Geodesic
Smith normal form of a multivariate matrix associated with partitions
Journal of Algebraic Combinatorics, Tome 41 (2015) no. 1, pp. 73-82.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Consideration of a question of E. R. Berlekamp [Inf. Control 6, 1--13 (1963; Zbl 0214.47201)] led L. Carlitz et al. [J. Comb. Theory, Ser. A 11, 258--271 (1971; Zbl 0227.05007)] to give a combinatorial interpretation of the entries of certain matrices of determinant 1 in terms of lattice paths. Here we generalize this result by refining the matrix entries to be multivariate polynomials, and by determining not only the determinant but also the Smith normal form of these matrices. A priori the Smith form need not exist but its existence follows from the explicit computation. It will be more convenient for us to state our results in terms of partitions rather than lattice paths.
Classification : 05A15, 05A17, 05A30, 05B20
Keywords: lattice paths, partitions, determinants, Smith normal form, $q$-Catalan numbers. 
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Bessenrodt, Christine; Stanley, Richard P. Smith normal form of a multivariate matrix associated with partitions. Journal of Algebraic Combinatorics, Tome 41 (2015) no. 1, pp. 73-82. http://geodesic.mathdoc.fr/item/JAC_2015__41_1_a6/