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

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

Part II: Explicit Expressions, in n, for Sorting probabilities relating cells in a random Standard Young tableau of rectangular shapes
[i.e. (Pr(Y[c1] > Y[c2]) - (Pr(Y[c2] > Y[c1])] ,
and their limits as n goes to infinity, for all cells c1 in the first row and all cells c2 not related to it (i.e. lying to the Southwest (to the left and below)

  • If you want to see a computer-generated article with explicit expressions (as rational functions of n) for the sorting probability of cell c1 vs. cell c2 in 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 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 computer-generated article with explicit expressions (as rational functions of n) for the sorting probability of cell c1 vs. cell c2 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 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 computer-generated article with explicit expressions (as rational functions of n) for the sorting probability of cell c1 vs. cell c2 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 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 computer-generated article with explicit expressions (as rational functions of n) for the sorting probability of cell c1 vs. cell c2 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 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 computer-generated article with explicit expressions (as rational functions of n) for the sorting probability of cell c1 vs. cell c2 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 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 computer-generated article with explicit expressions (as rational functions of n) for the sorting probability of cell c1 vs. cell c2 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 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 Greene-Nijenhuis-Wilf 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 so-called 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 2-rowed 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.

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