print(`Created:Oct, 2002.`):
print(`This version: Oct, 2002`):
print(`Written by Xiangdong Wen, wen@math.temple.edu`):
 print(`Please report bugs to wen@math.temple.edu`):
 print(`The most current version of this  package and paper`):
 print(` are  available from`):
 print(`http://www.math.temple.edu/~wen/`):
 print(`For a list of the procedures type Help(), for help with`):
 print(`a specific procedure, type Help(procedure_name)`):

Help:=proc()
if args=NULL then
 print(`Contains the following procedures:`):
 print(`sumall_west,sumall_lind,sumall_4n_1,sumall_4n_2,`):
print(`sumall_3n_1,sumall_3n_2,sumall_2n(),sumall_1n.`):
fi:

if nops([args])=1 and op(1,[args])=sumall_west then
print(`sumall_west(): gives the total weights for the chains of L(4,n) by Douglas B. West`):
print(`A Symmetric chain Decomposition of L(4,n)`):
print(`Europ.J.Combinatorics(1980)1,379-383`):
print(`example: sumall_west()  `):
fi:

if nops([args])=1 and op(1,[args])=sumall_lind then
print(`sumall_lind(): gives the total weights for the chains of L(3,n) by Bernt Lindstrom`):
print(`A Partition of L(3,n) into Saturated Symmetric chains`):
print(`Europ.J.Combinatorics(1980)1,61-63`):
print(`example: sumall_lind()  `):
fi:

if nops([args])=1 and op(1,[args])=sumall_4n_1 then
print(`sumall_4n_1(): gives the total weights for the chains of L(4,n) by the authors`):
print(`example: sumall_4n_1()  `):
fi:
if nops([args])=1 and op(1,[args])=sumall_4n_2 then
print(`sumall_4n_2(): gives the total weights for the chains of L(4,n) by the authors`):
print(`example: sumall_4n_2()  `):
fi:
if nops([args])=1 and op(1,[args])=sumall_3n_1 then
print(`sumall_3n_1(): gives the total weights for the chains of L(3,n) by the authors`):
print(`example: sumall_3n_1()  `):
fi:
if nops([args])=1 and op(1,[args])=sumall_3n_2 then
print(`sumall_3n_2(): gives the total weights for the chains of L(3,n) by the authors`):
print(`example: sumall_3n_2()  `):
fi:
if nops([args])=1 and op(1,[args])=sumall_2n then
print(`sumall_2n(): gives the total weights for the chains of L(2,n)`):
print(`example: sumall_2n()  `):
fi:
if nops([args])=1 and op(1,[args])=sumall_1n then
print(`sumall_1n(): gives the total weights for the chains of L(1,n)`):
print(`example: sumall_1n()  `):
fi:
end:









