The Wide Partition Conjecture

Let $M$ be a matroid. An $M$-tableau of shape $\lambda$ is a Young diagram of shape $\lambda$ with an element of $M$ in each cell. A partition $\lambda$ satisfies Rota's conjecture if, for every matroid $M$ and every sequence $(I_i)$ of independent sets of $M$ with $|I_i|=\lambda_i$ for all $i$, there is an $M$-tableau $T$ of shape $\lambda$ such that the elements in row $i$ are exactly $I_i$ and the elements in column $j$ form an independent set of $M$.

The Wide Partition Conjecture. A partition $\lambda$ satisfies Rota's conjecture if and only if it is wide.

The conjecture generalizes Rota's basis conjecture and was formulated as a possible route to proving it by induction. The paper explains that necessity is straightforward, while sufficiency is the substantive unresolved direction; the free-matroid case is treated separately.