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Encyclopedia Britannica



NUMBERS, PARTITION OF

This article appears in Volume V19, Page 865 of the Encyclopedia Britannica.

Encyclopedia Britannica - Main :: NEW-NUM
NUMBERS, PARTITION OF . 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


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