Q-Hook length formulas for signed labeled forests
Q-Hook length formulas for signed labeled forests
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签名标记森林的 Q-Hook 长度公式
DOI：
10.1016/j.aam.2013.05.001
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发表时间：
2013
影响因子：
1.1
通讯作者：
Guo Peter L.
中科院分区：
文献类型：
--
作者：
Chen William Y. C.;Gao Oliver X. Q.;Guo Peter L.
带符号标记的森林被定义为一个（平面）森林，标记为1,2，…，n以及与一些顶点相关的负号。有符号标记森林可以看作是有符号排列的扩展。我们定义了有符号标记森林的反转数、flag major index和R-major index，可以看作是Björner和Wachs引入的标记森林统计量的B类类似物。有符号标记森林的标志主指数基于Adin和Roichman定义的有符号排列的标志主指数。引入了基于自然序的有符号置换的r主指标，并证明了它可以用Reiner定义的主指标通过双射表示。然后定义了有符号标记森林的R-major指数。我们通过对给定森林的上述三个指标进行q计数的有符号标记得到了q钩长度公式，并证明了这些统计量在有符号标记森林上是等分布的。我们的公式可以看作是Björner和Wachs公式的B类类似物。我们还给出了关于偶符号标记森林的反转数的D型模拟。
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 a signed 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 a signed 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.