What is the homotopy type of the poset of nontrivial decompositions of $\mathbf{R}^n$? Consider the following partial order.  The objects are unordered tuples $\{V_1,\ldots,V_m\}$, where each $V_i \subseteq \mathbf{R}^n$ is a nontrivial linear subspace and $V_1 \oplus \cdots \oplus V_m = \mathbf{R}^n$.  We say that $\{V_1,\ldots,V_m\} \geq \{W_1,\ldots,W_i\}$ if for each $j=1,\ldots,m$ there exists $1 \leq k \leq i$ such that $V_j \subseteq W_k$.  (I.e. if $\{V_1,\ldots,V_m\}$ is a finer decomposition than $\{W_1,\ldots,W_i\}$.)  This partial order has a minimal element, $\{\mathbf{R}^n\}$, which is the trivial decomposition.  If we remove it, what is the homotopy type of this partial order?
Some things that are known about similar problems.

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*If we're working over $\mathbb{F}_1$, i.e. with subsets of $\{1,\ldots,n\}$, then there is both a minimal and a maximal object.  If we remove both of these then the homotopy type of the partial order is a wedge of $S^{n-3}$'s.


*If instead we work with ordered decompositions, then this is a barycentric subdivision of Ruth Charney's split building, so it is a wedge of $S^{n-2}$'s.
Edit: clarity and notation.
 A: Let me write $V$ for a finite-dimensional vector space over some field (the field will not play a role), and $\mathsf{P}(V)$ for the poset described in the question, which I consider as a category. Let me rephrase the order relation: $\{V_1, \ldots, V_n\} \geq \{W_1, \ldots, W_m\}$ if and only if each $W_j$ is a direct sum of $V_i$'s. The nerve of this poset is $(\mathrm{dim}(V)-2)$-dimensional, so to see that it is a wedge of spheres one must show that it is $(\mathrm{dim}(V)-3)$-connected. I think this is true.
Let $\mathsf{S}(V)$ denote the following category. It has objects given by pairs $([n], f : [n] \to Sub(V))$ consisting of a standard set $[n] := \{1,2,\ldots, n\}$ and a function from $[n]$ to the set of proper vector subspaces of $V$, such that $\bigoplus_{i \in [n]} f(i) = V$. A morphism in $\mathsf{S}(V)$ from $([n], f)$ to $([n'], f')$ is given by a surjection $e : [n] \to [n']$ such that $f'(i) = \bigoplus_{j \in e^{-1}(i)} f(j)$.
There is a functor $F : \mathsf{S}(V) \to \mathsf{P}(V)$ given by sending $([n], f)$ to the unordered collection $\{f(i)\}_{i \in [n]}$.
Now let me do something a bit odd: choose a total order $\prec$ on the set of vector subspaces of $V$. Then we can consider objects of $\mathsf{P}(V)$ as given by lists of subspaces $(V_1, \ldots, V_n)$ with $V_1 \prec V_2 \prec \cdots \prec V_n$, and such that $\bigoplus_{i=1}^n V_i = V$. Attempt to define a functor $G : \mathsf{P}(V) \to \mathsf{S}(V)$ by
$$G(V_1, \ldots, V_n) := ([n], i \mapsto V_i)$$
on objects. If $(V_1, \ldots, V_n) \geq (W_1, \ldots, W_m)$ in $\mathsf{P}(V)$ then each $W_j$ is a direct sum of $V_i$'s. Define a function $e : [n] \to [m]$ by $e(i) = j$ if $V_i \subset W_j$; this gives a morphism in $\mathsf{S}(V)$ from $([n], i \mapsto V_i)$ to $([n], j \mapsto W_j)$. To check that this is indeed  a functor suppose that $(W_1, \ldots, W_m) \geq (U_1, \ldots, U_\ell)$, with $e'(j) = k$ when $W_j \subset U_k$. Then we indeed have $e' e(i) = k$ when $V_i \subset W_{e(i)} \subset U_k$, so $G$ is indeeed a functor.
The conclusion of this discussion is that $\mathsf{P}(V)$ is a retract of $\mathsf{S}(V)$, so it suffices to show that $\mathsf{S}(V)$ is $(\mathrm{dim}(V)-3)$-connected.
The full name of $\mathsf{S}(V)$ is $\mathsf{S}^{E_\infty}(V)$, the $E_\infty$-splitting category defined by Galatius, Kupers, and I in Definition 17.17 of Cellular $E_k$-algebras (applied to the symmetric monoidal groupoid of finite-dimensional vector spaces; the way I have described it here is not identical to that definition, but is an equivalent category). It has a cousin $\mathsf{S}^{E_1}(V)$ which turns out to be precisely (the category of simplices of) Charney's split building. In particular $\mathsf{S}^{E_1}(V)$ is  $(\mathrm{dim}(V)-3)$-connected by Charney's theorem.
It would take too long to explain all the details here, but the theory developed in that paper shows that the collection of all $\Sigma^2 \mathsf{S}^{E_\infty}(V)$'s can be obtained from the collection of all $\Sigma^2 \mathsf{S}^{E_1}(V)$'s by an iterated bar construction, and in particular given Charney's connectivity result (for all $V$) it follows that the $\Sigma^2 \mathsf{S}^{E_\infty}(V)$ are also $(\mathrm{dim}(V)-1)$-connected, so the $\mathsf{S}^{E_\infty}(V)$ are homologically $(\mathrm{dim}(V)-3)$-connected. (It should not be hard to show it is simply-connected, by hand.) I'm happy to discuss the details by e-mail, if you like.
(Also, now that I believe it is true I expect there must be a more elementary way to deduce it from Charney's theorem.)
Edit:

