Non-crossing Trees, Quadrangular Dissections, Ternary Trees, and Duality-Preserving Bijections

Abstract

Using the theory of Properly Embedded Graphs developed in an earlier work we define an involutory duality on the set of labeled non-crossing trees that lifts the obvious duality in the set of unlabeled non-crossing trees. The set of non-crossing trees is a free ternary magma with one generator and this duality is an instance of a duality that is defined in any such magma. Any two free ternary magmas with one generator are isomorphic via a unique isomorphism that we call the structural bijection. Besides the set of non-crossing trees we also consider as free ternary magmas with one generator the set of ternary trees, the set of quadrangular dissections, and the set of flagged Perfectly Chain Decomposed Ditrees, and we give topological and/or combinatorial interpretations of the structural bijections between them. In particular the bijection from the set of quadrangular dissections to the set of non-crossing trees seems to be new. Further we give explicit formulas for the number of self-dual labeled and unlabeled non-crossing trees and the set of quadrangular dissections up to rotations and up to rotations and reflections.

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Notes

  1. 1.

    See for example [16] for the basic definitions of Fuss–Catalan numbers, called generalized Catalan numbers there.

  2. 2.

    All the relevant notions and terminology are reviewed in Sect. 2.5.

  3. 3.

    For an explanation of the term mind-body see Section 2.3 of [1].

  4. 4.

    Thanks to the anonymous referee of a previous version for pointing this out.

  5. 5.

    This definition was given for ternary trees in [6]. See also Remark 2.9.

  6. 6.

    Called “symmetric ternary trees” there.

  7. 7.

    See for example [27, sections 5.3 and 11.1], or any “Discrete Mathematics” textbook.

  8. 8.

    For general pegs these two arcs could be the same, but this cannot happen for nc-trees, except in the degenerate case of the tree with no edges.

  9. 9.

    First observed by Dénes in [5].

  10. 10.

    Alternatively we can obtain the peg as the total space of a branched covering of the disk, see [1, Section 4.4].

  11. 11.

    Recall that we use left and right exponential notation for conjugation. Since transpositions are involutions the distinction is mute in the current context; however, it is useful in more general contexts.

  12. 12.

    What we called c in Remark 2.16.

  13. 13.

    Essentially due to Moszowski [20].

  14. 14.

    This is different than the formula in [2]. The inconsistency is due to different conventions on how to multiply transpositions and how exactly the braid group acts. With our conventions Armstrong’s formula would give \(K^{-1}\) that corresponds to the action of \(\Delta _m^{-1}\). Of course, \(K^{-1}\) is also a complement in the lattice.

  15. 15.

    Actually when \(\Gamma \) is a tree they are required!

  16. 16.

    Directed Acyclic Graph. This observation is essentially due to [8]. The definition of medial digraphs was inspired in part from that paper.

  17. 17.

    Recall that a ditree is a digraph whose underlying graph is a tree.

  18. 18.

    That is a linear extension of the poset whose Hasse diagram is the dag.

  19. 19.

    The first one may even be called pointless.

  20. 20.

    Thanks to the anonymous referee that brought this work to my attention.

  21. 21.

    The enumeration of oriented unlabeled trees is not explicitly stated there but a formula can be deduced from the calculations.

  22. 22.

    It follows that so is the triple \(\left( \mathcal {N}_m, \nu _q,\mathrm {C}_{m+1} \right) \).

References

  1. 1.

    N. Apostolakis. A duality for labeled graphs and factorizations with applications to graph embeddings and Hurwitz enumeration, April 2018. arXiv: 1804.01214.

  2. 2.

    D. Armstrong. Generalized noncrossing partitions and combinatorics of Coxeter groups. Mem. Amer. Math. Soc., 202(949):x+159, 2009.

  3. 3.

    Ph. Chassaing and G. Schaeffer. Random planar lattices and integrated superBrownian excursion. Probab. Theory Related Fields, 128(2):161–212, 2004.

    MathSciNet  Article  Google Scholar 

  4. 4.

    J. Cigler. Some remarks on Catalan families. European J. Combin., 8(3):261–267, 1987.

    MathSciNet  Article  Google Scholar 

  5. 5.

    J. Dénes. The representation of a permutation as the product of a minimal number of transpositions, and its connection with the theory of graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl., 4:63–71, 1959.

  6. 6.

    E. Deutsch, S. Feretić, and M. Noy. Diagonally convex directed polyominoes and even trees: a bijection and related issues. Discrete Math., 256(3):645–654, 2002. LaCIM 2000 Conference on Combinatorics, Computer Science and Applications (Montreal, QC).

  7. 7.

    The Sage Developers. SageMath, the Sage Mathematics Software System (Version 8.0), 2017. http://www.sagemath.org.

  8. 8.

    S. Dulucq and J.G. Penaud. Cordes, arbres et permutations. Discrete Math., 117(1-3):89–105, 1993.

    MathSciNet  Article  Google Scholar 

  9. 9.

