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We study three operations on Riordan arrays. First, we investigate when the sum of Riordan arrays yields another Riordan array. We characterize the A-and Z-sequences of these sums of Riordan arrays, and also identify an analog for A-sequences when the sum of Riordan arrays does not yield a Riordan array. In addition, we define the new operations 'Der' and 'Flip' on Riordan arrays. We fully characterize the Riordan arrays resulting from these operations applied to the Appell and Lagrange subgroups of the Riordan group. Finally, we study the application of these operations to various known Riordan arrays, generating many combinatorial identities in the process.

We present a unifying framework in which both the ν-Tamari lattice, introduced by Préville-Ratelle and Viennot, and principal order ideals in Young's lattice indexed by lattice paths ν, are realized as the dual graphs of two combinatorially striking triangulations of a family of flow polytopes which we call the ν-caracol flow polytopes. The first triangulation gives a new geometric realization of the ν-Tamari complex introduced by Ceballos, Padrol and Sarmiento. We use the second triangulation to show that the h * -vector of the ν-caracol flow polytope is given by the ν-Narayana numbers, extending a result of Mészáros when ν is a staircase lattice path. Our work generalizes and unifies results on the dual structure of two subdivisions of a polytope studied by Pitman and Stanley.

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