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In number theory, the crank of a partition of an integer is a certain integer associated with the partition. The term was first introduced without a definition by Freeman Dyson in a 1944 paper published in Eureka, a journal published by the Mathematics Society of Cambridge University. Dyson then gave a list of properties this yet to be defined quantity should have. George E. Andrews and F.G. Garvan in 1988 discovered a definition for crank satisfying the properties hypothesized for it by Dyson.〔 ==Dyson's crank== Let ''n'' be a non-negative integer and let ''p''(''n'') denote the number of partitions of ''n'' (''p''(0) is defined to be 1). Srinivasa Ramanujan in a paper published in 1918 stated and proved the following congruences for the partition function ''p''(''n''), since known as Ramanujan congruences. * ''p''(5''n'' + 4) ≡ 0 (mod 5) * ''p''(7''n'' + 5) ≡ 0 (mod 7) * ''p''(11''n'' + 6) ≡ 0 (mod 11) These congruences imply that partitions of numbers of the form 5''n'' + 4 (respectively, of the forms 7''n'' + 5 and 11''n'' + 6 ) can be divided into 5 (respectively, 7 and 11) subclasses of equal size. The then known proofs of these congruences were based on the ideas of generating functions and they did not specify a method for the division of the partitions into subclasses of equal size. In his Eureka paper Dyson proposed the concept of the rank of a partition. The rank of a partition is the integer obtained by subtracting the number of parts in the partition from the largest part in the partition. For example, the rank of the partition λ = of 9 is 4 − 5 = − 1. Denoting by ''N''(''m'' , ''q'', ''n''), the number of partitions of ''n'' whose ranks are congruent to ''m'' modulo ''q'', Dyson considered ''N''(''m'' , 5 , 5 ''n'' + 4) and ''N''(''m'' , 7 , 7 ''n'' + 5) for various values of ''n'' and ''m''. Based on empirical evidences Dyson formulated the following conjectures known as ''rank conjectures''. For all non-negative integers ''n'' we have: * ''N''(0 , 5 , 5''n'' + 4) = ''N''(1 , 5 , 5''n'' + 4) = ''N''(2 , 5 , 5''n'' + 4) = ''N''(3 , 5 , 5''n'' + 4) = ''N''(4 , 5 , 5''n'' + 4). * ''N''(0 , 7 , 7''n'' + 5) = ''N''(1 , 7 , 7''n'' + 5) = ''N''(2 , 7 , 7''n'' + 5) = ''N''(3 , 7 , 7''n'' + 5) = ''N''(4 , 7 , 7''n'' + 5) = ''N''(5 , 7 , 7''n'' + 5) = ''N''(6 , 7 , 7 ''n'' + 5) Assuming that these conjectures are true, they provided a way of splitting up all partitions of numbers of the form 5''n'' + 4 into five classes of equal size: Put in one class all those partitions whose ranks are congruent to each other modulo 5. The same idea can be applied to divide the partitions of integers of the from 7''n'' + 6 into seven equally numerous classes. But the idea fails to divide partitions of integers of the form 11''n'' + 6 into 11 classes of the same size, as the following table shows. Partitions of the integer 6 ( 11''n'' + 6 with ''n'' = 0 ) divided into classes based on ranks || || || || || || || || || |- | || || || || || || || || || || |} Thus the rank cannot be used to prove the theorem combinatorially. However, Dyson wrote, '' I hold in fact :'' * '' that there exists an arithmetical coefficient similar to, but more recondite than, the rank of a partition; I shall call this hypothetical coefficient the "crank" of the partition and denote by M(m , q , n) the number of partitions of n whose crank is congruent to m modulo q;'' * ''that M(m , q , n) = M(q − m , q , n);'' * ''that M(0 , 11 , 11n + 6) = M(1 , 11 , 11n + 6) = M(2 , 11 , 11n + 6) = M(3 , 11 , 11n + 6) = M(4 , 11 , 11n + 6);'' * ''that . . . '' ''Whether these guesses are warranted by evidence, I leave it to the reader to decide. Whatever the final verdict of posterity may be, I believe the "crank" is unique among arithmetical functions in having been named before it was discovered. May it be preserved from the ignominious fate of the planet Vulcan. '' 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Crank of a partition」の詳細全文を読む スポンサード リンク
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