# OK to post homework
# Robert Dougherty-Bliss
# 2021-04-12
# Assignment 24

# 1.

# I believe it!

# 2.

read "C24.txt":

TabPairs := proc(n, m)
    shapes := Pn(n):
    res := []:
    for s in shapes do
        Ts := TAB(s, m):
        STs := SYT(s, m):
        res := [op(res), seq(seq([T, S], S in STs), T in Ts)]:
    od:

    res:
end:

# 3.

# Recap: RS(pi) takes a PERMUTATION (of [n]) and outputs a pair of SYT. Now we
# want to make it work for WORDS (with alphabet [n]).

# RS inserts every element of the permutation into the tableaux with the
# "bumping" procedure. The OLD bumping procedure said "it is legal to insert x
# in a row if there an element > x in the row," which preserved row-and column
# strictness. However, this assumed that everything being inserted was unique:
# if x > everything in a row, then it is safe to put it in the next row. But if
# x = something in a row, then putting it into the next row might break column
# strictness.

# The NEW bumping procedure says "it is legal to insert x in a row if there is
# an element >= x in the row." This way, we aren't going to run into the column
# strictness problem.

# George's homework used the old bumping procedure, but we fixed it to work in
# the general case in class (I think).