Crapo's Basic Properties of the Beta Invariant is a result in [matroid theory](/page/Matroid%20Theory). It formalizes the assertion that let be a finite matroid on ground set with rank , and let be the dual matroid. For every positive rank finite matroid , define its beta invariant by \begin{align } \beta(P)=( 1)^{r P(E(P)) 1}\chi P'(1), \end{align } where is the.