A symmetric chain decomposition of L(5,n)
A symmetric chain decomposition of L(5,n)
Nine parallel chains of L(5,n)
chain1
chain1
In[]:=
chain1[n_,i_,k_,j_,p_]:=Join[Table[{p,k,j+k+t,1+i+j+2k,1+i+2j+2k},{t,0,1+i+p}],Table[{p,k+t,1+i+j+k+p,1+i+j+2k,1+i+2j+2k},{t,1,1+i+j}],Table[{p+t,1+i+j+k,1+i+j+k+p,1+i+j+2k,1+i+2j+2k},{t,1,1+i+k-p}],Table[{1+i+k,1+i+j+k,1+i+j+k+p,1+i+j+2k,1+i+2j+2k+t},{t,1,-1-i-2j-3k+n+p}],Table[{1+i+k,1+i+j+k,1+i+j+k+p,1+i+j+2k+t,-k+n+p},{t,1,-1-i-j-3k+n}],Table[{1+i+k,1+i+j+k,1+i+j+k+p+t,-k+n,-k+n+p},{t,1,-1-i-2j-2k+n-p}],Table[{1+i+k,1+i+j+k+t,-j-k+n,-k+n,-k+n+p},{t,1,k-2(1+i+j+2k)+n}],Table[{1+i+k+t,-1-i-j-2k+n,-j-k+n,-k+n,-k+n+p},{t,1,k-2(1+i+j+2k)+n}]];parallelChain1[n_,i_,k_,j_]:=Table[chain1[n,i,k,j,p],{p,0,k}];allParallelChain1[n_]:=Flatten[Table[parallelChain1[n,i,k,j],{i,0,Floor[(n-2)/2]},{j,0,Floor[(n-2-2i)/2]},{k,0,Floor[(n-2-2i-2j)/3]}],2];
chain2
chain2
In[]:=
chain2[n_,i_,k_,j_,p_]:=Join[Table[{k-p,k,j+k,1+i+j+2k,1+i+2j+2k+t},{t,1,k-2(1+i+j+2k)+n}],Table[{k-p,k,j+k,1+i+j+2k+t,-1-i-k+n},{t,1,k-2(1+i+j+2k)+n}],Table[{k-p,k,j+k+t,-1-i-j-k+n,-1-i-k+n},{t,1,-1-i-2j-2k+n-p}],Table[{k-p,k+t,-1-i-j-k+n-p,-1-i-j-k+n,-1-i-k+n},{t,1,-1-i-j-3k+n}],Table[{k-p+t,-1-i-j-2k+n,-1-i-j-k+n-p,-1-i-j-k+n,-1-i-k+n},{t,1,-1-i-2j-3k+n+p}],Table[{-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},{t,1,1+i+k-p}],Table[{-1-i-2j-2k+n,-1-i-j-2k+n,-1-i-j-k+n-p,-1-i-j-k+n+t,n-p},{t,1,1+i+j}],Table[{-1-i-2j-2k+n,-1-i-j-2k+n,-1-i-j-k+n-p+t,-k+n,n-p},{t,1,i+p}]];parallelChain2[n_,i_,k_,j_]:=(Table[chain2[n,i,k,j,p],{p,k,0,-1}]);allParallelChain2[n_]:=Flatten[Table[parallelChain2[n,i,k,j],{i,0,Floor[(n-3)/2]},{j,0,Floor[(n-3-2i)/2]},{k,0,Floor[(n-3-2i-2j)/3]}],2];
chain3
chain3
In[]:=
