#-----------------------------------------------------------------
# This maple program computes given a right hand side of a consistency
# table computes ALL the vertices of the polytope transportation polytope.
#
# INPUT: a valid feasible righthandside, circuits, matrix of graph
#
# OUTPUT: A rational function with the right-handside as parameters.
#-----------------------------------------------------------------

interface(quiet=true):
with(linalg):
with(networks):
with(simplex):

#--------------------------------------------------------
# This procedure Gale dualizes the chambers.
#--------------------------------------------------------

realchambers:=proc(B,cellsofchamber)
local elements,thechamber:

elements:={$1..coldim(B)}:
RETURN(convert(map(x->(elements minus x),cellsofchamber),list)):
end:

#------------------------------------------------------------
# Produces unoriented circuits (just as dependent sets no sign).
#---------------------------------------------------------------

Tounsigned_circuits:=proc(circuits)

RETURN(map(x->op(1,x) union op(2,x),circuits)):

end:

#---------------------------------------------------------------
# Turns the elements of the circuit into two groups of variables
#---------------------------------------------------------------

zcircuits:=proc(circuits)
local nova,c:

nova:=[]:
for c in circuits do
nova:=[op(nova),[map(x->z[x],op(1,c)),map(x->z[x],op(2,c))]]:
od:
RETURN(nova):
end:

#---------------------------------------------------------------------------
# This subroutine finds the coordinate of the vertex of the polytope associated
# with a given base and righthandside of the equation.
#-----------------------------------------------------------------------

coordinates_vertex:=proc(B,righthandside,base)
local const,aux,i,vaux,mataux,y,tempo,tempo2,lista:


aux:=[]: 
for i from 1 to coldim(B) do
  if member(i,base) then
    aux:=[op(aux),y.i]:
  else
    aux:=[op(aux),0]:
  fi:
od:
lista:=convert(base,list):
tempo:=map(x->y.x,lista):
mataux:=submatrix(B,[$1..rowdim(B)],lista):
vaux:=vector(righthandside);
#print(`caca`,linsolve(mataux,vaux),`cama`,lista);
const:=convert(linsolve(mataux,vaux),list);
tempo2:={}:
for i from 1 to nops(const) do
  tempo2:=tempo2 union {op(i,tempo)=op(i,const)}:
od:
RETURN(subs(tempo2,aux));
end:


#-------------------------------------------------------------
# This next subroutine changes a List into a monomial in the
# variables z[i], where the indices run on the size of the list.
#
# EXAMPLE: ListToMonomial(coordinates_vertex(U,[b1,b2,b3],{1,2,6}));
#-------------------------------------------------------------

ListToMonomial:=proc(v)
local tempo,i:
global x:

tempo:=1:
for i from 1 to nops(v) do
tempo:=tempo*(z[i]^op(i,v)):
od:
RETURN(tempo);
end:


#---------------------------------------------------------
# This little subroutine computes the support of a list in fact
#---------------------------------------------------------

support:=proc(l)

local i,neg,pos;
neg:={};
pos:={};
for i from 1 to nops(l) do
  if op(i,l)>0 then
    pos:=pos union {i}:
  elif op(i,l)<0 then
    neg:=neg union {i}:
 fi
od;
RETURN([pos,neg]);
end:

#------------------------------------------------------------
# This procedure goes from the Matrix to the undirected graph
# data structure where we can very fast compute the circuits.
#------------------------------------------------------------

FrommatrixTOgraph:=proc(U)
local edgy,i:
global G:

new(G):
addvertex({$1..rowdim(U)},G):
for i from 1 to coldim(U) do
 edgy:=op(1,support(convert(convert(submatrix(U,[$1..rowdim(U)],[i]),vector),list))):
 addedge(edgy,names=z[i],G):
od:
end:


#----------------------------------------------------------------------
# first an simple auxiliary statement that given a list of indexes that are not
# zero creates a 0,1 vector indicating those indices.

indicator:=proc(c,length) 
local i,v:

v:=[]:
for i from 1 to length do
 if member(i,c) then
  v:=[op(v),1]:
 else
  v:=[op(v),0]:
 fi:
od:
RETURN(v);
end:

indicatordouble:=proc(pos,neg,length) 
local i,v:

v:=[]:
for i from 1 to length do
 if member(i,pos) then
  v:=[op(v),1]:
 elif member(i,neg) then
  v:=[op(v),-1]:
 else
  v:=[op(v),0]:
 fi:
od:
RETURN(v);
end:


#-----------------------------------------------------------------------------
# We compute for a given vertex v of the polytope P_b (with b in the
# chamber) Brions series of its cone K_v (all those polytopes are 
# normally equivalent). This gives one term in Brion's rational function 
# associated to the polytope Pb. The right hand side b is undetermined so far.
# At the same time we are determinining the new adjacent vertices to v!!
#--------------------------------------------------------------------------

BrionsVertexseries:=proc(nodo,circuits,circuitos,circuitsnunu,U)
local d,enotation,auxrhs,numerator, denominator,finalists,e,l,f,adenom,bdenom,bector,vv1,vv2,vvr,entrada,nuevobector:
global G,newneighbors:

base:=op(1,nodo):
bector:=op(2,nodo):
auxrhs:=[]:
for d from 1 to nops(vertices(G))-1 do
auxrhs:=[op(auxrhs),b.d]:
od:

enotation:=map(x->z[x],base):
#print(`esto es enotation`,enotation);
numerator:=ListToMonomial(bector):
denominator:=1:
finalists:={}:
#print(`los circuitos`,circuitos);
for e in (edges(G) minus enotation) do
  # print(`el aristo`,e,`ciclo`, fundcyc(enotation union {e},G));
   if member(fundcyc(enotation union {e},G),circuitos,'l') then
     if member(e,op(1,op(l,circuits))) then
       finalists:=finalists union {op(l,circuitsnunu)}:
     else 
       finalists:=finalists union {[op(2,op(l,circuitsnunu)),op(1,op(l,circuitsnunu))]}:
     fi:
   fi:
od:
#print(`estos fueron`,finalists);
#print( `antes`,newneighbors);
for f in finalists do

# This part creates the piece of f for the bb series.


    adenom:=ListToMonomial(indicator(op(1,f),nops(edges(G))));
    bdenom:=ListToMonomial(indicator(op(2,f),nops(edges(G))));
    denominator:=denominator*(1-(adenom/bdenom));