###################################################################
###################################################################
Sum_of_one_Segment:=proc(n,arr1,arr2,n_lowerbound,n_upperbound,num_pars)local s,y,size,a,b,r_upper,z,c;
 size:=nops(arr1);
 a:=[n-arr1[size],seq((arr1[size-t]-arr1[size-t-1]),t=0..size-2),arr1[1]];
 b:=[n-arr2[size],seq((arr2[size-t]-arr2[size-t-1]),t=0..size-2),arr2[1]];
 for s from 1 to size+1 do
 if (a[s]-b[s]<>0)then r_upper:=a[s]-b[s];
 fi;od;
 for s from 1 to size+1 do
 if (a[s]-b[s]<>0)then z[s]:=r*(a[s]-b[s])/(r_upper);
 else z[s]:=0;
 fi;
 od;
 for s from 1 to size+1 do
 c[s]:=b[s]+z[s];
 od;
 if num_pars=4 then normal(sum(sum(sum(sum(sum(sum(expression(c,size+1),r=0..infinity),n=n_lowerbound..n_upperbound),j=0..infinity),i=0..infinity),k=0..infinity),l=0..infinity)-sum(sum(sum(sum(sum(sum(expression(c,size+1),r=r_upper..infinity),n=n_lowerbound..n_upperbound),j=0..infinity),i=0..infinity),k=0..infinity),l=0..infinity));
 elif num_pars=3 then
 normal(sum(sum(sum(sum(sum(expression(c,size+1),r=0..infinity),n=n_lowerbound..n_upperbound),j=0..infinity),i=0..infinity),k=0..infinity)-sum(sum(sum(sum(sum(expression(c,size+1),r=r_upper..infinity),n=n_lowerbound..n_upperbound),j=0..infinity),i=0..infinity),k=0..infinity));
 elif num_pars=2 then
 normal(sum(sum(sum(sum(expression(c,size+1),r=0..infinity),n=n_lowerbound..n_upperbound),j=0..infinity),i=0..infinity)-sum(sum(sum(sum(expression(c,size+1),r=r_upper..infinity),n=n_lowerbound..n_upperbound),j=0..infinity),i=0..infinity));
 elif num_pars=1 then
 normal(sum(sum(sum(expression(c,size+1),r=0..infinity),n=n_lowerbound..n_upperbound),i=0..infinity)-sum(sum(sum(expression(c,size+1),r=r_upper..infinity),n=n_lowerbound..n_upperbound),i=0..infinity)); 
 elif num_pars=0 then
 normal(sum(sum(expression(c,size+1),r=0..infinity),n=n_lowerbound..n_upperbound)-sum(sum(expression(c,size+1),r=r_upper..infinity),n=n_lowerbound..n_upperbound));
 fi:
 end:
 ###################################################################
###################################################################
 expression:=proc(k,size)
 local t,s;
 t:=1;
 for s from 0 to size-1 do
 t:=t*x[s]^k[s+1];
 od;
 t;
 end:
 
 ###################################################################
###################################################################
 Sum_of_one_chain:=proc(n,arr,n_lower_bound,n_upper_bound,num)
 local s,t,y,z,result;
 result:=0;
 t:=arr[1];t[1]:=t[1]+1;
 result:=result+Sum_of_one_Segment(n,t,arr[1],n_lower_bound,n_upper_bound,num);
 for s from 1 to nops(arr)-1 do
 result:=result+Sum_of_one_Segment(n,arr[s],arr[s+1],n_lower_bound,n_upper_bound,num);
 od;
 normal(result);
 end:
 
###################################################################
###################################################################  
sumall_4n_1:=proc()
local a,b:
 a:=Sum_of_one_chain(n,[
                 [n-2*k-2*j-i-1,n-2*k-j-i-1,n-k-j-1,n-k],
                 [n-2*k-2*j-i-1,n-2*k-j-i-1,n-k-i-j-1,n-k],
                 [n-2*k-2*j-i-1,n-2*k-j-i-1,n-k-i-j-1,n-k-i-1],
                 [k,          n-2*k-j-i-1,n-k-i-j-1,n-k-i-1],
                 [k,          k+j,      n-k-i-j-1,n-k-i-1],
                 [k,          k+j,      2*k+j+i+1,n-k-i-1], 
                 [k,k+j,2*k+i+j+1,2*k+2*j+i+2]],2*i+2*j+3*k+3,infinity,3):
 
