Editing Matroid (section)

====Beta invariant====
The ''beta invariant'' of a matroid, introduced by [[Henry Crapo (mathematician)|Crapo]] (1967), may be expressed in terms of the characteristic polynomial <math> p </math> as an evaluation of the derivative<ref name=Wh87123>{{harvp|White|1987|p=123}}</ref>
:<math> \beta(M) = (-1)^{r(M)-1} p_M'(1) </math>
or directly as<ref name=Wh87124>{{harvp|White|1987|p=124}}</ref>
:<math> \beta(M) = (-1)^{r(M)} \sum_{X \subseteq E} (-1)^{|X|} r(X)</math>.
The beta invariant is non-negative, and is zero if and only if <math> M </math> is disconnected, or empty, or a loop.  Otherwise it depends only on the lattice of flats of <math> M</math>. If <math> M </math> has no loops and coloops then <math> \beta( M ) = \beta( M^* )</math>.<ref name=Wh87124/>