# now we must find the base associated to f

entrada:=min(op(map(x->op(x,bector),op(2,f)))):
#print(`estos son las entradas escojo la mas pequena`,entrada);
vv1:=vector(indicator(op(1,f),nops(edges(G)))):
vv2:=vector(indicator(op(2,f),nops(edges(G)))):
vvr:=matadd(vv1,vv2,entrada,-entrada);
#print(matadd(vvr,vector(bector),1,1),bector,`si si como no`,vvr);
 nuevobector:=convert(matadd(vvr,vector(bector),1,1),list):
 newneighbors:=newneighbors union {[op(1,support(nuevobector)),nuevobector]}:
 od;

 RETURN(numerator/denominator);
end:


FromtopcomTOmaple:=proc(chamber)
local newchamber,c:

newchamber:={}:
for c in chamber do
 newchamber:=newchamber union {map(x->x+1,c)}:
od:
print(nops(newchamber));
RETURN(newchamber):
end:

#----------------------------------------------------------
# This subroutine finds an initial minimal feasible solution which is
# a root for the spanning graph of the enumeration.
#-----------------------------------------------------------

rootenum:=proc(A,b)
local constraints,varvector,value:

varvector:=map(i->x.i,[$1..coldim(A)]):
constraints:=geneqns(A,varvector,vector(b)):
value:=subs(minimize(x1,constraints,NONNEGATIVE),varvector):
print(value);
RETURN([op(1,support(value)),value]):
end:

#------------------------------------------------------------------------------
# This subroutine finds the limit of the univariate BB series when the
# auxiliary parameter t tends to zero. The answer is the number of lattice
# points inside the polytope. This works rightnow only for 4x4 tables as
# the substitution of generic values has to be done for each polytope
# more or less independently.
#------------------------------------------------------------------------------

limiting:=proc(f,a)
local l,s1,s2,s3,s4,s5,s6,s7,s8,s9,s10,s11,s12,s13,s14,s15,s16,cosa:

cosa:=f:
save cosa,bbseries4x4:
print(`saving series`);
s1:=subs(z[1]=exp(2*t),f):
s2:=subs(z[2]=exp(23*t),s1):
s3:=subs(z[3]=exp(11*t),s2):
s4:=subs(z[4]=exp(t),s3):
s5:=subs(z[5]=exp(17*t),s4):
s6:=subs(z[6]=exp(3*t),s5):
s7:=subs(z[7]=exp(t),s6):
s8:=subs(z[8]=exp(2*t),s7):
s9:=subs(z[9]=exp(5*t),s8):
s10:=subs(z[10]=exp(7*t),s9):
s11:=subs(z[11]=exp(2*t),s10):
s12:=subs(z[12]=exp(23*t),s11):
s13:=subs(z[13]=exp(11*t),s12):
s14:=subs(z[14]=exp(t),s13):
s15:=subs(z[15]=exp(17*t),s14):
s16:=subs(z[16]=exp(3*t),s15):
print(`voy a empezar el calculo del limite`);
series(s16,t=0,5);
l:=limit(s16,t=0);
print(`the number of lattice points is`,l);
RETURN(l);
end:

#---------------------------------------------------------------
# Finally we compute the Barvinok-Brion series associated to the
# family of convex polytopes in question as we go along generating 
# the vertices in a DFS fashion.
#---------------------------------------------------------------

PolytopeLatticepointseries:=proc(U,circuits,righthandside)
local i,temp,result,allbases,rhside,V,C,new,allvertices,tobevisited,visited_nodes,nonrepeated:
global G,newneighbors:

# These are just initialization steps.

FrommatrixTOgraph(U):
tobevisited:={rootenum(submatrix(U,[$1..rowdim(U)-1],[$1..coldim(U)]),righthandside)}:
print(tobevisited);
allvertices:={}:
visited_nodes:={}:
V:=submatrix(U,[$1..rowdim(U)-1],[$1..coldim(U)]):
temp:=0:

# Now we run through the general framework of finding series and for each found
# vertex.

while tobevisited <> {} do
 new:=op(1,tobevisited):
 newneighbors:={}:
 temp:=temp+BrionsVertexseries(new,zcircuits(circuits),Tounsigned_circuits(zcircuits(circuits)),circuits,V);
 allvertices:=(allvertices union newneighbors) union {new}:
# print(visited_nodes);
 nonrepeated:=newneighbors minus visited_nodes:
 tobevisited:=(tobevisited minus {new}) union nonrepeated:
 visited_nodes:=visited_nodes union {new}:
 print(`so far I have`,nops(allvertices),`and left`,nops(tobevisited));
od:
  print(`We finish generating the BBKLP rational function. Stored  in serie45`);
  save temp,serie45;
  print(`FINI`);
end:


# The matrix U is the incidence matrix of K_45 

U:=matrix([[1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0], [0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1], [1, 0, 0, 0, 0, 1, 0,
0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 1, 0, 0, 0,
0, 1, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1,
0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0,
0, 0, 1, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0,
1]]);

