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In mathematics, particularly in the fields of number theory and combinatorics, the rank of a partition of a positive integer is a certain integer associated with the partition. In fact at least two different definitions of rank appear in the literature. The first definition, with which most of this article is concerned, is that the rank of a partition is the number obtained by subtracting the number of parts in the partition from the largest part in the partition. The concept was introduced by Freeman Dyson in a paper published in the journal Eureka. It was presented in the context of a study of certain congruence properties of the partition function discovered by the Indian mathematical genius Srinivasa Ramanujan. A different concept, sharing the same name, is used in combinatorics, where the rank is taken to be the size of the Durfee square of the partition. ==Definition== By a ''partition'' of a positive integer ''n'' we mean a finite multiset λ = of positive integers satisfying the following two conditions: * λk ≥ . . . ≥ λ2 ≥ λ1 > 0. * ''λ''''k'' + . . . + λ2 + λ1 = ''n''. If ''λ''''k'', . . . , ''λ''2, ''λ''1 are distinct, that is, if * ''λ''''k'' > . . . > λ2 > λ1 > 0 the partition ''λ'' is called a ''strict partition'' of ''n''. The integers ''λ''''k'', λ''k'' − 1, ..., ''λ''1 are the ''parts'' of the partition. The number of parts in the partition ''λ'' is ''k'' and the largest part in the partition is ''λ''''k''. The rank of the partition ''λ'' (whether ordinary or strict) is defined as ''λ''''k'' − ''k''.〔 The ranks of the partitions of ''n'' take the following values and no others:〔 :''n'' − 1, ''n'' −3, ''n'' −4, . . . , 2, 1, 0, −1, −2, . . . , −(''n'' − 4), −(''n'' − 3), −(''n'' − 1). The following table gives the ranks of the various partitions of the number 5. Ranks of the partitions of the integer 5 || 4 || 2 || 2 |- | || 3 || 2 || 1 |- | || 3 || 3 || 0 |- | || 2 || 3 || −1 |- | || 2 || 4 || −2 |- | || 1 || 5 || −4 |} 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「rank of a partition」の詳細全文を読む スポンサード リンク
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