To run

sigs[n_]:=DeleteDuplicates[Sort/@((Rule@@@Partition[#,2])&/@Catenate[Permutations@*Reverse/@Select[IntegerPartitions[n],EvenQ@*Length]])]

In[]:=

sigs[4]

In[]:=

{{13},{31},{22},{11,11}}

Out[]=

allall=Table[Echo@Export[$PhysicsDataDirectory<>"/MWCausalInvariance-k3/"<>ToString[n]<>".wxf",EchoFunction[Length][ParallelMapMonitored[#TimeConstrained[Do[If[TotalCausalInvariantQ[#,t],Return[t]],{t,20}],5]&,EnumerateSubstitutionSystemRules[Echo@#,3]]]&/@sigs[n]],{n,15}]

In[]:=

/Users/sw/Dropbox/Physics/Data/MWCausalInvariance-k3/1.wxf

{11}

/Users/sw/Dropbox/Physics/Data/MWCausalInvariance-k3/2.wxf

{12}

{21}

/Users/sw/Dropbox/Physics/Data/MWCausalInvariance-k3/3.wxf

{13}

{31}

{22}

{11,11}

/Users/sw/Dropbox/Physics/Data/MWCausalInvariance-k3/4.wxf

{14}

{41}

{23}

{32}

{11,12}

{11,21}

/Users/sw/Dropbox/Physics/Data/MWCausalInvariance-k3/5.wxf

{15}

{51}

{24}

{42}

{33}

{11,13}

{11,31}

{11,22}

{12,12}

{12,21}

{21,21}

{11,11,11}

122

/Users/sw/Dropbox/Physics/Data/MWCausalInvariance-k3/6.wxf

{16}

196

{61}

196

{25}

196

{52}

196

{34}

196

{43}

196

{11,14}

205

{11,41}

205

{11,23}

205

{11,32}

205

{12,13}

205

{12,31}

205

{13,21}

LinkObject

:Unable to communicate with closed link LinkObject[50026@10.20.50.7,50027@10.20.50.7,1087,64].

KernelObject

:Subkernel connected through KernelObject

Name: \macpro-basement

KernelID: \54

 appears dead.

Parallel`Developer`QueueRun

::req

:Requeueing evaluations {2265} assigned to KernelObject

Name: \macpro-basement

State: \defunct

.

LaunchKernels

:Kernel KernelObject

Name: \macpro-basement

State: \defunct

 resurrected as KernelObject

Name: \macpro-basement

KernelID: \69

.

205

{21,31}

205

{12,22}

196

{21,22}

196

{11,11,12}

244

{11,11,21}

244

/Users/sw/Dropbox/Physics/Data/MWCausalInvariance-k3/7.wxf

{17}

574

{71}

574

{26}

574

{62}

574

{35}

574

{53}

574

{11,15}

610

{11,51}

610

{44}

574

{11,24}

574

{11,42}

574

{12,14}

574

{12,41}

574

{14,21}

LinkObject

:Unable to communicate with closed link LinkObject[49943@10.20.50.7,49944@10.20.50.7,1062,39].

KernelObject

:Subkernel connected through KernelObject

Name: \lambda

KernelID: \29

 appears dead.

Parallel`Developer`QueueRun

::req

:Requeueing evaluations {4776} assigned to KernelObject

Name: \lambda

State: \defunct

.

LaunchKernels

:Kernel KernelObject

Name: \lambda

State: \defunct

 resurrected as KernelObject

Name: \lambda

KernelID: \70

.

574

{21,41}

574

{11,33}

610

{13,13}

610

{13,31}

610

{31,31}

610

{12,23}

574

{12,32}

574

{13,22}

574

{21,23}

574

{21,32}

574

{22,31}

574

{11,11,13}

730

{11,11,31}

730

{22,22}

574

{11,11,22}

610

{11,12,12}

610

{11,12,21}

610

{11,21,21}

610

{11,11,11,11}

1094

/Users/sw/Dropbox/Physics/Data/MWCausalInvariance-k3/8.wxf

{18}

1681

{81}

1681

{27}

1681

{72}

1681

{36}

1681

{63}

1681

{11,16}

1708

{11,61}

1708

{45}

1681

{54}

1681

{11,25}

1708

{11,52}

1708

{12,15}

LinkObject

:Unable to communicate with closed link LinkObject[50024@10.20.50.7,50025@10.20.50.7,1086,63].

General

:Further output of LinkObject::linkd will be suppressed during this calculation.

KernelObject

:Subkernel connected through KernelObject

Name: \macpro-basement

KernelID: \53

 appears dead.

General

:Further output of KernelObject::rdead will be suppressed during this calculation.

Parallel`Developer`QueueRun

::req

:Requeueing evaluations {10179} assigned to KernelObject

Name: \macpro-basement

State: \defunct

.

General

:Further output of Parallel`Developer`QueueRun::req will be suppressed during this calculation.

LaunchKernels

:Kernel KernelObject

Name: \macpro-basement

State: \defunct

 resurrected as KernelObject

Name: \macpro-basement

KernelID: \71

.

General

:Further output of LaunchKernels::clone will be suppressed during this calculation.

1708

{12,51}

1708

{15,21}

1708

{21,51}

1708

{11,34}

1708

{11,43}

1708

{13,14}

1708

{13,41}

1708

{14,31}

1708

{31,41}

1708

{12,24}

1681

{12,42}

1681

{14,22}

1681

{21,24}

1681

{21,42}

1681

{22,41}

1681

{11,11,14}

1825

{11,11,41}

1825

{12,33}

1708

{13,23}

1708

{13,32}

1708

{21,33}

1708

{23,31}

1708

{31,32}

1708

{22,23}

1681

{22,32}

1681

{11,11,23}

1825

{11,11,32}

1825

{11,12,13}

1825

{11,12,31}

1825

{11,13,21}

1825

{11,21,31}

1825

{11,12,22}

1708

{11,21,22}

1708

{12,12,12}

1708

{12,12,21}

1708

{12,21,21}

1708

{21,21,21}

1708

{11,11,11,12}

2188

{11,11,11,21}

2188

/Users/sw/Dropbox/Physics/Data/MWCausalInvariance-k3/9.wxf

{19}

5002

{91}

5002

{28}

5002

{82}

5002

{37}

5002

{73}

5002

{11,17}

5110

{11,71}

5110

{46}

5002

{64}

5002

{11,26}

5002

{11,62}

5002

{12,16}

5002

{12,61}

5002

{16,21}

5002

{21,61}

5002

{55}

5002

{11,35}

5110

{11,53}

5110

{13,15}

5110

{13,51}

5110

{15,31}

5110

{31,51}

5110

{12,25}

5002

{12,52}

5002

{15,22}

5002

{21,25}

5002

{21,52}

5002

{22,51}

5002

{11,11,15}

5470

{11,11,51}

5470

{11,44}

5002

{14,14}

5002

{14,41}

5002

{41,41}

5002

{12,34}

5002

{12,43}

5002

{13,24}

5002

{13,42}