circuits:={[{2, 9, 13, 20}, {3, 7, 15, 19}], [{2, 10, 14, 16}, {5, 9, 11, 17}
], [{8, 14, 1}, {4, 6, 13}], [{8, 14, 20, 1}, {3, 10, 11, 19}], [{3, 6, 20},
{5, 8, 16}], [{9, 12, 18, 1}, {4, 7, 13, 16}], [{10, 12, 19, 1}, {4, 7, 11,
20}], [{15, 19, 1}, {4, 11, 20}], [{9, 15, 18, 1}, {5, 6, 13, 19}], [{8, 15,
17, 1}, {3, 10, 12, 16}], [{2, 8, 15, 16}, {3, 6, 12, 20}], [{9, 13, 20, 1},
{5, 8, 11, 19}], [{8, 15, 16}, {10, 11, 18}], [{2, 10, 11, 19}, {4, 7, 15, 16}
], [{8, 12, 19, 1}, {4, 7, 11, 18}], [{13, 17, 1}, {3, 12, 16}], [{2, 13, 20
}, {5, 12, 18}], [{8, 14, 1}, {3, 9, 11}], [{4, 10, 11, 17}, {5, 6, 12, 19}],
[{8, 19, 1}, {4, 6, 18}], [{13, 17, 1}, {2, 11, 18}], [{8, 15, 19, 1}, {5,
6, 14, 18}], [{8, 15, 19}, {9, 13, 20}], [{4, 7, 15, 16}, {5, 9, 11, 17}], [{3
, 9, 11, 20}, {5, 8, 14, 16}], [{7, 14, 1}, {2, 9, 11}], [{8, 14, 20, 1}, {3
, 9, 15, 16}], [{12, 19, 1}, {2, 14, 16}], [{2, 10, 14, 18}, {5, 8, 12, 19}],
[{9, 15, 18, 1}, {4, 8, 11, 20}], [{8, 15, 19, 1}, {3, 9, 11, 20}], [{3, 15,
19}, {5, 14, 18}], [{2, 10, 14}, {5, 9, 12}], [{10, 12, 18, 1}, {3, 6, 15, 17
}], [{2, 10, 11, 18}, {3, 7, 15, 16}], [{4, 7, 11, 20}, {5, 9, 12, 16}], [{3,
10, 11, 17}, {5, 6, 12, 18}], [{9, 12, 20, 1}, {4, 7, 15, 16}], [{4, 7, 15},
{5, 9, 12}], [{3, 6, 15, 19}, {4, 10, 13, 16}], [{7, 15, 19, 1}, {2, 9, 11, 
20}], [{9, 12, 20, 1}, {5, 6, 14, 17}], [{6, 15, 18}, {10, 13, 16}], [{9, 13,
20, 1}, {3, 6, 15, 19}], [{2, 10, 13, 19}, {3, 9, 15, 17}], [{6, 15, 18}, {8,
11, 20}], [{2, 9, 15, 18}, {3, 10, 12, 19}], [{9, 18, 1}, {3, 6, 19}], [{17,
1}, {2, 16}], [{6, 14, 20}, {9, 15, 16}], [{8, 15, 17, 1}, {5, 7, 13, 16}],
[{8, 12, 20, 1}, {3, 7, 15, 16}], [{3, 10, 12, 19}, {5, 7, 14, 18}], [{3, 20}
, {5, 18}], [{7, 13, 19, 1}, {4, 6, 12, 18}], [{3, 9, 11, 20}, {5, 6, 13, 19}
], [{9, 20}, {10, 19}], [{4, 6, 13, 20}, {5, 9, 11, 18}], [{8, 14, 20}, {10, 
13, 19}], [{2, 9, 15, 18}, {5, 7, 13, 19}], [{3, 9, 20}, {4, 10, 18}], [{8, 19
, 1}, {3, 9, 16}], [{3, 6, 14, 20}, {4, 8, 15, 16}], [{7, 15, 18, 1}, {2, 8,
11, 20}], [{3, 10, 17}, {5, 7, 18}], [{9, 12, 1}, {4, 7, 11}], [{3, 7, 11, 20
}, {5, 8, 12, 16}], [{8, 15, 19, 1}, {4, 6, 13, 20}], [{2, 9, 13, 20}, {4, 10
, 12, 18}], [{4, 12, 20}, {5, 14, 17}], [{2, 9, 15}, {5, 7, 14}], [{9, 13, 20}
, {10, 14, 18}], [{4, 6, 20}, {5, 9, 16}], [{3, 9, 11, 20}, {4, 6, 15, 18}], [
{8, 12, 1}, {3, 7, 11}], [{10, 14, 18, 1}, {3, 9, 11, 20}], [{7, 15, 19, 1}
, {5, 9, 12, 16}], [{3, 7, 15, 16}, {5, 8, 11, 17}], [{8, 12, 19, 1}, {2, 9,
13, 16}], [{7, 13, 19, 1}, {3, 9, 12, 16}], [{9, 1}, {4, 6}], [{9, 15, 18, 
1}, {5, 8, 14, 16}], [{2, 9, 15, 18}, {4, 10, 13, 17}], [{2, 9, 11, 20}, {5,
7, 14, 16}], [{2, 10, 13, 19}, {4, 8, 12, 20}], [{3, 6, 14}, {4, 8, 11}], [{3,
6, 14, 20}, {5, 9, 13, 16}], [{10, 13, 17, 1}, {2, 8, 11, 20}], [{9, 15, 17,
1}, {4, 10, 12, 16}], [{2, 8, 15, 19}, {4, 7, 13, 20}], [{9, 13, 1}, {3, 6,
14}], [{9, 12, 20, 1}, {2, 10, 14, 16}], [{7, 14, 18}, {8, 12, 19}], [{7, 15,
18, 1}, {2, 10, 13, 16}], [{2, 9, 15, 16}, {4, 6, 12, 20}], [{6, 15, 19}, {9,
11, 20}], [{9, 15, 17, 1}, {2, 10, 11, 19}], [{3, 7, 15}, {5, 8, 12}], [{14,
18, 1}, {3, 11, 19}], [{2, 9, 16}, {4, 6, 17}], [{7, 13}, {8, 12}], [{2, 19},
{4, 17}], [{9, 15, 1}, {5, 6, 14}], [{7, 14, 20, 1}, {2, 9, 15, 16}], [{10,
12, 18, 1}, {2, 8, 15, 16}], [{2, 8, 15, 19}, {5, 9, 13, 17}], [{2, 9, 11, 20
}, {4, 6, 15, 17}], [{4, 10, 11, 18}, {5, 8, 14, 16}], [{7, 13, 1}, {2, 8, 11
}], [{2, 9, 15, 16}, {5, 7, 11, 19}], [{4, 7, 11, 20}, {5, 6, 14, 17}], [{2, 8