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*As @inna says in the comments, the functor $F : \mathsf{S}(V) \to \mathsf{P}(V)$ is actually an equivalence, because there is a natural isomorphism $\mathrm{Id} \Rightarrow F \circ G$ given by applying the unique permutation necessary to put things in the order given by $\prec$.


*Let me try to give some references for what I said in the final paragraph. All references are to Cellular $E_k$-algebras:
Writing $\mathsf{G}$ for the symmetric monoidal category of finite-dimensional vector spaces and linear isomorphisms, we work in the category $\mathsf{sSet}_*^\mathsf{G}$ of functors from $\mathsf{G}$ to pointed simplicial sets. This is again symmetric monoidal by Day convolution, and the functor
$$\mathbb{t}(V) = \begin{cases}
S^0 & V \neq 0\\
* & V=0
\end{cases}$$
is a nonnital commutative monoid, and hence also a nonunital $E_\infty$-algebra. (In the paper this object is called $\underline{*}_{>0}$.)
There is another character involved: the derived $E_k$-indecomposables $Q^{E_k}_\mathbb{L}(\mathbb{t})$ for $1 \leq k \leq \infty$, which are again objects of $\mathsf{sSet}_*^\mathsf{G}$. For $V$ an object of $\mathsf{G}$ we write $H_{V,d}^{E_k}(\mathbb{t}) := H_d(Q^{E_k}_\mathbb{L}(\mathbb{t})(V))$.
The ingredients I have in mind are now:
a. Combining Proposition 17.4 and Lemma 17.10 shows that
$$\Sigma Q^{E_1}_\mathbb{L}(\mathbb{t})(V) \simeq \Sigma^2 \mathsf{S}^{E_1}(V)$$
(in fact by Section 17.5 this can be desuspended once) and hence Charney's theorem shows that $H_{V,d}^{E_1}(\mathbb{t})=0$ for $d < \dim(V)-1$.
b. Theorem 14.4 (this is the application of the bar construction result, but is packaged so one doesn't explicitly have to think about that) applied with $\rho(V) := \mathrm{dim}(V)$ shows that $H_{V,d}^{E_\infty}(\mathbb{t})=0$ for $d < \dim(V)-1$ too.
c. Corollary 17.23 shows that $Q^{E_\infty}_\mathbb{L}(\mathbb{t})(V) \simeq \Sigma \mathsf{S}^{E_\infty}(V)$ so the above translates to $\tilde{H}_*(\mathsf{S}^{E_\infty}(V))=0$ for $* \leq \mathrm{dim}(V)-3$.
