Proof

weight of vector

In[]:=
weight[{a1_,a2_,a3_,a4_,a5_},n_]:=(x0)^(n-a5)(x1)^(a5-a4)(x2)^(a4-a3)(x3)^(a3-a2)(x4)^(a2-a1)(x5)^a1

sumAllChain1Weight

In[]:=
sumChain1Weight1[n_,i_,k_,j_,p_]:=​​Sum[weight[{p,k,j+k+t,1+i+j+2k,1+i+2j+2k},n],{t,0,1+i+p}];sumChain1Weight2[n_,i_,k_,j_,p_]:=Sum[weight[{p,k+t,1+i+j+k+p,1+i+j+2k,1+i+2j+2k},n],{t,1,1+i+j}];sumChain1Weight3[n_,i_,k_,j_,p_]:=Sum[weight[{p+t,1+i+j+k,1+i+j+k+p,1+i+j+2k,1+i+2j+2k},n],{t,1,1+i+k-p}];sumChain1Weight4[n_,i_,k_,j_,p_]:=Sum[weight[{1+i+k,1+i+j+k,1+i+j+k+p,1+i+j+2k,1+i+2j+2k+t},n],{t,1,-1-i-2j-3k+n+p}];​​sumChain1Weight5[n_,i_,k_,j_,p_]:=Sum[weight[{1+i+k,1+i+j+k,1+i+j+k+p,1+i+j+2k+t,-k+n+p},n],{t,1,-1-i-j-3k+n}];sumChain1Weight6[n_,i_,k_,j_,p_]:=Sum[weight[{1+i+k,1+i+j+k,1+i+j+k+p+t,-k+n,-k+n+p},n],{t,1,-1-i-2j-2k+n-p}];sumChain1Weight7[n_,i_,k_,j_,p_]:=Sum[weight[{1+i+k,1+i+j+k+t,-j-k+n,-k+n,-k+n+p},n],{t,1,k-2(1+i+j+2k)+n}];sumChain1Weight8[n_,i_,k_,j_,p_]:=Sum[weight[{1+i+k+t,-1-i-j-2k+n,-j-k+n,-k+n,-k+n+p},n],{t,1,k-2(1+i+j+2k)+n}];
In[]:=
sumAllChain1Weight1=Sum[Sum[Sum[Sum[Sum[sumChain1Weight1[n,i,k+p,j,p],{n,2+2i+2j+3k+3p,Infinity}],{p,0,Infinity}],{k,0,Infinity}],{i,0,Infinity}],{j,0,Infinity}];
In[]:=
sumAllChain1Weight2=Sum[Sum[Sum[Sum[Sum[sumChain1Weight2[n,i,k+p,j,p],{n,2+2i+2j+3k+3p,Infinity}],{p,0,Infinity}],{k,0,Infinity}],{i,0,Infinity}],{j,0,Infinity}];
In[]:=
sumAllChain1Weight3=Sum[Sum[Sum[Sum[Sum[sumChain1Weight3[n,i,k+p,j,p],{n,2+2i+2j+3k+3p,Infinity}],{p,0,Infinity}],{k,0,Infinity}],{i,0,Infinity}],{j,0,Infinity}];
In[]:=
sumAllChain1Weight4=Sum[Sum[Sum[Sum[Sum[sumChain1Weight4[n,i,k+p,j,p],{n,2+2i+2j+3k+3p,Infinity}],{p,0,Infinity}],{k,0,Infinity}],{i,0,Infinity}],{j,0,Infinity}];
