On Noncrossing and Plane Tree-Like Structures

Year 2021, Volume: 4 Issue: 2, 89 - 99, 30.06.2021

Isaac Owino OKOTH

https://doi.org/10.33434/cams.803065

Abstract

Mathematical trees are connected graphs without cycles, loops and multiple edges. Various trees such as Cayley trees, plane trees, binary trees, $d$-ary trees, noncrossing trees among others have been studied extensively. Tree-like structures such as Husimi graphs and cacti are graphs which posses the conditions for trees if, instead of vertices, we consider their blocks. In this paper, we use generating functions and bijections to find formulas for the number of noncrossing Husimi graphs, noncrossing cacti and noncrossing oriented cacti. We extend the work to obtain formulas for the number of bicoloured noncrossing Husimi graphs, bicoloured noncrossing cacti and bicoloured noncrossing oriented cacti. Finally, we enumerate plane Husimi graphs, plane cacti and plane oriented cacti according to number of blocks, block types and leaves.

Keywords

noncrossing tree, plane trees, tree-like structure, Husimi graph, cactus, enumeration

Supporting Institution

Maseno University

Thanks

We thenk DergiPark for allowing us to publish in your journal.

References

• [1] M. B´ona, M. Bousquet, G. Labelle, P. Leroux, Enumeration of m-ary cacti, Adv. Appl. Math., 24 (1) (2000), 22-56.
• [2] P. Flajolet, M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204 (1-3) (1999), 203-229.
• [3] G. W. Ford, G. E. Uhlenbeck, Combinatorial Problems in the Theory of Graphs, I, Proc. Nat. Acad. Sciences, 42 (1956), 122-128.
• [4] F. Harary, G. E. Uhlenbeck, On the number of Husimi trees, Proc. Nat. Aca. Sci., 39 (1953), 315-322.
• [5] K. Husimi, Note on Mayers’ theory of cluster integrals, J. Chem. Phys., 18 (1950), 682-684.
• [6] S. Kim, S. Seo, H. Shin, Refined enumeration of vertices among all rooted rooted d-trees, (2018), arXiv:1806.06417.
• [7] P. Leroux. Enumerative problems inspired by Mayer’s theory of cluster integrals, Electron. J. Combin., 11 (2004).
• [8] J. E. Mayer, Equilibrium Statistical Mechanics, The international encyclopedia of physical chemistry and chemical physics, Pergamon Press, Oxford, 1968.
• [9] M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180 (1-3) (1998), 301-313.
• [10] I. O. Okoth, Combinatorics of oriented trees and tree-like structures, PhD Thesis, Stellenbosch University, 2015.
• [11] J. H. Przytycki, A. S. Sikora, Polygon Dissections and Euler, Fuss, Kirkman, and Cayley Numbers, J. Combin. Theory, Ser. A, 92 (1) (2000), 68-76.
• [12] C. Springer, Factorizations, Trees, and Cacti, Proceedings of the Eighth International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC), University of Minnesota,(1996), 427-438.
• [13] R. P. Stanley. Enumerative Combinatorics. Vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999.
• [14] E. Tzanaki, Polygon dissections and some generalizations of cluster complexes, J. Combin. Theory, Ser. A, 113(6) (2006), 1189-1198.