## Higher combinatorics: proceedings of the NATO Advanced Study Institute held in Berlin (West Germany), September 1-10, 1976 |

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Page 119

... and less illuminating, terms. One can even say that it was essentially proved (

though in analytic rather than geometrical terms) by Mirsky and Perfect who in

1967 established the linear representability of transversal matroids by assigning

indeterminate coordinates exactly as described above [29 - and see 28, Theorem

7.1.3]. 2. Characterizations of transversal matroids To study the finer structure of

transversal matroids we need to look at the

X is ...

... and less illuminating, terms. One can even say that it was essentially proved (

though in analytic rather than geometrical terms) by Mirsky and Perfect who in

1967 established the linear representability of transversal matroids by assigning

indeterminate coordinates exactly as described above [29 - and see 28, Theorem

7.1.3]. 2. Characterizations of transversal matroids To study the finer structure of

transversal matroids we need to look at the

**cyclic flats**. In a given matroid M a setX is ...

Page 120

It is implicit in the definition of a simplicial matroid £3 that a cyclic set of rank k in

S or in the restriction SJT must lie in a k-face of V; moreover, a cyclic k-flat must

be a k-face (or, more precisely, the intersection of that face with E or T as the case

may be). This fact not only shows that the number of cyclic k-flats in a transversal

matroid of rank r cannot exceed (. ) as was proved by Brualdi and Mason [10] but

also imposes a restriction on the way the

It is implicit in the definition of a simplicial matroid £3 that a cyclic set of rank k in

S or in the restriction SJT must lie in a k-face of V; moreover, a cyclic k-flat must

be a k-face (or, more precisely, the intersection of that face with E or T as the case

may be). This fact not only shows that the number of cyclic k-flats in a transversal

matroid of rank r cannot exceed (. ) as was proved by Brualdi and Mason [10] but

also imposes a restriction on the way the

**cyclic flats**fit together. THEOREM 2.Page 121

of all families of cyclic sets, but, as in the case of Theorem 2, it is not difficult to

show that it is enough to consider

and require the condition only for a very restricted set of families of

DEFINITION. Call a set A a critical flat if it is the intersection of all the

which properly contain it; then let F-],...,Fjt be the minimal (in terms of set inclusion

) members of the set of all

lpF(J)-pA.

of all families of cyclic sets, but, as in the case of Theorem 2, it is not difficult to

show that it is enough to consider

**cyclic flats***. In fact, it is possible to go furtherand require the condition only for a very restricted set of families of

**cyclic flats**.DEFINITION. Call a set A a critical flat if it is the intersection of all the

**cyclic flats**which properly contain it; then let F-],...,Fjt be the minimal (in terms of set inclusion

) members of the set of all

**cyclic flats**properly containing A and put 6A = I (-1)|ji+lpF(J)-pA.

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### Contents

THE ROLE OF EULERIAN SERIES | 3 |

DISTRIBUTIONS EULERIENNES ET MAHONIENNES SUR LE GROUPE | 27 |

COHENMACAULAY COMPLEXES | 48 |

Copyright | |

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### Other editions - View all

Higher Combinatorics: Proceedings of the NATO Advanced Study Institute held ... M. Aigner Limited preview - 2012 |

Higher Combinatorics: Proceedings of the NATO Advanced Study Institute held ... M. Aigner No preview available - 2011 |

### Common terms and phrases

Aigner algebraic biplane block chain partition characterization Cohen-Macaulay colour combinatorial geometry complete class configuration construction contains cyclic flats defined denoted Dilworth truncation distributive lattices elements eulérienne example finite lattice finite partially ordered free matroid function Germany Gorenstein graph group G h-vector Hence Higher Combinatorics hyperplane identity incidence incidence matrix indep independent induced infinite Inst integers IRISE irreducible isomorphic J.A. Thas lemma linear lines Math matrix matroid join maximal arc modular cut modular lattices nombres number of partitions O-sequence obtain parallel parameters s,t partial geometries partial lattice partially ordered set perfect codes permutation group plane of order points polynomial poset primitive Proc projective plane proof q-analog quadrangle with parameters rank relation representation resp restriction result Rogers-Ramanujan identities S.E. Payne simplicial Steiner system strict gammoids strong map subquadrangle subset subspace t-transitive transitive transversal matroids unique Univ V-code vector space weak maps