In[]:=
sumAllChain1Weight5=Sum[Sum[Sum[Sum[Sum[sumChain1Weight5[n,i,k+p,j,p],{n,2+2i+2j+3k+3p,Infinity}],{p,0,Infinity}],{k,0,Infinity}],{i,0,Infinity}],{j,0,Infinity}];
In[]:=
sumAllChain1Weight6=Sum[Sum[Sum[Sum[Sum[sumChain1Weight6[n,i,k+p,j,p],{n,2+2i+2j+3k+3p,Infinity}],{p,0,Infinity}],{k,0,Infinity}],{i,0,Infinity}],{j,0,Infinity}];
In[]:=
sumAllChain1Weight7=Sum[Sum[Sum[Sum[Sum[sumChain1Weight7[n,i,k+p,j,p],{n,2+2i+2j+3k+3p,Infinity}],{p,0,Infinity}],{k,0,Infinity}],{i,0,Infinity}],{j,0,Infinity}];
In[]:=
sumAllChain1Weight8=Sum[Sum[Sum[Sum[Sum[sumChain1Weight8[n,i,k+p,j,p],{n,2+2i+2j+3k+3p,Infinity}],{p,0,Infinity}],{k,0,Infinity}],{i,0,Infinity}],{j,0,Infinity}];
In[]:=
sumAllChain1Weight=Simplify[sumAllChain1Weight1+sumAllChain1Weight2+sumAllChain1Weight3+sumAllChain1Weight4+sumAllChain1Weight5+sumAllChain1Weight6+sumAllChain1Weight7+sumAllChain1Weight8]
Out[]=
x0x4(-1+x0x1x3x4)
(-1+x0)(-1+x0x3)(-1+x1x3)(-1+x0x4)(-1+x1x4)(-1+x0x2x4)(-1+x0x3x5)
+
x3x4x5
(-1+x3)(-1+x4)(-1+x2x4)(-1+x3x5)(-1+x0x3x5)(-1+x1x3x5)
+
x3
2
x5
(-1+x4)(-1+x2x4)(-1+x5)(-1+x3x5)(-1+x0x3x5)(-1+x1x3x5)
+
x0x5(-1+
2
x0
x2x4x5)
(-1+x0)(-1+x0x4)(-1+x1x4)(-1+x0x2x4)(-1+x0x5)(-1+x0x2x5)(-1+x0x3x5)
+
x0(-x3+x2(-1+x0x3(1+x5)))
(-1+x0)(-1+x0x2)(-1+x0x3)(-1+x1x3)(-1+x0x2x4)(-1+x0x2x5)(-1+x0x3x5)
+
x1x5(1+
2
x0
x1x3
2
x5
+x0x1x5(-1+x3(-1+(-1+x1)x5)))
(-1+x0)(-1+x1)(-1+x1x4)(-1+x0x5)(-1+x1x5)(-1+x0x2x5)(-1+x0x3x5)(-1+x1x3x5)
+
x2x5(1+
2
x1
x2x4x5+x1x2(-x5+x4(-1+(-1+x2)x5)))
(-1+x1)(-1+x2)(-1+x1x4)(-1+x2x4)(-1+x1x5)(-1+x2x5)(-1+x0x2x5)(-1+x1x3x5)
+
x3x5(1+x0
2
x2
x3
2
x5
+x2x3x5(-1+x0(-1+(-1+x3)x5)))
(-1+x2)(-1+x3)(-1+x2x4)(-1+x2x5)(-1+x0x2x5)(-1+x3x5)(-1+x0x3x5)(-1+x1x3x5)

