Pieri formula for power sum symmetric polynomial

Question

I know Pieri's formula for elementary symmetric polynomials and for complete homogeneous symmetric polynomials, but is there an analogue for power sum symmetric polynomial? It seems that it should be similar, but the summation should be carried out by other partitions. Maybe someone knows where to read about it?

Answer 1

Macdonald p.48:

$s_\mu p_r=\sum_\lambda(-1)^{\mathrm{ht}(\lambda-\mu)}s_\lambda$

where the sum is over all partitions $\lambda\supset\mu$ such that $\lambda-\mu$ is a border strip of length $r$, and $\mathrm{ht}(\theta)$ is the height of a border strip $\theta$.