### Abstract

Given a family of sets L, where the sets in L admit k 'degrees of freedom', we prove that not all (k+1)-dimensional posets are containment posets of sets in L. Our results depend on the following enumerative result of independent interest: Let P(n, k) denote the number of partially ordered sets on n labeled elements of dimension k. We show that log P(n, k)∼nk log n where k is fixed and n is large.

Original language | English (US) |
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Pages (from-to) | 11-16 |

Number of pages | 6 |

Journal | Order |

Volume | 5 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 1988 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Geometry and Topology
- Computational Theory and Mathematics

### Keywords

- AMS subject classifications (1980): 06A10 (primary), 14N10 (secondary)
- Partially ordered set
- containment order
- degrees of freedom
- partial order dimension

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## Cite this

Alon, N., & Scheinerman, E. R. (1988). Degrees of freedom versus dimension for containment orders.

*Order*,*5*(1), 11-16. https://doi.org/10.1007/BF00143892