Some applications of $S$-restricted set partitions (Q2416506)

Stirlings numbers of the second kind find attention of many researchers due to their arithmetical properties and analytical properties of their generating function. Although these numbers were generalized in different ways combinatorially and some ideas were already mentioned in early works, recently many research groups have started showing interest in this topic. In this paper, authors have presented many relations and applications for the combinatorial sequence that counts the possible partition of a finite set with the restriction that the size of each block is contained in a given set. They have considered some applications of these generalizations and have modified some well known identities of the S-restricted Stirling and Bell numbers. An extension of Fubini numbers and some new identities have also been obtained. Several methods from elementary combinatorics to iterated integrals have been involved in this study. One of the main applications presented here is of the S-restricted partitions to the study of the lonesum matrices. Finally a generalization of the poly-Bernoulli numbers is given.