A study of a family of generating functions of Nelsen-Schmidt type and some identities on restricted barred preferential arrangements

A preferential arrangement of a set Xn={1,2,…,n} is a partition of the set Xn together with a linear ordering of the blocks. Separating the blocks of a preferential arrangement with vertical bars results in a barred preferential arrangement. Nelsen and Schmidt proposed a family of generating functions Pk,1(x)=ekx2−ex (where k∈N0={0,1,2,3,…}); which for k=0 and for k=2 gives respectively, the exponential generating function for the number of preferential arrangements of Xn and that of the number of chains in the power set of Xn. They then asked “could there be combinatorial structures associated with either Xn or the power set of Xn whose integer sequences are generated by members of the family Pk,1(x) for other values of k ?” In this study we propose an answer to this question by showing how the enumerations of barred preferential arrangements can be associated with members of the family Pk,1(x) of generating functions for all values of k in positive integers. Further we use a notion of restricted barred preferential arrangements, enumerations of which then leads to a more general family of generating functions Pk,j(x)=ekx(2−ex)j for k,j in positive integers.