By Shalosh B. Ekhad and Doron Zeilberger

If you want to see a computergenerated article with lots of explicit expressions (as rational functions of n) for the
probability distribution of the occupant of cell [1,i]
in a (uniformlyat) randomgenerated Young tableau of rectangular shape with 2 rows and n columns (i.e. of shape [n,n]) for
all i between 2 and 40, then
the input gives the
output.

If you want to see an abbreviated version of the above, only with the LIMITING distribution, as n goes to infinity, then
but going all way to i=60,
the input gives the
output.

If you want to see a nice plot for the limiting occupancy distribution of the cell [1,40] in a random standard Young tableau of shape [n,n], here is the plot

If you want to see a computergenerated article with lots of explicit expressions (as rational functions of n) for the
probability distribution of the occupant of cell [1,i]
in a (uniformlyat) randomgenerated Young tableau of rectangular shape with 3 rows and n columns (i.e. of shape [n,n,n]) for
all i between 2 and 40, then
the input gives the
output.

If you want to see an abbreviated version of the above, only with the LIMITING distribution, as n goes to infinity, then
the input gives the
output.

If you want to see a nice plot for the limiting occupancy distribution of the cell [1,40] in a random standard Young tableau of shape [n,n,n], here is the plot

If you want to see a computergenerated article with lots of explicit expressions (as rational functions of n) for the
probability distribution of the occupant of cell [1,i]
in a (uniformlyat) randomgenerated Young tableau of rectangular shape with 4 rows and n columns (i.e. of shape [n,n,n,n]) for
all i between 2 and 13, then
the input gives the
output.

If you want to see an abbreviated version of the above, (but for i from 2 to 10) only with the LIMITING distribution, as n goes to infinity, then
the input gives the
output.

If you want to see a nice plot for the limiting occupancy distribution of the cell [1,10] in a random standard Young tableau of shape [n,n,n,n], here is the plot

If you want to see a computergenerated article with lots of explicit expressions (as rational functions of n) for the
probability distribution of the occupant of cell [1,i]
in a (uniformlyat) randomgenerated Young tableau of rectangular shape with 5 rows and n columns (i.e. of shape [n,n,n,n,n]) for
all i between 2 and 7, then
the input gives the
output.

If you want to see an abbreviated version of the above, only with the LIMITING distribution, as n goes to infinity, then
the input gives the
output.

If you want to see a nice plot for the limiting occupancy distribution of the cell [1,7] in a random standard Young tableau of shape [n,n,n,n,n], here is the plot

If you want to see a computergenerated article with lots of explicit expressions (as rational functions of n) for the
probability distribution of the occupant of cell [1,i]
in a (uniformlyat) randomgenerated Young tableau of rectangular shape with 6 rows and n columns (i.e. of shape [n,n,n,n,n,n]) for
all i between 2 and 6, then
the input gives the
output.

If you want to see an abbreviated version of the above, only with the LIMITING distribution, as n goes to infinity, (but only from i=2 to i=5) then
the input gives the
output.

If you want to see a nice plot for the limiting occupancy distribution of the cell [1,6] in a random standard Young tableau of shape [n,n,n,n,n,n], here is the plot

If you want to see a computergenerated article with lots of explicit expressions (as rational functions of n) for the
probability distribution of the occupant of cell [1,i]
in a (uniformlyat) randomgenerated Young tableau of rectangular shape with 7 rows and n columns (i.e. of shape [n,n,n,n,n,n,n]) for
all i between 2 and 3, then
the input gives the
output.

If you want to see an abbreviated version of the above, only with the LIMITING distribution, as n goes to infinity, then
the input gives the
output.

If you want to see a nice plot for the limiting occupancy distribution of the cell [1,3] in a random standard Young tableau of shape [n,n,n,n,n,n,n], here is the plot

If you want to see a computergenerated article with lots of explicit expressions (as rational functions of n) for the
probability distribution of the occupant of cell [1,i]
in a (uniformlyat) randomgenerated Young tableau of rectangular shape with 8 rows and n columns (i.e. of shape [n,n,n,n,n,n,n,n]) for
all i between 2 and 3, then
the input gives the
output.

If you want to see an abbreviated version of the above, only with the LIMITING distribution, as n goes to infinity, then
the input gives the
output.

If you want to see confirmation by simulation, using the beautiful GreeneNijenhuisWilf algorithm to generate a random Standard Young Tableau
the input gives the
output.

If you want to see a computergenerated article with explicit expressions (as rational functions of n) for the
sorting probability of cell c1 vs. cell c2
in
in a (uniformlyat) randomgenerated Young tableau of rectangular shape with 2 rows and n columns (i.e. of shape [n,n]) for
all cells c1=[1,i] between i=2 and i=60, and all cells c2 below it and to its left (i.e. c2=[2,j] with j < i)
the input gives the
output.

If you want to see an abbreviated version of the above, only with the LIMITING sorting probabilities, as n goes to infinity, then
the input gives the
output.

