CHAP, ill] PARTITIONS. Ill 



to the next. Here the entries for x = 2 give the number of partitions of 

 3 = 1 + 2, 4, 5, into 2 distinct parts; and are the sums of the units 

 in the respective columns in the accompanying scheme of units arranged by 



1 1 1 1 1 1 1 1 



1 1 1 1 1 1 



11 11 



1 1 



twos. Apply the same process to these numbers for x = 2, taking them 

 by threes: 



112 233 445 



112 233 ... 



112 ... 



Summing the columns, we obtain the number of partitions of 6 = 1 + 2 + 3, 

 7, 8, into x = 3 distinct parts. Taking these by fours, we get similarly 

 the series for x = 4. This property shows that 



(t, x) = (t - x, x) + (t, x - 1), 



if (t, x) is the tth term of the series for x. 



To pass to the number of partitions into x distinct terms of 1, , p, 

 we must delete the partition (p + 1) + 1 of p + 2, and (p + 1) + 2, 

 (p + 2) + 1 of p + 3, etc. Thus the number of terms hi the " first series 

 of subtraction " is 



s = l: 1 1 1 1 1 1 .. 

 x = 2: 123456 . 

 x = 3: 12469 12... 



any line of which is formed from the preceding line as in the former problem. 

 Thus (t, x + 1) = (t x, x + 1) + (t, x). But for x = 2 we counted the 

 partition of 2p + 2 into parts each p + 1. Hence we must correct our 

 subtractive series by employing the "first additive series 



" 



x = 2: 112233- ; x = 3: 1246912.-.; 



leading to (t, x + 2) = (t x, x + 2) + (t, x + 1). Then we have a 

 second subtractive series, etc. The general one of these difference equa- 

 tions is 



(t, x + X) = (t - x, x + X) + (t, x + X - 1). 



It has the integral a 'II, where n = n^ia^+A, if a y+A = 1/(1 a~ j }. Thus 

 H = 1/D, D = (I - ar l )(l - a~ 2 ) (!- ar x ). If 



1/D = 1 + A'a~ l + A"a- z + -, 



we find (as by Paoli) that A' = r, 2 A" = A'r + r', , where r (m) is the 



sum of the (m + l)th powers of the roots of D = 0. The general integral 



is (t, x + X) = 0(0 + A'<f>(t - 1) + A"4>(t - 2) + .-.. For X = 0, we 



find by using x = 1 (cf. Paoli) that (t, x) = A (t ~ l) . For general X, write 



