Positive definite matrices and Catalan numbers, revisited

Shapiro, Louis W.

doi:10.1090/S0002-9939-1984-0728375-5

Positive definite matrices and Catalan numbers, revisited

Author: Louis W. Shapiro
Journal: Proc. Amer. Math. Soc. 90 (1984), 488-496
MSC: Primary 05A15; Secondary 05C50
DOI: https://doi.org/10.1090/S0002-9939-1984-0728375-5
MathSciNet review: 728375
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Abstract: In this note a combinatorial correspondence is used to prove that the number of positive definite, tridiagonal, integral matrices of determinant 1 whose sub and super diagonals consist solely of ones is ${C_n} = (_n^{2n})/(n + 1)$ . The correspondence is then further used to count such matrices by trace and also by number of ones on the main diagonal. Other related correspondences and results are given including those for determinant equal to $2,3,4{\rm {and5}}$ .

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