Vector weighted Stirling numbers and an application in graph theory
Abstract
We introduce \textit{vector weighted Stirling numbers}, which are a generalization of ordinary Stirling numbers and restricted Stirling numbers. Some relations between vector weighted Stirling numbers and ordinary Stirling numbers and some of their applications are stated. Moreover, as an application of vector weighted Stirling numbers of the second kind in graph theory, we compute the number of maximal independent sets of different sizes in k-intersection graphs.
Keywords
Full Text:
PDFDOI: http://dx.doi.org/10.5614/ejgta.2021.9.1.20
References
S. Barati, B. Benyi, A. Jafarzadeh, and D. Yaqubi, Mixed restricted Stirling numbers. Acta Math. Hungar. 158 (2019), 159–172.
B. Benyi, M. Mendez, J.L. Ramirez, and T. Wakhare, Restricted r-Stirling numbers and their combinatorial applications, Applied Mathematics and Computation., 348, (2019), 186-205.
A.Z. Broder, The r-Stirling numbers, Discrete Math. 49 (1984), 241–259.
L. Carlitz, Weighted Stirling numbers of the first and second kind, I. Fibonacci Quart. 18 (1980), 147–162.
J.Y. Choi and J.D.H. Smith, On the combinatorics of multi-restricted numbers, Ars Combin. 75 (2005), 45–63.
L. Comtet, Advanced Combinatorics D. Reidel Publishing Co., The Netherlands, 1974.
J. Engbers, D. Galvin, and C. Smyth, Restricted Stirling and Lah number matrices and their inverses, J. Combin. Theory, Series A 161 (2019), 271–298.
F. Esmaeeli, A. Erfanian, and F. Mansoori, k-Intersection graph of a finite set, Math. Interdisc. Res. 4 (2019), 305–317.
I. Gessel and R.P. Stanley, Stirling Polynomials, J. Combin. Theory, Series A 24 (1978), 24–33.
C. Godsil and G. Royle. Algebraic Graph Theory. Springer, 2001.
M. Griffiths and I. Mezo, A Generalization of Stirling Numbers of the Second Kind via a Special Multiset, J. Integer Seq, 13 (2010), Article 10.2.5.
I. Gutman and F. Harary, Generalizations of the matching polynomial, Utilitas Math. 24 (1983), 97–106.
W. Lang. Combinatorial interpretation of generalized Stirling numbers. J. Integer Seq 12 (2009), Article 09.3.3.
V. Pita Ruiz and J. Claudio de, Generalized Stirling numbers and hyper-sums of powers of binomial coefficients. Electron. J. Combin. 21(1) (2014), Paper 1.10.
A. Schreiber, Multivariate Stirling polynomials of the first and second kind, Discrete Math. 338 (2015), 2462–2484.
R. Stanley, Enumerative Combinatorics I, Cambridge University Press, 1997.
J.H. Van Lint and R.M. Wilson, A Course in Combinatorics, Cambridge University Press, 1996.
D.B. West, Introduction to Graph Theory, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 2000.
Refbacks
- There are currently no refbacks.
ISSN: 2338-2287
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.