区间序的物种
The species of interval orders
AI总结：
该研究在虚拟物种环中建立区间序物种的表达式，通过代数与双射方法证明相关物种恒等式，还揭示Glaisher的T数与特定24色区间序的计数关系。
AI中文摘要：
我们证明了，在虚拟物种环中，区间序的物种$\boldsymbol{\textrm{I}}$可表示为$\boldsymbol{\textrm{I}}=\boldsymbol{\textrm{sum}}_{\boldsymbol{m}\boldsymbol{\neq}\boldsymbol{0}}(\boldsymbol{-1})^{\boldsymbol{m}}\boldsymbol{\times}\boldsymbol{\textrm{prod}}_{\boldsymbol{i}\boldsymbol{=}\boldsymbol{1}}^{\boldsymbol{m}}\boldsymbol{((\boldsymbol{E}^{\boldsymbol{-1}})^{\boldsymbol{i}}\boldsymbol{-}\boldsymbol{1})}$，其中$\boldsymbol{E}^{\boldsymbol{-1}}$是集合物种$\boldsymbol{E}$的乘法逆元。该右侧是Claesson和Hannah引入的带符号选票矩阵的虚拟物种，他们已证明其带符号基数可计数带标签区间序；我们将此结果加强为物种恒等式，并通过代数方法和双射方法（利用自然的符号反转对合）两次证明。$\boldsymbol{\textrm{I}}$的循环指数级数可特化为带标签和不带标签区间序的生成级数；我们将区间序的自同构群描述为杨子群，并证明恒等式$\boldsymbol{\textrm{I}}=\boldsymbol{\textrm{R}}\boldsymbol{\times}\boldsymbol{E}_{\boldsymbol{+}}$，其中$\boldsymbol{\textrm{R}}$是刚性区间序的物种。此外，我们还证明Glaisher的T数$\boldsymbol{T}_{\boldsymbol{n}}$可计数集合$\boldsymbol{[n]}$上无孤立元素颜色为24的24色区间序。
英文摘要：
We show that, in the ring of virtual species, \[ \mathcal{I}=\sum_{m\geq 0}(-1)^m\prod_{i=1}^{m}\bigl((E^{-1})^i-1\bigr), \] where $\mathcal{I}$ is the species of interval orders and $E^{-1}$ is the multiplicative inverse of the species $E$ of sets. The right-hand side is the virtual species of signed ballot matrices introduced by Claesson and Hannah. They showed that its signed cardinality counts labeled interval orders. We strengthen this to a species identity, which we prove twice: first algebraically and then bijectively, using a natural sign-reversing involution. The cycle index series of $\mathcal{I}$ specializes to the generating series for labeled and unlabeled interval orders. We describe the automorphism group of an interval order as a Young subgroup and prove the identity $\mathcal{I}=\mathcal{R}\circ E_+$, where $\mathcal{R}$ is the species of rigid interval orders. We also show that Glaisher's T-number $T_n$ counts the $24$-colored interval orders on $[n]$ in which no isolated element has color $24$.
