Generating Functions for Inverted Semistandard Young Tableaux and Generalized Ballot Numbers

An inverted semistandard Young tableau is a row-standard tableau along with a collection of inversion pairs that quantify how far the tableau is from being column semistandard. Such a tableau with precisely k inversion pairs is said to be a k-inverted semistandard Young tableau. Building upon earlier work by Fresse and the author, this paper develops generating functions for the numbers of kinverted semistandard Young tableaux of various shapes λ and contents μ. An easily-calculable generating function is given for the number of k-inverted semistandard Young tableaux that “standardize” to a fixed semistandard Young tableau. For m-row shapes λ and standard content μ, the total number of k-inverted standard Young tableaux of shape λ is then enumerated by relating such tableaux to m-dimensional generalizations of Dyck paths and counting the numbers of “returns to ground” in those paths. In the rectangular specialization of λ = nm this yields a generating function that involves m-dimensional analogues of the famed Ballot numbers. Our various results are then used to directly enumerate all k-inverted semistandard Young tableaux with arbitrary content and two-row shape λ = a1b1, as well as all k-inverted standard Young tableaux with two-column shape λ = 2n.