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We establish a tantalizing symmetry of certain numbers refining the Narayana numbers. In terms of Dyck paths, this symmetry is interpreted in the following way: if $$w_{n,k,m}$$ is the number of Dyck paths of semilength $$n$$ with $$k$$ occurrences of $UD$ and $$m$$ occurrences of $UUD$, then $$w_{2k+1,k,m}=w_{2k+1,k,k+1-m}$$. We give a combinatorial proof of this fact, relying on the cycle lemma, and showing that the numbers $$w_{2k+1,k,m}$$ are multiples of the Narayana numbers. We prove a more general fact establishing a relationship between the numbers $$w_{n,k,m}$$ and a family of generalized Narayana numbers due to Callan. A closed-form expression for the even more general numbers $$w_{n,k_{1},k_{2},\ldots, k_{r}}$$ counting the semilength-$$n$$ Dyck paths with $$k_{1}$$ $UD$-factors, $$k_{2}$$ $UUD$-factors, $$\ldots$$, and $$k_{r}$$ $$U^{r}D$$-factors is also obtained, as well as a more general form of the discussed symmetry for these numbers in the case when all rise runs are of certain minimal length. Finally, we investigate properties of the polynomials $$W_{n,k}(t)= \sum_{m=0}^k w_{n,k,m} t^m$$, including real-rootedness, $$\gamma$$-positivity, and a symmetric decomposition. 

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.