chain3[n_,i_,j_,u_,k_,p_]:=Join[{{2j,i+2j+p,i+2j+2k+u,2i+4j+2k+u,2i+4j+4k+2u}},Apply[Join,Table[{{2j+t,i+2j+p,-1+i+2j+2k+t+u,2i+4j+2k+u,2i+4j+4k+2u},{2j+t,i+2j+p,i+2j+2k+t+u,2i+4j+2k+u,2i+4j+4k+2u}},{t,1,i}]],Table[{i+2j,i+2j+p,2i+2j+2k+u,2i+4j+2k+u,2i+4j+4k+t+2u},{t,1,-2i-6j-4k+n-2u}],Table[{i+2j,i+2j+p,2i+2j+2k+u,2i+4j+2k+t+u,-2j+n},{t,1,-3i-6j-4k+n+p-2u}],Table[{i+2j,i+2j+p,2i+2j+2k+t+u,-i-2j-2k+n+p-u,-2j+n},{t,1,-3i-4j-4k+n-2u}],Table[{i+2j,i+2j+p+t,-i-2j-2k+n-u,-i-2j-2k+n+p-u,-2j+n},{t,1,-3i-6j-2k+n-p-u}],Table[{i+2j+t,-2i-4j-2k+n-u,-i-2j-2k+n-u,-i-2j-2k+n+p-u,-2j+n},{t,1,-3i-6j-4k+n-2u}]];parallelChain3[n_,i_,j_,u_,k_]:=Table[chain3[n,i,j,u,k,p],{p,0,2k+u}];allParallelChain3[n_]:=Flatten[Table[parallelChain3[n,i,j,u,k],{u,0,1},{j,0,Floor[(n-2u-1)/6]},{k,0,Floor[(n-2u-6j-1)/4]},{i,0,Floor[(n-2u-1-6j-4k)/3]}],3];
chain4
chain4
In[]:=
chain4[n_,i_,j_,u_,k_,p_]:=Join[Apply[Join,Table[{{2j+t,1+i+2j+p,i+2j+2k+t+u,2+2i+4j+2k+u,3+2i+4j+4k+2u},{2j+t,1+i+2j+p,1+i+2j+2k+t+u,2+2i+4j+2k+u,3+2i+4j+4k+2u}},{t,1,i}]],{{1+i+2j,1+i+2j+p,1+2i+2j+2k+u,2+2i+4j+2k+u,3+2i+4j+4k+2u}},Table[{1+i+2j,1+i+2j+p,1+2i+2j+2k+u,2+2i+4j+2k+u,3+2i+4j+4k+t+2u},{t,1,-4-2i-6j-4k+n-2u}],Table[{1+i+2j,1+i+2j+p,1+2i+2j+2k+u,2+2i+4j+2k+t+u,-1-2j+n},{t,1,-3-3i-6j-4k+n+p-2u}],Table[{1+i+2j,1+i+2j+p,1+2i+2j+2k+t+u,-1-i-2j-2k+n+p-u,-1-2j+n},{t,1,-2-3i-4j-4k+n-2u}],Table[{1+i+2j,1+i+2j+p+t,-1-i-2j-2k+n-u,-1-i-2j-2k+n+p-u,-1-2j+n},{t,1,-3-3i-6j-2k+n-p-u}],Table[{1+i+2j+t,-2-2i-4j-2k+n-u,-1-i-2j-2k+n-u,-1-i-2j-2k+n+p-u,-1-2j+n},{t,1,-4-3i-6j-4k+n-2u}]];parallelChain4[n_,i_,j_,u_,k_]:=Table[chain4[n,i,j,u,k,p],{p,0,2k+u}];allParallelChain4[n_]:=Flatten[Table[parallelChain4[n,i,j,u,k],{u,0,1},{j,0,Floor[(n-2u-4)/6]},{k,0,Floor[(n-2u-6j-4)/4]},{i,0,Floor[(n-2u-4-6j-4k)/3]}],3];
chain5
chain5
In[]:=
chain5[n_,i_,j_,u_,k_,p_]:=Join[Table[{2j,1+i+2j+p,1+i+2j+2k+u,2+2i+4j+2k+u,2+2i+4j+4k+t+2u},{t,1,-3-3i-6j-4k+n-2u}],Table[{2j,1+i+2j+p,1+i+2j+2k+u,2+2i+4j+2k+t+u,-1-i-2j+n},{t,1,-3-3i-6j-4k+n+p-2u}],Table[{2j,1+i+2j+p,1+i+2j+2k+t+u,-1-i-2j-2k+n+p-u,-1-i-2j+n},{t,1,-3-3i-4j-4k+n-2u}],Table[{2j,1+i+2j+p+t,-2-2i-2j-2k+n-u,-1-i-2j-2k+n+p-u,-1-i-2j