Bounded Littlewood identities with
fixed number of odd rows or odd columns
(30 pages)
Abstract.
A Littlewood identity is an identity equating a sum of Schur functions
with an infinite product. A bounded Littlewood identity is one where
the sum is taken over the partitions with a bounded number of
rows or columns. The price to pay is that the infinite product has to
be replaced by a determinant.
The focus of this article is on refinements of such bounded Littlewood
identities where one also prescribes the number of odd-length rows or columns
of the partitions. Goulden [Discrete Math. 99 (1992), 69-77]
had given such a refinement in which the number of columns is bounded
and the number of odd-length rows is prescribed. We provide
refinements where the number of columns is bounded and the number of
odd-length columns is prescribed. Furthermore, we present new formulations
of such bounded Littlewood identities involving skewing operators.
As corollaries we obtain non-standard formulas for numbers of standard
Young tableaux with restricted shapes as above.
In the last part of the article we discuss combinatorial
interpretations of such identities in terms of up-down tableaux.
As corollaries, we obtain identities between numbers of standard Young
tableaux and numbers of (marked) vacillating tableaux.
The following versions are available:
Back to Christian Krattenthaler's
home page.