Here is a plot of the limiting sorting probability, as n goes to infinity of
the cell [1,60] vs. the cells [2,j] for j from 1 to 59.

If you want to see a computergenerated article with explicit expressions (as rational functions of n) for the
sorting probability of cell c1 vs. cell c2
in a (uniformlyat) randomgenerated Young tableau of rectangular shape with 3 rows and n columns (i.e. of shape [n,n,n]) for
all cells c1=[1,i] between i=2 and i=36, and all cells c2 below it and to its left (i.e. c2=[2,j], and c2=[3,j] with j < i)
the input gives the
output.

If you want to see an abbreviated version of the above, only with the LIMITING sorting probabilities (but only up to i=30), as n goes to infinity, then
the input gives the
output.

Here is a plot of the limiting sorting probability, as n goes to infinity of
the cell [1,36] vs. the cells [2,j] for j from 1 to 35.

Here is a plot of the limiting sorting probability, as n goes to infinity of
the cell [1,36] vs. the cells [3,j] for j from 1 to 35.

If you want to see a computergenerated article with explicit expressions (as rational functions of n) for the
sorting probability of cell c1 vs. cell c2
in a (uniformlyat) randomgenerated Young tableau of rectangular shape with 4 rows and n columns (i.e. of shape [n,n,n,n]) for
all cells c1=[1,i] between 2 and 16, and all cells c2 below it and to its left (i.e. c2=[2,j], c2=[3,j], and c2=[4,j], with j < i)
the input gives the
output.

If you want to see an abbreviated version of the above, only with the LIMITING sorting probabilities, as n goes to infinity, then
the input gives the
output.

If you want to see a computergenerated article with explicit expressions (as rational functions of n) for the
sorting probability of cell c1 vs. cell c2
in a (uniformlyat) randomgenerated Young tableau of rectangular shape with 5 rows and n columns (i.e. of shape [n,n,n,n,n]) for
all cells c1=[1,i] between 2 and 8, and all cells c2 below it and to its left (i.e. c2=[2,j], c2=[3,j], c2=[4,j], and c2=[5,j], with j < i)
the input gives the
output.

If you want to see an abbreviated version of the above, only with the LIMITING sorting probabilities, as n goes to infinity, then
the input gives the
output.

If you want to see a computergenerated article with explicit expressions (as rational functions of n) for the
sorting probability of cell c1 vs. cell c2
in a (uniformlyat) randomgenerated Young tableau of rectangular shape with 6 rows and n columns (i.e. of shape [n,n,n,n,n,n]) for
all cells c1=[1,i] between 2 and 5, and all cells c2 below it and to its left (i.e. c2=[2,j], c2=[3,j], c2=[4,j], c2=[5,j], and c2=[6,j], with j < i)
the input gives the
output.

If you want to see an abbreviated version of the above, only with the LIMITING sorting probabilities, as n goes to infinity, then
the input gives the
output.

If you want to see a computergenerated article with explicit expressions (as rational functions of n) for the
sorting probability of cell c1 vs. cell c2
in a (uniformlyat) randomgenerated Young tableau of rectangular shape with 7 rows and n columns (i.e. of shape [n,n,n,n,n,n,n]) for
all cells c1=[1,i] between 2 and 4, and all cells c2 below it and to its left (i.e. c2=[2,j], c2=[3,j], c2=[4,j], c2=[5,j], c2=[6,j], and c2=[7,j], with j < i)
the input gives the
output.

If you want to see an abbreviated version of the above, only with the LIMITING sorting probabilities, as n goes to infinity, then
the input gives the
output.

If you want to see confirmation by simulation, using the beautiful GreeneNijenhuisWilf algorithm to generate a random Standard Young Tableau
the input gives the
output.

If you want to see the MINIMAL sorting probabilities, as well as the pair of cells that achieve them for Standard Young Tableaux of shape [n,n]
(the socalled Catalan poset) from n=2 to n=1000
the input gives the
output.

If you want to see the MINIMAL sorting probabilities, as well as the pair of cells that achieve them for Standard Young Tableaux of shape [n,n,n]
(i.e. 3 by n rectangular shape) from n=2 to n=32
(but only with one of the participant cells in the first row, hence these are upper bounds to the true values, but possibly sharp)
the input gives the
output.

If you want to see the MINIMAL sorting probabilities, as well as the pair of cells that achieve them for Standard Young Tableaux of shape [n,n,n,n]
(i.e. 4 by n rectangular shape) from n=2 to n=12
(but only with one of the participant cells in the first row, hence these are upper bounds to the true values, but possibly sharp)
the input gives the
output.
Part III

If you want to see the Statistics of the Limiting Occupancy of the cell [1,i] in a 2rowed
Standard Young tableau of shape [n,n], as n goes to infinity, and its Asympotic behavior as i goes to infinity
the input gives the
output .
Note that this took less than 8 seconds.

If you are willing to wait a bit longer, you can get to the first ten moments, see
the input that gives the
output .
Note that this took less than 134 seconds.