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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 اصل مقاله (423.88 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2026.31448.2856 | ||
| نویسنده | ||
| S. Bera* | ||
| Faculty of Mathematics, Dhirubhai Ambani University, Gandhinagar, India | ||
| 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 | ||
| مراجع | ||
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