2014/12/19 by Daniel Yaqubi, Madjid Mirzavaziri, Yaqubi, Daniel +3
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities
paper · pdf · doi:10.48550/arxiv.1412.6239
The Stirling number of the second kind n\brace k counts the number of ways to partition a set of n labeled balls into k non-empty unlabeled cells. As an extension of this, we consider b1+b2+…+bn balls with b1 balls labeled 1, b2 balls labeled 2, …, bn balls labeled n and c1+c2+…+ck cells with c1 cells labeled 1, c2 cells labeled 2, …, ck cells labeled k and then we called the number of ways to partition the set of these balls into non-empty cells of these types as the mixed partition numbers. As an application, we give a new statement of the r-Stirling numbers of second kind and r-Bell numbers. We also introduce the r-mixed Stirling number of second kind and r-mixed Bell numbers. Finally, for a positive integer m we evaluate the number of ways to write m as the form m1⋅ m2⋅…⋅ mk, where k\geqslant 1 and mi's are positive integers greater than 1.