$P_n$ is given as $$P_n(f) = \sum_{\lambda \,\vdash\, n} (-1)^{\lambda_1} \prod \binom{\lambda_j}{\lambda_{j+1}} f_{j}^{\lambda_j}$$ where

* the sum is over partitions $\lambda = \lambda_1 \ge \lambda_2 \ge \cdots$ of $n$;
* $\lambda_{j+1} = 0$ if $j+1$ is greater than the number of parts in $\lambda$;
* I've adjusted the index from $f_{j+1}$ to $f_j$ because it appears that the increment was a hack to 1-index the variables in a 0-indexed programming language.

Then in the section *Prototypic examples* the text says 

> So let's define $P^k_n(f)$ as the $P$ transform of $f$ restricted to the partitions of $n$ with largest part $k$

Therefore $$P^k_n(f) = \sum_{\lambda' \,\vdash\, n-k} (-1)^{k} \binom{k}{\lambda'_1} f_1^k \prod \binom{\lambda'_j}{\lambda'_{j+1}} f_{j+1}^{\lambda_j}$$