  Associated to any vector configuration A is a toric ideal encoded by vectors
in the kernel of A. Each toric ideal has two special generating sets: the
universal Gr\"obner basis and the Graver basis. While the former is generally a
proper subset of the latter, there are cases for which the two sets coincide.
The most prominent examples among them are toric ideals of unimodular matrices.
Equality of universal Gr\"obner basis and Graver basis is a combinatorial
property of the toric ideal (or, of the defining matrix), providing interesting
information about ideals of higher Lawrence liftings of a matrix. Nonetheless,
a general classification of all matrices for which both sets agree is far from
known. We contribute to this task by identifying all cases with equality within
two families of matrices; namely, those defining rational normal scrolls and
those encoding homogeneous primitive colored partition identities.
