The z factor in polyphase decomposition
The polyphase components of $h[n]$ are

$$e_k[n]=h[Mn+k]\tag{1}$$

Note that if $h[n]$ starts at $n=0$, then so do all polyphase components $e_k[n]$. In order to get back the original sequence $h[n]$ from the polyphase components $e_k[n]$, we need to insert $M-1$ zeros between the samples of $e_k[n]$, and shift each component back to its original position, i.e. shift $e_k[n]$ by $k$ samples. In the frequency domain this corresponds to

$$H(z)=\sum_{k=0}^{M-1}E_k(z^M)z^{-k}\tag{2}$$

Filtering an input signal with $\mathcal{Z}$-transform $X(z)$ results in

$$Y(z)=H(z)X(z)=\sum_{k=0}^{M-1}X(z)E_k(z^M)z^{-k}\tag{3}$$

which corresponds to the structure in figure $4.37$ in Oppenheim and Schafer. The terms $E_k(z^M)z^{-k}$ can clearly be implemented either by delaying the input or the output of the polyphase components:

$$X(z)\left[E_k(z^M)z^{-k}\right]=\left[X(z)z^{-k}\right]E_k(z^M)$$