, 15}, {3, 10, 12}], [{4, 10, 13}, {5, 8, 14}], [{8, 15, 1}, {5, 6, 13}], [{4
, 8, 15, 16}, {5, 9, 11, 18}], [{2, 8, 11, 20}, {5, 7, 13, 16}], [{12, 20}, {
15, 17}], [{9, 13, 17, 1}, {3, 6, 12, 19}], [{9, 13, 20, 1}, {4, 10, 11, 18}
], [{2, 10, 14}, {4, 7, 15}], [{3, 6, 15}, {5, 8, 11}], [{3, 10, 14, 17}, {5,
9, 12, 18}], [{8, 14, 17, 1}, {3, 7, 11, 19}], [{9, 17, 1}, {2, 6, 19}], [{2
, 10, 13}, {5, 8, 12}], [{7, 20, 1}, {5, 6, 17}], [{4, 8, 12, 20}, {5, 7, 14,
18}], [{8, 14, 20, 1}, {4, 10, 13, 16}], [{6, 13, 20}, {8, 15, 16}], [{3, 12,
20}, {5, 13, 17}], [{9, 13, 17, 1}, {2, 6, 14, 18}], [{7, 15}, {10, 12}], [{
12, 20, 1}, {5, 11, 17}], [{10, 13, 19, 1}, {4, 8, 11, 20}], [{10, 12, 19, 
1}, {2, 9, 15, 16}], [{3, 10, 14, 16}, {4, 8, 11, 20}], [{10, 14, 17, 1}, {4
, 6, 12, 20}], [{3, 10}, {5, 8}], [{2, 8, 15, 16}, {3, 10, 11, 17}], [{2, 20},
{5, 17}], [{2, 6, 15}, {5, 7, 11}], [{8, 12, 20, 1}, {2, 10, 13, 16}], [{7, 
15, 19, 1}, {5, 6, 14, 17}], [{2, 8, 11, 20}, {3, 10, 12, 16}], [{3, 10, 12},
{5, 7, 13}], [{3, 15}, {5, 13}], [{2, 10, 13, 16}, {5, 8, 11, 17}], [{8, 14, 
17, 1}, {2, 6, 13, 19}], [{10, 12, 1}, {5, 7, 11}], [{15, 1}, {5, 11}], [{8
, 20, 1}, {3, 10, 16}], [{2, 10, 11, 19}, {5, 9, 12, 16}], [{10, 14, 1}, {4,
6, 15}], [{10, 13, 17, 1}, {3, 7, 15, 16}], [{2, 6, 13, 20}, {3, 7, 15, 16}],
[{2, 9, 13, 20}, {4, 8, 15, 17}], [{9, 15, 18}, {10, 13, 19}], [{2, 10, 14, 16
}, {4, 7, 11, 20}], [{2, 10}, {5, 7}], [{8, 12, 19, 1}, {4, 6, 13, 17}], [{2,
15, 19}, {5, 14, 17}], [{7, 15, 16}, {10, 11, 17}], [{7, 14, 20, 1}, {5, 6, 
12, 19}], [{10, 19, 1}, {5, 9, 16}], [{9, 15, 18, 1}, {3, 10, 11, 19}], [{4,
10, 12, 18}, {5, 8, 14, 17}], [{8, 15, 17, 1}, {5, 6, 12, 18}], [{4, 6, 15, 
18}, {5, 9, 13, 16}], [{7, 13, 20, 1}, {3, 10, 12, 16}], [{4, 15, 17}, {5, 12
, 19}], [{13, 20, 1}, {3, 15, 16}], [{10, 12, 1}, {2, 6, 15}], [{10, 14, 18,
1}, {5, 9, 13, 16}], [{2, 14, 20}, {5, 12, 19}], [{7, 11, 20}, {10, 12, 16}],
[{11, 20}, {15, 16}], [{8, 15, 19, 1}, {3, 10, 14, 16}], [{7, 20, 1}, {2, 10
, 16}], [{4, 10, 11}, {5, 6, 14}], [{3, 10, 11, 17}, {5, 7, 13, 16}], [{7, 13,
19, 1}, {3, 6, 14, 17}], [{10, 13, 19, 1}, {5, 9, 11, 18}], [{3, 10, 14, 16}
, {5, 9, 11, 18}], [{13, 1}, {3, 11}], [{3, 9, 15, 17}, {4, 7, 13, 20}], [{4,
8, 15, 16}, {5, 6, 13, 19}], [{9, 15, 17, 1}, {4, 7, 11, 20}], [{2, 8, 15, 16
}, {5, 7, 11, 18}], [{10, 13, 19, 1}, {5, 8, 14, 16}], [{3, 11, 20}, {5, 13,
16}], [{2, 10, 11}, {5, 6, 12}], [{2, 10, 13, 16}, {3, 6, 15, 17}], [{14, 17,
1}, {2, 11, 19}], [{7, 13, 20, 1}, {2, 10, 11, 18}], [{7, 19, 1}, {4, 6, 17
}], [{14, 17, 1}, {4, 12, 16}], [{9, 12, 18, 1}, {3, 6, 14, 17}], [{7, 15, 
1}, {5, 6, 12}], [{8, 12, 1}, {2, 6, 13}], [{4, 6, 13, 20}, {5, 8, 14, 16}],
[{2, 8, 15, 16}, {5, 6, 13, 17}], [{4, 10, 12}, {5, 7, 14}], [{4, 15}, {5, 14}
], [{2, 9, 11, 18}, {4, 6, 13, 17}], [{2, 8}, {3, 7}], [{2, 14, 18}, {4, 13, 
17}], [{2, 8, 11}, {3, 6, 12}], [{8, 19}, {9, 18}], [{12, 18}, {13, 17}], [{7,
13, 16}, {8, 11, 17}], [{2, 6, 14}, {4, 7, 11}], [{2, 18}, {3, 17}], [{6, 12},
{7, 11}], [{8, 12, 19}, {9, 13, 17}], [{3, 11, 19}, {4, 13, 16}], [{7, 13, 19}
, {9, 12, 18}], [{2, 8, 19}, {3, 9, 17}], [{3, 9, 11}, {4, 6, 13}], [{3, 14},
{4, 13}], [{2, 9, 18}, {3, 7, 19}], [{2, 6, 14, 18}, {3, 9, 12, 16}], [{2, 9,
11, 18}, {3, 6, 12, 19}], [{2, 14, 18}, {3, 12, 19}], [{3, 6, 19}, {4, 8, 16}]
, [{2, 13}, {3, 12}], [{3, 7, 11, 19}, {4, 8, 12, 16}], [{6, 13, 19}, {8, 14,
