# Serial symmetric exchanges

Let *M* be a matroid, and let *A* and *B* be two bases of *M*. A subset *X* of *A* and a subset *Y* of *B*, both of size *k*, form a **serial symmetric exchange** with respect to *A* and *B* if there is an ordering [math]x_1,\dots,x_k[/math] of *X* and an ordering [math]y_1,\dots,y_k[/math] of *Y* such that both [math]A \setminus \{x_1,\dots,x_i\} \cup \{y_1,\dots,y_i\}[/math] and [math]B \setminus \{y_1,\dots,y_i\} \cup \{x_1,\dots,x_i\}[/math] are bases for every [math]i \in \{1,\dots,k\}[/math].

Is it true that for any matroid *M*, any two bases *A* and *B*, and any [math]X \subseteq A[/math], there exists [math]Y \subseteq B[/math] such that *X* and *Y* form a serial symmetric exchange with respect to *A* and *B*?

## Remarks

This conjecture was proposed by Kotlar and Ziv ^{[1]}. If true, it would imply the conjecture on Cyclic orderings of matroids for *k=2*, and the conjecture on Equitability of matroids. Kotlar and Ziv proved that the conjecture is true if [math]|X|=2[/math].

## References

- ↑ D. Kotlar, R. Ziv,
*On Serial Symmetric Exchanges of Matroid Bases*, arXiv link