1876.] Verification in the Partition of Numbers. 257 



of partitions in which the number of terms is uneven. For the case 

 7i = 9, the partitions which contain only uneven elements without repe- 

 titions are two in number, viz. 9 and 1+3 + 5 ; and of the rest, fourteen 

 consist of an even, and fourteen of an uneven number of terms. 

 We have, as above, 



(f) — — — -L_ — =1+*.1+*M+* 



l_t.l-t\l-t° 



whence 



-_l + t+f(l + t) + t\l + t)(l + f) 

 + (•(1 + 0(1 + 0(1 +* 3 )+&c; 



i+^+^i+o+^a+oa+o+Ac, 



i-t\i-t 5 .i-f. 



and we obtain 



XVII. P(3, 5, 7 . . .)n= number of partitions without repetitions into 

 the elements 1, 2, 3 . . ., in which the two greatest parts are consecutive 

 (t. e. differ by unity). 



And also, by multiplying (e) by 1 — t, we find that each side of XVII. 

 = Q(3, 5, 6, 7, 9 . . .)n ; so that if all the partitions without repetitions 

 of a number n be written down, then 



XVIII. the number of partitions not involving 1, 2, 4, 8, 16 ... = the 

 number of partitions in which the two greatest parts are consecutive. 



In the case of » = 9, XVII. gives 2=2; and XVIII. of course also 

 gives 2=2, there being only two partitions without repetitions in which 

 1, 2 . . . are not involved, viz. 9 and 6 + 3, and two in which the tAvo 

 greatest parts are consecutive, viz. 4 + 3 + 2 and 5 + 4. 



Euler's identity, 



l-t.l-f.l-t 3 ...=l-t-~t 2 + t 5 + f-t l2 -&e. 

 (where the exponents are the pentagonal numbers), gives the theorem that 



XIX. Q even n-Q uneven ™=(- l) m or 0, 



according as n is or is not of the form j(3m 2 +m). Thus we see that, 

 considering only partitions without repetitions, the number of the parti- 

 tions of a non-pentagonal number into an even number of parts is always 

 equal to the number of its partitions into an uneven number ; and that 

 if the number be pentagonal the numbers of even and uneven partition- 

 ments differ only by unity. If n=d, Q even n = Q uneven ?i===4. 

 We deduce from (S) that 



1 + t .l + t 3 .1 + f . . .{l-t* .l-f .l-t l \ . .}=l + t+t 3 + t e +t 10 +&c.', 



and by Euler's identity, just quoted, the quantity in { } 



== l-.t i -f + f + f a -t ia -&c; 

 whence 



XX. Q(l,3,5...>-Q(l,3,5...)(n-4)-Q(l,3,5...)(n-8) 



+ Q(1, 3, 5 . . .)(n-20) + &c. = l or 0, 



