The number ${n \choose k}$ of $k$-element subsets of an $n$-element set and the number $\left( {n \choose k} \right)$ of $k$-element **multisets** of an $n$-element set satisfy the reciprocity formula


$\displaystyle {-n \choose k} = (-1)^k \left( {n \choose k} \right)$


when extended to negative integer indices, for example by applying the usual recurrence relations to all integers. There's an interesting way to think about the "negative cardinalities" involved here using Euler characteristic, which is due to Schanuel; see, for example, [this paper of Jim Propp](http://arxiv.org/abs/math/0203289). Another (related?) way to think about this relationship is in terms of the symmetric and exterior algebras; see, for example, [this blog post](http://qchu.wordpress.com/2009/11/06/set-multiset-duality-and-supervector-spaces/).


The number $S(n, k)$ of $k$-block partitions of a set with $n$ elements and the number $c(n, k)$ of permutations of a set with $n$ elements with $k$ cycles satisfy a well-known inverse matrix relationship, but they also satisfy the reciprocity formula


$c(n, k) = S(-k, -n)$


when extended to negative integer indices, again by applying the usual recurrence relations. 


**Question:** Are there any known highbrow interpretations of this reciprocity formula? 



Highbrow interpretations of Stirling number reciprocity
Supplementary Exercise 3.2(d,e) on page 313 of my book *Enumerative Combinatorics*, vol. 1, second printing, shows that this Stirling number reciprocity is a special case of the reciprocity theorem for order polynomials (Exercise 3.61(a)). Thus it is related to a lot of "highbrow" math, such as the reciprocity between a Cohen-Macaulay ring and its canonical module. (For the basic properties of canonical modules, see Section I.12 of my other book *Combinatorics and Commutative Algebra,* second ed.)