 b:=Sum_of_one_chain(n,[
                     [n-2*k-2*j-i,n-2*k-j-i,n-k-j,n-k],
                     [k+i,n-2*k-j-i,n-k-j,n-k],
                     [k+i,k+j+i,n-k-j,n-k],
                     [k+i,k+j+i,2*k+j+i,n-k] ,
                     [k+i,k+j+i,2*k+j+i,2*k+2*j+i],
                     [k,k+j+i,2*k+j+i,2*k+2*j+i], 
                     [k,k+j,2*k+i+j,2*k+2*j+i]],2*i+2*j+3*k,infinity,3):
 normal(a+b);
end:
###################################################################
###################################################################
sumall_4n_2:=proc()
local a,b:
 a:=Sum_of_one_chain(n,[
                 [n-2*k-2*j-i,n-2*k-j-i,n-k-j,n-k],
                 [n-2*k-2*j-i,n-2*k-j-i,n-k-i-j,n-k],
                 [n-2*k-2*j-i,n-2*k-j-i,n-k-i-j,n-k-i],
                 [k,          n-2*k-j-i,n-k-i-j,n-k-i],
                 [k,          k+j,      n-k-i-j,n-k-i],
                 [k,          k+j,      2*k+j+i,n-k-i], 
                 [k,k+j,2*k+i+j,2*k+2*j+i]],2*i+2*j+3*k,infinity,3): 
 b:=Sum_of_one_chain(n,[
                     [n-2*k-2*j-i-2,n-2*k-j-i-1,n-k-j,n-k],
                     [k+i+1,n-2*k-j-i-1,n-k-j,n-k],
                     [k+i+1,k+j+i+1,n-k-j,n-k],
                     [k+i+1,k+j+i+1,2*k+j+i+1,n-k] ,
                     [k+i+1,k+j+i+1,2*k+j+i+1,2*k+2*j+i+1],
                     [k,k+j+i+1,2*k+j+i+1,2*k+2*j+i+1], 
                     [k,k+j+1,2*k+i+j+1,2*k+2*j+i+1]],2*i+2*j+3*k+3,infinity,3):
 normal(a+b);
end:
###################################################################
###################################################################
sumall_3n_1:=proc()
local a,b:
 a:=Sum_of_one_chain(n,[[n-i-3*j,n-i-2*j,n-j],
                     [i+j,n-i-2*j,n-j] ,
                     [i+j,i+2*j,n-j],
                     [i+j,i+2*j,i+3*j], 
                     [j,i+2*j,i+3*j]],2*i+4*j,infinity,2):
 b:=Sum_of_one_chain(n,[[n-i-3*j-1,n-i-2*j-1,n-j-1],
                     [n-i-3*j-1,n-i-2*j-1,n-i-j-1] ,
                     [j,n-i-2*j-1,n-i-j-1],
                     [j,i+2*j+1,n-i-j-1], 
                     [j,i+2*j+1,i+3*j+2]], 2*i+4*j+3,infinity,2):
 normal(a+b);
end:
###################################################################
###################################################################
sumall_3n_2:=proc()
local a,b:
 a:=Sum_of_one_chain(n,[[n-i-3*j-2,n-i-2*j-1,n-j],
                     [i+j+1,n-i-2*j-1,n-j] ,
                     [i+j+1,i+2*j+1,n-j],
                     [i+j+1,i+2*j+1,i+3*j+1], 
                     [j+1,i+2*j+1,i+3*j+1]],2*i+4*j+3,infinity,2):
 b:=Sum_of_one_chain(n,[[n-i-3*j,n-i-2*j,n-j],
                     [n-i-3*j,n-i-2*j,n-i-j] ,
                     [j,n-i-2*j,n-i-j],
                     [j,i+2*j,n-i-j], 
                     [j,i+2*j,i+3*j]], 2*i+4*j,infinity,2):
 normal(a+b);
end:  
###################################################################
###################################################################
sumall_2n:=proc()  
local a:
a:=Sum_of_one_chain(n,[
                     [n-i,n-i],
                     [i,n-i], 
                     [i,i]],2*i,infinity,1);
end:
###################################################################
###################################################################
sumall_1n:=proc()
local a:
a:=Sum_of_one_chain(n,[[n], 
                     [0]],0,infinity,0);
