Combinatorial statistics of Dyson and Andrews–Garvan modulo $11$

Among the most famous results in the theory of partitions are Ramanujan’s congruences for modulo $5,7$ and $11$. In 1944, Freeman Dyson conjectured combinatorial statistic called rank which explains the congruences for modulo $5,7$ but it fails to explain the congruence modulo $11$. However, Andrews and Garvan discovered the so-called crank, which is a combinatorial statistic explains all three Ramanujan’s congruences. Using recent results of Frank Garvan and Rishabh Sarma we established how exactly rank fails to explain Ramanujan’s congruence modulo $11$ in terms of theta functions and proved many arithmetic properties of rank and crank modulo $11$.