  The $F$-signature is a numerical invariant defined by the number of free
direct summands in the Frobenius push-forward, and it measures singularities in
positive characteristic. It can be generalized by focussing on the number of
non-free direct summands. In this paper, we provide several methods to compute
the (generalized) $F$-signature of a Hibi ring which is a special class of
toric rings. In particular, we show that it can be computed by counting the
elements in the symmetric group satisfying certain conditions. As an
application, we also give the formula of the (generalized) $F$-signature for
some Segre products of polynomial rings.
