Distribution Of The Hollow Nested Order Statistics Of The Shape Of Approximately Rectangular Trapezoid

Order statistics model is an important tool in the enumeration of the SYT-type chart.The distribution of complex order statistics model can be obtained by counting the enumeration of the complex SYT-type chart which corresponds to the complex order statistics model.Let ?\?{(i0,j0)} be the SYT of shape ? truncated by ? whose upper left cell is(i0,j0),where ? and ? are partitions of integers.In this paper,the enumeration formula of the SYT-type chart of shape(nm\?k)\((n-2)m-2\?k-1)|{(2,2)},(n,m?2)is obtained by calculating the multiple integrals of the corresponding nested order statistics and by the combined properties,and the result is a complex double combination equation.Besides,the corresponding probability distribution on U(0,1)is also obtained which is recorded as P((nm\?k)\((n-2)m-2\?k-1)|{(2,2)}).Finally,by changing the values n,m,k in the enumeration formula,we directly obtain the formula of the SYT-type chart of the shape(nm\(n-2)(m-2))|{(2,2)},and some other main results are as follows.1.The formula of the SYT-type chart of shape ?n+2((H)(n?2)is H?n+2(H)(n?2)=((3n+1)!)/(n!(2n+3)!)(n2+3n+6)Which is called convolution of Catalan triangle,and where n?2.2.The formula of the SYT-type chart of shape(n3\(1,1))(H)is#12 Where C(n)is related to the Catalan number,and n?2.