A binary Steinhaus triangle is a triangle of zeroes and ones that points down and with the same local rule as the Pascal triangle modulo 2. A binary Steinhaus triangle is said to be rotationally symmetric, horizontally symmetric or dihedrally symmetric if it is invariant under the 120 degrees rotation, the horizontal reflection or both, respectively. The first part of this paper is devoted to the study of linear subspaces of rotationally symmetric, horizontally symmetric and dihedrally symmetric binary Steinhaus triangles. We obtain simple explicit bases for each of them by using elementary properties of the binomial coefficients. A Steinhaus graph is a simple graph with an adjacency matrix whose upper-triangular part is a binary Steinhaus triangle. A Steinhaus graph is said to be even or odd if all its vertex degrees are even or odd, respectively. One of the main results of this paper is the existence of an isomorphism between the linear subspace of even Steinhaus graphs and a certain linear subspace of dihedrally symmetric binary Steinhaus triangles. This permits us to give, in the second part of this paper, an explicit basis for even Steinhaus graphs and for the vector space of parity-regular Steinhaus graphs; i.e., the linear subspace of Steinhaus graphs that are even or odd. Finally, in the last part of this paper, we consider the generalized Pascal triangles, that are triangles of zeroes and ones, that point up now, and always with the same local rule as the Pascal triangle modulo 2. New simple bases for each linear subspace of symmetric generalized Pascal triangles are deduced from the results of the first part.

翻译：binary Steinhaus 三角形是一个由零组成的三角形和一个点点点和与 Pascal 三角形相同的本地规则的三角形。 2 我们通过使用双声调系数的基本属性获得每个三角形的简单清晰基础。 如果在120 度旋转、 水平反射或两者下都是不固定的, 则Steinhaus 三角形是一个双向、 水平反射或双向的三角形。 本文的第一部分用于研究旋转对称、 水平对称和双向对齐的线性亚空间。 本文的主要结果之一, 总是通过使用 即使是 Steinhaus 三角形的直径基点, 我们的直径直径直径三角形图是一个简单的图形图, 其上角部分是双向的 Steinha 三角形。 Steinha 图形的直径直线性平面图表示, 其所有螺旋度均匀或奇异。 本文的主要结果之一, 在于, 直径直径平面的直径直径直径直径直径平面的直径直径直径直径直径直径平面, 。 直平面的平面平面图显示, 。