    J. A. Eidswick. Short factorizations of permutations into transpositions. Discrete Math., 73(3):239–243, 1989.

    MathSciNet  Article  Google Scholar 

  10. 10.

    S.P Eu and T.S. Fu. The cyclic sieving phenomenon for faces of generalized cluster complexes. Adv. in Appl. Math., 40(3):350–376, 2008.

  11. 11.

    The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.7.8, 2015.

  12. 12.

    R. L. Graham, D. E. Knuth, and O. Patashnik. Concrete mathematics. Addison-Wesley Publishing Company, Reading, MA, second edition, 1994. A foundation for computer science.

  13. 13.

    J. L. Gross and T. W. Tucker. Topological Graph Theory. Dover Books on Mathematics Series. Dover Publications, 1987.

    MATH  Google Scholar 

  14. 14.

    F. Harary, E. M. Palmer, and R. C. Read. On the cell-growth problem for arbitrary polygons. Discrete Math., 11:371–389, 1975.

    MathSciNet  Article  Google Scholar 

  15. 15.

    M. C. Herando. Complejidad de Estructuras Geométricas y Combinatorias. PhD thesis, Universitat Politècnica de Catalunya, 1999.

  16. 16.

    P. Hilton and J. Pedersen. Catalan numbers, their generalization, and their uses. Math. Intelligencer, 13(2):64–75, 1991.

    MathSciNet  Article  Google Scholar 

  17. 17.

    D. E. Knuth. Donald Knuth’s 20th Annual Christmas Tree Lecture: \(3/2\)–ary Trees. https://www.youtube.com/watch?v=P4AaGQIo0HY, 2014. Accessed 07/13/2018.

  18. 18.

    S. K. Lando and A. K. Zvonkin. Graphs on surfaces and their applications, volume 141 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 2004. With an appendix by Don B. Zagier, Low-Dimensional Topology, II.

  19. 19.

    J. Q. Longyear. Graphs and permutations. In Graph theory and its applications: East and West (Jinan, 1986), volume 576 of Ann. New York Acad. Sci., pages 385–388. New York Acad. Sci., New York, 1989.

  20. 20.

    P. Moszkowski. A solution to a problem of Dénes: a bijection between trees and factorizations of cyclic permutations. European J. Combin., 10(1):13–16, 1989.

    MathSciNet  Article  Google Scholar 

  21. 21.

    G. Musiker, R. Schiffler, and L. Williams. Positivity for cluster algebras from surfaces. Adv. Math., 227(6):2241–2308, 2011.

    MathSciNet  Article  Google Scholar 

  22. 22.

    M. Noy. Enumeration of noncrossing trees on a circle. Discrete Math., 180(1–3):301 – 313, 1998. Proceedings of the 7th Conference on Formal Power Series and Algebraic Combinatorics.

  23. 23.

    J. H. Przytycki and A. S. Sikora. Polygon dissections and Euler, Fuss, Kirkman, and Cayley numbers. J. Combin. Theory Ser. A, 92(1):68–76, 2000.

    MathSciNet  Article  Google Scholar 

  24. 24.

    S. Rao and J. Suk. Dihedral sieving phenomena. Discrete Math., 343(6):111849, 12, 2020.

  25. 25.

    V. Reiner, D. Stanton, and D. White. The cyclic sieving phenomenon. J. Combin. Theory Ser. A, 108(1):17–50, 2004.

    MathSciNet  Article  Google Scholar 

  26. 26.

    R. W. Robinson. Counting graphs with a duality property. In Combinatorics (Swansea, 1981), volume 52 of London Math. Soc. Lecture Note Ser., pages 156–186. Cambridge Univ. Press, Cambridge-New York, 1981.

  27. 27.

    K. Rosen. Discrete Mathematics and Its Applications. McGraw-Hill, seventh edition, 2012.

  28. 28.

    F. Ruskey. Generating linear extensions of posets by transpositions. J. Comb. Theory, Ser. B, 54(1):77–101, 1992.

  29. 29.

    G. Schaeffer. Conjugaison d’arbres et cartes combinatoires aléatoires. PhD thesis, Université Bordeaux I, 1998.

  30. 30.

    N. J. A. Sloane, editor. The On-Line Encyclopedia of Integer Sequences. Published electronically at https://oeis.org.

  31. 31.

    Z. Stier, J. Wellman, and Z.X. Xu. Dihedral sieving on cluster complexes, 2019.

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Correspondence to Nikos Apostolakis.

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The Computer Algebra Systems Sage [7] and GAP [11] were used extensively to confirm calculations and check conjectures at several stages of this project. I would like to thank Cormac O’Sullivan for valuable comments. Finally, I would also like to thank the referees of this and earlier versions of the paper for invaluable comments and for suggesting interesting connections with the literature.

Communicated by Matjaz Konvalinka.

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Apostolakis, N. Non-crossing Trees, Quadrangular Dissections, Ternary Trees, and Duality-Preserving Bijections. Ann. Comb. 25, 345–392 (2021). https://doi.org/10.1007/s00026-021-00531-w

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