In[]:=
skrules
Out[]=
{s[x_][y_][z_]x[z][y[z]],k[x_][y_]x}
In[]:=
AllCombinators[4]
Out[]=
{s[s][s][s],s[s[s][s]],s[s[s]][s],s[s[s[s]]],s[s][s[s]],s[s][s][k],s[s[s][k]],s[s[s]][k],s[s[s[k]]],s[s][s[k]],s[s][k][s],s[s[k][s]],s[s[k]][s],s[s[k[s]]],s[s][k[s]],s[s][k][k],s[s[k][k]],s[s[k]][k],s[s[k[k]]],s[s][k[k]],s[k][s][s],s[k[s][s]],s[k[s]][s],s[k[s[s]]],s[k][s[s]],s[k][s][k],s[k[s][k]],s[k[s]][k],s[k[s[k]]],s[k][s[k]],s[k][k][s],s[k[k][s]],s[k[k]][s],s[k[k[s]]],s[k][k[s]],s[k][k][k],s[k[k][k]],s[k[k]][k],s[k[k[k]]],s[k][k[k]],k[s][s][s],k[s[s][s]],k[s[s]][s],k[s[s[s]]],k[s][s[s]],k[s][s][k],k[s[s][k]],k[s[s]][k],k[s[s[k]]],k[s][s[k]],k[s][k][s],k[s[k][s]],k[s[k]][s],k[s[k[s]]],k[s][k[s]],k[s][k][k],k[s[k][k]],k[s[k]][k],k[s[k[k]]],k[s][k[k]],k[k][s][s],k[k[s][s]],k[k[s]][s],k[k[s[s]]],k[k][s[s]],k[k][s][k],k[k[s][k]],k[k[s]][k],k[k[s[k]]],k[k][s[k]],k[k][k][s],k[k[k][s]],k[k[k]][s],k[k[k[s]]],k[k][k[s]],k[k][k][k],k[k[k][k]],k[k[k]][k],k[k[k[k]]],k[k][k[k]]}
In[]:=
SKEvolveList[s[k][k][s],10]
Out[]=
{s[k][k][s],k[s][k[s]],s,s,s,s,s,s,s,s,s}
In[]:=
In[]:=
LeafCount/@nonterms
Out[]=
{8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8}
In[]:=
ParallelEvaluate[1]
CloudConnect
:Connecting to a cloud running an earlier version of the Wolfram Engine: 12.1
LinkObject
:Unable to communicate with closed link LinkObject[62903@10.20.50.7,62904@10.20.50.7,334,35].
LinkObject
:Unable to communicate with closed link LinkObject[62905@10.20.50.7,62906@10.20.50.7,335,36].
LinkObject
:Unable to communicate with closed link LinkObject[62907@10.20.50.7,62908@10.20.50.7,336,37].
General
:Further output of LinkObject::linkd will be suppressed during this calculation.
Parallel`Developer`ConnectKernel
::failinit
:12 of 12 kernels failed to initialize.
Out[]=
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
In[]:=
ResourceFunction["ParallelMapMonitored"][ListStepPlot[ResourceFunction["ParallelMapMonitored"][SKFixedPointEvolveList[#,2000]&,nonterms],PlotLabel#]&,nonterms]
In[]:=
LeafCount/@SKEvolveList[s[s][s][s[s]][s][s],100]
Out[]=
{7,8,8,11,11,11,12,17,25,33,41,50,59,87,115,149,187,215,243,272,301,389,417,426,441,450,459,468,483,492,501,529,558,587,675,684,699,708,717,726,741,750,759,768,777,792,817,842,851,860,875,900,925,934,943,958,983,1008,1017,1026,1041,1066,1091,1100,1115,1124,1133,1142,1152,1162,1193,1235,1277,1325,1399,1408,1433,1464,1505,1514,1524,1534,1565,1607,1649,1697,1771,1780,1805,1836,1877,1886,1901,1910,1919,1928,1938,1948,1979,2021,2063}
In[]:=
ListLogPlot[%,JoinedTrue]
Out[]=
In[]:=
SKEvolveList[s[s][s][s[s]][s][s],20]
Out[]=
In[]:=
CombinatorTree/@%
Out[]=

,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,

In[]:=
Position[#,s[_][_][_]]&/@SKEvolveList[s[s][s][s[s]][s][s],20]//Column
Out[]=
{{0,0}}
{{0}}
{{0}}
{{}}
{{0}}
{{0}}
{{}}
{{}}
{{1}}
{{1,0}}
{{1,0}}
{{1,0,1},{1}}
{{1,1,0},{1}}
{{1,0,1,0},{1,0},{1,1,1,0}}
{{1,0,0,1,0},{1,0,1,1,0},{1,0,1},{1,0},{1,1,1,0}}
{{1,0,0,1,1,0},{1,0,0,1},{1,0,1,0,0},{1,0,1,1,1,0},{1,0,1,1},{1,1,1,0},{1}}
{{1,0,0,1,0},{1,0,0},{1,0,1,1,0},{1,1,0,0,0},{1,1,0,1,1,0},{1,1,0,1},{1,1,1,1,0}}
{{1,0,0,0,1,0},{1,0,0,0},{1,0,0,1,1,0},{1,0,1,1,0},{1,1,0,0,0},{1,1,0,1,1,0},{1,1,0,1},{1,1,1,1,0}}
{{1,0,0,0,0,1,0},{1,0,0,0,1,1,0},{1,0,0,0},{1,0,0,1,1,0},{1,0,1,1,0},{1,1,0,0,0},{1,1,0,1,1,0},{1,1,0,1},{1,1,1,1,0}}
{{1,0,0,0,0,1,1,0},{1,0,0,0,1,0,0},{1,0,0,0,1,1,1,0},{1,0,0,1,1,0},{1,0,0},{1,0,1,1,0},{1,1,0,0,0},{1,1,0,1,1,0},{1,1,0,1},{1,1,1,1,0}}
{{1,0,0,0,0,1,0},{1,0,0,0,1,1,0},{1,0,0,1,0,0,0},{1,0,0,1,0,1,1,0},{1,0,0,1,1,1,0},{1,0,0},{1,0,1,1,0},{1,1,0,0,0},{1,1,0,1,1,0},{1,1,0,1},{1,1,1,1,0}}

More on the S combinator

More S forms

Random eruptions:
Larger scale growth
Highly regular growth

Larger scale growth

Random Eruptions

Busy Beavers

Halting times

All MWs

S,K branched termination