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
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q-Hook length formulas for forests. / Björner, Anders; Galloway, Michelle L.
In: Journal of Combinatorial Theory, Series A, Vol. 52, No. 2, 1989, p. 165-187.Research output: Contribution to journal › Article
}
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 -