Note on enumeration of partitions contained in a given shape

AbstractCarlitz, Handa, and Mohanty proved determinantal formulas for counting partitions contained in a fixed bounding shape by area. Gessel and Viennot introduced a combinatorial method for proving such formulas by interpreting the determinants as counting suitable configurations of signed lattice paths. This note describes an alternative combinatorial approach that uses sign-reversing involutions to prove matrix inversion results. Combining these results with the classical adjoint formula for the inverse of a matrix, we obtain a new derivation of the Handa–Mohanty determinantal formula