# algorithms – inequivalent vertex weights on finite poset

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 $$|P| My goal is to obtain them for every $$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.