Determinants, Paths, and Plane Partitions

In studying representability of matroids, Lindström [42] gave a combinatorial interpretation to certain determinants in terms of disjoint paths in digraphs. In a previous paper [25], the authors applied this theorem to determinants of binomial coefficients. Here we develop further applications. As in [25], the paths under consideration are lattice paths in the plane. Our applications may be divided into two classes: first are those in which a determinant is shown to count some objects of combinatorial interest, and second are those which give a combinatorial interpretation to some numbers which are of independent interest. In the first class are formulas for various types of plane partitions, and in the second class are combinatorial interpretations for Fibonomial coefficients, Bernoulli numbers, and the less-known Salié and Faulhaber numbers (which arise in formulas for sums of powers, and are closely related to Genocchi and Bernoulli numbers). Other enumerative applications of disjoint paths and related methods can be found in [14], [26], [19], [51–54], [57], and [67].

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