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A signed labeled forest is defined as a (plane) forest labeled by {1,2,...,n} along with minus signs associated to some vertices. Signed labeled forestscan be viewed as an extension of signed permutations. We define the inversionnumber, the flag major index and the R-major index on signed labeled forests.They can be considered as type B analogues of the indices for labeled forestsintroduced by Bjorner and Wachs. The flag major index for signed labeledforests is based on the flag major index on signed permutations introduced byAdin and Roichman, whereas the R-major index for signed labeled forests isbased on the R-major index that we introduce for signed permutations, which isclosely related to the major defined by Reiner. We obtain q-hook lengthformulas by q-counting signed labelings of a given forest with respect to theabove indices, from which we see that these three indices are equidistributedfor signed labeled forests. Our formulas for the major indices and theinversion number are type B analogues of the formula due to Bjorner and Wachs.We also give a type D analogue with respect to the inversion number ofeven-signed labeled forests.

q-Hook Length Formulas for Signed Labeled Forests