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 $2,3,4{\rm {and5}}$ .

[1] R. Alter, Some remarks and results on Catalan numbers, Congressus Numer., vol. 3, Utilitas Math., Winnipeg, Man., 1975, pp. 109-132. MR 0329910 (48:8250)
[2] L. Euler, Opera Omnia 26 (1953), xvi-xviii.
[3] M. Gardner, Catalan numbers: An integer sequence that materializes in unexpected places, Mathematical Games, Scientific American 234 (1976), 120-5, 132.
[4] H. W. Gould, Research bibliography of two special number sequences, 1239 College Ave., Morgantown, W. Va. 26505, 1976. MR 0401633 (53:5460)
[5] H. Izbicki, Uber Unterbaumes eines Baumes, Monatsh. Math. 74 (1970), 56-62. MR 0263706 (41:8307)
[6] D. Knuth, Fundamental algorithms, 2nd ed., vol. 1, Addison-Wesley, Reading, Mass., 1973, pp. 532-3.
[7] F. T. Leighton and M. Newman, Positive definite matrices and Catalan numbers, Proc. Amer. Math. Soc. 79 (1980), 177-181. MR 565333 (81c:15019)
[8] J. W. Moon, Counting labeled trees, Canad. Math. Mono., vol. 1, Canad. Math. Congress, 1970, pp. 39-46. MR 0274333 (43:98)
[9] J. Riordan, A note on Catalan parentheses, Amer. Math. Monthly 80 (1973), 904-906. MR 0335291 (49:73)
[10] L. Shapiro, A short proof of an identity of Touchard's concerning Catalan numbers, J. Combin. Theory Ser. A 20 (1976), 375-6. MR 0406819 (53:10605)
[11] J. van Lint (ed.), Combinatorial theory seminar (Eindhoven Univ. of Technology, 1974), Lecture Notes in Math., vol. 382, Springer-Verlag, Berlin and New York, 1974, pp. 21-27. MR 0351823 (50:4311)