34 research outputs found
Balanced simplices
An additive cellular automaton is a linear map on the set of infinite
multidimensional arrays of elements in a finite cyclic group
. 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 with the same multiplicity. For any
additive cellular automaton of dimension or higher, the existence of
infinitely many balanced simplices of appearing in
such orbits is shown, and this, for an infinite number of values . 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.Comment: 33 pages ; 11 figures ; 1 tabl
On the problem of Molluzzo for the modulus 4
We solve the currently smallest open case in the 1976 problem of Molluzzo on
, namely the case . This amounts to constructing,
for all positive integer congruent to or , a sequence of
integers modulo of length generating, by Pascal's rule, a Steinhaus
triangle containing with equal multiplicities.Comment: 12 pages ; 3 figures ; 3 tables, Integers : Electronic Journal of
Combinatorial Number Theory, State University of West Georgia, Charles
University, and DIMATIA, 2012, 12, pp.A1
Symmetric binary Steinhaus triangles and parity-regular Steinhaus graphs
52 pages, 24 figures.A binary Steinhaus triangle is a triangle of zeroes and ones that points down and with the same local rule than the Pascal triangle modulo 2. A binary Steinhaus triangle is said to be rotationally symmetric, horizontally symmetric or dihedrally symmetric if it is invariant under the 120 degrees rotation, the horizontal reflection or both, respectively. The first part of this paper is devoted to the study of linear subspaces of rotationally symmetric, horizontally symmetric and dihedrally symmetric binary Steinhaus triangles. We obtain simple explicit bases for each of them by using elementary properties of the binomial coefficients. A Steinhaus graph is a simple graph with an adjacency matrix whose upper-triangular part is a binary Steinhaus triangle. A Steinhaus graph is said to be even or odd if all its vertex degrees are even or odd, respectively. One of the main results of this paper is the existence of an isomorphism between the linear subspace of even Steinhaus graphs and a certain linear subspace of dihedrally symmetric binary Steinhaus triangles. This permits us to give, in the second part of this paper, an explicit basis for even Steinhaus graphs and for the vector space of parity-regular Steinhaus graphs, that is the linear subspace of Steinhaus graphs that are even or odd. Finally, in the last part of this paper, we consider the generalized Pascal triangles, that are triangles of zeroes and ones, that point up now, and always with the same local rule than the Pascal triangle modulo 2. New simple bases for each linear subspace of symmetric generalized Pascal triangles are deduced from the results of the first part
On a problem of Molluzzo concerning Steinhaus triangles in finite cyclic groups
Let be a finite sequence of length in .
The \textit{derived sequence} of is the sequence of length
obtained by pairwise adding consecutive terms of . The collection of
iterated derived sequences of , until length 1 is reached, determines a
triangle, the \textit{Steinhaus triangle generated by the sequence
}. We say that is \textit{balanced} if its Steinhaus triangle
contains each element of with the same multiplicity.
An obvious necessary condition for to be the length of a balanced sequence
in is that divides the binomial coefficient
. It is an open problem to determine whether this condition on
is also sufficient. This problem was posed by Hugo Steinhaus in 1963 for
and generalized by John C. Molluzzo in 1976 for . So far, only
the case has been solved, by Heiko Harborth in 1972. In this paper, we
answer positively Molluzzo's problem in the case for all .
Moreover, for every odd integer , we construct infinitely many balanced
sequences in . This is achieved by analysing the
Steinhaus triangles generated by arithmetic progressions. In contrast, for any
even with , it is not known whether there exist infinitely many
balanced sequences in . As for arithmetic progressions,
still for even, we show that they are never balanced, except for exactly 8
cases occurring at and .Comment: 29 pages, 10 figure
On Ramsey numbers of complete graphs with dropped stars
Let be the smallest integer such that for any -coloring (say,
red and blue) of the edges of , , there is either a red
copy of or a blue copy of . Let be the complete graph on
vertices from which the edges of are dropped. In this note we
present exact values for and new upper bounds
for in numerous cases. We also present some results for
the Ramsey number of Wheels versus .Comment: 9 pages ; 1 table in Discrete Applied Mathematics, Elsevier, 201
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 , that are Steinhaus triangles containing all the elements of
with the same multiplicity. For every odd number ,
we build an orbit in , by the linear cellular automaton
generating the Pascal triangle modulo , which contains infinitely many
balanced Steinhaus triangles. This orbit, in , is
obtained from an integer sequence called the universal sequence. We show that
there exist balanced Steinhaus triangles for at least of the admissible
sizes, in the case where 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 odd. We prove the existence of balanced generalized Pascal triangles
for at least of the admissible sizes, in the case where is an odd
prime power, and the existence of balanced lozenges for all admissible sizes,
in the case where is a square-free odd number.Comment: 30 pages ; 10 figure
Regular Steinhaus graphs of odd degree
A Steinhaus matrix is a binary square matrix of size which is symmetric,
with diagonal of zeros, and whose upper-triangular coefficients satisfy
for all . Steinhaus matrices
are determined by their first row. A Steinhaus graph is a simple graph whose
adjacency matrix is a Steinhaus matrix. We give a short new proof of a theorem,
due to Dymacek, which states that even Steinhaus graphs, i.e. those with all
vertex degrees even, have doubly-symmetric Steinhaus matrices. In 1979 Dymacek
conjectured that the complete graph on two vertices is the only regular
Steinhaus graph of odd degree. Using Dymacek's theorem, we prove that if
is a Steinhaus matrix associated with a regular
Steinhaus graph of odd degree then its sub-matrix is a multi-symmetric matrix, that is a doubly-symmetric matrix where each
row of its upper-triangular part is a symmetric sequence. We prove that the
multi-symmetric Steinhaus matrices of size whose Steinhaus graphs are
regular modulo 4, i.e. where all vertex degrees are equal modulo 4, only depend
on parameters for all even numbers , and on
parameters in the odd case. This result permits us
to verify the Dymacek's conjecture up to 1500 vertices in the odd case.Comment: 16 page