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Cubic realizations of Tamari interval lattices. (English) Zbl 1436.05010

Summary: We introduce cubic coordinates, which are integer words encoding intervals in the Tamari lattices. Cubic coordinates are in bijection with interval-posets, themselves known to be in bijection with Tamari intervals. We show that in each degree the set of cubic coordinates forms a lattice, isomorphic to the lattice of Tamari intervals. Geometric realizations are naturally obtained by placing cubic coordinates in space, highlighting some of their properties. Finally, we consider the cellular structure of these realizations.

MSC:

05A15 Exact enumeration problems, generating functions
06B99 Lattices
06A07 Combinatorics of partially ordered sets
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References:

[1] F. Bergeron and L.-F. Préville-Ratelle. “Higher trivariate diagonal harmonics via generalized Tamari posets”.J. Combin.3.3 (2012), pp. 317-341.Link. · Zbl 1291.05213
[2] O. Bernardi and N. Bonichon. “Intervals in Catalan lattices and realizers of triangulations”. J. Combin. Theory Ser. A116.1 (2009), pp. 55-75.Link. · Zbl 1161.06001
[3] M. Bousquet-Mélou, E. Fusy, and L.-F. Préville-Ratelle. “The number of intervals in the m-Tamari lattices”.Electronic J. Combin.18.2 (2012), Art. P31, 26 pp.Link. · Zbl 1262.05005
[4] F. Chapoton. “Sur le nombre d’intervalles dans les treillis de Tamari”.Sém. Lothar. Combin. 55(2006), Art. B55f, 18 pp.Link. · Zbl 1207.05011
[5] F. Chapoton. “Une note sur les intervalles de Tamari”.Ann. Math. Blaise Pascal25.2 (2018), pp. 299-314.Link. · Zbl 1507.06001
[6] G. Châtel and V. Pons. “Counting smaller elements in the Tamari andm-Tamari lattices”. J. Combin. Theory Ser. A134(2015), pp. 58-97.Link. · Zbl 1315.05143
[7] W. Fang and L.-F. Préville-Ratelle. “The enumeration of generalized Tamari intervals”. European J. Combin.61(2017), pp. 69-84.Link. · Zbl 1352.05191
[8] S. Giraudo. “Combinatoire algébrique des arbres”. Section 4.4.3. PhD thesis. Université Paris-Est Marne-la-Vallée, 2011.
[9] J. M. Pallo. “Enumerating, ranking and unranking binary trees”.Comput. J.29.2 (1986), pp. 171-175.Link. · Zbl 0585.68066
[10] L.-F. Préville-Ratelle and X. Viennot. “The enumeration of generalized Tamari intervals”. Trans. Amer. Math. Soc.369.7 (2017), pp. 5219-5239.Link. · Zbl 1433.05323
[11] B. Rognerud. “Exceptional and modern intervals of the Tamari lattice”.To appear in Sém. Lothar. Combin.(2019). · Zbl 1420.19002
[12] D. Tamari. “The algebra of bracketings and their enumeration”.Nieuw Arch. Wisk. (3)10 (1962), pp. 131-146. · Zbl 0109.24502
[13] W.
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