Note on enumeration of partitions contained in a given shape

Carlitz, 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. Let M = (M1 ≤ M2 ≤ · · · ≤ Ms) and N = (N1 ≤ N2 ≤ · · · ≤ Ns) be two fixed partitions such that M ⊆ N , i.e., Mi ≤ Ni for all i. Handa and Mohanty [3] proved the following determinantal formula that enumerates partitions lying between M and N by area: ∑ λ:M⊆λ⊆N q|λ| = det ([ Ni −Mj + 1 j − i+ 1 ] q q(j−i)(j−i+1)/2+(j−i+1)Mj )