Magic \(N\)-cubes form a free monoid. (English) Zbl 0885.05007

Adler, Allan

Electron. J. Comb. 4, No. 1, Research paper R15, 14 p. (1997); printed version J. Comb. 4, No. 1, 147-160 (1997).

Summary: We prove a conjecture stated in an earlier paper [A. Adler and S. R. Li, Am. Math. Mon. 84, 618-627 (1977; Zbl 0389.05018)]. The conjecture states that with respect to a rather natural operation, the set of \(N\)-dimensional magic cubes forms a free monoid for every integer \(N>1\). A consequence of this conjecture is a certain identity of formal Dirichlet series. These series and the associated power series are shown to diverge. Generalizations of the underlying ideas are presented. We also prove variants of the main results for magic cubes with remarkable power sum properties.

MSC:

05A15	Exact enumeration problems, generating functions
05B15	Orthogonal arrays, Latin squares, Room squares
05A19	Combinatorial identities, bijective combinatorics

Keywords:

free monoid; magic cubes; identity; power series

Citations:

Zbl 0389.05018