EXPECTED VALUE AND STANDARD DEVIATION OF PRODUCT-SUMS OF PERMUTATION

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

Enumeration of bigrassmannian permutations below a permutation in Bruhat order

In theory of Coxeter groups, bigrassmannian elements are well known as elements which have precisely one left descent and precisely one right descent. In this article, we prove formulas on enumeration of bigrassmannian permutations weakly below a permutation in Bruhat order in the symmetric groups. For the proof, we use equivalent characterizations of bigrassmannian permutations by Lascoux-Schutzenberger and Reading.Comment: 7 pages