  I show for the specific case of the scalar field spectrum on regular
tessellations of the sphere that the first two terms of the heat--kernel
expansion are related to the first two terms of the Ehrhart (quasi)polynomial.
In trying to make this relation precise, I consider degeneracies as partition
denumerants and show the connection of the group theory expressions with
Popoviciu's theorem and with the notion of Sylvester waves. General denumerants
are considered and the first wave, i.e. the polynomial part, is written using
the A-genus multiplicative sequence. It is pointed out that Sylvester in effect
did the same thing and that he had also obtained Ehrhart reciprocity. I derive
an algebraically neat form for the second wave which involves the combination
of two multiplicative sequences.
