Representation of Dyck words in tensors that zipper merge contiguous integer compositions

Let $0<k\in\mathbb{Z}$. We zipper-merge integer compositions with sums $k$ and $k+1$, equal number of parts and initial entries equal at least to 1 and 2, respectively. This yields bitstrings with two initial zeros, $k-1$ remaining zeros and $k$ ones. Tensors whose entries are such bitstrings contain unique representations of all Dyck words of length $2k$. If rows and columns of such tensors are disposed in descending lexicographic order, then their entries not representing Dyck words form disjoint unions of descending staircases corresponding to strict lower triangular submatrices.