+n},{t,1,-3-3i-6j-2k+n-p-u}],Table[{2j+t,-2-2i-4j-2k+n-u,-2-2i-2j-2k+n-u,-1-i-2j-2k+n+p-u,-1-i-2j+n},{t,1,-2-2i-6j-4k+n-2u}],Apply[Join,Table[{{-2-2i-4j-4k+n-2u,-2-2i-4j-2k+n-u,-2-2i-2j-2k+n+t-u,-1-i-2j-2k+n+p-u,-2-i-2j+n+t},{-2-2i-4j-4k+n-2u,-2-2i-4j-2k+n-u,-2-2i-2j-2k+n+t-u,-1-i-2j-2k+n+p-u,-1-i-2j+n+t}},{t,1,i}]],{{-2-2i-4j-4k+n-2u,-2-2i-4j-2k+n-u,-1-i-2j-2k+n-u,-1-i-2j-2k+n+p-u,-1-2j+n}}];parallelChain5[n_,i_,j_,u_,k_]:=Table[chain5[n,i,j,u,k,p],{p,0,2k+u}];allParallelChain5[n_]:=Flatten[Table[parallelChain5[n,i,j,u,k],{u,0,1},{j,0,Floor[(n-2u-4)/6]},{k,0,Floor[(n-2u-6j-4)/4]},{i,0,Floor[(n-2u-4-6j-4k)/3]}],3];
chain6
chain6
In[]:=
chain6[n_,i_,j_,u_,k_,p_]:=Join[Table[{1+2j,2+i+2j+p,2+i+2j+2k+u,4+2i+4j+2k+u,5+2i+4j+4k+t+2u},{t,1,-7-3i-6j-4k+n-2u}],Table[{1+2j,2+i+2j+p,2+i+2j+2k+u,4+2i+4j+2k+t+u,-2-i-2j+n},{t,1,-6-3i-6j-4k+n+p-2u}],Table[{1+2j,2+i+2j+p,2+i+2j+2k+t+u,-2-i-2j-2k+n+p-u,-2-i-2j+n},{t,1,-5-3i-4j-4k+n-2u}],Table[{1+2j,2+i+2j+p+t,-3-2i-2j-2k+n-u,-2-i-2j-2k+n+p-u,-2-i-2j+n},{t,1,-6-3i-6j-2k+n-p-u}],Table[{1+2j+t,-4-2i-4j-2k+n-u,-3-2i-2j-2k+n-u,-2-i-2j-2k+n+p-u,-2-i-2j+n},{t,1,-6-2i-6j-4k+n-2u}],Apply[Join,Table[{{-5-2i-4j-4k+n-2u,-4-2i-4j-2k+n-u,-4-2i-2j-2k+n+t-u,-2-i-2j-2k+n+p-u,-2-i-2j+n+t},{-5-2i-4j-4k+n-2u,-4-2i-4j-2k+n-u,-3-2i-2j-2k+n+t-u,-2-i-2j-2k+n+p-u,-2-i-2j+n+t}},{t,1,i}]],{{-5-2i-4j-4k+n-2u,-4-2i-4j-2k+n-u,-3-i-2j-2k+n-u,-2-i-2j-2k+n+p-u,-1-2j+n}}];parallelChain6[n_,i_,j_,u_,k_]:=Table[chain6[n,i,j,u,k,p],{p,0,2k+u}];allParallelChain6[n_]:=Flatten[Table[parallelChain6[n,i,j,u,k],{u,0,1},{j,0,Floor[(n-2u-7)/6]},{k,0,Floor[(n-2u-6j-7)/4]},{i,0,Floor[(n-2u-7-6j-4k)/3]}],3];
chain7
chain7
In[]:=
