Triangular powers of integers from determinants of binomial coefficient matrices

✍ Scribed by L.J. Ratliff Jr; D.E. Rush

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 812 KB
• Volume: 291
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(98)10244-6

✦ Synopsis

Formulas arc obtained for the determinants of certain matrices whose entries are zero and either binomial coefficients or their negatives. A consequence is that. for all integers II ?: 2 and k ;~2, there exists an (II -I)(k -I) x (n -I)(k -1) matrix M(n, k) whose entries arc the alternating binomial coefficients (-1 r" (~) and zeros such that det(M (II, k)) == ±kij,-I, where I n -_, is the (11 -I)th triangular number. Further, if we form the infinite matrixr' whose kth row is (~). (~), (;) ~... , then each of the above mentioned determinants is, up to sign, the determinant of un 11 y. 11 submatrix A of.J} obtained by selecting the initial 11 columns, and some choice of It rows or ;il. The matrices :-1 (11, k), and others that we will consider also have the unexpected property that det(INf(n, k)1) == Idct{M (II l k))I, where IAt; den-ites the matrix obtained from A1 by replacing each entry with its absolute value.