Parsimony and the rank of a flattening matrix

Abstract: The standard models of sequence evolution on a tree determine probabilities for every character or site pattern. A flattening is an arrangement of these probabilities into a matrix, with rows corresponding to all possible site patterns for one set $A$ of taxa and columns corresponding to all site patterns for another set $B$ of taxa. Flattenings have been used to prove difficult results relating to phylogenetic invariants and consistency and also form the basis of several methods of phylogenetic inference. We prove that the rank of the flattening equals $r^{\ell_T(A|B)}$, where $r$ is the number of states and $\ell_T(A|B)$ is the parsimony length of the binary character separating $A$ and $B$. This result corrects an earlier published formula and opens up new applications for old parsimony theorems. Since completing this work, we have learnt that an equivalent result has been proved much earlier by Casanellas and Fern\'andez-S\'anchez, using a different proof strategy.