Balanced simplices

Jonathan Chappelon (1)

1 IMAG - Institut Montpelliérain Alexander Grothendieck

An additive cellular automaton is a linear map on the set of infinite multidimensional arrays of elements in a finite cyclic group $\mathbb{Z}/m\mathbb{Z}$. In this paper, we consider simplices appearing in the orbits generated from arithmetic arrays by additive cellular automata. We prove that they constitute a source of balanced simplices, that are simplices containing all the elements of $\mathbb{Z}/m\mathbb{Z}$ with the same multiplicity. For any additive cellular automaton of dimension $1$ or higher, the existence of infinitely many balanced simplices of $\mathbb{Z}/m\mathbb{Z}$ appearing in such orbits is shown, and this, for an infinite number of values $m$. The special case of the Pascal cellular automata, the cellular automata generating the multidimensional simplices of Pascal, is studied in detail.

Mots clés: Balanced simplices; additive cellular automaton; Pascal cellular automaton; Steinhaus triangles; Pascal triangles; Molluzzo's problem

Domaines: Combinatoire [math.CO]; Théorie des nombres [math.NT]

Jonathan Chappelon. Balanced simplices. Advances in Applied Mathematics, 2015, 62, pp.74-117. ⟨10.1016/j.aam.2014.09.007⟩. ⟨hal-00949618v1⟩