The twelvefold way, the nonintersecting circles problem, and partitions of multisets
The twelvefold way, the nonintersecting circles problem, and partitions of multisets
Let n be a nonnegative integer and A = {a1, . . . , ak} be a multiset with k positive integers such that a1 ⩽ · · · ⩽ ak . In this paper, we give a recursive formula for partitions and distinct partitions of positive integer n with respect to a multiset A . We also consider the extension of the twelvefold way. By using this notion, we solve the nonintersecting circles problem, which asks to evaluate the number of ways to draw n nonintersecting circles in the plane regardless of their sizes. The latter also enumerates the number of unlabeled rooted trees with n + 1 vertices.
___
- [1] Aigner M. Combinatorial Theory. Berlin, Germany: Springer, 1979.
- [2] Andrews G. The Theory of Partitions (Encyclopaedia of Mathematics and Its Applications). Cambridge, UK:
Cambridge University Press, 1976.
- [3] Blecher M. Properties of integer partitions and plane partitions. PhD, University of the Witwatersrand, Johannesburg,
South Africa, 2012.
- [4] Comtet L. Advanced Combinatorics: The Art of Finite and Infinite Expansions. Amsterdam, the Netherlands:
Springer, 1974.
- [5] Hille E. A problem in factorisatio numerorum. Acta Arithmetica 1936; 2: 134-144. doi: 10.4064/aa-2-1-134-144
- [6] Knopfmacher A, Munagi A. Successions in integer partitions. Ramanujan Journal 2009; 18: 239-255. doi:
10.1007/s11139-008-9140-2
- [7] MacMahon PA. Memoir on the theory of the compositions of numbers. Philosophical Transactions of the Royal
Society London (A) 1893; 184: 835-901. doi: 10.1098/rsta.1893.0017
- [8] MacMahon PA. The enumeration of the partitions of multipartite numbers. Mathematical Proceedings of the
Cambridge Philosophical Society 1927; 22: 951–963. doi: 10.1017/S0305004100014547
- [9] Mansour T. Combinatorics of Set Partitions. Boca Raton, FL, USA: Chapman & Hall/CRC, Taylor & Francis
Group, 2012.
- [10] Sloane NJ. The On-line Encyclopedia of Integer Sequences. https://oeis.org/, 2010.
- [11] Stanley R. Enumerative Combinatorics, Volume 1. Cambridge, UK: Cambridge University Press, 1999.