At the moment I am studying a proof by Gel'fand, Rybnikov and Stone (Theorem 4 in the paper "Projective Orientations of Matroids"). To be more precise, I am try ing to describe a new presentation by generators and relations for the inner Tutte groups of a matroid.
I find a result. However, it would be nice to compute some examples in the uniform case. Here is my problem.
Let $U_{d}(m)$ be the uniform matroid of rank $d$ with ground set $E$ of $m$ elements and let us write $\mathfrak{C}$ for its set of circuits. Let us fix an arbitrary enumeration $\{C_{d}\}_{d\in D}$ for $\mathfrak{C}$ and a total order $<$ on $D$.
Let us consider the set $\mathbb{W}$ of symbols $[C_{i_{1}}C_{i_{2}}|C_{i_{3}}C_{i_{4}}]$ where $C_{i_{h}}\in\mathfrak{C}$ for $1\leq h \leq 4$ with the extra conditions:
1) $|C_{i_{1}}\cup C_{i_{2}}\cup C_{i_{3}}\cup C_{i_{4}}|=d+2$
2) $i_{1}\neq i_{3}$ and $i_{1}\neq i_{4}$ and $i_{2}\neq i_{3}$ and $i_{2}\neq i_{4}$
In a similar way, let us write $\mathcal{W}$ for the set of symbols $(C_{j_{1}}C_{j_{2}}|C_{j_{3}}C_{j_{4}})$ where $C_{j_{k}}\in\mathfrak{C}$ for $1\leq k \leq 4$ with the extra conditions:
1) $|C_{j_{1}}\cup C_{j_{2}}\cup C_{j_{3}}\cup C_{j_{4}}|=d+2$
2) $j_{1}<j_{2}$ and $j_{3}<j_{4}$ and $j_{1}<j_{3}$
$\textbf{Problem:}$ Compute the cardinality of $\mathbb{W}$ and $\mathcal{W}$.