آمار
| تعداد نشریات | 6 |
| تعداد شماره‌ها | 131 |
| تعداد مقالات | 1,631 |
| تعداد مشاهده مقاله | 1,879,788 |
| تعداد دریافت فایل اصل مقاله | 1,741,101 |
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 | ||
|
[1] M. Aigner, A Course in Enumeration, vol. 238, Springer, 2007.
[2] A. Ayyer, Determinants and perfect matchings, J. Combin. Theory Ser. A 120 (2013), no. 1, 304–314. https://doi.org/10.1016/j.jcta.2012.08.007
[3] S. Bera and S.K. Mukherjee, Combinatorial proofs of some determinantal identities, Linear Multilinear Algebra 66 (2018), no. 8, 1659–1667. https://doi.org/10.1080/03081087.2017.1366970
[4] I.M. Gessel and G. Viennot, Binomial determinants, paths, and hook length formulae, Adv. Math. 58 (1985), no. 3, 300–321. https://doi.org/10.1016/0001-8708(85)90121-5
[5] I. Heller and C.B. Tompkins, An extension of a theorem of Dantzig’s, Linear inequalities and related systems 38 (1956), 247–254.
[6] A.J. Hoffman and J.B. Kruskal, Integral boundary points of convex polyhedra, Linear Inequalities and Related Systems (H.W. Kuhn and A.W. Tucker, eds.), Annals of Mathematics Studies, vol. 38, Princeton University Press, Princeton, 1956, pp. 223–246. [7] , Integral boundary points of convex polyhedra, 50 Years of Integer Programming, 1958–2008 (Michael J¨unger et al., eds.), Springer-Verlag, 2010, pp. 49–50.
[8] C. Krattenthaler, Advanced determinant calculus, S´em. Lothar. Combin. 42 (1999), B42q.
[9] C. Krattenthaler, Advanced determinant calculus: A complement, Linear Algebra Appl. 411 (2005), 68–166. https://doi.org/10.1016/j.laa.2005.06.042
[10] P.A. MacMahon, Memoir on the theory of the partition of numbers, Philosophical Transactions of the Royal Society of London. Series A 187 (1897), 619–673.
[11] J.R. Stembridge, Nonintersecting paths, pfaffians and plane partitions, Adv. Math. 83 (1990), no. 1, 96–131. https://doi.org/10.1016/0001-8708(90)90070-4 [12] T. Zaslavsky, Signed graphs, Discrete Appl. Math. 4 (1982), no. 1, 47–74. https://doi.org/10.1016/0166-218X(82)90033-6
[13] D. Zeilberger, A combinatorial proof of Newton’s identities , Discrete Math. 49 (1984), no. 3, 319. https://doi.org/10.1016/0012-365X(84)90171-7 | ||
|
آمار تعداد مشاهده مقاله: 147 تعداد دریافت فایل اصل مقاله: 86 |
||