Suppose that $P$ and $C$ are two unordered partitions of $[n]$, the set of positive integers from 1 to $n$. Let $c(C,P,x)$ be the number of functions $f$ from $[n]$ to $[x]$ for which

(1) $f(a)=f(b)$ if $a$ and $b$ belong to the same part of $C$,

(2) $f(a)≠f(b)$ if $a$ and $b$ belong to the same part of $P$.

**Conjecture:** The sum $\sum (-1)^{p-1} (p-1)! c(C,P,x)$ is independent of $C$. Here the sum is over all partitions $P$, and $p$ denotes the number of parts of $P$.

Consider the case where $C$ has a single part. Then $c(C,P,x)$ is zero unless $P$ is the trivial partition into $n$ parts, and the sum is $(-1)^{n-1} (n-1)! x$. Thus the conjecture is that this is always the value of the alternating sum, for arbitrary $C$.

This conjecture arises from the same context as questions "Stirling number identity via homology" and "Polynomials akin to Bell polynomials," asked by the same user.