TY - JOUR
T1 - Bases for permutation groups and matroids
AU - Cameron, P. J.
AU - Fon-Der-Flaass, D. G.
PY - 1995/1/1
Y1 - 1995/1/1
N2 - In this paper, we give two equivalent conditions for the irredundant bases of a permutation group to be the bases of a matroid. (These are deduced from a more general result for families of sets.) If they hold, then the group acts geometrically on the matroid, in the sense that the fixed points of any element form a flat. Some partial results towards a classification of such permutation groups are given. Further, if G acts geometrically on a perfect matroid design, there is a formula for the number of G-orbits on bases in terms of the cardinalities of flats and the numbers of G-orbits on tuples. This reduces, in a particular case, to the inversion formula for Stirling numbers.
AB - In this paper, we give two equivalent conditions for the irredundant bases of a permutation group to be the bases of a matroid. (These are deduced from a more general result for families of sets.) If they hold, then the group acts geometrically on the matroid, in the sense that the fixed points of any element form a flat. Some partial results towards a classification of such permutation groups are given. Further, if G acts geometrically on a perfect matroid design, there is a formula for the number of G-orbits on bases in terms of the cardinalities of flats and the numbers of G-orbits on tuples. This reduces, in a particular case, to the inversion formula for Stirling numbers.
UR - http://www.scopus.com/inward/record.url?scp=21844519186&partnerID=8YFLogxK
U2 - 10.1016/0195-6698(95)90035-7
DO - 10.1016/0195-6698(95)90035-7
M3 - Article
AN - SCOPUS:21844519186
SN - 0195-6698
VL - 16
SP - 537
EP - 544
JO - European Journal of Combinatorics
JF - European Journal of Combinatorics
IS - 6
ER -