poincare(Arrangement) -- compute the Poincaré polynomial of an arrangement

The Poincaré polynomial $\pi({\mathcal A},t)$ of a central arrangement of rank $r$ equals $t^r\,T(1+t^{-1},0)$, where $T(x,y)$ is the Tutte polynomial. Alternatively, \[ \pi({\mathcal A},t)=\sum_F\mu(\widehat{0},F)(-t)^{r(F)}, \] where the sum is over all flats $F$, the function $\mu$ denotes the Möbius function of the intersection lattice, and $r(F)$ is the rank of the flat $F$. The characteristic polynomial of an (essential) arrangement is closely related and defined by \[ \chi({\mathcal A},t)=t^r\pi({\mathcal A},-t^{-1}). \]

If ${\mathcal A}$ is an arrangement defined over the complex numbers, a classical theorem of Brieskorn-Orlik-Solomon asserts that $\pi({\mathcal A},t)$ is also the Poincaré polynomial of the complement of the union of hyperplanes. In certain interesting cases, the Poincaré polynomial factors into linear factors. This is the case if ${\mathcal A}$ is the set of reflecting hyperplanes associated with a real or complex reflection group, in which case the (co)exponents of the reflection group appear as the linear coefficients of the factors.

More generally (since reflection arrangements are free), if the module of logarithmic derivations $D({\mathcal A})$ on $\mathcal A$ is free, Terao's Factorization Theorem states that the Poincaré polynomial factors as a product $\prod_{i=1}^r(1+m_i t)$, where the $m_i$'s are the degrees of the generators of the graded free module $D({\mathcal A})$.

The Poincaré polynomial appears in various enumerative contexts as well. If ${\mathcal A}$ is an arrangement defined over the real numbers, then $\pi({\mathcal A},1)$ equals the number of connected components in the complement of the union of hyperplanes. Similarly, $d/dt[\pi({\mathcal A},t)]$ evaluated at $t=1$ counts the number of bounded components in the complement of the decone of ${\mathcal A}$.

If ${\mathcal A}$ is a non-central arrangement, the Poincaré polynomial $\pi({\mathcal A},t)$ equals $\pi(c{\mathcal A},t)/(1+t)$, where $c{\mathcal A}$ denotes the cone of ${\mathcal A}$.