Experimenting with Standard Young Tableaux

By Shalosh B. Ekhad and Doron Zeilberger

.pdf    .tex

Written: March 29, 2023

In fond memory of Albert Nijenhuis and Herbert Wilf, and in honor of Curtis Greene [pic[Philadelphia, 2006; From left: HW, AN, CG] ]

Using Symbolic Computation with Maple, we can discover lots of (rigorously-proved!) facts about Standard Young Tableaux, in particular the distribution of the entries in any specific cell, and the sorting probabilities.

# Maple package

• SYT.txt, a Maple package to experiment with Standard Young tableaux.

# Sample Input and Output for SYT.txt

## Part I: Explicit Expressions, in n, for Occupancy Distribution (and their limits) of the cells in the first row of Rectangular Shapes with n columns and up to 7 rows

• If you want to see a computer-generated article with lots of explicit expressions (as rational functions of n) for the probability distribution of the occupant of cell [1,i] in a (uniformly-at) random-generated 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 computer-generated article with lots of explicit expressions (as rational functions of n) for the probability distribution of the occupant of cell [1,i] in a (uniformly-at) random-generated 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 computer-generated article with lots of explicit expressions (as rational functions of n) for the probability distribution of the occupant of cell [1,i] in a (uniformly-at) random-generated 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 computer-generated article with lots of explicit expressions (as rational functions of n) for the probability distribution of the occupant of cell [1,i] in a (uniformly-at) random-generated 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 computer-generated article with lots of explicit expressions (as rational functions of n) for the probability distribution of the occupant of cell [1,i] in a (uniformly-at) random-generated 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 computer-generated article with lots of explicit expressions (as rational functions of n) for the probability distribution of the occupant of cell [1,i] in a (uniformly-at) random-generated 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 computer-generated article with lots of explicit expressions (as rational functions of n) for the probability distribution of the occupant of cell [1,i] in a (uniformly-at) random-generated 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 Greene-Nijenhuis-Wilf algorithm to generate a random Standard Young Tableau

the input gives the output.