  It is known that, for any $k$-list assignment $L$ of a graph $G$, the number
of $L$-list colorings of $G$ is at least the number of the proper $k$-colorings
of $G$ when $k>(m-1)/\ln(1+\sqrt{2})$. In this paper, we extend the Whitney's
broken cycle theorem to $L$-colorings of signed graphs, by which we show that
if $k> \binom{m}{3}+\binom{m}{4}+m-1$ then, for any $k$-assignment $L$, the
number of $L$-colorings of a signed graph $\Sigma$ with $m$ edges is at least
the number of the proper $k$-colorings of $\Sigma$. Further, if $L$ is $0$-free
(resp., $0$-included) and $k$ is even (resp., odd), then the lower bound
$\binom{m}{3}+\binom{m}{4}+m-1$ for $k$ can be improved to
$(m-1)/\ln(1+\sqrt{2})$.
