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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 $k$-inverted semistandard Young tableau of various shapes $λ$ and contents $μ$. An easily-calculable generating function is given for the number of $k$-inverted semistandard Young tableau 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 tableau of shape $λ$ are 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 $λ= n^m$ 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 $λ= a^1 b^1$, as well as all $k$-inverted standard Young tableaux with two-column shape $λ=2^n$.