#Ok to post homework
#Tifany Tong, December 13th, 2020, HW #25

# 1.) 
# [9,6,5,3,1,1,1]

# Ferrers Diagram:

# *********
# ******
# *****
# ***
# *
# *
# *

# Conjugate Partition: [7,4,4,3,3,2,1,1,1]

# 2.) pnk(151,10) = 79811865

# 3.) [[6,5,3,1,1,1,1,1],-1]

# 6-3(-1)-8-1  = 6+3-9 = 0
# -1+1 = 0

# [[6,3,2,2,2,2,2],0] = BZ([[6,5,3,1,1,1,1,1],-1])


# 4.)

# pnFast(10000);
# 36167251325636293988820471890953695495016030339315650422081868605887952568754066420592310556052906916435144

# [seq(pnFast(i), i = 1 .. 10000)][10000];
# 36167251325636293988820471890953695495016030339315650422081868605887952568754066420592310556052906916435144

# pnFast(20000);
# 252114813812529697916619533230470452281328949601811593436850314108034284423801564956623970731689824369192324789351994903016411826230578166735959242113097

# [seq(pnFast(i), i = 1 .. 20000)][20000];
# 252114813812529697916619533230470452281328949601811593436850314108034284423801564956623970731689824369192324789351994903016411826230578166735959242113097

# It can cache the previous results as all the previous results have to be calculated with the seq command before approaching the final value. 

# 5.) 

pnFastMod := proc(n, m) local i:
return [seq(pnFast(i), i = 1 .. n)][n] mod m:
end:

# pnFastMod(10^5, 101) = 89

# 6.) 
with(ListTools):
check := proc(L) local i, ans:
for i from 2 to nops(L) do
     if L[i] = L[i - 1] then 
	return false: 
     fi:
od:
return true:
end:

InvGlashier := proc(L) local i, curr1, curr2, curr3, curr, co:
co := L:
while check(co) = false do 
    for i from 2 to nops(co) do 
	if co[i] = co[i - 1] then 
	    curr1 := [op(co[1 .. i - 2])]:
            curr2 := [op(curr1), 2*co[i]]:
            curr3 := [op(curr2), co[i + 1 .. nops(co)]]:
            curr3 := Flatten(curr3):
            co := sort(curr3, `>`): 
            break:
	fi: 
    od: 
od:
return sort(co):
end:

# With this maple code I am able to get InvGlashier(Glashier(sort(L))) = sort(L).  


# 7.)

InvSyl := proc(L) local i, r, m, L1, M1, a1:
option remember:
if L = [] then RETURN([]): fi:
if nops(L) = 1 then RETURN([seq(1, i = 1 .. L[1])]): fi:
for i to nops(L) while 1 < L[i] do: od:
r := i - 1: m := nops(L) - r:
if L[1] mod 2 = 1 then a1 := 1/2*L[1] - 1/2:
else a1 := 1/2*L[1]: fi:
L1 := [seq(L[i] - 2, i = 2 .. r)]:
M1 := InvSyl(L1):
[a1 + r + m, a1 + r - 1, op(M1)]:
end:

# Having difficulties with this procedure.