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PARTITION OF NUMBERS

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Originally appearing in Volume V19, Page 865 of the 1911 Encyclopedia Britannica.
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PARTITION OF NUMBERS. This mathematical subject, created by Euler, though relating essentially to positive integer numbers, is scarcely regarded as a part of the Theory of Numbers (see NUMBER). We consider in it a number as made up by the addition of other numbers: thus the partitions of the successive numbers 1, 2, 3, 4, 5, 6, &c., are as follows: I; 2 , I I ; 3, 2I, III; 4, 31, 22, 21I, III1; 5, 41, 32, 311, 221, 2II1, 1I1I1; 6, 51, 42, 411, 33, 32I, 311I, 222, 22II, 2IIII, IIIIII. These are formed each from the preceding ones; thus, to form the partitions of 6 we take first 6; secondly, 5 prefixed to each of the partitions of I (that is, 51); thirdly, 4 prefixed to each of the partitions of 2 (that is, 42, 411); fourthly, 3 prefixed to each of the partitions of 3 (that is, 33, 321, 3111); fifthly, 2 prefixed, not to each of the partitions of 4, but only to those partitions which begin with a number not exceeding 2 (that is, 222, 2211, 2111 I) ; and lastly, r prefixed to all the partitions of 5 which begin with a number not exceeding 1 (that is, 1r11 1I); and so in other cases. The method gives all the partitions of. a number, but we may consider different classes of partitions: the partitions into a given number of parts, or into not more than a given number of parts; or the partitions into given parts, either with repetitions or without repetitions, &c. It is possible, for any particular class of partitions, to obtain methods more or less easy for the formation of the partitions either of a given Thus a partition of 6 is 42; writing this in the form ) II II' and summing the columns instead of the lines, we obtain the conjugate partifion 2211; evidently, starting from 2211, the conjugate partition is 42. If we form all the partitions of 6 into not more than three parts, these are 6, 51, 42, 33, 411, 321, 222, and the conjugates are
End of Article: PARTITION OF NUMBERS
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I want to know more aboutthe convergent series of P(n) which was estimated first by Hardy& Ramanujan then improved by Rademacher . thank you.
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