  The permanental polynomial of a graph $G$ is
$\pi(G,x)\triangleq\mathrm{per}(xI-A(G))$. From the result that a bipartite
graph $G$ admits an orientation $G^e$ such that every cycle is oddly oriented
if and only if it contains no even subdivision of $K_{2,3}$, Yan and Zhang
showed that the permanental polynomial of such a bipartite graph $G$ can be
expressed as the characteristic polynomial of the skew adjacency matrix
$A(G^e)$. In this paper we first prove that this equality holds only if the
bipartite graph $G$ contains no even subdivision of $K_{2,3}$. Then we prove
that such bipartite graphs are planar. Further we mainly show that a
2-connected bipartite graph contains no even subdivision of $K_{2,3}$ if and
only if it is planar 1-cycle resonant. This implies that each cycle is oddly
oriented in any Pfaffian orientation of a 2-connected bipartite graph
containing no even subdivision of $K_{2,3}$. As applications, permanental
polynomials for some types of bipartite graphs are computed.
