  In this paper we describe the facets cone associated to transversal
polymatroid presented by $\mathcal{A} =
\{\{1,2\},\{2,3\},...,\{n-1,n\},\{n,1\}\}.$ Using the Danilov-Stanley theorem
to characterize the canonicale module, we deduce that the base ring associated
to this polymatroid is Gorenstein ring. Also, starting from this polymatroid we
describe the transversal polymatroids with Gorenstein base ring in dimension 3
and with the help $\it Normaliz$ in dimension 4.
