Consider a finite graded poset $\Gamma$ and assign to each maximal element $z\in \Gamma$ a variable $\mu(z)$. I want to solve the system of equations (minimally, I want to compute its rank, ideally, obtain an explicit basis)
$$\sum_{w\leqslant z} \mu(z) = 0.$$
That is, for each $w\in \Gamma$, the weight assigned to the maximal elements that lie over it must sum to zero. (I'm excluding the maximal elements themselves, of course.)
Note that one can write any linear system in terms of a bipartite graph by assigning the equations into one group of vertices of the graph, the variables to the other, and joining an equation to a variable if such variable appears in the chosen equation. One then chooses $\mu(z)$ accordingly and the equations above are precisely those of the linear system (consider a bipartite graph as a poset in the obvious way). In such a sense, this problem is as broad as it gets.
The following extra information may (or may not) help narrowing it down.
To be more concrete, the problem arises as follows. Fix a combinatorial species $P:\mathsf{Set}^\times\longrightarrow \mathsf{Set}$ (a functor) with restrictions, that is:
- For each finite set $I$ and each subset $S$ of $I$, there is an arrow $$(?)_S : P(I)\to P(S)$$
- These arrows are compatible in the sense they form a sheaf over finite sets and injections of subsets, and bijections:
- If $T\subseteq S\subseteq I$ then $((?)_S)_T = (?)_T$
- For every $I$, $(-)_I ={\rm id}_I$
- For every bijection $\sigma : I \longrightarrow J$ and every subset $S$ of $I$, $$(P(\sigma)(?))_{\sigma(S)} = P(\sigma)((?)_{S})$$
Using this data one can form, for any finite set $I$, a poset $\Gamma(I)$ as follows. The underlying set of $\Gamma(I)$ is the collection of $P$-structures on subsets of $I$:
$$\Gamma(I) = \bigcup_{S\subseteq I} P(S).$$
We define a partial order on $\Gamma(I)$ so that $z\leqslant w$ if
- The support of $z$ is contained in that of $w$: $z\in P(S)$ and $w\in P(T)$ with $S\subseteq T$,
- $z$ is obtained by restricting $w$ to $S$: $z= (w)_S$
Note that $\Gamma(I)$ is graded by the cardinality of the support of a structure. Thus, for each subset $S$ of $I$ I want to solve the system of equations
$$\sum_{z:(z)_S=w} \mu(z) = 0$$
as $z$ ranges through $P(I)$. To give even a more concrete example, I have computed a few values of the dimension $d_n$ ($n=\# I$) of solutions of the above for $L$ the species of linear orders, and $d_n$ coincides with the derrangement sequence $1,0,1,2,9,44,\ldots$.
Variants of this poset have been studied here
https://arxiv.org/pdf/0902.4011v3.pdf
for example. One option is inverting (after suitably defining $\mu(w)$ for every other $w\in \Gamma$) the equations above to obtain recurrences, but I would expect that the calculation of the Möbius function of an arbitrary poset like the above is not easy at all (as the cited paper shows).
One can fix a ring $k$ and linealize the category of species with restrictions by postcomposition with the free functor to $k$-modules. This category is in fact the category of (left) comodules over a fixed (linealized) species, the exponential species, and it is monoidal with a particular product (the Cauchy product). The problem above amounts to computing the cotensor product of $P$ with the unit for this monoidal structure.