Konferans Bildirisi
BibTex RIS Kaynak Göster

On Noncrossing and Plane Tree-Like Structures

Yıl 2021, , 89 - 99, 30.06.2021

Isaac Owino Okoth

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

Öz

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.

Anahtar Kelimeler

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

Destekleyen Kurum

Maseno University

Teşekkür

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

Kaynakça

  • [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.

Yıl 2021, , 89 - 99, 30.06.2021

Isaac Owino Okoth

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

Öz

Kaynakça

  • [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.
Toplam 14 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Makaleler
Yazarlar

Isaac Owino Okoth

Yayımlanma Tarihi 30 Haziran 2021
Gönderilme Tarihi 30 Eylül 2020
Kabul Tarihi 14 Haziran 2021
Yayımlandığı Sayı Yıl 2021

Kaynak Göster

APA Okoth, I. O. (2021). On Noncrossing and Plane Tree-Like Structures. Communications in Advanced Mathematical Sciences, 4(2), 89-99. https://doi.org/10.33434/cams.803065
AMA Okoth IO. On Noncrossing and Plane Tree-Like Structures. Communications in Advanced Mathematical Sciences. Haziran 2021;4(2):89-99. doi:10.33434/cams.803065
Chicago Okoth, Isaac Owino. “On Noncrossing and Plane Tree-Like Structures”. Communications in Advanced Mathematical Sciences 4, sy. 2 (Haziran 2021): 89-99. https://doi.org/10.33434/cams.803065.
EndNote Okoth IO (01 Haziran 2021) On Noncrossing and Plane Tree-Like Structures. Communications in Advanced Mathematical Sciences 4 2 89–99.
IEEE I. O. Okoth, “On Noncrossing and Plane Tree-Like Structures”, Communications in Advanced Mathematical Sciences, c. 4, sy. 2, ss. 89–99, 2021, doi: 10.33434/cams.803065.
ISNAD Okoth, Isaac Owino. “On Noncrossing and Plane Tree-Like Structures”. Communications in Advanced Mathematical Sciences 4/2 (Haziran 2021), 89-99. https://doi.org/10.33434/cams.803065.
JAMA Okoth IO. On Noncrossing and Plane Tree-Like Structures. Communications in Advanced Mathematical Sciences. 2021;4:89–99.
MLA Okoth, Isaac Owino. “On Noncrossing and Plane Tree-Like Structures”. Communications in Advanced Mathematical Sciences, c. 4, sy. 2, 2021, ss. 89-99, doi:10.33434/cams.803065.
Vancouver Okoth IO. On Noncrossing and Plane Tree-Like Structures. Communications in Advanced Mathematical Sciences. 2021;4(2):89-9.

Creative Commons License   The published articles in CAMS are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License..