Binary binomial equivalence via hyperplane arrangements

Rigo and Salimov (2015) proved that the number of binary $2$-binomial equivalence classes of words of length $n$ is the $n^{\text{th}}$ cake number. We give a geometric explanation of this identity by constructing an explicit arrangement of $n$ planes in three-dimensional space whose chambers are naturally indexed by these equivalence classes. This arrangement is the three-dimensional member of an infinite family of hyperplane arrangements. In dimensions $1$, $2$, and $3$, the corresponding quotients recover abelian equivalence, a natural intermediate equivalence between abelian and $2$-binomial equivalence, and binary $2$-binomial equivalence itself. In higher dimensions, the same family realises natural refinements of $2$-binomial equivalence. We also determine the sizes of the resulting classes. Each size is given by a coefficient of a suitable Gaussian binomial coefficient. This yields the full class-size distribution for binary $2$-binomial equivalence and stabilisation results for the number of classes of any fixed cardinality.