The Exact Sorting Probability Distirbution of cell, [1, 2],  vs. the cell ,

    [2, 1], in a random Standard Young Tableau of shape, [30, 20],

    using the exact symbolic expression



Compared to  the approximation obtained by simulation via the Greene-Nijenhu\

    is-Wilf algorithm, 10000, times



                              By Shalosh B. Ekhad



The exact value of the prob. that the occupant of cell, [1, 2],

    is larger than the occupant of cell , [2, 1],

    minus the prob. that the opposite is true for symbolic shape, [3 n, 2 n],

     is



-1/5*(13*n-9)/(5*n-1)


                   BTW this took, 0.063, seconds to compute



We are interested in what happens when n=, 10, i.e. for the specific shape,

    [30, 20]





                                  the value is



                                 -0.4938775510



Note that there are, 16723268860760, Standard Young Tableaux of shape, [30, 20],
    so it would be impractical to compute this directly. Let's do it by simu\
    lation



Now let's compare it with the approximation that you get by generating, 10000,

    standard Young tableau of that shape, namely, [30, 20],

    using the Green-Nijenhuis-Algorithm





                                 -0.5052000000



                        BTW this took, 60.982,  seconds



                                  The ratio is



                                  0.9775881849



                        -------------------------------





The Exact Sorting Probability Distirbution of cell, [1, 5],  vs. the cell ,

    [2, 1], in a random Standard Young Tableau of shape, [40, 40, 40, 40],

    using the exact symbolic expression



Compared to  the approximation obtained by simulation via the Greene-Nijenhu\

    is-Wilf algorithm, 1000, times



                              By Shalosh B. Ekhad



The exact value of the prob. that the occupant of cell, [1, 5],

    is larger than the occupant of cell , [2, 1],

    minus the prob. that the opposite is true for symbolic shape, [n, n, n, n],

     is



3/2*(19*n^3-11*n^2-46*n+54)/(-3+4*n)/(-1+2*n)/(-1+4*n)


                   BTW this took, 62.590, seconds to compute



We are interested in what happens when n=, 40, i.e. for the specific shape,

    [40, 40, 40, 40]





                                  the value is



                                  0.9101678078



Note that there are, 24419603473357360677983537133480859658819470213611794390\

    97180159042317121445706376000, Standard Young Tableaux of shape,

    [40, 40, 40, 40], so it would be impractical to compute this directly. Le\
    t's do it by simulation



Now let's compare it with the approximation that you get by generating, 1000,

    standard Young tableau of that shape, namely, [40, 40, 40, 40],

    using the Green-Nijenhuis-Algorithm





                                  0.904000000



                        BTW this took, 25.979,  seconds



                                  The ratio is



                                  1.006822796