More cranks and \(t\)-cores (Q2730739)

Let \(p(n)\) be the number of partitions of \(n\) and consider the results NEWLINE\[NEWLINEp(5n+4)\equiv 0\pmod 5,\quad p(7n+5) \equiv 0\pmod 7,NEWLINE\]NEWLINE NEWLINE\[NEWLINEp(11n+6) \equiv 0\pmod {11},\quad p(25n+24)\equiv 0\pmod {25}NEWLINE\]NEWLINE NEWLINE\[NEWLINEp(49n+47)\equiv 0\pmod {49}, \quad p(121n+116) \equiv 0\pmod {121}.NEWLINE\]NEWLINE These are the first six of Ramanujan's partition congruences. Dyson's rank gives a criterion for splitting the partitions of \(5n+4\) into 5 equal classes. It also splits the partitions of \(7n+5\) into 7 equal classes, but it does not split the partitions of \(11n+6\) into 11 equal classes. A statistic called the crank was discovered by Andrews and Garvan, which divides the partitions of \(11n+6\) into 11 equal classes, as well as giving new interpretations of the mod 5 and 7 results. Later, more cranks were found by Garvan, Kim and Stanton, for the mod 5, 7, 11 and 25 congruences.NEWLINENEWLINENEWLINEIn this paper, a crank for the mod 49 congruence is given. A crank for the mod 121 congruence has not yet been found, and the author believes that if there is one, then it must be more complicated.