A further look at a generalization of Waring's formula (Q2240589)

In this paper, the author considers elementary symmetric functions \(e_k\) with equal exponents \(r\) in terms of the elementary symmetric functions \(e_\lambda\). More precisely, the coefficients \(A_\lambda^{(k,r)}\) defined by \[ e_{k}(x_1^r, x_2^r, \dots )=(-1)^{k(r+1)} \sum_{\lambda \vdash kr} A_\lambda^{(k,r)} e_\lambda (x_1,x_2,\dots) \] are studied. Here, the left hand side is nothing but the plethysm \(e_k \circ p_r\) with a power-sum symmetric function. \textit{J. Konvalina} [J. Comb. Theory, Ser. A 75, No. 2, 281--294 (1996; Zbl 0857.05094)] gave an expression for the coefficient \(A_\lambda^{(k,r)}\) by the use of refinements of \(\lambda\). In this paper, an alternative expression for \(A_\lambda^{(k,r)}\) is obtained in terms of Möbius functions of set partitions.