16}], [{7, 19}, {9, 17}], [{6, 14, 17}, {9, 12, 16}], [{2, 9, 18}, {4, 8, 17}]
, [{11, 18}, {13, 16}], [{6, 12, 19}, {9, 11, 17}], [{12, 19}, {14, 17}], [{2,
6, 13, 19}, {4, 8, 12, 16}], [{3, 7, 14}, {4, 8, 12}], [{7, 14, 18}, {9, 13, 
17}], [{2, 9, 13}, {3, 7, 14}], [{6, 13, 17}, {8, 12, 16}], [{2, 6, 13, 19}, {
4, 7, 11, 18}], [{2, 13, 16}, {3, 11, 17}], [{2, 8, 14, 16}, {4, 7, 11, 18}],
[{2, 6, 13, 19}, {3, 7, 14, 16}], [{8, 11, 19}, {9, 13, 16}], [{3, 9, 17}, {4,
7, 18}], [{3, 6, 12, 19}, {4, 7, 13, 16}], [{8, 14}, {9, 13}], [{3, 9, 11, 17}
, {4, 6, 12, 18}], [{3, 14, 17}, {4, 12, 18}], [{7, 11, 18}, {8, 12, 16}], [{3
, 7, 14, 16}, {4, 8, 11, 17}], [{2, 9, 13}, {4, 8, 12}], [{2, 6, 19}, {4, 7, 
16}], [{2, 8, 14, 16}, {4, 6, 13, 17}], [{2, 9, 13, 16}, {3, 7, 11, 19}], [{3,
9}, {4, 8}], [{6, 18}, {8, 16}], [{2, 6, 14, 18}, {4, 7, 13, 16}], [{2, 11, 18
}, {3, 12, 16}], [{6, 13, 17}, {7, 11, 18}], [{2, 14, 16}, {4, 11, 17}], [{3,
9, 12}, {4, 7, 13}], [{11, 19}, {14, 16}], [{6, 12, 18}, {7, 13, 16}], [{2, 8,
16}, {3, 6, 17}], [{2, 9, 13, 16}, {4, 8, 11, 17}], [{3, 7, 11, 19}, {4, 6, 13
, 17}], [{2, 11, 19}, {4, 12, 16}], [{6, 14}, {9, 11}], [{2, 6, 18}, {3, 7, 16
}], [{2, 9}, {4, 7}], [{2, 9, 11}, {4, 6, 12}], [{2, 14}, {4, 12}], [{8, 14, 
17}, {9, 12, 18}], [{3, 9, 11, 17}, {4, 7, 13, 16}], [{6, 17}, {7, 16}], [{3,
14, 16}, {4, 11, 18}], [{2, 8, 14, 16}, {3, 6, 12, 19}], [{2, 13, 19}, {4, 12,
18}], [{2, 9, 13, 16}, {3, 6, 14, 17}], [{2, 8, 11, 19}, {4, 7, 13, 16}], [{3,
9, 16}, {4, 6, 18}], [{3, 6, 14, 17}, {4, 7, 11, 18}], [{2, 6, 13, 19}, {3, 9,
11, 17}], [{11, 17}, {12, 16}], [{2, 8, 11, 19}, {3, 9, 12, 16}], [{6, 12, 19}
, {7, 14, 16}], [{7, 13, 19}, {8, 14, 17}], [{6, 14, 18}, {8, 11, 19}], [{2, 8
, 19}, {4, 7, 18}], [{6, 13}, {8, 11}], [{2, 8, 14}, {3, 9, 12}], [{2, 8, 11,
19}, {4, 6, 12, 18}], [{3, 7, 19}, {4, 8, 17}], [{2, 8, 11, 19}, {3, 6, 14, 17
}], [{3, 6, 14, 17}, {4, 8, 12, 16}], [{8, 14, 16}, {9, 11, 18}], [{7, 11, 19}
, {9, 12, 16}], [{3, 6, 12, 19}, {4, 8, 11, 17}], [{2, 6, 14, 18}, {3, 7, 11,
19}], [{3, 7, 14, 16}, {4, 6, 12, 18}], [{2, 9, 11, 18}, {4, 8, 12, 16}], [{2,
6, 13}, {3, 7, 11}], [{7, 14}, {9, 12}], [{2, 8, 14, 16}, {3, 9, 11, 17}], [{6
, 19}, {9, 16}], [{6, 14, 17}, {7, 11, 19}], [{2, 9, 11, 18}, {3, 7, 14, 16}],
[{2, 6, 14, 18}, {4, 8, 11, 17}], [{3, 9, 12, 16}, {4, 7, 11, 18}], [{6, 14, 
18}, {9, 13, 16}], [{7, 18}, {8, 17}], [{6, 13, 19}, {9, 11, 18}], [{13, 19},
{14, 18}], [{2, 9, 13, 16}, {4, 6, 12, 18}], [{6, 12, 18}, {8, 11, 17}], [{7,
14, 16}, {9, 11, 17}], [{3, 6, 15, 19}, {4, 8, 11, 20}], [{2, 13, 19}, {3, 14,
17}], [{3, 9, 12, 16}, {4, 6, 13, 17}], [{3, 19}, {4, 18}], [{2, 8, 14}, {4, 7
, 13}], [{6, 14, 20}, {10, 11, 19}], [{7, 13, 20}, {10, 12, 18}], [{3, 12, 19}
, {4, 13, 17}], [{2, 11, 20}, {5, 12, 16}], [{2, 6, 15, 18}, {3, 7, 11, 20}],
[{10, 13, 19, 1}, {3, 6, 14, 20}], [{3, 10, 19}, {5, 9, 18}], [{10, 17, 1},
{5, 7, 16}], [{2, 6, 20}, {5, 7, 16}], [{12, 1}, {2, 11}], [{7, 15, 18, 1},
{5, 8, 12, 16}], [{7, 15, 18, 1}, {3, 10, 11, 17}], [{3, 6, 15, 19}, {5, 8, 
14, 16}], [{8, 14, 17, 1}, {4, 6, 12, 18}], [{10, 19, 1}, {4, 6, 20}], [{2,
8, 15, 19}, {4, 10, 12, 18}], [{4, 7, 20}, {5, 9, 17}], [{3, 10, 12, 16}, {5,
6, 13, 17}], [{4, 7, 15, 16}, {5, 6, 12, 19}], [{4, 10, 11, 17}, {5, 7, 14, 16
}], [{3, 9, 15, 16}, {5, 6, 14, 18}], [{2, 10, 19}, {4, 7, 20}], [{7, 13, 1},
{3, 6, 12}], [{12, 20, 1}, {2, 15, 16}], [{7, 13, 19, 1}, {2, 8, 14, 16}], [
