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Orthogonal polynomials and Smith normal form. (English) Zbl 1395.05029

Summary: Smith normal form evaluations found by C. Bessenrodt and R. P. Stanley [J. Algebr. Comb. 41, No. 1, 73–82 (2015; Zbl 1307.05009)] for some Hankel matrices of \(q\)-Catalan numbers are proven in two ways. One argument generalizes the Bessenrodt-Stanley results for the Smith normal form of a certain multivariate matrix that refines one studied by E. R. Berlekamp [Inf. Control 6, 1–13 (1963; Zbl 0214.47201); Comput. Math. Appl. 39, No. 11, 77–88 (2000; Zbl 0953.05010)], and L. Carlitz et al. [J. Comb. Theory, Ser. A 11, 258–271 (1971; Zbl 0227.05007)]. The second argument, which uses orthogonal polynomials, generalizes to a number of other Hankel matrices, Toeplitz matrices, and Gram matrices. It gives new results for \(q\)-Catalan numbers, \(q\)-Motzkin numbers, \(q\)-Schröder numbers, \(q\)-Stirling numbers, \(q\)-matching numbers, \(q\)-factorials, \(q\)-double factorials, as well as generating functions for permutations with eight statistics.

MSC:

05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
33C90 Applications of hypergeometric functions
15A21 Canonical forms, reductions, classification
05A17 Combinatorial aspects of partitions of integers
05A15 Exact enumeration problems, generating functions
05A30 \(q\)-calculus and related topics
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References:

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