What is the crank of a partition?
نویسندگان
چکیده
The crank of an integer partition is an integer statistic which yields a combinatorial explanation of Ramanujan’s three famous congruences for the partition function. The existence of such a statistic was conjectured by Freeman Dyson in 1944 and realized 44 years later by George Andrews and Frank Gravan. Since then, groundbreaking work by Ken Ono, Karl Mahlburg, and others tells us that not only are there infinitely many congruences for the partition function, but infinitely many of them are explained by the crank. In this note, we will introduce the partition function, define the crank of a partition, and show how the crank underlies congruences of the partition function. Along the way, we will recount the fascinating history behind the crank and its older brother, the rank. This exposition ends with the recent work of Karl Mahlburg on crank function congruences. Andrews and Ono [3] have a good quick survey from which to begin.
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