{14, 1}, {4, 11}], [{3, 6, 15, 19}, {5, 9, 11, 18}], [{3, 6, 12, 20}, {5, 7,
13, 16}], [{2, 10, 13, 19}, {5, 9, 12, 18}], [{4, 8, 12, 20}, {5, 9, 13, 17}],
[{7, 1}, {2, 6}], [{7, 14, 20}, {9, 15, 17}], [{4, 11, 20}, {5, 14, 16}], [{2
, 10, 14, 18}, {4, 7, 13, 20}], [{2, 15, 16}, {5, 11, 17}], [{8, 20}, {10, 18}
], [{7, 15, 18}, {10, 13, 17}], [{2, 10, 14, 16}, {4, 6, 15, 17}], [{3, 10, 19
}, {4, 8, 20}], [{3, 7, 15, 19}, {4, 10, 13, 17}], [{9, 18, 1}, {4, 8, 16}],
[{2, 6, 13, 20}, {5, 8, 12, 16}], [{10, 13, 17, 1}, {5, 8, 12, 16}], [{2, 6,
13, 20}, {3, 10, 11, 17}], [{7, 15, 19, 1}, {4, 10, 11, 17}], [{2, 6, 15, 19}
, {4, 7, 11, 20}], [{2, 10, 13, 19}, {3, 7, 14, 20}], [{2, 15}, {5, 12}], [{7,
14, 18, 1}, {2, 9, 13, 16}], [{2, 8, 14, 20}, {3, 9, 15, 17}], [{7, 14, 20, 
1}, {2, 10, 11, 19}], [{8, 1}, {3, 6}], [{7, 14, 20, 1}, {4, 10, 12, 16}],
[{14, 20, 1}, {4, 15, 16}], [{6, 12, 20}, {7, 15, 16}], [{3, 10, 12, 19}, {4,
7, 13, 20}], [{3, 15, 19}, {4, 13, 20}], [{7, 13, 19, 1}, {2, 9, 11, 18}], [{
8, 15, 17, 1}, {2, 10, 11, 18}], [{2, 10, 18}, {3, 7, 20}], [{2, 6, 15, 19},
{5, 7, 14, 16}], [{7, 19, 1}, {2, 9, 16}], [{10, 14, 17, 1}, {4, 7, 15, 16}]
, [{2, 6, 14, 20}, {4, 7, 15, 16}], [{8, 14, 17, 1}, {2, 9, 11, 18}], [{8, 15
, 19, 1}, {5, 9, 13, 16}], [{3, 9, 15, 16}, {4, 6, 13, 20}], [{4, 10}, {5, 9}
], [{13, 19, 1}, {4, 11, 18}], [{2, 9, 15, 16}, {4, 10, 11, 17}], [{9, 13, 20
, 1}, {3, 10, 14, 16}], [{3, 9, 15, 16}, {5, 8, 11, 19}], [{9, 15, 17, 1}, {
5, 6, 12, 19}], [{9, 12, 18, 1}, {4, 8, 11, 17}], [{3, 10, 11, 19}, {5, 6, 14
, 18}], [{7, 14, 20, 1}, {4, 6, 15, 17}], [{9, 11, 20}, {10, 14, 16}], [{3, 7
, 15, 19}, {5, 8, 14, 17}], [{10, 1}, {5, 6}], [{7, 15, 19, 1}, {2, 10, 14,
16}], [{3, 10, 11, 19}, {4, 6, 13, 20}], [{12, 19, 1}, {4, 11, 17}], [{10, 14
, 18, 1}, {4, 8, 15, 16}], [{8, 17, 1}, {2, 6, 18}], [{7, 14, 18, 1}, {4, 6
, 13, 17}], [{3, 15, 17}, {5, 12, 18}], [{8, 15, 19, 1}, {4, 10, 11, 18}], [{
6, 20}, {10, 16}], [{2, 8, 15, 19}, {3, 9, 12, 20}], [{13, 20}, {15, 18}], [{3
, 7, 14, 20}, {5, 9, 13, 17}], [{6, 15, 17}, {10, 12, 16}], [{4, 8, 15}, {5, 9
, 13}], [{20, 1}, {5, 16}], [{4, 10, 12, 18}, {5, 7, 13, 19}], [{4, 15, 18},
{5, 13, 19}], [{8, 14, 17, 1}, {3, 9, 12, 16}], [{3, 7, 15, 16}, {5, 6, 12, 
18}], [{4, 8, 15, 17}, {5, 9, 12, 18}], [{2, 9, 15}, {4, 10, 12}], [{2, 10, 14
, 16}, {5, 6, 12, 19}], [{2, 8, 14, 20}, {5, 9, 12, 18}], [{8, 12, 19, 1}, {3
, 9, 11, 17}], [{10, 14, 18, 1}, {3, 6, 15, 19}], [{4, 6, 15}, {5, 9, 11}], [
{3, 10, 14}, {5, 9, 13}], [{19, 1}, {4, 16}], [{3, 7, 20}, {5, 8, 17}], [{10,
14, 1}, {5, 9, 11}], [{3, 9, 15, 17}, {5, 7, 14, 18}], [{2, 8, 15}, {5, 7, 13
}], [{3, 10, 14, 17}, {4, 8, 12, 20}], [{14, 18, 1}, {4, 13, 16}], [{7, 14, 
20, 1}, {5, 9, 11, 17}], [{2, 10, 11, 19}, {4, 6, 12, 20}], [{10, 12, 18, 1}
, {3, 7, 11, 20}], [{8, 15, 19}, {10, 14, 18}], [{15, 18, 1}, {3, 11, 20}], [
{10, 13, 17, 1}, {5, 7, 11, 18}], [{2, 6, 13, 20}, {5, 7, 11, 18}], [{12, 18,
1}, {2, 13, 16}], [{8, 12, 20, 1}, {5, 6, 13, 17}], [{7, 13, 20, 1}, {2, 8,
15, 16}], [{2, 9, 11, 20}, {4, 10, 12, 16}], [{3, 10, 14}, {4, 8, 15}], [{3, 6
, 12, 20}, {5, 8, 11, 17}], [{2, 14, 20}, {4, 15, 17}], [{3, 7, 14, 20}, {5, 8
, 12, 19}], [{9, 20, 1}, {4, 10, 16}], [{18, 1}, {3, 16}], [{7, 14, 18, 1},
{3, 9, 11, 17}], [{13, 20, 1}, {5, 11, 18}], [{9, 15, 17, 1}, {2, 6, 14, 20}
], [{9, 13, 20, 1}, {4, 8, 15, 16}], [{10, 14, 18, 1}, {4, 6, 13, 20}], [{4,
6, 15, 18}, {5, 8, 11, 19}], [{10, 13, 17, 1}, {2, 6, 15, 18}], [{10, 12, 19,