sumAllChain2Weight

In[]:=
sumChain2Weight1[n_,i_,k_,j_,p_]:=Sum[weight[{k-p,k,j+k,1+i+j+2k,1+i+2j+2k+t},n],{t,1,k-2(1+i+j+2k)+n}];sumChain2Weight2[n_,i_,k_,j_,p_]:=Sum[weight[{k-p,k,j+k,1+i+j+2k+t,-1-i-k+n},n],{t,1,k-2(1+i+j+2k)+n}];sumChain2Weight3[n_,i_,k_,j_,p_]:=Sum[weight[{k-p,k,j+k+t,-1-i-j-k+n,-1-i-k+n},n],{t,1,-1-i-2j-2k+n-p}];sumChain2Weight4[n_,i_,k_,j_,p_]:=Sum[weight[{k-p,k+t,-1-i-j-k+n-p,-1-i-j-k+n,-1-i-k+n},n],{t,1,-1-i-j-3k+n}];sumChain2Weight5[n_,i_,k_,j_,p_]:=Sum[weight[{k-p+t,-1-i-j-2k+n,-1-i-j-k+n-p,-1-i-j-k+n,-1-i-k+n},n],{t,1,-1-i-2j-3k+n+p}];sumChain2Weight6[n_,i_,k_,j_,p_]:=Sum[weight[{-1-i-2j-2k+n,-1-i-j-2k+n,-1-i-j-k+n-p,-1-i-j-k+n,-1-i-k+n+t},n],{t,1,1+i+k-p}];sumChain2Weight7[n_,i_,k_,j_,p_]:=Sum[weight[{-1-i-2j-2k+n,-1-i-j-2k+n,-1-i-j-k+n-p,-1-i-j-k+n+t,n-p},n],{t,1,1+i+j}];sumChain2Weight8[n_,i_,k_,j_,p_]:=Sum[weight[{-1-i-2j-2k+n,-1-i-j-2k+n,-1-i-j-k+n-p+t,-k+n,n-p},n],{t,1,i+p}];
In[]:=
sumAllChain2Weight1=Sum[Sum[Sum[Sum[Sum[sumChain2Weight1[n,i,k+p,j,p],{n,3+2i+2j+3k+3p,Infinity}],{p,0,Infinity}],{k,0,Infinity}],{i,0,Infinity}],{j,0,Infinity}];
In[]:=
sumAllChain2Weight2=Sum[Sum[Sum[Sum[Sum[sumChain2Weight2[n,i,k+p,j,p],{n,3+2i+2j+3k+3p,Infinity}],{p,0,Infinity}],{k,0,Infinity}],{i,0,Infinity}],{j,0,Infinity}];
In[]:=
sumAllChain2Weight3=Sum[Sum[Sum[Sum[Sum[sumChain2Weight3[n,i,k+p,j,p],{n,3+2i+2j+3k+3p,Infinity}],{p,0,Infinity}],{k,0,Infinity}],{i,0,Infinity}],{j,0,Infinity}];
In[]:=
sumAllChain2Weight4=Sum[Sum[Sum[Sum[Sum[sumChain2Weight4[n,i,k+p,j,p],{n,3+2i+2j+3k+3p,Infinity}],{p,0,Infinity}],{k,0,Infinity}],{i,0,Infinity}],{j,0,Infinity}];
In[]:=
sumAllChain2Weight5=Sum[Sum[Sum[Sum[Sum[sumChain2Weight5[n,i,k+p,j,p],{n,3+2i+2j+3k+3p,Infinity}],{p,0,Infinity}],{k,0,Infinity}],{i,0,Infinity}],{j,0,Infinity}];
In[]:=
sumAllChain2Weight6=Sum[Sum[Sum[Sum[Sum[sumChain2Weight6[n,i,k+p,j,p],{n,3+2i+2j+3k+3p,Infinity}],{p,0,Infinity}],{k,0,Infinity}],{i,0,Infinity}],{j,0,Infinity}];
In[]:=
sumAllChain2Weight7=Sum[Sum[Sum[Sum[Sum[sumChain2Weight7[n,i,k+p,j,p],{n,3+2i+2j+3k+3p,Infinity}],{p,0,Infinity}],{k,0,Infinity}],{i,0,Infinity}],{j,0,Infinity}];
In[]:=
sumAllChain2Weight8=Sum[Sum[Sum[Sum[Sum[sumChain2Weight8[n,i,k+p,j,p],{n,3+2i+2j+3k+3p,Infinity}],{p,0,Infinity}],{k,0,Infinity}],{i,0,Infinity}],{j,0,Infinity}];
In[]:=
sumAllChain2Weight=Simplify[sumAllChain2Weight1+sumAllChain2Weight2+sumAllChain2Weight3+sumAllChain2Weight4+sumAllChain2Weight5+sumAllChain2Weight6+sumAllChain2Weight7+sumAllChain2Weight8]
Out[]=
x0x1x2
(-1+x0)(-1+x1)(-1+x0x2)(-1+x1x3)(-1+x0x2x4)(-1+x0x2x5)
+
x0
2
x2
(-1+x1)(-1+x2)(-1+x0x2)(-1+x1x3)(-1+x0x2x4)(-1+x0x2x5)
+
x0x4(x3+x4-(1+x0+x1)x3x4+x0x1
2
x3
2
x4
)
(-1+x3)(-1+x0x3)(-1+x1x3)(-1+x4)(-1+x0x4)(-1+x1x4)(-1+x0x2x4)(-1+x0x3x5)
+
x2
2
x5
(-1+x1x2x4x5)
(-1+x1x4)(-1+x2x4)(-1+x5)(-1+x1x5)(-1+x2x5)(-1+x0x2x5)(-1+x1x3x5)
+
x1
2
x5
(-1+x0x1x3
2
x5
)
(-1+x1x4)(-1+x5)(-1+x0x5)(-1+x1x5)(-1+x0x2x5)(-1+x0x3x5)(-1+x1x3x5)
+
x0x5(x4+x5-(1+x0+x0x2)x4x5+
2
x0
x2
2
x4
2
x5
)
(-1+x4)(-1+x0x4)(-1+x1x4)(-1+x0x2x4)(-1+x5)(-1+x0x5)(-1+x0x2x5)(-1+x0x3x5)
+
x0x3(x2+x3+
2
x0
2
x2
2
x3
x5-x2x3(1+x0+x0x5))
(-1+x2)(-1+x0x2)(-1+x3)(-1+x0x3)(-1+x1x3)(-1+x0x2x4)(-1+x0x2x5)(-1+x0x3x5)
+
x2x3
3
x5
(-1+x0(-1+x2x5+x3x5))
(-1+x2x4)(-1+x5)(-1+x2x5)(-1+x0x2x5)(-1+x3x5)(-1+x0x3x5)(-1+x1x3x5)

