q-Analogs of Riordan Arrays

The Riordan group consisting of proper Riordan arrays shows up naturally in a variety of combinatorial settings. In this chapter, we define the q-analog of a Riordan array called a q-Riordan array and denoted as (g, f) q. It is defined by using a pair of Eulerian generating functions g, f of the form ∑n≥0anzn[n]q! where [ n] q! = 1 (1 + q) (1 + q+ q2) ⋯ (1 + q+ ⋯ + qn-1). A q-analog appears naturally in several contexts of combinatorics, quantum group theory, and so on. We establish some algebraic properties for q-Riordan arrays. Noticing that [ n] q! reduces to n! upon setting q= 1, we see that a q-Riordan array reduces to an exponential Riordan array for the case of q= 1. This suggests that q-Riordan arrays might be useful not only for studying enumeration problem [2, 10, 11] but also for defining q-analogs of some orthogonal polynomials [3, 4]. Indeed, it is shown that q-Riordan arrays associated to the counting functions may be applied to the enumeration problem on set partitions by block inversions. This notion also leads us to find q-analogs of the composition formula and the exponential formula, respectively.