Unimodular matrices and lattice paths enumeration via Pascal’s triangle

	Unimodular matrices and lattice paths enumeration via Pascal’s triangle
Communications in Combinatorics and Optimization
مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 30 خرداد 1405
نوع مقاله: Original paper
شناسه دیجیتال (DOI): 10.22049/cco.2026.31448.2856
نویسنده
S Bera^*
Faculty of mathematics, DAU, Gandhinagar
چکیده
This article investigates a remarkable combinatorial identity involving a distinguished family of matrices whose entries are defined via binomial coefficients. Specifically, we consider a class of $ n \times n $ matrices parameterized by a positive integer $ m $, where each entry reflects a structured pattern derived from Pascal's triangle, particularly the diagonals corresponding to figurate numbers such as triangular, tetrahedral, and higher-dimensional simplex numbers. We establish, by means of a bijective argument, that the determinant of any such matrix is identically equal to $ 1 $, independent of the specific values of $ m $ and $ n $, provided that $ 2 \leq m \leq n $. This result unveils a profound connection between classical binomial identities and the enumeration of lattice paths in grid graphs.
کلیدواژه‌ها
Unimodular matrix؛ $k$-simplex number؛ Lattice path enumeration

آمار تعداد مشاهده مقاله: 2

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