This is an example in some notes that I worked on. It's a bit involved, and I don't know how to simplify the approach further, so let me just offer a sketch (in particular, I want to try to ignore a bunch of boundary cases which result in awkward constructions that I don't know how to explain well at the moment), which at first glance, looks similar to the approaches sketched above. If a more detailed version is useful, please feel free to contact me by email.

Consider the category $C$ of ordered nonempty finite sets and where $Hom_C(S,T)$ is the set of surjections $f \colon T \to S$ such that $\min f^{-1}(i) < \min f^{-1}(j)$ whenever $i<j$. The idea behind this definition: we might want to work with all surjections but "kill" the bijections, and this is one such way to do it which is closed under composition.

Now consider the category of functors from $C$ to $k$-vector spaces (call these functors $C$-modules), where $k$ is a field. The basic projective $C$-modules $P_n$ are given by

$P_n(S) = k[Hom_C([n],S)]$

where $k[X]$ is the vector space with basis $X$ and $[n] = \{1,\dots,n\}$. Let $k_n$ be the simple $C$-module which assigns $k$ to all sets of size $n$ and $0$ otherwise. We can consider the minimal projective resolution of $k_m$. I claim that it looks like this:

$\cdots \to P_{m+2}^{\oplus c(m+2,m)} \to P_{m+1}^{\oplus c(m+1,m)} \to P_m \to k_m \to 0.$

If you evaluate this sequence on a set of size $n>m$, you will get the desired complex because $\dim_k P_i([n]) = S(n,i)$.

How do we construct such a projective resolution? It is enough to calculate the Ext groups between the different $k_m$ since $Hom(P_n,k_m)$ is $1$-dimensional if and only if $n=m$ and is $0$ otherwise. To do this calculation, we can reduce to calculating the homology of certain simplicial complexes: given $n<m$, define a simplicial complex whose $i-1$ simplices ($i>0$) are chains of morphisms

$[n] \to S_1 \to \cdots \to S_i \to [m]$

and a simplex contains another if it can be obtained by composing morphisms (this is related to the nerve construction of a category except I fix the endpoints). I want to consider the augmented simplicial cochain complex of this, but with a weird caveat: when $i=0$, I want to allow "multiple" empty simplices, which are indexed by the set $Hom_C([n],[m])$. (This is defined this weird way just to incorporate boundary cases correctly.) Anyway, let $H^i$ be the cohomology of this weird complex with coefficients in $k$. Then the claim is that

$Ext^i(k_n, k_m) = H^{i-2}$ for $i>0$.

This follows from modifying an argument of Cibils in

http://dx.doi.org/10.1016/0022-4049(89)90058-3

(the relevant result is Proposition 2.1).

Anyway, the final punchline: these nerve-like things I've introduced are order complexes of truncations of the partition lattice. Specifically, we look at set partitions of the set $[m]$ which have at least $n$ parts and less than $m$ parts. It can be shown that this poset (with a min and max adjoined) is EL-shellable and hence its order complex is homotopy equivalent to a wedge of spheres of maximal dimension (in this case $m - n - 2$ -- let me ignore discussing the case $m-n$ being small). One can calculate the number of spheres from general poset topology tools -- but instead we could appeal to the fact this wedge of spheres fact implies that a complex like I mentioned exists (i.e., a linear resolution and modulo calculating the ranks) and that the ranks have to be the cycle numbers $c$ in order for the Euler characteristic to work out.

positiveintegers...) $\endgroup$ – Mariano Suárez-Álvarez Oct 27 '11 at 20:25