# OK to post homework
# Aurora Hiveley, 1/30/25, Assignment 3

Help := proc(): print(`PC(n,p,K), NuCGx(n,x), NuCGk(n,k)`): end:

## copied from Dr. Z's website
PC:=proc(n,p,K) local i:evalf(coeff(add(IsCo(RG(n,p)),i=1..K),true,1)/K):end:

# With n=10,20,30, and p=1/10, 2/10, ... and K=200 experiment and try to estimate the threshhold for connectivity, 
# i.e. when it changes being very unlikely to very likely.

# [seq(PC(10,i/10,200), i=1..10)];
# output: [0.005, 0.14, 0.625, 0.94, 0.995, 1, 1, 1, 1, 0]
# [seq(PC(20,i/10,200), i=1..10)];
# output: [0.065, 0.71, 0.98, 1, 1, 1, 1, 1, 1, 1, 0]
# [seq(PC(30,i/10,200), i=1..10)];
# output: [0.19, 0.955, 0.99, 1, 1, 1, 1, 1, 1, 0]

# as n increases, a smaller p is needed to make the probability very likely
# for n=10, it shifts at ~ p = 4/10
# for n=20, it shifts at ~ p = 3/10 
# for n=30, it shifts at ~ p = 2/10 




## NuCG(x)
# NuCGx(n,x): inputs a pos. integer n and a variable x, and outputs a list of length n whose i-th entry is P[i](x)
NuCGx:=proc(n,x) local f,i:
f:=log(add((1+x)^(i*(i-1)/2)*z^i/i!,i=0..n)):
f:=taylor(f,z=0,n+1);
[seq(i!*coeff(f,z,i),i=1..n)]:
end:



## NuCGk(n,k)
# NuCGk(n,k): inputs a positive integer n and a non-negative integer k and outputs the sequence of coefficients of x^(n-1+k) of P[n](x)
NuCGk := proc(n,k) option remember:
coeff(NuCGx(n,x)[n],x,n-1+k):
end:

# examples: 
# [seq(NuCGk(n,0), n=1..8)];
# output: [1, 1, 3, 16, 125, 1296, 16807, 262144] --> A000272
# [seq(NuCGk(n,1), n=1..8)];
# [0, 0, 1, 15, 222, 3660, 68295, 1436568] --> A-57500
# the OEIS has entries for sequences up to (and including) k=12







### copied from C3.txt 
#C3.txt: Jan.30, 2025
Help3:=proc(): print(`AM(G), Neis(G), IsCo(G), NuCG(n), NuCGc(n) `): end:

#Neis(G): inputs a graph G=[n,E] and outputs a list of length n, N, such that
#N[i] is the set of neighbors of vertexi
Neis:=proc(G) local n,E,N,i,e:
n:=G[1]: 
E:=G[2]:
for i from 1 to n do
 N[i]:={}:
od:
for e in E do
N[e[1]]:=N[e[1]] union {e[2]}:
N[e[2]]:=N[e[2]] union {e[1]}:
od:
[seq(N[i],i=1..n)]:
end:

#C(G,i): The connected component of vertex i
CC:=proc(G,i) local n,C1,C2,C3,N,i1:
if not (type(G,list) and nops(G)=2 and type(G[1],integer) and G[1]>=0 and type(G[2],set)) then
  RETURN(FAIL):
fi:
n:=G[1]:
N:=Neis(G):

C1:={i}:
C2:=C1 union { seq(op(N[i1]), i1 in C1)}:
while C1<>C2 do
 C3:=C2 union { seq(op(N[i1]), i1 in C2)}:
C1:=C2:
C2:=C3:
od:
C2:
end:

#IsCo(G): Is the graph G connected?
IsCo:=proc(G): evalb(nops(CC(G,1))=G[1]):end:

#NuCG(n): the first n terms of the sequence enumerating CONNECTED labeled graphs on n vertices
#OEIS A001187  
NuCG:=proc(n) local i,g:
[seq(coeff(add(IsCo(g), g in Graphs(i)),true,1),i=1..n)]:
end:

