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

TL;DR

This paper introduces a tensor-based representation of Dyck words using zipper-merged integer compositions, revealing structured patterns and disjoint unions related to lexicographic ordering.

Contribution

It presents a novel tensor construction that encodes all Dyck words of a given length through zipper-merged compositions and analyzes their structural properties.

Findings

Tensor entries uniquely represent Dyck words of length 2k.

Disjoint unions of staircases correspond to non-Dyck word entries.

Structured patterns emerge in tensor arrangements based on lexicographic order.

Abstract

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.