Double Riordan Arrays and Their Compressions, Fourth International Symposium, on Riordan Arrays and Related Topics

Inspired by the Fibonacci tree shown in the recent book, Catalan Numbers, by Richard Stanley, we present a combinatorial way to construct a type of lower triangle matrices related to the double Riordan arrays by using the succession rule, an ECO technique. We call those lower triangle matrices the compressions of the double Riordan arrays. The Pascal-Fibonacci triangle constructed from the Fibonacci tree shown in the Stanley’s book is an example of the compressions of the double Riordan arrays. We give sequence characterizations of the double Riordan arrays, subgroups of the double Riordan group, and the compressions of the double Riordan arrays. Some connections of the Riodan arrays and the double Riordan arrays as well as their compressions are established. The properties of the double Riordan arrays and their compressions such as those related to their P matrices and stabilizers are studied. The applications of the the linear map induced by the double Riordan arrays and their compressions are presented.