NuCGc:=proc(n) local f,z,i:
#The number of labeled graphs on n vertices is 2^binomial(n,2)
f:=log(add(2^binomial(i,2)*z^i/i!,i=0..n)):
f:=taylor(f,z=0,n+1):
[seq(i!*coeff(f,z,i),i=1..n)]:
end:








## adjacency matrices Code by Aurora Hiveley
# AM(G): inputs a graph [n,E] and outputs the adjacency matrix, represented as a list of length n of lists of length n, 
# such that M[i][j]=1 if {i,j} belongs to E and 0 otherwise. 
# For example AM([2,{{1,2}}]); should output [[0,1],[1,0]] .

AM := proc(G) local n,E,e,M:
n := G[1]:
E := G[2]:

M := [[0$n]$n]:     # initialize n x n matrix of all 0's

for e in E do
    M[e[1]][e[2]]++:
    M[e[2]][e[1]]++:
od:

M:
end:



#C2.txt: Jan. 27, 2025
Help2:=proc(): print(`LC(p), RG(n,p), Cliques(G,k) `):end:
with(combinat):
#LC(p): inputs a rational number between 0 and 1 and outputs true with prob. p
LC:=proc(p) local a,b,ra:
if not (type(p,fraction) and p>=0 and p<=1) then
 RETURN(FAIL):
fi:
a:=numer(p):
b:=denom(p):
ra:=rand(1..b)():
if ra<=a then
 true:
else
 false:
fi:
end:

RG:=proc(n,p) local E,i,j:
E:={}:
for i from 1 to n do
 for j from i+1 to n do
   if LC(p) then
    E:=E union {{i,j}}:
   fi:
 od:
od:
[n,E]:
end:

#Cliques(G,k): inputs a graph G and a pos. integer k outputs the set of
#k-cliques
Cliques:=proc(G,k) local n, E,S,i,c,C:
n:=G[1]:
E:=G[2]:
S:={}:
C:=choose({seq(i,i=1..n)},k):
for c in C do
 if choose(c,2) minus E={} then
  S:=S union {c}:
 fi:
od:
S:
end:

###From C1
#C1.txt: Jan. 23, 2025 Exp Math (Dr. Z.)
Help1:=proc(): print(`Graphs(n), Tri(G) , TotTri(G) `):   end:



#An undirected  graph is a set of vertices V and a set of edges
#[V,E]   and edge e={i,j} where i and j belong to V
#Our vertices are labeled {1,2,...,n}
#Our data structure is [n,E] where E is the set of edges
[3,{{1,2},{1,3},{2,3}}];

#If there are n vertices how many (undirected) graphs there

#Graphs(n): inputs a non-neg. integer and outputs the set of ALL
#graphs on {1,...,n} 
Graphs:=proc(n) local i,j,S,E,s:
E:={seq(seq({i,j},j=i+1..n),   i=1..n)};
S:=powerset(E):
{seq([n,s],s in S)}:
end:

#Tri(G): inputs a graph [n,E] and outputs the set of all triples {i,j,k}
#such {{i,j},{i,k},{j,k}} is a subset of E
Tri:=proc(G) local n,S,E,i,j,k:
n:=G[1]:
E:=G[2]:
#S is the set of love triangles
S:={}:
for i from  1 to n do
 for j from i+1 to n do
   for k from j+1 to n do
    #if member({i,j},E) and member({i,k},E), and member({j,k},E) then
     if {{i,j},{i,k},{j,k}} minus E={} then
       S:=S union {{i,j,k}}:
    fi:
   od:
 od:
od:

S:
end:

#Comp(G): the complement of G=[n,E]
Comp:=proc(G) local n,i,j,E:
n:=G[1]:
E:=G[2]:
[n,{seq(seq({i,j},j=i+1..n),   i=1..n)} minus E]:
end:
#Tot(G): the total number of love triangles and hate triangles
TotTri:=proc(G)
nops(Tri(G))+nops(Tri(Comp(G))):
end: