A Survey Of The Riordan Group

This project is all about a Survey of the Riordan group which is intimately related to the Riordan arrays in particular to the Fundamental Theorem of Riordan Arrays (FTRA) in solving enumerative problems. We focus on counting the average number of points on thex- axis of Dyck paths and the average number of hills in Dyck paths using the Catalan numbers, Fine numbers, and Schrőder numbers by switching between sequences and generating functions. The project also gives a unified presentation about tackling combinatorial identities, and finally introduce the group nature of Riordan arrays under matrix multiplication(*) defined by(g(z) f.(z))* (h(z).l(z))=(g(z)h(f(z))l(f(z))) , where g (z).f(z)andl(z) ares(an.k)nk.>0 generating functions in a proper Riordan array such that the column of a combinatorial sequence of defined byank=(zn)d(z)h(z)k

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