Identifier
Values
['A',1] => ([],1) => 1
['A',2] => ([(0,2),(1,2)],3) => 0
['B',2] => ([(0,3),(1,3),(3,2)],4) => 0
['G',2] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => 0
['A',3] => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6) => 0
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Description
The alternating sum of the coefficients of the Poincare polynomial of the poset cone.
For a poset $P$ on $\{1,\dots,n\}$, let $\mathcal K_P = \{\vec x\in\mathbb R^n| x_i < x_j \text{ for } i < _P j\}$. Furthermore let $\mathcal L(\mathcal A)$ be the intersection lattice of the braid arrangement $A_{n-1}$ and let $\mathcal L^{int} = \{ X \in \mathcal L(\mathcal A) | X \cap \mathcal K_P \neq \emptyset \}$.
Then the Poincare polynomial of the poset cone is $Poin(t) = \sum_{X\in\mathcal L^{int}} |\mu(0, X)| t^{codim X}$.
This statistic records its $Poin(-1)$.
Map
to root poset
Description
The root poset of a finite Cartan type.
This is the poset on the set of positive roots of its root system where $\alpha \prec \beta$ if $\beta - \alpha$ is a simple root.