An explicit arrangement of n hyperplanes in 3-space indexes binary 2-binomial equivalence classes of binary words of length n, with class sizes given by coefficients of Gaussian binomials.
Steiner, Einige Gesetze \" u ber die Theilung der Ebene und des Raumes , J.\ Reine Angew.\ Math.\ 1 (1826), 349--364
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Binary binomial equivalence via hyperplane arrangements
An explicit arrangement of n hyperplanes in 3-space indexes binary 2-binomial equivalence classes of binary words of length n, with class sizes given by coefficients of Gaussian binomials.