1}, {5, 7, 14, 16}], [{15, 19, 1}, {5, 14, 16}], [{10, 14, 18, 1}, {5, 8, 
11, 19}], [{2, 10, 16}, {5, 6, 17}], [{3, 6, 14, 20}, {4, 10, 11, 18}], [{3, 9
, 15, 17}, {5, 8, 12, 19}], [{7, 18, 1}, {2, 8, 16}], [{2, 10, 14, 18}, {3, 7
, 15, 19}], [{3, 7, 14, 20}, {4, 8, 15, 17}], [{9, 12, 20}, {10, 14, 17}], [{9
, 13, 1}, {4, 8, 11}], [{10, 12, 18, 1}, {2, 6, 13, 20}], [{4, 10, 18}, {5,
8, 19}], [{4, 8, 11, 20}, {5, 9, 13, 16}], [{2, 10, 11, 18}, {3, 6, 12, 20}],
[{2, 10, 11, 18}, {5, 6, 13, 17}], [{4, 7, 13, 20}, {5, 8, 14, 17}], [{10, 18,
1}, {3, 6, 20}], [{2, 9, 15, 18}, {5, 8, 14, 17}], [{4, 6, 12, 20}, {5, 9, 11
, 17}], [{9, 15}, {10, 14}], [{13, 19, 1}, {3, 14, 16}], [{2, 6, 14, 20}, {5,
7, 11, 19}], [{4, 6, 12, 20}, {5, 7, 14, 16}], [{9, 12, 18, 1}, {3, 7, 11, 19
}], [{4, 8, 11, 20}, {5, 6, 14, 18}], [{2, 6, 15, 18}, {5, 8, 11, 17}], [{10,
13, 1}, {3, 6, 15}], [{3, 9, 15, 16}, {4, 10, 11, 18}], [{2, 8, 15, 19}, {5,
7, 14, 18}], [{2, 10, 13, 16}, {3, 7, 11, 20}], [{3, 14, 20}, {5, 13, 19}], [{
3, 10, 14, 17}, {4, 7, 15, 18}], [{6, 15}, {10, 11}], [{3, 9, 12, 20}, {5, 8,
14, 17}], [{14, 20, 1}, {5, 11, 19}], [{4, 10, 17}, {5, 7, 19}], [{8, 12, 19,
1}, {3, 7, 14, 16}], [{10, 14, 17, 1}, {2, 6, 15, 19}], [{4, 7, 15, 18}, {5,
8, 12, 19}], [{15, 17, 1}, {5, 12, 16}], [{10, 12, 18, 1}, {5, 8, 11, 17}],
[{2, 8, 11, 20}, {5, 6, 12, 18}], [{3, 7, 14, 20}, {4, 10, 12, 18}], [{2, 9, 
13, 20}, {3, 10, 14, 17}], [{4, 10, 12, 16}, {5, 7, 11, 19}], [{10, 14, 17, 1
}, {2, 9, 11, 20}], [{4, 15, 16}, {5, 11, 19}], [{2, 10, 13, 19}, {4, 7, 15, 
18}], [{2, 9, 13, 20}, {5, 8, 12, 19}], [{7, 18, 1}, {3, 6, 17}], [{6, 13, 20
}, {10, 11, 18}], [{3, 9, 12, 20}, {4, 10, 13, 17}], [{7, 13, 20}, {8, 15, 17}
], [{9, 12, 20, 1}, {4, 10, 11, 17}], [{6, 15, 19}, {10, 14, 16}], [{8, 14, 
20}, {9, 15, 18}], [{2, 10, 11, 18}, {5, 8, 12, 16}], [{8, 12, 19, 1}, {2, 6,
14, 18}], [{12, 18, 1}, {3, 11, 17}], [{7, 15, 19}, {9, 12, 20}], [{2, 9, 20}
, {4, 10, 17}], [{8, 15, 17, 1}, {3, 7, 11, 20}], [{9, 15, 17, 1}, {5, 7, 14
, 16}], [{3, 9, 20}, {5, 8, 19}], [{2, 6, 15, 19}, {4, 10, 12, 16}], [{10, 18,
1}, {5, 8, 16}], [{2, 9, 15, 18}, {4, 8, 12, 20}], [{2, 6, 15, 18}, {5, 7, 13
, 16}], [{9, 13, 17, 1}, {2, 8, 11, 19}], [{3, 9, 15}, {4, 10, 13}], [{9, 13,
17, 1}, {4, 8, 12, 16}], [{3, 10, 14, 16}, {5, 6, 13, 19}], [{7, 13, 20, 1},
{5, 8, 11, 17}], [{9, 15, 1}, {4, 10, 11}], [{3, 9, 12, 20}, {5, 7, 13, 19}],
[{8, 14, 20, 1}, {5, 9, 11, 18}], [{10, 12, 18, 1}, {5, 7, 13, 16}], [{15, 
18, 1}, {5, 13, 16}], [{9, 12, 20, 1}, {5, 7, 11, 19}], [{2, 9, 15, 18}, {3,
7, 14, 20}], [{3, 10, 12, 16}, {5, 7, 11, 18}], [{7, 14, 18, 1}, {3, 6, 12, 
19}], [{3, 15, 16}, {5, 11, 18}], [{2, 9, 20}, {5, 7, 19}], [{10, 13, 19, 1},
{4, 6, 15, 18}], [{3, 10, 14, 16}, {4, 6, 15, 18}], [{2, 8, 14, 20}, {4, 10, 
13, 17}], [{4, 10, 16}, {5, 6, 19}], [{4, 8, 20}, {5, 9, 18}], [{3, 9, 12, 20}
, {4, 7, 15, 18}], [{8, 15, 17, 1}, {2, 6, 13, 20}], [{9, 13, 17, 1}, {3, 7,
14, 16}], [{4, 7, 15, 18}, {5, 9, 13, 17}], [{9, 12, 20, 1}, {2, 6, 15, 19}],
[{2, 9, 13, 20}, {5, 7, 14, 18}], [{9, 15, 17}, {10, 12, 19}], [{2, 8, 11, 20}
, {3, 6, 15, 17}], [{2, 9, 15, 16}, {5, 6, 14, 17}], [{2, 6, 15, 18}, {3, 10,
12, 16}], [{4, 10, 13, 16}, {5, 6, 14, 18}], [{8, 12, 20}, {10, 13, 17}], [{9,
12, 1}, {2, 6, 14}], [{4, 7, 13, 20}, {5, 9, 12, 18}], [{2, 6, 15, 19}, {5, 9
, 11, 17}], [{4, 10, 11, 18}, {5, 6, 13, 19}], [{4, 6, 15, 17}, {5, 9, 12, 16}
