  Let $(G,h)$ be a nilpotent Lie group endowed with a left invariant Riemannian
metric, $\mathfrak{g}$ its Euclidean Lie algebra and $Z(\mathfrak{g})$ the
center of $\mathfrak{g}$. By using an orthonormal basis adapted to the
splitting
  $\mathfrak{g}=(Z(\mathfrak{g})\cap[\mathfrak{g},\mathfrak{g}])\oplus
O^+\oplus (Z(\mathfrak{g})\cap[\mathfrak{g},\mathfrak{g}]^\perp)\oplus
  O^-$, where $O^+$ (resp. $O^-$) is the orthogonal of
$Z(\mathfrak{g})\cap[\mathfrak{g},\mathfrak{g}]$ in
$[\mathfrak{g},\mathfrak{g}]$ (resp. is the orthogonal of
$Z(\mathfrak{g})\cap[\mathfrak{g},\mathfrak{g}]^\perp$ in
$[\mathfrak{g},\mathfrak{g}]^\perp$), we show that the signature of the Ricci
operator of $(G,h)$ is determined by the dimensions of the vector spaces
$Z(\mathfrak{g})\cap[\mathfrak{g},\mathfrak{g}],$
$Z(\mathfrak{g})\cap[\mathfrak{g},\mathfrak{g}]^\perp$ and the signature of a
symmetric matrix of order
$\dim[\mathfrak{g},\mathfrak{g}]-\dim(Z(\mathfrak{g})\cap[\mathfrak{g},\mathfrak{g}])$.
This permits to associate to $G$ a subset $\mathbf{Sign}(\mathfrak{g})$ of
$\mathbf{N}^3$ depending only on the Lie algebra structure, easy to compute and
such that, for any left invariant Riemannian metric on $G$, the signature of
its Ricci operator belongs to $\mathbf{Sign}(\mathfrak{g})$. We show also that
for any nilpotent Lie group of dimension less or equal to 6,
$\mathbf{Sign}(\mathfrak{g})$ is actually the set of signatures of the Ricci
operators of all left invariant Riemannian metrics on $G$. We give also some
general results which support the conjecture that the last result is true in
any dimension.
