Homomesy in products of two chains. (English) Zbl 1319.05151

Propp, James; Roby, Tom

Electron. J. Comb. 22, No. 3, Research Paper P3.4, 29 p. (2015).

Summary: Many invertible actions \(\tau\) on a set \(\mathcal{S}\) of combinatorial objects, along with a natural statistic \(f\) on \(\mathcal{S}\), exhibit the following property which we dub homomesy: the average of \(f\) over each \(\tau\)-orbit in \(\mathcal{S}\) is the same as the average of \(f\) over the whole set \(\mathcal{S}\). This phenomenon was first noticed by D. I. Panyushev [Eur. J. Comb. 30, No. 2, 586–594 (2009; Zbl 1165.06001)] in the context of the rowmotion action on the set of antichains of a root poset; D. Armstrong et al. [Trans. Am. Math. Soc. 365, No. 8, 4121–4151 (2013; Zbl 1271.05011)] proved Panyushev’s conjecture. We describe a theoretical framework for results of this kind that applies more broadly, giving examples in a variety of contexts. These include linear actions on vector spaces, sandpile dynamics, Suter’s action on certain subposets of Young’s Lattice, Lyness 5-cycles, promotion of rectangular semi-standard Young tableaux, and the rowmotion and promotion actions on certain posets. We give a detailed description of the latter situation for products of two chains.

MSC:

05E18	Group actions on combinatorial structures
05E10	Combinatorial aspects of representation theory
06A11	Algebraic aspects of posets

Keywords:

antichains; Ballot theorems; homomesy; Lyness 5-cycle; orbit; order ideals; Panyushev complementation; permutations; poset; product of chains; promotion; rowmotion; sandpile; Suter’s symmetry; toggle group; Young’s lattice; Young tableaux

Citations:

Zbl 1165.06001; Zbl 1271.05011

References:

[1]	D. Armstrong, Generalized noncrossing partitions and combinatorics of Coxeter groups, Mem. Amer. Math. Soc. 202 (2006), no. 949. · Zbl 1191.05095
[2]	D. Armstrong, C. Stump and H. Thomas, A uniform bijection between nonnest ing and noncrossing partitions, Trans. Amer. Math. Soc. 365 (2013), no. 8, 4121-4151.arXiv:1101.1277v2 · Zbl 1271.05011
[3]	E. Barbier, Generalisation du probleme resolu par M. J. Bertrand, Comptes´ Rendus de l’Academie des Sciences, 105 (1887), 407.
[4]	J. Bloom, O. Pechenik, and D. Saracino, Proofs and generalizations of a homo mesy conjecture of Propp and Roby,arXiv:1308.0546v2. · Zbl 1322.05136
[5]	A. Brouwer and A. Schrijver, On the period of an operator, defined on antichains, Math. Centrum report ZW 24/74 (1974). · Zbl 0282.06003
[6]	P. Cameron and D.G. Fon-Der-Flaass, Orbits of Antichains Revisited, Europ. J. Comb. 16 (1995), 545-554. · Zbl 0831.06001
[7]	A. Dvoretzky and T. Motzkin, A problem of arrangements, Duke Math. J. 14 (1947), 305-313. · Zbl 0030.16701
[8]	D. Einstein and J. Propp, Combinatorial, piecewise-linear, and birational ho momesy for products of two chains, 2013.arXiv:1310.5294v1
[9]	S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002), 497-529. · Zbl 1021.16017
[10]	D.G. Fon-Der-Flaass, Orbits of Antichains in Ranked Posets, Europ. J. Comb. 14 (1993), 17-22. the electronic journal of combinatorics 22(3) (2015), #P3.4 28 · Zbl 0777.06002
[11]	R.L. Graham, D.E. Knuth, and O. Patashnik, Concrete Mathematics: A foun dation for computer science, 2nd Ed., Addison-Wesley, 1994. · Zbl 0836.00001
[12]	A. Holroyd and J. Propp, Rotor walks and Markov chains, in Algorithmic Prob ability and Combinatorics, American Mathematical Society (2010), 105-126. [H+08]A. Holroyd, L. Levine, K. M´esz´aros, Y. Peres, J. Propp, and D. Wilson, Chip-Firing and Rotor-Routing on Directed Graphs, Progress in Probability, 60 (2008), 331-364. For an updated version see arXiv:0801.3306v4. · Zbl 1173.82339
[13]	A. N. Kirillov and A. D. Berenstein, Groups generated by involutions, Gelfand Tsetlin patterns and combinatorics of Young tableaux, Algebra i Analiz 7 (1995), 92-152; English transl., St. Petersburg Math. J. 7 (1996), no. 1, 77-127; also available (as of July 2013) at http://math.uoregon.edu/ arkadiy/bk1.pdf · Zbl 0848.20007
[14]	L. Levine and J. Propp, What is a . . . sandpile?, Notices of the AMS 57(8) (2010), 976-979. · Zbl 1203.82061
[15]	D.I. Panyushev, On orbits of antichains of positive roots, Europ. J. Comb. 30(2) (2009), 586-594.arXiv:0711.3353v2 · Zbl 1165.06001
[16]	J. Propp, The combinatorics of frieze patterns and Markoff numbers, 2005. arXiv:math/0511633v4 · Zbl 1435.05018
[17]	V. Reiner, D. Stanton, and D. White, The cyclic sieving phenomenon, J. Com bin. Theory Ser. A 108 (2004), 17-50. · Zbl 1052.05068
[18]	V. Reiner, D. Stanton, and D. White, What is. . . cyclic sieving?, Notices of the AMS 61(2) (2014), 169-171. · Zbl 1338.05012
[19]	M. Renault, Four Proofs of the Ballot Theorem, Math. Mag. 80 (2007), 345-352. · Zbl 1144.05303
[20]	B. Rhoades, Cyclic sieving, promotion, and representation theory, J. Combin. Theory Ser. A 117 (2010), 38-76.arXiv:1005.2568v1 · Zbl 1230.05289
[21]	R. Stanley, Enumerative Combinatorics Volume 1, 2nd Ed., no. 49 in Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2011.
[22]	R. Stanley, Enumerative Combinatorics Volume 2, no. 62 in Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1999.
[23]	R. Stanley, Promotion and Evacuation, Electronic J. Comb. 16(2) (2009), #R9. · Zbl 1169.06002
[24]	R. Suter, Young’s Lattice and Dihedral Symmetries, Europ. J. Comb. 23 (2002), 233-238. · Zbl 1001.05120
[25]	J. Striker and N. Williams, Promotion and rowmotion, Europ. J. Comb. 33 (2012), 1919-1942. · Zbl 1260.06004
[26]	Wikipedia article Bloch group, downloaded August 28, 2014;http://en. wikipedia.org/wiki/Bloch_group; see the section titled “Bloch-Wigner func tion”. the electronic journal of combinatorics 22(3) (2015), #P3.4 29