q-Hook length formulas for forests

Anders Björner, Michelle L. Wachs

Research output: Contribution to journalArticlepeer-review

34 Scopus citations


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 languageEnglish (US)
Pages (from-to)165-187
Number of pages23
JournalJournal of Combinatorial Theory, Series A
Issue number2
StatePublished - Nov 1989

ASJC Scopus subject areas

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


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