# Matthew Russell
# Experimental Math
# Homework 23
# I give permission to post this

read `public_html/em12/hw22.txt`:
read `public_html/em12/C22.txt`:
read `public_html/em12/C13.txt`:

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# Write a procedure PT() (PT for Periodic Table), that inputs nothing,
# and outputs the set of Conway's 92 elements. Verify that indeed you got 92 of them.
# Hint: Start with [3], and keep applying SplitUp(L), of the previous homework assignment,
# that is based on the true/false procedure
# split(w1,w2)
# that you should use as a black box.

PT:=proc() local els, needtodo, n, S:
option remember:

els:={}:
needtodo:={[3]}:
while needtodo<>{} do
	n:=needtodo[1]:
	needtodo:=needtodo minus {n}:
	els:=els union {n}:
	S:=SplitUp(JC1(n)):
	needtodo:=needtodo union (convert(S,set) minus els):
od:

return els:

end:

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# Write a procedure Mat(), that inputs nothing, and outputs the 0-1 matrix such that
# entry (i,j) has a 1 if and only if the "molecule" (that may be an atom)
# JC1(w)
# applied to atom i, has atom j in it.

Mat:=proc() local P, M, i, curr, j, k:

P:=PT():
M:=Matrix(nops(P)):
for i from 1 to nops(P) do
	curr:=convert(SplitUp(JC1(P[i])),multiset):
	for j from 1 to nops(P) do
		for k in curr do
			if P[j]=k[1] then
				M[i,j]:=k[2]:
			fi:
		od:
	od:
od:

return M:

end:

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# By using the built-in linalg packgage (or LinearAlgebra) find the numerical
# estimate of Conway's constant, by finding the largest eigenvalue of Mat().

GetConwayConstant:=proc() local n, M, eigvals, curr, marker, i, now:

n:=nops(PT()):
M:=Mat():
eigvals:=LinearAlgebra[Eigenvalues](M):
curr:=0:
marker:=0:
for i from 1 to n do
	now:=abs(evalf(eigvals[i])):
	if now>curr then
		curr:=now:
		marker:=i:
	fi:
od:

return evalf(eigvals[marker]):

end:

# This gives 1.30357726903430.

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# Using LIF(P,u,N), and exponential generatingfunctionolgy, write a procedure,
# RLT1(S,N),
# that inputs a set of positive integers S, and a positive integer N,
# and outputs the list of size N whose i-th entry is the number of rooted labelled trees
# on i vertices where each vertex either has outdegree 0 (a leaf) or an outdegree that belongs to S.

# Coming up soon

# Find RLT1({1},20), RLT1({1,2},20), RLT1({1,2,3},20), RLT1({1,3},20).
# Are any of these in Sloane?

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# Using LIF(P,u,N), and exponential generatingfunctionolgy, write a procedure,
# RLT2(S,N) ,
# that inputs a set of positive integers S, and a positive integer N,
# and outputs the list of size N whose i-th entry is the number of rooted labelled trees
# on i vertices where each vertex either has outdegree 0 (a leaf) or an outdegree that DOES NOT belong to S.

# Coming up soon

# Find RLT2({1},20), RLT2({1,2},20), RLT2({1,2,3},20), RLT2({1,3},20).
# Are any of these in Sloane?