The Kontsevich graph orientation morphism revisited

Abstract: The orientation morphism $Or(\cdot)(P)\colon \gamma\mapsto\dot{P}$ associates differential-polynomial flows $\dot{P}=Q(P)$ on spaces of bi-vectors $P$ on finite-dimensional affine manifolds $N^d$ with (sums of) finite unoriented graphs $\gamma$ with ordered sets of edges and without multiple edges and one-cycles. It is known that $d$-cocycles $\boldsymbol{\gamma}\in\ker d$ with respect to the vertex-expanding differential $d=[{\bullet}!!{-}!{-}!!{\bullet},\cdot]$ are mapped by $Or$ to Poisson cocycles $Q(P)\in\ker\,[![ P,{\cdot}]!]$, that is, to infinitesimal symmetries of Poisson bi-vectors $P$. The formula of orientation morphism $Or$ was expressed in terms of the edge orderings as well as parity-odd and parity-even derivations on the odd cotangent bundle $\Pi T^* N^d$ over any $d$-dimensional affine real Poisson manifold $N^d$. We express this formula in terms of (un)oriented graphs themselves, i.e. without explicit reference to supermathematics on $\Pi T^* N^d$.