144 HISTORY OF THE THEORY OF NUMBERS. [CHAP, in 



and two similar relations. There is a list of unsolved questions on parti- 

 tions by Sylvester. 1376 



P. G. Tait 138 considered in connection with knots of order n those 

 partitions of 2n with no part > n and no part < 2. After the largest part 

 is removed, the numbers left form the partitions pi, p*+\, -, pL-z, where 

 p r , is the number of partitions of s with no part > r and none < 2. If 

 r > s, pi = pi. If r < s, the above argument shows that 



pi = p r s _ r + pl- 1 ^ + . . . + p]_ 2 . 



There is a table of values of pi for r ^ 17, s ^ 32. 



E. Pascal 139 used n numerical functions /,-(x) which increase when x 

 increases. Let the difference of two values of /i for two successive integral 

 values of Xi be unity. If x k -i < x k and 







every number is expressible in the form /i(xi) + + /n(#n) in one and 

 but one way. As corollaries, every number N can be expressed in one and 

 but one way as a sum of n decreasing binomial coefficients: 



N = (#1)1 + (#2)2 + ' + (Xn)n, Xk < Xk+l', 



also as a sum of n increasing binomial coefficients: 



N = [2l t + [31, + + \ji + l] Xn , x k < Xk+i. 



E. Sadun 140 considered the number s(n, r) of sets of integral solutions ^ 

 of the pair of equations, in which r ^ n, 



Xi + X 2 + + Xn = r, Xi + 2X 2 + + n\ n = n. 



Set S(n) = s(n, !)++ s(n t n). If r == [n/2], S(n - r) = s(n, r). 

 For r =i n, the pair of equations have as many solutions as the equation 



i + 2 2 + + ra r = n 

 has integral solutions ^ with a r > 0, or as the system 



ai + 2 2 + + (r - IH-i = n - tr (t = 1, 2, , [n/r]) 



has solutions. Hence we can compute s(n, r). For r = 1, the equation 

 is ai = n, ai > 0, whence s(n, 1) = 1. For r = 2, the system is 



whence s(n, 2) = [n/2]. Finally, he identified s(n, r) with a function 

 connected with a linear differential equation of order n. 



P. A. MacMahon 141 employed symmetric functions as an instrument for 

 the study of partitions and other problems of combinations. He considered 

 n objects specified by (pqr- ), p + q + = n, meaning that p objects 



137b Math. Quest. Educ. Times, 45, 1886, 133-7. One is proved by Sharp, 47, 1887, 139-140. 



138 Trans. Roy. Soc. Edinburgh, 32, 1887, 340-2. 



133 Giornale di Mat., 25, 1887, 45-9. 



140 Annali di Mat., (2), 15, 1887-8, 209-221. 



Proc. London Math. Soc., 19, 1887-8, 220-256. Cf. 28, 1896-7, 9-10. 



