Delannoy--Steinhaus triangles over $\mathbb{Z}/2\mathbb{Z}$: weight spectrum, balanced triangles, and extremal values

Abstract

A Delannoy--Steinhaus triangle is obtained from a finite sequence by a recurrence governed by the Delannoy numbers. We introduce this construction over $\mathbb{Z}/2\mathbb{Z}$ and study its weight distribution. The relevant Delannoy coefficients are all odd, which reduces every entry to the parity of a consecutive interval of the generating sequence. Encoding these interval parities by prefix parities yields a weight formula depending only on the numbers of zeros and ones in the prefix-parity sequence. We use this formula to determine the complete weight spectrum and the exact multiplicity of each weight. As a consequence, we characterize and enumerate the balanced triangles: a balanced triangle generated by a binary sequence of length $n$ exists if and only if $n+1$ is a perfect square. We also determine the canonical-vector weights, the minimum nonzero weight, the {second-smallest nonzero weight}, the maximum weight, and the average weight.