Finite and infinite dimensional Lie group structures on Riordan groups

Diameter of io-decomposable Riordan graphs of the Bell type

“…. Thus we can see that the four vertex pairs (1,5), (1,6), (1,7) and (1,8) in CG r 8 have maximal distance 3 i.e., the four vertex pairs (8,4), (8,3), (8,2) and (8, 1) in CG 8 have the maximal distance 3.…”

Commutators and commutator subgroups of the Riordan group

“…(3) The case i = 1 have already been discussed in (2). Now to see the general case i > 1 see that in (8),…”

“…The arguments followed herein do not apply for rings, since we usually need to take multiplicative inverses. See (17) and (8).…”

Riordan graphs I: Structural properties

“…In this section, we introduce the notion of a (0, 1)-Riordan matrix of order n, which will play an important role in defining Riordan graphs with n vertices in the next section. First note [10] that the Riordan group R over κ = R or C can be described as an inverse limit of an inverse sequence of groups (R n ) n∈N of finite matrices. So R n is a subgroup of the classical Lie group GL(n, κ), and elements in R n are n×n Riordan matrices denoted by (g, f ) n which are obtained from (infinite) Riordan matrices (g, f ) by taking their leading principal submatrix of order n. Now consider Riordan matrices over the finite field κ = Z 2 .…”

“…In particular, Riordan matrices found applications in the context of the computation of combinatorial sums [34]. Also, see [10] for a recent paper about Lie theory on the Riordan group, which is the set of invertible Riordan matrices.…”