2 research outputs found
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 inversion pairs is said
to be a -inverted semistandard Young tableau. Building upon earlier work by
Fresse and the author, this paper develops generating functions for the numbers
of -inverted semistandard Young tableau of various shapes and
contents . An easily-calculable generating function is given for the
number of -inverted semistandard Young tableau that "standardize" to a fixed
semistandard Young tableau. For -row shapes and standard content
, the total number of -inverted standard Young tableau of shape
are then enumerated by relating such tableaux to -dimensional
generalizations of Dyck paths and counting the numbers of "returns to ground"
in those paths. In the rectangular specialization of this
yields a generating function that involves -dimensional analogues of the
famed Ballot numbers. Our various results are then used to directly enumerate
all -inverted semistandard Young tableaux with arbitrary content and two-row
shape , as well as all -inverted standard Young tableaux
with two-column shape
Inversions of Semistandard Young Tableaux
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 -inverted Young tableau is a row-standard
tableau with precisely inversion pairs, and may be interpreted as a
generalization of (column-standard) Young tableau. Inverted Young tableau 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 -inverted tableaux of a given shape is
invariant under permutation of content. We then enumerate -inverted Young
tableaux for a variety of shapes and contents, and generalize an earlier result
that places -inverted Young tableaux of a general shape in bijection with
-inverted Young tableaux of a variety of related shapes