EXPECTED VALUE AND STANDARD DEVIATION OF PRODUCT-SUMS OF PERMUTATION

Abstract

In this paper we introduce the expected value and the standard deviation of productsums of permutations. We know that the product-sums of permutations are: H 5(:7) = :E i:T(i) i=1 where 5(:7) denotes the value of product-sums in :7. It is show11 that every element occurs at least once from Rn to Qn where Qn stands for the sum of the squares of the first natural n(n -c- 1)(n .:.. 'l) numhers and Rn = 6 - for the product-sums :corresponding to the permutation n(n 1) ... 2L and hence product-sum Qn corresponding to 12 ... (n - l)n is the maximal, and Rn corresponding to n ... 21 is the minimal one. A mode for the production of the productsums, is indicated. T • f I 1 ,r(5(' n(n + If d D" ~( n~(n -:- Inn 1) 1 1t1S urtlcrtlat.rl :7») .1: an -(~:7»= ·~-:-147-A-:"'.---'-wlere .1I(5(:7» is the expected value and D(5(:-r» is the standard deyiation of product-mms. We also introduced the complement permutations so that :-r(i) .:.. :-r'(i) = n -:- 1 where :-r' is a complement of :-r since:-r (:-r(I), ... , :-r(n» and :7" = (:-r'(I), ...• :-r'(n» denotes the images of (1, ... , n). Of course we will do work in the probability field where each permutation has the same probability.