### 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) |
---|---|

Pages (from-to) | 165-187 |

Number of pages | 23 |

Journal | Journal of Combinatorial Theory, Series A |

Volume | 52 |

Issue number | 2 |

DOIs | |

State | Published - 1989 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Journal of Combinatorial Theory, Series A*,

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

**q-Hook length formulas for forests.** / Björner, Anders; Galloway, Michelle L.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory, Series A*, vol. 52, no. 2, pp. 165-187. https://doi.org/10.1016/0097-3165(89)90028-9

}

TY - JOUR

T1 - q-Hook length formulas for forests

AU - Björner, Anders

AU - Galloway, Michelle L

PY - 1989

Y1 - 1989

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=38249004360&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38249004360&partnerID=8YFLogxK

U2 - 10.1016/0097-3165(89)90028-9

DO - 10.1016/0097-3165(89)90028-9

M3 - Article

AN - SCOPUS:38249004360

VL - 52

SP - 165

EP - 187

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 2

ER -