AI 中文总结
本文研究 $\boldsymbol{\text{Z}/2\boldsymbol{\text{Z}}}$ 上的 Delannoy–Steinhaus 三角形，推导其权重公式，确定完整权重谱，刻画平衡三角形存在条件，并计算各类极值权重。
AI 中文摘要
Delannoy–Steinhaus 三角形由有限序列经 Delannoy 数控制的递推关系生成，本文引入该结构在 $\boldsymbol{\text{Z}/2\boldsymbol{\text{Z}}}$（模 2 整数环）上的版本并研究其权重分布。相关 Delannoy 系数均为奇数，这使得三角形的每个元素仅等于生成序列中某连续区间元素的奇偶性；通过前缀奇偶性编码这些区间奇偶性，可得到仅依赖前缀奇偶序列中 0 和 1 数量的权重公式。利用该公式确定完整权重谱及每个权重的精确重数，据此刻画并计数平衡三角形：长度为 $n$ 的二元序列生成的平衡三角形存在当且仅当 $n+1$ 为完全平方数；此外还确定了标准向量权重、最小非零权重、次小非零权重、最大权重及平均权重。
英文摘要
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.
Comments12 pages
