Combinatorial Stokes formulae

For an n -dimensional pseudomanifold whose vertices get labels from a finite set, there is a “combinatorial Stokes” formula, found by Ky Fan, which links the number of simplices getting n different labels on the boundary with the number of simplices getting n + 1 different labels. In 1998, a generalization of this formula was proved by Lee and Shih taking into account the possibility of putting several labels on each vertex. We re-prove and generalize this latter combinatorial Stokes formula in a rather simple and natural way. Furthermore, some applications of the combinatorial Stokes formula of Fan are given; one of them provides a new combinatorial proof of Schrijver’s theorem about Kneser graphs.