112 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



apd 



00 



exp - xM = E Mk{-xY/k\ 







= {I- a) \ du g-^" Z (n + i)-i^-e-"("+i)-^ (21) 



JQ 



= (1 -a) Z all + a:(n + 1)]"' 



Hence 



ikT, = fc!(l - a) Z (n + 1)V (22) 



These moments are expressible in terms of polynomials associated with 

 the distribution of permutations into classes according to the number of 

 readings left to right necessary to find the elements in standard order. ^ 

 Indeed the ratio 



n{a) =Mu{l- af/h\ 



has the recurrence relation 



n+i(a) = {ka + l)n{a) + a(l - a)r\{a) (23) 



and the first few values are as follows 



ri = 1 n = I -\- ^a -\- a 



r2 = 1 + a r4 = 1 + lla + lla' + a 



r5 = 1 + 26a + 66a' + 26a' + a 



Notice that rk{0) = \, rk{l) = /c!, just as for the precise results. 



6. EXPONENTIAL SUMS 



The shape of the delay curves, from direct calculation, and also from 

 Mellor's results, suggests representation in exponential sums. If 



Fiu) = Ai e-^'-"^""'' + A2 e-<^-"^«/^« + . . . (24) 



then 



M, ^^ ~ "^' = yl,xj + A^t+... (25) 



by a simple calculation. For k exponentials, 2k moments (including 



