A q-analogue of the Riordan group

The Riordan group consisting of Riordan matrices shows up naturally in a variety of combinatorial settings. In this paper, we define a q-Riordan matrix to be a q  -analogue of the (exponential) Riordan matrix by using the Eulerian generating functions of the form _∑n?0fnzⁿ/n_!q${\sum }_{n?0}{f}_n{z}^n/n{!}_q$. We first prove that the set of q-Riordan matrices forms a loop (a quasigroup with an identity element) and find its loop structures. Next, it is shown that q-Riordan matrices associated to the counting functions may be applied to the enumeration problem on set partitions by block inversions. This notion leads us to find q-analogues of the composition formula and the exponential formula, respectively.