Distributions And A Pplications Of Three Kinds Of Hollow Truncated Nested Order Statistics

Order statistics is a type of statistics that plays an important role in probability statistics and statistical inference,which is widely used in many fields.The enumeration of SYT(Standard Young tableaux)is an important issue in enumerative combinatorics.Based on the nested order statistics of shape(n3)\(n-3)|{(2,2)},this paper combines nested order statistics with SYT enumeration problem to study the distribution of three types of truncated nested order statistics.The order statistics of shape(n3)\(n-3)|{(2,2)},marked as S,consists of three sets of independent order statistics(?1:n(1),?2:n(1),…,?n:n(1),(?1:3(2),?2:3(2),?3:3(2))and(?1:n(3),?2:n(3),…,?n:n(3)).The SYT enumeration method is applied to the research on the distribution of nested order statistics in this paper.By using the method of SYT deformation,multiple integration and combinatorial identities,the SYT enumeration formulas of the three types S\(1),S\(2)and S\(2,1)are derived as follows:(?)hs\(2)=C3(n-1)-n-1,hs\(2,1)=C3(n-2)-1 where(?)is the third convolution of Catalan number.By using the SYT enumeration formulas,the distributions of nested order statistics of type S\(1),S\(2)and S?(2,1)are calculated as follows:(?)Finally,this paper discusses the application of these three types of nested order statistics in the voting problem for three candidates.