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.