q-Hook length formulas for signed labeled forests

Abstract

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 forests can be viewed as an extension of signed permutations. We define the inversion number, 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 forests introduced by Björner and Wachs. The flag major index for signed labeled forests is based on the flag major index on signed permutations introduced by Adin and Roichman, whereas the R-major index for signed labeled forests is based on the R-major index that we introduce for signed permutations, which is closely related to the major defined by Reiner. We obtain q-hook length formulas by q-counting signed labelings of a given forest with respect to the above indices, from which we see that these three indices are equidistributed for signed labeled forests. Our formulas for the major indices and the inversion number are type B analogues of the formula due to Björner and Wachs. We also give a type D analogue with respect to the inversion number of even-signed labeled forests.

Sorting index and Mahonian–Stirling pairs for labeled forests

“…We will use W B (F ) and W D (F ) to denote the set of all signed labelings and the set of all even-signed labelings of a forest F , respectively. The type B and type D analogues of the inversion number of a labeled forest introduced by Björner and Wachs [2] was proposed by Chen et al [4]. The definition is as follows.…”

“…Note that if a signed labeling w is in fact in W(F ), then inv B (F, w) = inv(F, w). Chen et al [4] showed that for a forest F with n vertices w∈W B (F )…”

“…The set of all signed labelings of F will be denoted by W B (F ). Chen et al [4] extended the notion of inversions and major index to signed labeled forests, the latter one in two different ways. The inversion number inv B for signed labelings is motivated by the length function for signed permutations, while the major indices fmaj and rmaj are based on the major indices of signed permutations introduced by Adin and Roichman [1] and Reiner [11], respectively.…”

“…The inversion number inv B for signed labelings is motivated by the length function for signed permutations, while the major indices fmaj and rmaj are based on the major indices of signed permutations introduced by Adin and Roichman [1] and Reiner [11], respectively. The authors in [4] showed that w∈W B (F ) q fmaj(F,w) = w∈W B (F ) q rmaj(F,w) = w∈W B (F )…”

Tree hook length formulae, Feynman rules and B-series

“…In 2005, Seo [26] developed a bijection between these two combinatorial classes. Other combinatorial works on hook length formulae include: [2,25,12,7,14,30,5,29,15,6,9,18]. Kuba and Panholzer [19] discovered a general identity of hook length series in the form of a recurrence relation on the coefficients of the hook length series.…”