### Abstract

We present two q-analogues of a hook length formula of Knuth for the number of linear extensions of a partially ordered set whose Hasse diagram is a rooted forest. These q-analogues give formulas for the inversion index and the major index generating functions over permutations which correspond to linear extensions of a labeled forest. They generalize and unify several other q-formulas appearing in the literature. For linear forests all of these formulas reduce to MacMahon's classical formula for "q-counting" multiset permutations according to the major index and inversion index. We also extend MacMahon's formula in another direction by q-counting all labelings of a fixed forest according to two very natural statistics on labeled forests which generalize the major index and inversion index on permutations.

Original language | English (US) |
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Pages (from-to) | 165-187 |

Number of pages | 23 |

Journal | Journal of Combinatorial Theory, Series A |

Volume | 52 |

Issue number | 2 |

DOIs | |

State | Published - Nov 1989 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Cite this

*Journal of Combinatorial Theory, Series A*,

*52*(2), 165-187. https://doi.org/10.1016/0097-3165(89)90028-9