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Combinatorial Interpretation of Raney Numbers and Tree Enumerations

DOI: 10.4236/ojdm.2015.51001    3,514 Downloads   4,071 Views   Citations

ABSTRACT

A new combinatorial interpretation of Raney numbers is proposed. We apply this combinatorial interpretation to solve several tree enumeration counting problems. Further a generalized Catalan triangle is introduced and some of its properties are proved.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Pah, C. and Wahiddin, M. (2015) Combinatorial Interpretation of Raney Numbers and Tree Enumerations. Open Journal of Discrete Mathematics, 5, 1-9. doi: 10.4236/ojdm.2015.51001.

References

[1] Penson, K.A. and Zyczkowski, K. (2011) Product of Ginibre Matrices: Fuss-Catalan and Raney Distributions. Physical Review E, 83, Article ID: 061118. http://dx.doi.org/10.1103/PhysRevE.83.061118
[2] Mlotkowski, W., Penson, K.A. and Zyczkowski, K. (2013) Densities of the Raney Distributions. Documenta Mathematica, 18, 1573-1596.
[3] Jeurissen, R.H. (2008) Raney and Catalan. Discrete Mathematics, 308, 6298-6307.
http://dx.doi.org/10.1016/j.disc.2007.11.068
[4] Dziemiaczuk, M. (2014) Enumerations of Plane Trees with Multiple Edges and Raney Lattice Paths. Discrete Mathematics, 337, 9-24. http://dx.doi.org/10.1016/j.disc.2014.07.024
[5] Gould, H.W. (1972) Combinatorial Identities: A Standardized Set of Tables Listing 500 Binomial Coefficient Summations. Morgantown.
[6] (2011) The On-Line Encyclopedia of Integer Sequences. Sequence A196678. http://oeis.org
[7] Graham, R.L., Knuth, D.E. and Patashnik, O. (1994) Concrete Mathematics. Addison-Wesley, Boston.
[8] Hilton, P. and Pedersen, J. (1991) Catalan Numbers, Their Generalization, and Their Uses. Mathematical Intelligencer, 13, 64-75. http://dx.doi.org/10.1007/BF03024089
[9] Koshy, T. (2008) Catalan Numbers with Applications. Oxford University Press, USA.
http://dx.doi.org/10.1093/acprof:oso/9780195334548.001.0001
[10] Stanley, R.P. (2013) Catalan Addendum to Enumerative Combinatorics. Vol. 2.
[11] Stanley, R.P. (1999) Enumerative Combinatorics. Vol. 2, Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9780511609589
[12] Baxter, R.J. (1982) Exactly Solved Models in Statistical Mechanics. Academic Press, London.
[13] Goltsev, A.V., Dorogovtsev, S.N. and Mendes, J.F.F. (2008) Critical Phenomena in Complex Networks. Reviews of Modern Physics, 80, 1275. http://dx.doi.org/10.1103/RevModPhys.80.1275
[14] Tamm, U. (2001) Some Aspects of Hankel Matrices in Coding Theory and Combinatorics. The Electronic Journal of Combinatorics, 8, 1-31.
[15] Lee, K., Jung, W.S., Park, J.S. and Choi, M.Y. (2000) Statistical Analysis of the Metropolitan Seoul Subway System: Network Structure and Passenger Flows. Physica A, 387, 6231-6234. http://dx.doi.org/10.1016/j.physa.2008.06.035
[16] Aval, J.C. (2008) Multivariate Fuss-Catalan Numbers. Discrete Mathematics, 308, 4660-4669.
http://dx.doi.org/10.1016/j.disc.2007.08.100
[17] Shapiro, L.W. (1976) A Catalan Triangle. Discrete Mathematics, 14, 83-90.
http://dx.doi.org/10.1016/0012-365X(76)90009-1
[18] Merlini, D., Sprugnoli, R. and Verri, M.C. (2006) Lagrange Inversion: When and How. Acta Applicandae Mathematica, 94, 233-249. http://dx.doi.org/10.1007/s10440-006-9077-7
[19] Pah, C.H. (2008) An Application of Catalan Number on Cayley Tree of Order 2: Single Polygon Counting. Bulletin of the Malaysian Mathematical Sciences Society, 31, 175-183.
[20] Pah, C.H. (2010) Single Polygon Counting on Cayley Tree of Order 3. Journal of Statistical Physics, 140, 198-207.
http://dx.doi.org/10.1007/s10955-010-9989-5
[21] Sasvari, Z. (1999) Inequalities for Binomial Coefficients. Journal of Mathematical Analysis and Applications, 236, 223-226. http://dx.doi.org/10.1006/jmaa.1999.6420
[22] Mukhomedov, F., Pah, C.H. and Saburov, M. (2010) Single Polygon Counting for m Fixed Nodes in Cayley Tree: Two Extremal Cases. Preprint. http://arxiv.org/abs/1004.2305

  
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