Here are the definitions of the Crapo beta invariant I know:
My definition of the Crapo's beta invariant of a matroid from the book "Combinatorial Geometries" from page 123 and 124 is as follows:
$$\beta(M) = (-1)^{r(M) - 1} \frac{d}{d \lambda} p(M; 1),$$ which equals $$(-1)^{r(M) - 1} \sum_F \mu_M(\emptyset, F)[r(M) - r(F)],$$ so that $$\beta(M) = (-1)^{r(M)} \sum_{F\in L} \sum_F \mu_M(\emptyset, F) r(F).$$ And finally, if we know that the characteristic polynomial of the matroid $M$ has the boolean expression $$p(M; \lambda) = \sum_{X \subseteq E} (-1)^{|X|} \lambda^{r(M) - r(X)}$$ then we could equally well define the Crapo beta invariant as $$\beta(M) = (-1)^{r(M)} \sum_{X\subseteq E} (-1)^{|X|} r(X).$$
But why always the Crapo beta invariant value greater than or equal zero?
I know that it satisfies the following properties
1- $\beta(M) > 0$ if and only if $M$ is connected and is not a loop.
2- If $e \in E$ is neither a loop nor an isthmus, then $$ \beta(M) = \beta (M - e) + \beta (M /e).$$
3- $\beta (M^*) = \beta (M)$ except when $M$ is an isthmus or a loop.
But still, I cannot prove why the beta invariant must be nonnegative.
Can someone clarify this to me please?
Let's take a look at the foundational properties of Crapo's beta invariant:
Connectedness and Loops:
When $\beta(M) > 0$ that indicates that the matroid $M$ is connected and is not a loop. Inherently, a loop has no contribution to the beta invariant, as it is an element that is dependent on itself, reflecting in a zero or negative value. So a connected, non-loop matroid having a positive beta invariant in a consequence of its definition, this reflects the underlying structure of the matroid.
Deletion and Contraction:
Unless the element being removed or contracted is a loop or an isthmus (in which case there are specific rules that apply), these operations do not inherently introduce any negative quantities, thus preserving the non-negative nature of the invariant.
Duality:
Again, except for loops and isthmuses, the equality of a matroid's beta invariant and its dual $M^*$ further supports the non-negative nature. Duality doesn't inherently introduce negativity, so it aligns with the non-negative nature of the beta invariant.
Conclusion:
Much like other combinatorial invariants that count specific features, the Crapo beta invariant is defined in such a way as to measure certain properties of a matroid that are inherently non-negative (think connectivity, number of interdependent sets).
Now, one could, potentially, construct a variant of the invariant that allows negative values in specific conditions. However, in standard matroid theory and considering the original definition, like many other combinatorial invariants, the Crapo beta invariant counts features of the structure that cannot be negative, thus remaining, itself, inherently non-negative.