30 pages ; 10 figuresInternational audienceIn this paper, we partially solve an open problem, due to J.C. Molluzzo in 1976, on the existence of balanced Steinhaus triangles modulo a positive integer $n$, that are Steinhaus triangles containing all the elements of $\mathbb{Z}/n\mathbb{Z}$ with the same multiplicity. For every odd number $n$, we build an orbit in $\mathbb{Z}/n\mathbb{Z}$, by the linear cellular automaton generating the Pascal triangle modulo $n$, which contains infinitely many balanced Steinhaus triangles. This orbit, in $\mathbb{Z}/n\mathbb{Z}$, is obtained from an integer sequence called the universal sequence. We show that there exist balanced Steinhaus triangles for at least $2/3$ of the admissible sizes, in the case where $n$ is an odd prime power. Other balanced Steinhaus figures, such as Steinhaus trapezoids, generalized Pascal triangles, Pascal trapezoids or lozenges, also appear in the orbit of the universal sequence modulo $n$ odd. We prove the existence of balanced generalized Pascal triangles for at least $2/3$ of the admissible sizes, in the case where $n$ is an odd prime power, and the existence of balanced lozenges for all admissible sizes, in the case where $n$ is a square-free odd number
Let S be a numerical semigroup and let (Z, ≤S) be the (locally finite) poset induced by S on the set of integers Z defined by x ≤S y if and only if y − x ∈ S for all integers x and y. In this paper, we investigate the Möbius function associated to (Z, ≤S) when S is an arithmetic semigroup.
Abstract. In this paper, we investigate the Möbius function µ S associated to a (locally finite) poset arising from a semigroup S of Z m . We introduce and develop a new approach to study µ S by using the Hilbert series of S. The latter enables us to provide formulas for µ S when S belongs to certain families of semigroups. Finally, a characterization for a locally finite poset to be isomorphic to a semigroup poset is given.
Abstract. An additive cellular automaton is a linear map on the set of infinite multidimensional arrays of elements in a finite cyclic group Z/mZ. 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 Z/mZ with the same multiplicity. For any additive cellular automaton of dimension 1 or higher, the existence of infinitely many balanced simplices of Z/mZ 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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