An Enumerative Geometry for Magic and Magilatin
Labellings

Matthias Beck^{1} and Thomas Zaslavsky^{2}

beck@math.sfsu.edu

zaslav@math.binghamton.edu

Annals of Combinatorics 10 (4) p.395-498 December, 2006

Abstract:

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.

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