Thirty-six full matrix forms of the Pascal triangle: Derivation and symmetry relations

Abstract

For all 2≤n∈N, the four vertices (00),(n0),(2nn),(nn) of the Pascal Triangle expanded from level 0 to level 2n define the greatest embedded rhomboid sub-block denoted n−GRSB in this paper. The n−GRSB is canonically partitioned into two triangular sub-blocks G and g, with respective vertex sets {(00),(n0),(nn)} and {(n+11),(2nn),(n+1n)}. The G-sub-block (resp. g-sub-block) has twelve distinct triangular matrix arrangements, numbered from 1 to 12 and designated here G-matrix set (resp. g-matrix set): three northeast, three northwest, three southwest and three distinct triangular southeast arrangements. From the n−GRSB we define thirty-six full matrix forms of the Pascal triangle (FP-matrices for short) simply adding pairwise complementary subblocks of the G- and g-matrices. We then identify and present the invariant groups underlying two significant partitions of the FP-matrix set. The insight gained from a previous study of the twelve G-matrices led us to derive the 36 full matrix forms presented in this paper. Several papers in the literature have dealt with some matrix forms of the Pascal Triangle. Only two of these are so far encountered in the literature. Our work is the first to focus on the hitherto little known 36 full matrix forms as mathematical objects in their own right. As novelty, this paper presents, for the first time, the set of the thirty-six full Pascal matrices. This work focuses on a systematic study of matrix forms derived from the Pascal Triangle, on the individual properties of these forms, their applications, and on the groups of transformations that structure their relations.

Keywords

• Pascal matrix
• Bipartition
• Circulant matrix
• Transformation matrix
• Symmetry transformation