I am trying to guess the value of the beta invariant of the partition lattice $\pi_4$ if I know the following information:
For any matroid $M,$ I know that
1- $\beta(M) \geq 0.$
2- $\beta(M) > 0$ if and only if $M$is connected and is not a loop.
3- If $e \in E$ is neither a coloop nor a loop,
$$\beta (M) = \beta (M - e) + \beta (M /e).$$
4- $\beta (M^*) = \beta (M)$ except when $M$ is a coloop or a loop.
I also tried to calculate it using: If $e$ is not a loop $\beta(M) = (-1)^{r(M) -1} \sum_{F\in L, e \notin F} \mu_M(\emptyset, F)$ but I got that the answer is 3 and by a little online Googling I understood that the answer is 1. In my calculations I used the idea that the lattice of flats of the matroid $M(K_4)$ is isomorphic to the partition lattice $\pi_4$ and I know from the drawing of $K_4$ that it does not have loops so I assumed also that $\pi_4$ also does not have loops, that is why I used the formula above in calculations. I also know that the Hasse Diagram of $\pi_4$ is the one given in the picture below:
Any clarification on how to just guess it from the 4 points above will be greatly appreciated!
For a strategy to tackle this problem that I think could work is try to compute the beta invariant of a minor of $M(K_4)$ (note that these correspond to cycle matroids of graphic minors of $K_4$). Then you could combine these using the recursive property of Crapo's beta invariant in bullet point 3. Perhaps that is easier than to directly compute it? Note that using the recursive structure only is not sufficient, since you have no bullet point telling you an exact non-zero value for $\beta$.
However, I do think the answer is probably that $\beta(M(K_4))=2$. A result from Sooyeon Lee's PhD thesis (https://egrove.olemiss.edu/etd/1772/) implies that if $M$ a wheel, which $M(K_4)$ is, then $\beta(M) = r(M)-1$. Since $r(M(K_4)) = 3$, this would imply the above. There is probably more a lot more useful information in the thesis, would you be interested.