Jonathan Chappelon

On the Möbius function of the locally finite poset associated with a numerical semigroup

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≤Sy 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.

On the Problem of Molluzzo for the Modulus 4

Abstract. We solve the currently smallest open case in the 1976 problem of Molluzzo on , namely the case . This amounts to constructing, for all positive integers n congruent to 0 or 7 mod 8, a sequence of integers modulo 4 of length n generating, by Pascals rule, a Steinhaus triangle containing 0, 1, 2, 3 with equal multiplicities.

Complete Kneser Transversals

Abstract Let k , d , λ ⩾ 1 be integers with d ⩾ λ . Let m ( k , d , λ ) be the maximum positive integer n such that every set of n points (not necessarily in general position) in R d has the property that the convex hulls of all k -sets have a common transversal ( d − λ ) -plane. It turns out that m ( k , d , λ ) is strongly connected with other interesting problems, for instance, the chromatic number of Kneser hypergraphs and a discrete version of Rados centerpoint theorem. In the same spirit, we introduce a natural discrete version m ⁎ of m by considering the existence of complete Kneser transversals . We study the relation between them and give a number of lower and upper bounds of m ⁎ as well as the exact value in some cases. The main ingredient for the proofs are Radons partition theorem as well as oriented matroids tools. By studying the alternating oriented matroid we obtain the asymptotic behavior of the function m ⁎ for the family of cyclic polytopes.

On Ramsey numbers of complete graphs with dropped stars

Let r ( G , H ) be the smallest integer N such that for any 2 -coloring (say, red and blue) of the edges of K n , n ź N , there is either a red copy of G or a blue copy of H . Let K n - K 1 , s be the complete graph on n vertices from which the edges of K 1 , s are dropped. In this note we present exact values for r ( K m - K 1 , 1 , K n - K 1 , s ) and new upper bounds for r ( K m , K n - K 1 , s ) in numerous cases. We also present some results for the Ramsey number of Wheels versus K n - K 1 , s .

Möbius function of semigroup posets through Hilbert series

In this paper, we investigate the Mobius 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.

Balanced simplices

An additive cellular automaton is a linear map on the set of infinite multidimensional arrays of elements in a finite cyclic group Z / m 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 Z / m Z with the same multiplicity. For any additive cellular automaton of dimension 1 or higher, the existence of infinitely many balanced simplices of Z / m 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.

A universal sequence of integers generating balanced Steinhaus figures modulo an odd number

In 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 Z/nZ with the same multiplicity. For every odd number n, we build an orbit in Z/nZ, by the linear cellular automaton generating the Pascal triangle modulo n, which contains infinitely many balanced Steinhaus triangles. This orbit, in Z/nZ, 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.

Codimension two and three Kneser Transversals

Let

Two-Player Tower of Hanoi

k,d,\lambda \geqslant 1

Ramsey for complete graphs with a dropped edge or a triangle

be integers with