sumAllChain3Weight

In[]:=
sumChain3Weight1[n_,i_,j_,u_,k_,p_]:=weight[{2j,i+2j+p,i+2j+2k+u,2i+4j+2k+u,2i+4j+4k+2u},n];sumChain3Weight2[n_,i_,j_,u_,k_,p_]:=Sum[weight[{1+2j+t,i+2j+p,i+2j+2k+t+u,2i+4j+2k+u,2i+4j+4k+2u},n],{t,0,-1+i}];​​sumChain3Weight3[n_,i_,j_,u_,k_,p_]:=Sum[weight[{1+2j+t,i+2j+p,1+i+2j+2k+t+u,2i+4j+2k+u,2i+4j+4k+2u},n],{t,0,-1+i}];sumChain3Weight4[n_,i_,j_,u_,k_,p_]:=Sum[weight[{i+2j,i+2j+p,2i+2j+2k+u,2i+4j+2k+u,2i+4j+4k+t+2u},n],{t,1,-2i-6j-4k+n-2u}];sumChain3Weight5[n_,i_,j_,u_,k_,p_]:=Sum[weight[{i+2j,i+2j+p,2i+2j+2k+u,2i+4j+2k+t+u,-2j+n},n],{t,1,-3i-6j-4k+n+p-2u}];sumChain3Weight6[n_,i_,j_,u_,k_,p_]:=Sum[weight[{i+2j,i+2j+p,2i+2j+2k+t+u,-i-2j-2k+n+p-u,-2j+n},n],{t,1,-3i-4j-4k+n-2u}];sumChain3Weight7[n_,i_,j_,u_,k_,p_]:=Sum[weight[{i+2j,i+2j+p+t,-i-2j-2k+n-u,-i-2j-2k+n+p-u,-2j+n},n],{t,1,-3i-6j-2k+n-p-u}];sumChain3Weight8[n_,i_,j_,u_,k_,p_]:=Sum[weight[{i+2j+t,-2i-4j-2k+n-u,-i-2j-2k+n-u,-i-2j-2k+n+p-u,-2j+n},n],{t,1,-3i-6j-4k+n-2u}];
In[]:=
sumAllChain3Weight1=Sum[Sum[Sum[Sum[Sum[Sum[sumChain3Weight1[n,i,j,u,k,p],{p,0,2k+u}],{n,2u+1+6j+4k+3i,Infinity}],{k,0,Infinity}],{i,0,Infinity}],{j,0,Infinity}],{u,0,1}];
In[]:=
sumAllChain3Weight2=Sum[Sum[Sum[Sum[Sum[Sum[sumChain3Weight2[n,i,j,u,k,p],{p,0,2k+u}],{n,2u+1+6j+4k+3i,Infinity}],{k,0,Infinity}],{i,0,Infinity}],{j,0,Infinity}],{u,0,1}];

sumAllChain4Weight

sumAllChain5Weight

sumAllChain6Weight

sumAllChain7Weight

sumAllChain8Weight

sumAllChain9Weight

sumAllChainWeight