CHAP, ill] PARTITIONS. 107 



P. Paoli 17 noted (p. 38) that n can be separated into m positive integral 

 parts in (ll) ways, if different permutations are counted separately. The 

 number (p. 42) of partitions (different permutations not counted) of n 

 into m parts > is 0(1) + 0(w + 1) + 0(2ra + 1) + , where 0(j) is 

 the number of partitions of n j into m 1 parts. The number (p. 53) 

 of ways n can be divided into m distinct parts is X(m) + X(2m) + X(3m) + 

 , if X(j) is the number of ways n j can be divided into m 1 distinct 

 parts. There are (p. 63) as many divisions of n into m distinct parts as of 

 n m(m l)/2 into m parts equal or distinct. Let \{/, <}>, co be the number 

 of ways 2n + 1, 2n, 2n + 1 can be divided into 2m 1, 2m, 2m + 1 odd 

 parts, respectively; let $[r~\, 0[r], |V] be the corresponding numbers 

 when n is replaced by n r. Then 



co = + 0[2m + 1] + 0[4ra + 2] + 0[6m + 3] +'. 

 If we impose also the condition that the odd parts be distinct, we have 

 = ^(2m) + iK4m) + , = 0(2m) + 0(4ra + 1) + 0(6m + 2) + 



The number (p. 76) of ways 2n is a sum of m even parts is 0(1) + 0(m + 1) 

 + 0(2w + 1) + , if 0(r) is the number of ways 2(n r) is a sum of 

 m 1 even parts. The number (p. 79) of ways n is a sum of m parts is 

 the number of ways 2n is a sum of m even parts. The number (p. 80) of 

 ways n is a sum of m distinct parts is the number of ways pn is a sum of m 

 distinct parts multiples of p. The number P(n, m) of partitions of n into 

 parts ^ m is ZP(n j, m 1), summed for j = 0, m, 2m, . The 

 number (p. 85) of partitions of n into m parts equals P(n m, m). If 

 0, co denote the number of ways (m l}a + rb and ma -f- rb can be formed 

 additively from m and m 1 terms of the progression a, a + b, a + 2b, 

 , then o> = + 0(ra) + 0(2m) + , where 00') is derived from by 

 replacing r by r j. Similarly (p. 92) when only distinct terms of the 

 progression are used. If (p. 98) is the number of ways n is a, sum of 

 numbers chosen from a, a + b, -, a + (m 1)6, and co that for a, -, 

 a + nib, then 



co = + 0[a + mb~] + 0[2(a + ra&)] + -. 



Finally (p. 103) the number of ways n is a sum of terms of any given series 

 is discussed. He gave a more extended treatment in his next paper. 

 Paoli 18 treated linear difference equations with variable coefficients : 



Z(y, x) = A x Z(y - 1, x) + B x Z(y - 2, x) + + X x Z(y - x, x) + - - 

 + A' x Z(y, x - 1) + B' x Z(y - 1, x - 1) + + X' x Z(y - x, x - 1) + , 



where A x , are given functions of x, and y is a function of x. Let the 

 integral be 



where m, a, n, b, are constants. The condition that a v Va x shall be an 



17 Opuscula analytica, Liburni, 1780, Opusc. II (Meditationes Arith.), 1. 



18 Memorie di mat. e fis. societa Italiana, 2, 1784, 787-845. 