end:
###################################################################
###################################################################
sumall_west :=proc()
local a,b;
 a:=Sum_of_one_chain(n,[[n-3*i-j-2,n-2*i-j-1,n-i,n],
                     [j+1,n-2*i-j-1,n-i,n] ,
                     [j+1,i+j+1,n-i,n],
                     [j+1,i+j+1,2*i+j+1,n], 
                     [0,i+j+1,2*i+j+1,n], 
                     [0,i+j+1,2*i+j+1,3*i+j+1],
                     [0,i+1,2*i+j+1,3*i+j+1]],3*i+2*j+3,infinity,2):
 b:=Sum_of_one_chain(n,[[n-3*i-j,n-2*i-j,n-i,n],
                     [n-3*i-j,n-2*i-j,n-i-j,n] ,
                     [0,n-2*i-j,n-i-j,n],
                     [0,n-2*i-j,n-i-j,n-j], 
                     [0,i,n-i-j,n-j], 
                     [0,i,2*i+j,n-j],
                     [0,i,2*i+j,3*i+j]],3*i+2*j,infinity,2):
 normal((a+b)/(1-x[0]*x[4]));
end:
###################################################################
###################################################################
############################################################ 
 Sum_of_one_Segment_lind:=proc(n,arr1,arr2,init_n,init_i)
 local s,y,size,a,b,r_upper,z,c,expre;
 size:=nops(arr1);
 a:=[n-arr1[size],seq((arr1[size-t]-arr1[size-t-1]),t=0..size-2),arr1[1]];
 b:=[n-arr2[size],seq((arr2[size-t]-arr2[size-t-1]),t=0..size-2),arr2[1]];
 for s from 1 to size+1 do
 if (a[s]-b[s]<>0)then r_upper:=a[s]-b[s];
 fi;od;
 for s from 1 to size+1 do
 if (a[s]-b[s]<>0)then z[s]:=normal(r*(a[s]-b[s])/(r_upper));
 else z[s]:=0;
 fi;
 od;
 for s from 1 to size+1 do
 c[s]:=b[s]+z[s];
 ;od;
 expre:=subs(n=2*k+init_n,expression(c,size+1));
 r_upper:=subs(n=2*k+init_n,r_upper);
if(init_i<>infinity)
then 
normal(sum(sum(sum(expre,r=0..infinity),k=i+1-init_n..infinity),i=init_i..infinity)-sum(sum(sum(expre,r=r_upper..infinity),k=i+1-init_n..infinity),i=init_i..infinity));
else 
normal(sum(sum(expre,r=0..infinity),k=0..infinity)-sum(sum(expre,r=r_upper..infinity),k=0..infinity));
 fi;
 end:
 ###########################################################
 Sum_of_one_chain_lind:=proc(n,arr,init_n,init_i)
 local s,t,y,z,result;
 result:=0;
 t:=arr[1];t[1]:=t[1]+1;
 result:=result+Sum_of_one_Segment_lind(n,t,arr[1],init_n,init_i);
 for s from 1 to nops(arr)-1 do
 result:=result+Sum_of_one_Segment_lind(n,arr[s],arr[s+1],init_n,init_i);
 od;
 normal(result);
 end:
############################################################## 
sumall_lind:=proc()
local oc124,oc35,ec1245,ec36,ed124,ed35,ec;
oc124:=Sum_of_one_chain_lind(n,[[n-2*i,n,n],[1,2*i+1,n],[0,2*i,n],[0,i+1,n-i+1],[0,i,n-i],[0,i,i]],1,0);
oc35:=Sum_of_one_chain_lind(n,[[n-2*i-1,n,n],[0,2*i+1,n],[0,2*i,n-1],[0,i+1,n-i]],1,0);
ec1245:=Sum_of_one_chain_lind(n,[[n-2*i-1,n,n],[1,2*i+2,n],[1,2*i+1,n-1],[0,2*i,n-1],[0,i,n-i-1],[0,i,n-i-2],[0,i,i]],0,0);
ec36:=Sum_of_one_chain_lind(n,[[n-2*i,n,n],[1,2*i+1,n],[0,2*i+1,n-1],[0,i+1,n-i-1]],0,0);
ed124:=Sum_of_one_chain_lind(n,[[n-2*i,n-1,n-1],[1,2*i,n-1],[1,2*i-1,n-2],[1,i,n-i-1],[1,i,n-i-2],[1,i,i]],0,1);
ed35:=Sum_of_one_chain_lind(n,[[n-2*i+1,n-1,n-1],[2,2*i,n-1],[1,2*i,n-2],[1,i+1,n-i-1]],0,1);
ec:=Sum_of_one_chain_lind(n,[[0,n,n],[0,0,n]],0,infinity);
normal((oc124+oc35)/(1-x[0]*x[3])+(ec+ec1245+ec36+ed124+ed35)/(1-x[0]^2*x[3]^2));
end:
###############################################################