On Combinatorial Interpretations of Some Elements of the Riordan Group

Riordan arrays, equipped with Shapiro’s multiplication rule, form a group and this group provides an interesting algebraic framework to solve combinatorial problems. In this chapter, we provide combinatorial interpretations for Riordan arrays of the form \(R = (g(z), zg(z))\) , where \(g(z)\) is a generating function satisfying a functional equation \(g(z) = 1 + z*{g(z)}^k\) , where k is a positive integer greater than or equal to 2. The combinatorial interpretations are then used to obtain the inverses of these elements of the Bell subgroup of the Riordan group explicitly in terms of powers of \(g(z)\) .