Let $mgeq1$ and $P$ be an arbitrary poset with vertex set $V={v_1,dots,v_n}$, edge set $E,$ and set $O$ of orbits under $text{Aut}(P).$ Can we efficiently generate all inequivalent nonnegative weight assignments $alpha:Vtomathbb{N}$ such that $sum_{i=1}^nalpha(v_i)=m$ from this information alone or do we need all inequivalent $alpha$ with sum $leq m$ for all $|P|<n?$ My goal is to obtain them for every $P$ (up to isomorphism) with $|P|leq10$.

If an inductive procedure is needed, maybe it will involve something along the following lines.

Fix an orbit $oin O$ and weight assignment $alpha$ on $Vsetminus o$ with sum $m-m’.$ For each $vin o$ and orbit $o’neq o,$ let $N_alpha(v,o’,w)$ be the number of incoming edges to $v$ from vertices in $o’$ with weight $w.$ For each $v_i,v_jin o$ we set $v_isim_{o’}v_j$ iff $N_alpha(v_i,o’,w)=N_alpha(v_j,o’,w)$ for all $wgeq0,$ and $v_isim v_j$ iff $v_isim_{o’}v_j$ for all $o’neq o.$

Let the equivalence classes under $sim$ have cardinalities $a_1,dots,a_h.$ For each composition $(c_1,dots,c_h)$ of $m’$ with nonnegative parts (one part to each class), we compute all possible combinations of partitions of $c_i$ with $a_i$ nonnegative parts (one part to each vertex).

I think the weight assignments produced by these partitions will be inequivalent, as will the entire collection obtained by repeating the above for all inequivalent $alphatext{‘s}$ on $Vsetminus o,$ but major issues remain such as how do we choose $o,$ etc.