  For a finite abstract simplicial complex G with n sets, define the n x n
matrix K(x,y) which is the number of subsimplices in the intersection of x and
y. We call it the counting matrix of G. Similarly as the connection matrix L
which is L(x,y)=1 if x and y intersect and 0 else, the counting matrix K is
unimodular. Actually, K is always in SL(n,Z). The inverse of K has the Green
function entries K^(-1)(x,y)=w(x) w(y) |W^+(x) intersected W^+y|, where W^+(x)
is the star of x, the sets in G which contain x and w(x)=(-1)^dim(x). The
matrix K is always positive definite. The spectra of K and K^(-1) always agree
so that the matrix Q=K-K^(-1) has the spectral symmetry spec(Q)=-spec(Q) and
the zeta function z(s) summing l(k)^(-s) with eigenvalues l(k) of K satisfies
the functional equation z(a+ib)=z(-a+ib). The energy theorem in this case tells
that the sum of the matrix elements of K^(-1)(x,y) is equal to the number sets
in G. In comparison, we had in the connection matrix case the identity that the
sum of the matrix elements of L^(-1) is the Euler characteristic of G.
