In this paper we calculate the branching ratios (BRs) of the 32 charmless
hadronic $B_c \to A P$ decays ($A=a_1(1260),b_1(1235),K_1(1270),K_1(1400),
f_1(1285),f_1(1420),h_1(1170),h_1(1380)$) by employing the perturbative
QCD(pQCD) factorization approach. These considered decay channels can only
occur via annihilation type diagrams in the standard model. From the numerical
calculations and phenomenological analysis, we found the following results: (a)
the pQCD predictions for the BRs of the considered $B_c$ decays are in the
range of $10^{-6}$ to $10^{-8}$, while the CP-violating asymmetries are absent
because only one type tree operator is involved here; (b) the BRs of $\Delta S=
0$ processes are generally much larger than those of $\Delta S =1$ ones due to
the large CKM factor of $|V_{ud}/V_{us}|^2\sim 19$; (c) since the behavior for
$^1P_1$ meson is much different from that of $^3P_1$ meson, the BRs of $B_c \to
A(^1P_1) P$ decays are generally larger than that of $B_c \to A(^3P_1) P$
decays; (d) the pQCD predictions for the BRs of B_c \to (K_1(1270), K_1(1400))
\etap and $(K_1(1270), K_1(1400)) K$ decays are rather sensitive to the value
of the mixing angle $\theta_K$.Comment: 1+17 pges, 1 figure, refs. added and some clarifications made,
accepted for publication in Phys.Rev.