Bounded Littlewood identities with a fixed number of odd rows

The identities, often referred to as "bounded Littlewood identities", are determinantal formulas for the sum of Schur functions indexed by partitions with bounded height. They have interesting combinatorial consequences such as connections between standard Young tableaux of bounded height and lattice walks in a Weyl chamber. Goulden gave a refinement of two bounded Littlewood identities depending on the number of odd columns of the partitions.

In this talk, we provide a refinement of two bounded Littlewood identities depending on the number of odd rows of the partitions and give combinatorial interpretations using (marked) up-down tableaux. We also show that the number of standard Young tableaux with bounded height having exactly k rows of odd length equals the number of certain lattice walks. This is a collaboration with Jang Soo Kim, Christian Krattenthaler, and Soichi Okada.