The Size of the Largest Antichain in the Partition Lattice
Consider the posetΠnof partitions of ann-element set, ordered by refinement. The sizes of the various ranks within this poset are the Stirling numbers of the second kind. Leta=12−elog(2)/4. We prove the following upper bound for the ratio of the size of the largest antichain to the size of the large...
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| Veröffentlicht in: | Journal of combinatorial theory. Series A 1998-08, Vol.83 (2), p.188-201 |
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| container_title | Journal of combinatorial theory. Series A |
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| creator | Canfield, E.Rodney |
| description | Consider the posetΠnof partitions of ann-element set, ordered by refinement. The sizes of the various ranks within this poset are the Stirling numbers of the second kind. Leta=12−elog(2)/4. We prove the following upper bound for the ratio of the size of the largest antichain to the size of the largest rank:d(Πn⩽)S(n,Kn)⩽c2na(logn)−a−1/4,for suitable constantc2andn>1. This upper bound exceeds the best known lower bound for the latter ratio by a multiplicative factor ofO(1). |
| doi_str_mv | 10.1006/jcta.1998.2871 |
| format | Article |
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| title | The Size of the Largest Antichain in the Partition Lattice |
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