Symmetric-like cyphers with several steps
This is not possible with symmetric cryptography.
To show this, I will construct an asymmetric encryption algorithm from your scheme.
Consider $pub = (k_1, \dots, k_{n-2})$ as the public key and $prk = (k_{n-1}, k_n)$ as the private key.
The encryption algorithm will be $c = enc(pub, d) := P(k_{n-2},P(k_{n-3},\dots ,P(k_1,d)))$
The decryption algorithm will be $d = dec(prk, c) := P(k_{n}, P(k_{n-1}, c))$
Since there may be no relations in the sets $puk \cup \{k_{n-1} \}$ and $puk \cup \{k_{n} \}$ , the computation of prk from puk will be infeasible.
This is a property of an asymmetric cryptographic system.