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.

Disclosure

“eparation of this manuscript. Competing interests The authors have no relevant financial or non-financial interests to disclose. Declaration on the use of generative AI All mathematical content of this article is the authors’ own work. Generative AI (Claude, Anthropic) was used solely for language editing and formatting, under the authors’ full review and responsibility. References [1] S. Amrouche, H. Belbachir, and J. L. Ramírez. Unimodality, linear recurrences and combina- t”

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2026-08-23T05:21:35.808481Z

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