Abstract
A tableau inversion is a pair of entries from the same column of a row-standard tableau that lack the relative ordering necessary to make the tableau column-standard. An $i$-inverted Young tableau is a row-standard tableau with precisely $i$ inversion pairs, and may be interpreted as a generalization of (column-standard) Young tableaux. Inverted Young tableaux that lack repeated entries were introduced by Fresse to calculate the Betti numbers of Springer fibers in Type A, and were later developed as combinatorial objects in their own right by Beagley and Drube. This paper generalizes earlier notions of tableau inversions to row-standard tableaux with repeated entries, yielding an interesting new generalization of semistandard (as opposed to merely standard) Young tableaux. We develop a closed formula for the maximum numbers of inversion pairs for a row-standard tableau with a specific shape and content, and show that the number of $i$-inverted tableaux of a given shape is invariant under permutation of content. We then enumerate $i$-inverted Young tableaux for a variety of shapes and contents, and generalize an earlier result that places $1$-inverted Young tableaux of a general shape in bijection with $0$-inverted Young tableaux of a variety of related shapes.
Highlights
Consider the non-increasing sequence of positive integers λ = (λ1, λ2, . . . , λm), and let N = λ1 + . . . + λm
Inverted Young tableaux that lack repeated entries were introduced by Fresse to calculate the Betti numbers of Springer fibers in Type A, and were later developed as combinatorial objects in their own right by Beagley and Drube
A filling of a Young diagram Y is an assignment of positive integers to the boxes of Y such that integers strictly increase from left-to-right across each row and weakly increase from top-to-bottom down each column
Summary
In this paper we will need to consider a generalization of semistandard fillings where integers strictly increase from left-to-right across each row but no longer need to weakly increase down each column. We refer to such an array as a row-standard tableau. We retain our notation that ninv(τ ) denotes the total number of distinct inversion pairs in τ In this case, the row-standard τ qualifies as a semistandard Young tableau if and only if ninv(τ ) = 0.
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