.org/abs/2108.13227)

[11] Einstein, David; Farber, Miriam; Gunawan, Emily; Joseph, Michael; Macauley, Matthew; Propp, James; Rubinstein-Salzedo, Simon Noncrossing partitions, toggles, and homomesies, Electron. J. Combin., Volume 23 (2016) no. 3, Paper no. 3.52, 26 pages | DOI | MR | Zbl

[12] Einstein, David; Propp, James Combinatorial, piecewise-linear, and birational homomesy for products of two chains, Algebr. Comb., Volume 4 (2021) no. 2, pp. 201-224 | DOI | MR | Zbl

[13] Elizalde, Sergi Fixed points and excedances in restricted permutations, Electron. J. Combin., Volume 18 (2011) no. 2, Paper no. 29, 17 pages | MR | Zbl

[14] Grinberg, Darij; Roby, Tom Iterative properties of birational rowmotion II: rectangles and triangles, Electron. J. Combin., Volume 22 (2015) no. 3, Paper no. 3.40, 49 pages | MR | Zbl

[15] Grinberg, Darij; Roby, Tom Iterative properties of birational rowmotion I: generalities and skeletal posets, Electron. J. Combin., Volume 23 (2016) no. 1, Paper no. 1.33, 40 pages | MR | Zbl

[16] Haddadan, Shahrzad Some instances of homomesy among ideals of posets, Electron. J. Combin., Volume 28 (2021) no. 1, Paper no. 1.60, 23 pages | DOI | MR | Zbl

[17] Hopkins, Sam Cyclic sieving for plane partitions and symmetry, SIGMA Symmetry Integrability Geom. Methods Appl., Volume 16 (2020), Paper no. 130, 40 pages | DOI | MR | Zbl

[18] Hopkins, Sam Minuscule doppelgängers, the coincidental down-degree expectations property, and rowmotion, Experimental Mathematics (2020), pp. 1-29 | DOI

[19] Hopkins, Sam Order polynomial product formulas and poset dynamics (2020) (https://arxiv.org/abs/2006.01568, for the AMS volume on Open Problems in Algebraic Combinatorics to accompany the 2022 OPAC conference at U. Minnesota)

[20] Joseph, Michael Antichain toggling and rowmotion, Electron. J. Combin., Volume 26 (2019) no. 1, Paper no. 1.29, 43 pages | MR | Zbl

[21] Joseph, Michael; Roby, Tom Birational and noncommutative lifts of antichain toggling and rowmotion, Algebr. Comb., Volume 3 (2020) no. 4, pp. 955-984 | DOI | MR | Zbl

[22] Joseph, Michael; Roby, Tom A birational lifting of the Stanley–Thomas word on products of two chains, Discrete Math. Theor. Comput. Sci., Volume 23 (2021) no. 1, Paper no. 17, 20 pages | MR

[23] Kreweras, Germain Sur les éventails de segments, Cahiers du Bureau universitaire de recherche opérationnelle Série Recherche, Volume 15 (1970), pp. 3-41 | Numdam

[24] Lalanne, J.-C. Une involution sur les chemins de Dyck, European J. Combin., Volume 13 (1992) no. 6, pp. 477-487 | DOI | MR | Zbl

[25] Musiker, Gregg; Roby, Tom Paths to understanding birational rowmotion on products of two chains, Algebr. Comb., Volume 2 (2019) no. 2, pp. 275-304 | DOI | Numdam | MR | Zbl

[26] Okada, Soichi Birational rowmotion and Coxeter-motion on minuscule posets, Electron. J. Combin., Volume 28 (2021) no. 1, Paper no. 1.17, 30 pages | DOI | MR | Zbl

[27] Panyushev, Dmitri I. ad-nilpotent ideals of a Borel subalgebra: generators and duality, J. Algebra, Volume 274 (2004) no. 2, pp. 822-846 | DOI | MR | Zbl

[28] Panyushev, Dmitri I. On orbits of antichains of positive roots, European J. Combin., Volume 30 (2009) no. 2, pp. 586-594 | DOI | MR | Zbl

[29] Pon, Steven; Wang, Qiang Promotion and evacuation on standard Young tableaux of rectangle and staircase shape, Electron. J. Combin., Volume 18 (2011) no. 1, Paper no. 18, 18 pages | MR | Zbl

[30] Proctor, Robert A. Shifted plane partitions of trapezoidal shape, Proc. Amer. Math. Soc., Volume 89 (1983) no. 3, pp. 553-559 | DOI | MR | Zbl

[31] Proctor, Robert A. Odd symplectic groups, Invent. Math., Volume 92 (1988) no. 2, pp. 307-332 | DOI | MR | Zbl

[32] Proctor, Robert A. New symmetric plane partition identities from invariant theory work of De Concini and Procesi, European J. Combin., Volume 11 (1990) no. 3, pp. 289-300 | DOI | MR | Zbl

[33] Propp, James A spectral theory for combinatorial dynamics (2021) (https://arxiv.org/abs/2105.11568)

[34] Propp, James; Roby, Tom Homomesy in products of two chains, Electron. J. Combin., Volume 22 (2015) no. 3, Paper no. 3.4, 29 pages | MR | Zbl

[35] Roby, Tom Dynamical algebraic combinatorics and the homomesy phenomenon, Recent trends in combinatorics (IMA Vol. Math. Appl.), Volume 159, Springer, [Cham], 2016, pp. 619-652 | DOI | MR | Zbl

[36] Rush, David B.; Shi, XiaoLin On orbits of order ideals of minuscule posets, J. Algebraic Combin., Volume 37 (2013) no. 3, pp. 545-569 | DOI | MR | Zbl

[37] Rush, David B.; Wang, Kelvin On Orbits of Order Ideals of Minuscule Posets II: Homomesy (2015) (https://arxiv.org/abs/1509.08047)

[38] Schützenberger, Marcel P. Promotion des morphismes d’ensembles ordonnés, Discrete Math., Volume 2 (1972), pp. 73-94 | DOI | MR | Zbl

[39] Stanley, Richard P. Two poset polytopes, Discrete Comput. Geom., Volume 1 (1986) no. 1, pp. 9-23 | DOI | MR | Zbl

[40] Stanley, Richard P. Promotion and evacuation, Electron. J. Combin., Volume 16 (2009) no. 2, Paper no. 9, 24 pages | MR | Zbl

[41] Stanley, Richard P. Enumerative combinatorics. Volume 1, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, Cambridge, 2012, xiv+626 pages | MR | Zbl

[42] Stein, William A. et al. Sage Mathematics Software (Version 9.0) (2020) (http://www.sagemath.org)

[43] Striker, Jessica Rowmotion and generalized toggle groups, Discrete Math. Theor. Comput. Sci., Volume 20 (2018) no. 1, Paper no. 17, 26 pages | MR | Zbl

[44] Striker, Jessica; Williams, Nathan Promotion and rowmotion, European J. Combin., Volume 33 (2012) no. 8, pp. 1919-1942 | DOI | MR | Zbl

[45] Thomas, Hugh; Williams, Nathan Rowmotion in slow motion, Proc. Lond. Math. Soc. (3), Volume 119 (2019) no. 5, pp. 1149-1178 | DOI | MR | Zbl

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