2.
A
magic labelling of a set system is a labelling of its points by distinct positive integers so that every set of the system has the same sum,
the
magic sum. Examples are magic squares (the sets are the rows, columns, and diagonals) and semimagic squares (the same, but without
the diagonals). A
magilatin labelling is like a magic labelling but the values need be distinct only within each set. We show that the number of
n ×
n magic or magilatin labellings is a quasipolynomial function of the magic sum, and also of an upper bound on the entries in
the square. Our results differ from previous ones because we require that the entries in the square all be different from
each other, and because we derive our results not by
ad hoc reasoning but from a general theory of counting lattice points in rational inside-out polytopes. We also generalize from
set systems to rational linear forms.
Dedicated to the memory of Claudia Zaslavsky, 1917–2006
Received August 10, 2005
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