#Please do not post homework

#Guy Adami, 2026-03-29, Assignment 17


#_____________________________________

Conj:=proc(L) local k,C1,i1,L1,i:
	option remember:
	
	if L=[] then
		RETURN([]):
	fi:

	k:=nops(L):
	L1:=[seq(L[i]-1,i=1..k)]:

	for i1 from 1 to nops(L1) while L1[i1]>0 do od:
	
	i1:=i1-1:
	L1:=[op(1..i1,L1)]:
	C1:=Conj(L1):
	[k,op(C1)]:
end:

#_____________________________________

HLc:=proc(L,c) local i,j,L1:
	L1:=Conj(L):
	i:=c[1]:
	j:=c[2]:
	L[i]-j+L1[j]-i+1:
end:

#_____________________________________

#Cells(L): the set of n (=sum(L)) cells [i,j]in the shape L
Cells:=proc(L) local k,i,j:
	k:=nops(L):
	{seq(seq([i,j]   , j=1..L[i]  ), i=1..k)}
end:

#_____________________________________

Hook:=proc(L,c) local k,i,j,C,i1,j1,i2:
	k:=nops(L):
	i:=c[1]:
	j:=c[2]:

	if not (i>=1 and i<=k) then
  		RETURN(FAIL):
	fi:

	if not(j>=1 and j<=L[i]) then
  		RETURN(FAIL):
	fi:

	C:={seq([i,j1],j1=j..L[i])}:

	for i2 from i to k while j<=L[i2] do od:
	
	i2:=i2-1:
	C:=C union {seq([i1,j],i1=i..i2)}:
end:


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#Question 1



RandSYT:= proc(L) local  n, Y, L1, corners,Tcell,c, C, row, col, i,j:

	n:= add(i, i in L):
	L1:= L:
	Y:= []:
	#Makes a Young tableux of all zeros to be filled in later
	for i from 1 to nops(L) do
		Y := [op(Y), [seq(0, j = 1 .. L[i])]]:
	od:

		
	for i from n by -1 to 1 do
		corners:= []:
		#Get all current corners
		C:= Cells(L1):
		for c in C do
			if HLc(L1,c) = 1 then
				corners:= [op(corners),c]:
			fi:
		od:
		
		
		#Pick a random corner
		Tcell := corners[rand(1..nops(corners))()];
		row:= Tcell[1]:
		col:= Tcell[2]:
		
		#Fill in that element into Y
		Y[row][col]:=i:


		#Update the shape
		L1[row]:=L1[row]-1:

		#remove row if = 0
		if L1[row] = 0 then
            		L1:= subsop(row=NULL, L1):
        		fi:

	od:
	
	RETURN(Y);
end:


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Probability2:=proc(m,L) local L, i, r, c,R:
	r:=0:
	c:=0:

	for i from 1 to m do
		R:=RandSYT(L);
		if R[1][2] = 2 then
			r:=r+1:
		elif R[2][1] = 2 then
			c:=c+1:
		fi:
	od:
	r/m;
end:


#Running for 10000 random SYTs, the probability found was 5037/10000 or 50.37%