Sprague–Grundy values of Rook on general partitions

Determine the Sprague–Grundy values of the Rook game for general integer partitions beyond the partition families treated in the paper.

Background

The Rook game permits removing any positive number of cells from the first row or the first column of a Young diagram. The paper determines the Rook Sprague–Grundy values on rectangles and on a family of generalized staircases, including a formula derived in Corollary 3.20.

The complete behavior on arbitrary integer partitions is not established. A general solution would extend the known formulas beyond rectangles and selected generalized staircases.