], [{3, 7, 15, 19}, {4, 8, 12, 20}], [{9, 17, 1}, {4, 7, 16}], [{4, 8, 15, 17
}, {5, 7, 13, 19}], [{8, 12, 20, 1}, {3, 10, 11, 17}], [{10, 14, 17, 1}, {5,
9, 12, 16}], [{2, 6, 14, 20}, {5, 9, 12, 16}], [{3, 9, 11, 20}, {4, 10, 13, 16
}], [{2, 8, 14, 20}, {5, 7, 13, 19}], [{3, 10, 12, 19}, {5, 9, 13, 17}], [{2,
10, 13, 16}, {5, 6, 12, 18}], [{2, 8, 20}, {3, 10, 17}], [{10, 12, 19, 1}, {5
, 9, 11, 17}], [{2, 9, 11, 20}, {5, 6, 12, 19}], [{7, 13, 20, 1}, {3, 6, 15,
17}], [{9, 15, 18, 1}, {3, 6, 14, 20}], [{6, 15, 17}, {7, 11, 20}], [{2, 10,
19}, {5, 9, 17}], [{9, 20, 1}, {5, 6, 19}], [{8, 15}, {10, 13}], [{9, 13, 17,
1}, {4, 7, 11, 18}], [{2, 8, 14, 20}, {4, 7, 15, 18}], [{3, 10, 16}, {5, 6, 
18}], [{3, 9, 15, 17}, {4, 10, 12, 18}], [{2, 13, 20}, {3, 15, 17}], [{8, 15,
1}, {3, 10, 11}], [{2, 10, 14, 18}, {3, 9, 12, 20}], [{2, 10, 14, 18}, {5, 9,
13, 17}], [{15, 17, 1}, {2, 11, 20}], [{9, 13, 20, 1}, {5, 6, 14, 18}], [{8,
14, 17, 1}, {4, 7, 13, 16}], [{2, 8, 15, 19}, {3, 10, 14, 17}], [{3, 7, 11, 
20}, {5, 6, 13, 17}], [{7, 20}, {10, 17}], [{7, 15, 18}, {8, 12, 20}], [{7, 13
, 19, 1}, {4, 8, 11, 17}], [{10, 13, 1}, {5, 8, 11}], [{3, 10, 12, 19}, {4,
8, 15, 17}], [{2, 8, 14, 20}, {3, 10, 12, 19}], [{2, 10, 13}, {3, 7, 15}], [{2
, 10, 13, 19}, {5, 8, 14, 17}], [{8, 14, 20, 1}, {5, 6, 13, 19}], [{9, 12, 18
, 1}, {2, 8, 14, 16}], [{3, 7, 15, 19}, {5, 9, 12, 18}], [{10, 13, 19, 1}, {
3, 9, 15, 16}], [{7, 15, 1}, {2, 10, 11}], [{9, 15, 16}, {10, 11, 19}], [{2,
6, 14, 20}, {4, 10, 11, 17}], [{3, 6, 14, 20}, {5, 8, 11, 19}], [{4, 10, 13, 
17}, {5, 8, 12, 19}], [{7, 15, 19}, {10, 14, 17}], [{3, 6, 15, 17}, {5, 8, 12,
16}], [{8, 12, 20, 1}, {5, 7, 11, 18}], [{8, 17, 1}, {3, 7, 16}], [{8, 11, 
20}, {10, 13, 16}], [{2, 10, 14, 18}, {4, 8, 15, 17}], [{8, 14, 20, 1}, {4, 6
, 15, 18}], [{10, 13, 17, 1}, {3, 6, 12, 20}], [{2, 8, 20}, {5, 7, 18}], [{7,
15, 19, 1}, {4, 6, 12, 20}], [{6, 12, 20}, {10, 11, 17}], [{7, 14, 20}, {10,
12, 19}], [{14, 20}, {15, 19}], [{3, 10, 11, 19}, {5, 9, 13, 16}], [{10, 17, 
1}, {2, 6, 20}], [{7, 14, 1}, {4, 6, 12}], [{4, 6, 15, 17}, {5, 7, 11, 19}],
[{3, 10, 14, 17}, {5, 7, 13, 19}], [{8, 12, 20, 1}, {2, 6, 15, 18}], [{8, 15,
17}, {10, 12, 18}], [{9, 15, 18, 1}, {4, 10, 13, 16}], [{2, 15, 19}, {4, 12,
20}], [{10, 14, 17, 1}, {5, 7, 11, 19}], [{3, 14, 20}, {4, 15, 18}], [{2, 10,
18}, {5, 8, 17}], [{3, 10, 11, 19}, {4, 8, 15, 16}], [{8, 20, 1}, {5, 6, 18}]
, [{4, 10, 13, 17}, {5, 7, 14, 18}], [{7, 15, 18, 1}, {3, 6, 12, 20}], [{10,
12, 19, 1}, {4, 6, 15, 17}], [{7, 15, 18, 1}, {5, 6, 13, 17}], [{4, 20}, {5,
19}], [{3, 10, 11}, {5, 6, 13}], [{10, 12, 19, 1}, {2, 6, 14, 20}], [{3, 9, 
15}, {5, 8, 14}], [{3, 6, 15, 17}, {5, 7, 11, 18}], [{9, 12, 18, 1}, {2, 6, 
13, 19}], [{2, 15, 18}, {3, 12, 20}], [{4, 13, 20}, {5, 14, 18}], [{2, 10, 11,
19}, {5, 6, 14, 17}], [{2, 15, 18}, {5, 13, 17}], [{4, 10, 13, 16}, {5, 8, 11,
19}], [{7, 13, 20, 1}, {5, 6, 12, 18}], [{4, 10, 12, 16}, {5, 6, 14, 17}], [{
7, 14, 18, 1}, {2, 8, 11, 19}], [{7, 14, 18, 1}, {4, 8, 12, 16}]}:


#------------------first example---------------------------------

a:=[3046,5173,6116,10928,182,778,3635,9558]:

#---------------------second example------------------------------
#a:=[338106,574203,678876,1213008, 20202, 142746, 410755, 1007773]:
# the sum total is 2,804,193.
#-----------------------------------------------------
st:=time():
resultado:=PolytopeLatticepointseries(U,circuits,a):
print(time()-st());
quit;