Binary binomial equivalence via hyperplane arrangements

Abstract

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.