Chapoton triangles for nonkissing complexes
Algebraic Combinatorics, Volume 3 (2020) no. 6, pp. 1331-1363.

We continue the study of the nonkissing complex that was introduced by Petersen, Pylyavskyy, and Speyer and was studied lattice-theoretically by the second author. We introduce a theory of Grid–Catalan combinatorics, given the initial data of a nonkissing complex, and show how this theory parallels the well-known Coxeter–Catalan combinatorics. In particular, we present analogues of Chapoton’s F-triangle, H-triangle, and M-triangle and give both combinatorial and lattice-theoretic interpretations of the objects defining these polynomials. In our Grid–Catalan setting, we prove that our analogue of Chapoton’s F-triangle and H-triangle identity holds, and we conjecture that our analogue of Chapoton’s F-triangle and M-triangle identity also holds.

Received:
Accepted:
Published online:
DOI: 10.5802/alco.142
Classification: 05A19, 06A07, 06B10
Keywords: lattice, Catalan number, Cambrian lattice, noncrossing partition, nonkissing complex
Garver, Alexander 1; McConville, Thomas 2

1 Department of Mathematics University of Michigan
2 Department of Mathematics Kennesaw State University
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Garver, Alexander; McConville, Thomas. Chapoton triangles for nonkissing complexes. Algebraic Combinatorics, Volume 3 (2020) no. 6, pp. 1331-1363. doi : 10.5802/alco.142. https://alco.centre-mersenne.org/articles/10.5802/alco.142/

[1] Armstrong, Drew Generalized noncrossing partitions and combinatorics of Coxeter groups, Mem. Amer. Math. Soc., 202, American Mathematical Society, 2009 no. 949, x+159 pages | DOI | MR | Zbl

[2] Athanasiadis, Christos A. On some enumerative aspects of generalized associahedra, European J. Combin., Volume 28 (2007) no. 4, pp. 1208-1215 | DOI | MR | Zbl

[3] Barnard, Emily The canonical join complex, Electron. J. Combin., Volume 26 (2019) no. 1, Paper no. Paper No. 1.24, 25 pages | MR | Zbl

[4] Bessis, David The dual braid monoid, Ann. Sci. École Norm. Sup. (4), Volume 36 (2003) no. 5, pp. 647-683 | DOI | Numdam | MR | Zbl

[5] Björner, Anders; Wachs, Michelle L. Shellable nonpure complexes and posets. I, Trans. Amer. Math. Soc., Volume 348 (1996) no. 4, pp. 1299-1327 | DOI | MR | Zbl

[6] Björner, Anders; Wachs, Michelle L. Shellable nonpure complexes and posets. II, Trans. Amer. Math. Soc., Volume 349 (1997) no. 10, pp. 3945-3975 | DOI | MR | Zbl

[7] Brady, Thomas; Watt, Colum K(π,1)’s for Artin Groups of Finite Type, Geom. Dedicata, Volume 94 (2002), pp. 225-250 | DOI | MR | Zbl

[8] Chapoton, Frédéric Enumerative properties of generalized associahedra, Sém. Lothar. Combin., Volume 51 (2004/05), Paper no. Art. B51b, 16 pages | MR | Zbl

[9] Chapoton, Frédéric Sur le nombre de réflexions pleines dans les groupes de Coxeter finis, Bull. Belg. Math. Soc. Simon Stevin, Volume 13 (2006) no. 4, pp. 585-596 | DOI | MR | Zbl

[10] Chapoton, Frédéric Stokes posets and serpent nests, Discrete Math. Theor. Comput. Sci., Volume 18 (2016) no. 3, Paper no. Paper No. 18, 30 pages | MR | Zbl

[11] Day, Alan Congruence normality: the characterization of the doubling class of convex sets, Algebra Universalis, Volume 31 (1994) no. 3, pp. 397-406 | DOI | MR | Zbl

[12] Fomin, Sergey; Williams, Lauren K.; Zelevinsky, Andrei Introduction to Cluster Algebras Chapters 1–3 (2016) (arXiv preprint https://arxiv.org/abs/1608.05735)

[13] Fomin, Sergey; Zelevinsky, Andrei Y-systems and generalized associahedra, Ann. of Math. (2), Volume 158 (2003) no. 3, pp. 977-1018 | DOI | MR | Zbl

[14] Freese, Ralph; Ježek, Jaroslav; Nation, James B. Free lattices, Mathematical Surveys and Monographs, 42, American Mathematical Society, Providence, RI, 1995, viii+293 pages | DOI | MR | Zbl

[15] Garver, Alexander; McConville, Thomas Oriented flip graphs of polygonal subdivisions and noncrossing tree partitions, J. Combin. Theory Ser. A, Volume 158 (2018), pp. 126-175 | DOI | MR | Zbl

[16] Hudson, John F. P. Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969, ix+282 pages | MR | Zbl

[17] Kreweras, Germain Sur les partitions non croisées d’un cycle, Discrete Math., Volume 1 (1972) no. 4, pp. 333-350 | DOI | MR | Zbl

[18] Manneville, Thibault The serpent nest conjecture for accordion complexes, European J. Combin., Volume 67 (2018), pp. 230-238 | DOI | MR | Zbl

[19] McConville, Thomas Lattice structure of Grid–Tamari orders, J. Combin. Theory Ser. A, Volume 148 (2017), pp. 27-56 | DOI | MR | Zbl

[20] Petersen, T. Kyle; Pylyavskyy, Pavlo; Speyer, David E. A non-crossing standard monomial theory, J. Algebra, Volume 324 (2010) no. 5, pp. 951-969 | DOI | MR | Zbl

[21] Reading, Nathan Lattice congruences, fans and Hopf algebras, J. Combin. Theory Ser. A, Volume 110 (2005) no. 2, pp. 237-273 | DOI | MR | Zbl

[22] Reading, Nathan Noncrossing arc diagrams and canonical join representations, SIAM J. Discrete Math., Volume 29 (2015) no. 2, pp. 736-750 | DOI | MR | Zbl

[23] Reading, Nathan Lattice theory of the poset of regions, Lattice theory: special topics and applications. Vol. 2, Birkhäuser/Springer, Cham, 2016, pp. 399-487 | MR | Zbl

[24] Santos, Francisco; Stump, Christian; Welker, Volkmar Noncrossing sets and a Grassmann associahedron, Forum Math. Sigma, Volume 5 (2017), Paper no. Paper No. e5, 49 pages | DOI | MR | Zbl

[25] Sloane, Neil J. A. The on-line encyclopedia of integer sequences (OEIS) A060854 | MR | Zbl

[26] Stanley, Richard P. Catalan numbers, Cambridge University Press, New York, 2015, viii+215 pages | DOI | MR | Zbl

[27] Stein, William A. et al. Sage Mathematics Software (Version 7.3) (2016) (http://www.sagemath.org)

[28] Thiel, Marko On the H-triangle of generalised nonnesting partitions, European J. Combin., Volume 39 (2014), pp. 244-255 | DOI | MR

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