# OK to post homework
# Robert Dougherty-Bliss
# 2021-05-01
# Assignment 26

read "C26.txt":

with(LinearAlgebra):

# 1.

JT := proc(L, x)
    local n, M:
    n := nops(L):

    M := Matrix([seq([seq(hkn(L[i] + j - i, n, x), j=1..n)], i=1..n)]):

    Determinant(M):
end:

# Create the conjugate partition of L.
conjugateL := proc(L)
    Lp := L:
    conj := []:

    for k from 1 to L[1] do
        conj := [op(conj), nops(Lp)]:
        Lp := select(x -> x > 0, Lp - [1 $ nops(Lp)]):
    od:

    conj:
end:

JT2 := proc(L, x)
    local n, M, Lc:
    Lc := conjugateL(L):
    n := nops(Lc):

    # Note that n = nops(L) here, not nops(conjugate(L)).
    M := Matrix([seq([seq(ekn(Lc[i] + j - i, nops(L), x), j=1..n)], i=1..n)]):

    Determinant(M):
end:

# 2.

with(ListTools):

StartShape := proc(Y, k)
    local row, i, L:

    L := NULL:
    for row from 1 to nops(Y) do
        for i from 1 to nops(Y[row]) while Y[row][i] <= k do od:
        if i = 1 then
            break:
        fi:
        L := L, i - 1:
    od:

    [L]:
end:

# 3.

EstStart := proc(L, k, K)
    Ys := [seq(GNW(L), n=1..K)]:
    ps := NULL:
    for p in Pn(k) do
        haveShape := nops(select(Y -> StartShape(Y, k) = p, Ys)):
        ps := ps, [p, evalf(haveShape / K)]:
    od:

    [ps]:
end:

# Note: EstStart([n, n], 2, K) estimates the proportion of two-row tableaux
# that have 2 in the first row or second row, like we discussed in class.

# EstStart([9 $ 12], 5, 1000);

# [[[5], 0.005000000000]
# [[4, 1], 0.1080000000]
# [[3, 2], 0.1960000000]
# [[3, 1, 1], 0.3130000000]
# [[2, 2, 1], 0.2270000000]
# [[2, 1, 1, 1], 0.1410000000]
# [[1, 1, 1, 1, 1], 0.01000000000]]