#its ok to post
#TaerimKim,10/31/2020,Assignment 15

#1.Using the methods from KiG and KtG

RoG:=proc(k,n) local a, b, i, E, V, T1, pt, Moves, Neighs;

V := [seq(seq(seq([i, j], j = 1 .. n), i = 1 .. k))];

for i to nops(V) do
T1[V[i]] := i;
od;

E:=[]:    

for i from 1 to nops(V) do   
pt:=V[i]:
Moves:={seq([a, 0], a = 1 .. k),seq([-a, 0], a = 1 .. k),seq([0, b], b = 1 .. n),seq([0, -b], b = 1 .. n)};

Neighs := {seq(pt + m, m in Moves)};

Neighs := Neighs intersect convert(V, set);

E:=[op(E),{seq(T1[v],v in Neighs)}]:  
od:

E,V:  
end:

#now evaluate the first 4 terms of RoG(3,i) when i=1..4
seq(SAWnu(RoG(3, i)), i = 1 .. 4);
                         2, 6, 96, 3132


#############################################################################

#(With using the Comboprogect2.txt)
#2.
#total number of walks from [0,0] to [40,40] using these 4 atomic steps
NuW([40, 40], {[0, 1], [1, 0], [1, 1], [2, 2]});
                2382564832244243056285491057263

#total number of walks that does not go over x=y(which always satisfy x>=y)
NuGW([40, 40], {[0, 1], [1, 0], [1, 1], [2, 2]});
                 89322096703094945357683861273


##############################################################################

#(With using the Comboprogect3.txt)
#3. similar cases to the above 

seq(NuW([i, i, i], {[0, 0, 1], [0, 1, 0], [1, 0, 0], [1, 1, 1]}), i = 1 .. 20);
7, 115, 2371, 54091, 1307377, 32803219, 844910395, 22188235867, 

  591446519797, 15953338537885, 434479441772845, 

  11927609772412075, 329653844941016785, 9163407745486783435, 

  255982736410338609931, 7181987671728091545787, 

  202271071826031620236525, 5715984422606794841997001, 

  162016571360163769411597081, 4604748196249289767697705221

#under the condition that walks should satisfy (x>=y>=z)

seq(NuGW([i, i, i], {[0, 0, 1], [0, 1, 0], [1, 0, 0], [1, 1, 1]}), i = 1 .. 20);
2, 10, 88, 1043, 14778, 236001, 4107925, 76314975, 1491934038, 

  30389576308, 640286048416, 13877540824735, 308102204007536, 

  6983346070924707, 161156356282624227, 3778249609096250059, 

  89826197363219012470, 2162338803354415120414, 

  52637415804379149938876, 1294313658632145337351381