chain7[u_,w_,i_,k_,p_,q_]:=Join[Table[{1+i+t,2+2i+p+u+2w,3+3i+k+u+2w,4+4i+k+q+2u+4w,4+4i+2k+2u+4w},{t,0,u+2w}],Table[{1+i+u+2w,2+2i+p+u+2w,3+3i+k+t+u+2w,4+4i+k+q+2u+4w,4+4i+2k+2u+4w},{t,1,u+2w}],{{1+i+u+2w,2+2i+p+u+2w,3+3i+k+2u+4w,4+4i+k+q+2u+4w,5+4i+2k+2u+4w}},Table[{1+i+t+u+2w,2+2i+p+u+2w,3+3i+k+2u+4w,4+4i+k+q+2u+4w,5+4i+2k+2u+4w},{t,1,i}],Table[{1+2i+u+2w,2+2i+p+u+2w,3+3i+k+2u+4w,4+4i+k+q+2u+4w,5+4i+2k+t+2u+4w},{t,1,i+u+2w}],{{2+2i+u+2w,2+2i+p+u+2w,3+3i+k+2u+4w,4+4i+k+q+2u+4w,5+5i+2k+3u+6w}}];parallelChain7[u_,w_,i_,k_]:=Table[Join[Table[chain7[u,w,i,k,p+t,p],{t,0,k-2p}],Table[chain7[u,w,i,k,k-p,p+t],{t,1,k-2p}]],{p,0,Floor[k/2]}];allParallelChain7[n_]:=Flatten[Table[parallelChain7[Mod[n,2],w,i,Floor[(n-6-3Mod[n,2]-6w-6i)/2]],{w,0,Floor[(n-6-3Mod[n,2])/6]},{i,0,Floor[(n-6-3Mod[n,2]-6w)/6]}],2];
chain8
chain8
In[]:=
chain8[u_,w_,i_,k_,p_,q_]:=Join[Table[{1+i+t,4+2i+p-u+2w,5+3i+k-u+2w,8+4i+k+q-2u+4w,9+4i+2k-2u+4w},{t,0,2+i-u+2w}],Table[{3+2i-u+2w,4+2i+p-u+2w,5+3i+k-u+2w,8+4i+k+q-2u+4w,9+4i+2k+t-2u+4w},{t,1,i}],{{4+2i-u+2w,4+2i+p-u+2w,5+3i+k-u+2w,8+4i+k+q-2u+4w,9+5i+2k-2u+4w}},Table[{4+2i-u+2w,4+2i+p-u+2w,5+3i+k+t-u+2w,8+4i+k+q-2u+4w,9+5i+2k-2u+4w},{t,1,2-u+2w}],Table[{4+2i-u+2w,4+2i+p-u+2w,7+3i+k-2u+4w,8+4i+k+q-2u+4w,9+5i+2k+t-2u+4w},{t,1,1-u+2w}]];parallelChain8[u_,w_,i_,k_]:=Table[Join[Table[chain8[u,w,i,k,p+t,p],{t,0,k-2p}],Table[chain8[u,w,i,k,k-p,p+t],{t,1,k-2p}]],{p,0,Floor[k/2]}];allParallelChain8[n_]:=Flatten[Table[parallelChain8[Mod[n,2],w,i,Floor[(n-12+3Mod[n,2]-6w-6i)/2]],{w,0,Floor[(n-12+3Mod[n,2])/6]},{i,0,Floor[(n-12+3Mod[n,2]-6w)/6]}],2];
chain9
chain9
All paralle chains
All paralle chains
Convert parallel chains to symmetric chains
Parallel chains form rectangles. The perimeters of a rectangle contains up to 2 symmetric chains(from lowest coordinated corner to the highest coordinated corner). The corner points are in the same chain and they could be in either of the border chains. All symmetric chains are obtained by taking perimeters off rectangles recursively.
Symmetric Chain Decomposition of L(5,n)
SCDL5n[n] gives a sysmmetric chains decomposition of L(5,n)
SCDL5n[n] gives a sysmmetric chains decomposition of L(5,n)
abbrev version
abbrev version