Dissections of a «strange» function

The strange function of Kontsevich and Zagier is defined by This series is defined only when q is a root of unity, and provides an example of what Zagier has called a quantum modular form. In their recent work on congruences for the Fishburn numbers (n) (whose generating function is F(1-q)), Andrews and Sellers recorded a speculation about the polynomials which appear in the dissections of the partial sums of F(q). We prove that a more general form of their speculation is true. The congruences of Andrews-Sellers were generalized by Garvan in the case of prime modulus, and by Straub in the case of prime power modulus. As a corollary of our theorem, we reprove the known congruences for (n) modulo prime powers.