q-Hook Length Formulas for Signed Labeled Forests
William Y.C. Chen, Oliver X.Q. Gao and Peter L. Guo
Abstract: A signed labeled forest is defined as a (plane) forest labeled by 1, 2, ... ,n along with minus signs associated with some vertices. Signed labeled forests can be viewed as an extension of signed permutations. We define the inversion number, the flag major index and the R-major index of asigned labeled forest, which can be considered as type B analogues of the statistics for a labeled forest introduced by Björner and Wachs. The flag major index of asigned labeled forest is based on the flag major index of a signed permutation defined by Adin and Roichman. We introduce the R-major index of a signed permutation based on the natural order, and we show that it can be expressed by the major index defined by Reiner via a bijection. Then we define the R-major index of a signed labeled forest. We obtain q-hook length formulas by q-counting signed labelings of a given forest with respect to the above three indices and we show that these statistics are equidistributed over signed labeled forests. Our formulas can be viewed as type B analogues of the formulas due to Björner and Wachs. We also give a type D analogue with respect to the inversion number of an even-signed labeled forest.
AMS Classification: 05A15, 05A17, 05A19
Keywords: Singed labeled forest, q-hook length formula, coxeter groups of type B, coxeter groups of type D (F, w)-partition