## Abstract

Let P be a graded poset. Assume that x_{1},...,x_{m} are elements of rank k and y_{1},...,y_{m} are elements of rank l for some k<l. Further suppose x_{i}≤y_{i}, for 1≤i≤m. Lehman and Ron (J. Combin. Theory Ser. A 94 (2001) 399) proved that, if P is the subset lattice, then there exist m disjoint skipless chains in P that begin with the x's and end at the y's. One complication is that it may not be possible to have the chains respect the original matching and hence, in the constructed set of chains, x_{i} and y_{i} may not be in the same chain. In this paper, by introducing a new matching property for posets, called shadow-matching, we show that the same property holds for a much larger class of posets including the divisor lattice, the subspace lattice, the lattice of partitions of a finite set, the intersection poset of a central hyperplane arrangement, the face lattice of a convex polytope, the lattice of noncrossing partitions, and any geometric lattice.

Original language | English (US) |
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Pages (from-to) | 77-87 |

Number of pages | 11 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 108 |

Issue number | 1 |

DOIs | |

State | Published - Oct 2004 |

## Keywords

- Boolean lattice
- Disjoint chains
- Geometric lattice
- Matching property
- Noncrossing partitions
- Shadow-matching