A Schur function identity related to the -enumeration of self-complementary plane partitions

We give another proof for the ( − 1 ) -enumeration of self-complementary plane partitions with at least one odd side-length by specializing a certain Schur function identity. The proof is analogous to Stanleyʹs proof for the ordinary enumeration. In addition, we obtain enumerations of 180°-symmetric rhombus tilings of hexagons with a barrier of arbitrary length along the central line.