Note on Partitions

IN a paper published in the Messenger of Mathematics (May, 1878), Prof. Sylvester has given a rule for abbreviating the calculation of (w:i,j) (w -1:i,j) ; where, to fix the ideas, let (x:i,j) be regarded as the number of modes of coinposino x with j of the numbers 0, 1, 2, . . . The abbreviation consists in rejectino from the partitionis of w-all partitions whose highest number is not repeated and rejecting from the partitions of w -1. all partitions which do not contain i; the number of partitions thus rejected being shown to be the same in the two cases. This becomes even more obvious if we convert the above (i, j) partitions into (j, i) partitions: that is, replace each of the above partitions by a corresponding one consisting, of i of the numbers 0, 1., . ..j. This, as is well known, can be done by decomposing each number into a column of l's and then recomposing by rows. Now, if we do this, it is plain that those partitions whose highest number was not repeated become partitions containing,r 1; and that those partitions which did not contain i become partitions having less than the full number of parts, or, in other words, partitions containing 0. So that Prof. Sylvester's abbreviation is equivalent to rejecting from the partitions of w those partitions which contain 1, and from the partitions of w 1 those which contain 0. And it is plain that the number of partitions of w which contain 1 is equal to the number of partitions of w 1 which contain 0; for the two sets of partitions are interchanged by the interchange of 0 and 1. Obviously, instead of rejecting the partitions of w which contain 1 andi those of w -1 which contain 0, we may reject the partitions of w which contain m (where qn is any one of the numbers 1, 2, . i (or j)) and those of w -1 which contain m1; the reason being the same as above. 187