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66 RANDOM NUMBERS 3.3.2 Given any permutation on n elements, let Z,, = 1 if position i is the beginning of an ascending run of length p or more, and let Zpi = 0 otherwise. For example, consider the permutation (9) with n = 10; we have and all other Z s are zero. With this notation, R; = Z,I + Z,2 + . . . + z,n (12) is the number of runs of length > p, and Rp=R;-R;+l (13) is the number of runs of length p exactly. Our goal is to compute the mean value of Rp, and also the covariance covar(RP, R4) = mean((RP -mean(R,))(R, -mean(R, which measures the interdependence of Rp and R,. These mean values are to be computed as the average over the set of all n! permutations. Equations (12) and (13) show that the answers can be expressed in terms of the mean values of Z,, and of ZpiZql, so as the first step of the derivation we obtain the following results (assuming that i < j): Zpi = (P + hlMP + 1)!1 ifisn-p+l; otherwise. 3 (0 il: + wql(P + lY(q + l)!, ifi+p 1. Note that Z~iZ ~~ is either zero or one, so the summation consists of counting all permutations Ur Us . . . U, for which Z,, = .Z ,j = 1, that is, all permutations such that a-1 > u% < . < Uifp-1 > Ui+p < . . . < Ui+p+q-l. (15) The number of such permutations may be enumerated as follows: there are (P+q+l n ) ways to choose the elements for the positions indicated in (15); there
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