Balanced simplices

Abstract : 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 are 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 Pascal simplices, that are a generalization of the Pascal triangle into arbitrary dimension, is studied in detail.
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Jonathan Chappelon. Balanced simplices. Advances in Applied Mathematics, Elsevier, 2015, 62, pp.74-117. ⟨10.1016/j.aam.2014.09.007⟩. ⟨hal-00949618v2⟩

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