The Double Riordan Group

  • Dennis E. Davenport
  • Louis W. Shapiro
  • Leon C. Woodson


The Riordan group is a group of infinite lower triangular matrices that are defined by two generating functions, $g$ and $f$. The kth column of the matrix has the generating function $gf^k$. In the Double Riordan group there are two generating function $f_1$ and $f_2$ such that the columns, starting at the left, have generating functions using $f_1$ and $f_2$ alternately. Examples include Dyck paths with level steps of length 2  allowed at even height and also ordered trees with differing degree possibilities at even and odd height(perhaps representing summer and winter). The Double Riordan group is a generalization not of the Riordan group itself but of the checkerboard subgroup. In this context both familiar and far less familiar sequences occur such as the Motzkin numbers and the number of spoiled child trees. The latter is a slightly enhanced cousin of ordered trees which are counted by the Catalan numbers.