Abstract
A Dyck path of semilength n is a path from (0,0) to (2n,0), such that the only allowed steps are U=(1,1) and D=(1,−1). The constraint on a Dyck path is that it never falls below the x-axis. In this paper, we give the concept of (a1,b1;a2,b2)-Dyck paths, which generates the recursive matrix corresponding to π=(a1,a2,a1,a2,⋯), σ=(0,0,0,0,⋯) and τ=(b1,b2,b1,b2,⋯). Some new combinatorial identities related to 2×2 minors and 2×2 permanents of the Dπ,σ,τ are investigated. When the weighted parameters (a1,b1;a2,b2) are specialized, several interesting identities are obtained about Catalan numbers and other combinatorial sequences. Based on Motzkin paths, alternating sums of 2×2 minors of the odd Dyck matrix are studied.
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