Standard Order Rule Counting
Standard Order Rule Counting
Initializations
Initializations
IntegerToRuleSignatures
IntegerToRuleSignatures
IntegerToRuleSignatures[int_Integer]:=SortBy[Sort[Rule@@(({Length[#],First[#]}&/@Split[#])&/@#)&/@Select[Union[Flatten[Table[{ReverseSort[Take[#,k]],ReverseSort[Drop[#,k]]},{k,1,Length[#]-1}]&/@Flatten[Permutations/@Select[IntegerPartitions[int],Min[#]≥2&&Max[Last/@Tally[#]]≥2&],1],1]],Max[Last/@Tally[#[[1]]]]+Max[Last/@Tally[#[[2]]]]>2&]],Union[Join[Last/@First[#],Last/@Last[#]]]&];
IntegerToRuleSignaturesOnes[int_Integer]:=SortBy[Sort[Rule@@(({Length[#],First[#]}&/@Split[#])&/@#)&/@Select[Union[Flatten[Table[{ReverseSort[Take[#,k]],ReverseSort[Drop[#,k]]},{k,1,Length[#]-1}]&/@Flatten[Permutations/@Select[IntegerPartitions[int],Min[#]1&&Max[Last/@Tally[#]]≥2&],1],1]],Max[Last/@Tally[#[[1]]]]+Max[Last/@Tally[#[[2]]]]>2&]],Union[Join[Last/@First[#],Last/@Last[#]]]&];
IntegerToRuleSignaturesBoring[int_Integer]:=SortBy[Sort[Rule@@(({Length[#],First[#]}&/@Split[#])&/@#)&/@Select[Union[Flatten[Table[{ReverseSort[Take[#,k]],ReverseSort[Drop[#,k]]},{k,1,Length[#]-1}]&/@Flatten[Permutations/@Select[IntegerPartitions[int],Min[#]≥2&],1],1]],Max[Last/@Tally[#[[1]]]]+Max[Last/@Tally[#[[2]]]]==2&]],Union[Join[Last/@First[#],Last/@Last[#]]]&];
Small standard orders
Small standard orders
GrowStandardOrder[order_,s_]:=Module[{max},max=Min[Max[order]+1,s];Append[order,#]&/@Range[max]];
StartTup[alpha_,tupsize_]:=Nest[Flatten[GrowStandardOrder[#,alpha]&/@#,1]&,{{1}},tupsize-1];
StartTup[4,12]//Length
700075
BellB[12]
4213597
rhs=Table[Select[Union[Union[Partition[#,2]]&/@Tuples[Range[mm],{2edges}]],Length[#]edges&]//Length,{edges,1,4},{mm,3,6}]
{{9,16,25,36},{36,120,300,630},{84,560,2300,7140},{126,1820,12650,58905}}
Table[Binomial[mm^2,edges],{edges,1,4},{mm,3,6}]
{{9,16,25,36},{36,120,300,630},{84,560,2300,7140},{126,1820,12650,58905}}
4^8/4!//Round
2731
6^8
1679616
StandardOrderFromIndex
StandardOrderFromIndex
canonicalizer
canonicalizer
ConnectedWolframModelQ
ConnectedWolframModelQ
Apply Signature
Apply Signature
Bell Indexed Sampling (complete 8 to 10, sampled 11 to 20)
Bell Indexed Sampling (complete 8 to 10, sampled 11 to 20)
Bell Indexed Sampling Runs (¶ indicates a sample is stored above)
Bell Indexed Sampling Runs (¶ indicates a sample is stored above)
Analysis 9
Analysis 9
niner={{{1,3}}{{2,3}},{{2,3}}{{1,3}},{{1,2}}{{1,3},{2,2}},{{1,3}}{{3,2}},{{2,2}}{{1,3},{1,2}},{{3,2}}{{1,3}},{{1,3},{1,2}}{{2,2}},{{1,3},{2,2}}{{1,2}},{{1,5}}{{2,2}},{{2,2}}{{1,5}}};
Length[niner]
10
Monitor[Table[{niner[[j]],Length[Select[ApplyWolframRuleSignaturetoList[niner[[j]],#]&/@orders9,FindCanonicalWolframModel[#]===#&&ConnectedWolframModelQ[#,Automatic]&]]},{j,1,10}],j]
Analysis 10
Analysis 10
sigs=IntegerToRuleSignatures[10]
Monitor[Table[{sigs[[j]],Length[Select[ApplyWolframRuleSignaturetoList[sigs[[j]],#]&/@orders10,FindCanonicalWolframModel[#]===#&&ConnectedWolframModelQ[#,Automatic]&]]},{j,1,Length[sigs]}],{Length[sigs],j}]
Analysis 15
Analysis 15
Canonical count estimate.
Table[Length[IntegerToRuleSignaturesOnes[k]],{k,6,15}]
{29,58,107,190,324,536,865,1366,2120,3232}
Table[Length[IntegerToRuleSignatures[k]],{k,8,20}]
{7,10,22,32,58,86,141,206,318,456,679,956,1381}
IntegerToRuleSignaturesOnes[8]
IntegerToRuleSignatures[8]
{{{1,2}}{{3,2}},{{2,2}}{{2,2}},{{3,2}}{{1,2}},{{1,2}}{{2,3}},{{2,3}}{{1,2}},{{1,4}}{{2,2}},{{2,2}}{{1,4}}}
IntegerToRuleSignatures[9]
Round[BellB[15]/(2!3!)]
115246545
5809 canonical connected rules in the sample of 100000
The actual number is 79359764
Grand Analysis
Grand Analysis
Currently for connected rules only.
The exact count for the below is 79359764: