Based on a weighted version of the bijection between Dyck paths and 2-Motzkin paths, we find combinatorial interpretations of two identities related to the Narayana polynomials and the Catalan numbers. These interpretations answer two questions posed recently by Coker.

We give a combinatorial interpretation of a matrix identity on Catalan numbers and the sequence (1, 4, 4 2 , 4 3 , . . .) which has been derived by Shapiro, Woan and Getu by using Riordan arrays. By giving a bijection between weighted partial Motzkin paths with an elevation line and weighted free Motzkin paths, we find a matrix identity on the number of weighted Motzkin paths and the sequence (1, k, k 2 , k 3 , . . .) for any k ≥ 2. By extending this argument to partial Motzkin paths with multiple elevation lines, we give a combinatorial proof of an identity recently obtained by Cameron and Nkwanta. A matrix identity on colored Dyck paths is also given, leading to a matrix identity for the sequence (1, t 2 + t, (t 2 + t) 2 , . . .).

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Abstract. In this paper, we are mainly concerned with the enumeration of (2k + 1, 2k + 3)-core partitions with distinct parts. We derive the number and the largest size of such partitions, confirming two conjectures posed by Straub.