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