Fixed Point Combinator

Defining equation:
y[f]f[y[f]]

Y:

In[]:=
ssk(s(k(ss(s(ssk))))k)//.Application[a_,b_]a[b]
Out[]=
s[s][k][s[k[s[s][s[s[s][k]]]]][k]]
In[]:=
LeafCount/@SKOuterEvolveList[s[s][k][s[k[s[s][s[s[s][k]]]]][k]],10]
Out[]=
{12,20,20,20,20,20,20,20,20,20,20}
In[]:=
SKOuterEvolveList[s[s][k][s[k[s[s][s[s[s][k]]]]][k]][f],10]
Out[]=
{s[s][k][s[k[s[s][s[s[s][k]]]]][k]][f],s[s[k[s[s][s[s[s][k]]]]][k]][k[s[k[s[s][s[s[s][k]]]]][k]]][f],s[k[s[s][s[s[s][k]]]]][k][f][k[s[k[s[s][s[s[s][k]]]]][k]][f]],k[s[s][s[s[s][k]]]][f][k[f]][k[s[k[s[s][s[s[s][k]]]]][k]][f]],s[s][s[s[s][k]]][k[f]][k[s[k[s[s][s[s[s][k]]]]][k]][f]],s[k[f]][s[s[s][k]][k[f]]][k[s[k[s[s][s[s[s][k]]]]][k]][f]],k[f][k[s[k[s[s][s[s[s][k]]]]][k]][f]][s[s[s][k]][k[f]][k[s[k[s[s][s[s[s][k]]]]][k]][f]]],f[s[s[s][k]][k[f]][k[s[k[s[s][s[s[s][k]]]]][k]][f]]],f[s[s][k][k[s[k[s[s][s[s[s][k]]]]][k]][f]][k[f][k[s[k[s[s][s[s[s][k]]]]][k]][f]]]],f[s[k[s[k[s[s][s[s[s][k]]]]][k]][f]][k[k[s[k[s[s][s[s[s][k]]]]][k]][f]]][k[f][k[s[k[s[s][s[s[s][k]]]]][k]][f]]]],f[k[s[k[s[s][s[s[s][k]]]]][k]][f][k[f][k[s[k[s[s][s[s[s][k]]]]][k]][f]]][k[k[s[k[s[s][s[s[s][k]]]]][k]][f]][k[f][k[s[k[s[s][s[s[s][k]]]]][k]][f]]]]]}
In[]:=
Count[#,f,Infinity,HeadsTrue]&/@%410
Out[]=
{1,1,2,3,2,3,4,3,4,5,7}
In[]:=
Count[#,f,Infinity,HeadsTrue]&/@SKOuterEvolveList[s[s][k][s[k[s[s][s[s[s][k]]]]][k]][f],40]
Out[]=
{1,1,2,3,2,3,4,3,4,5,7,6,8,6,8,11,8,7,10,13,18,16,15,20,15,20,28,20,17,16,24,32,45,40,38,37,50,37,50,71,50}
In[]:=
ListLinePlot[%]
Out[]=
In[]:=
SKOuterEvolveList[s[s][k][s[k[s[s][s[s[s][k]]]]][k]][s],40]
Out[]=
{s[s][k][s[k[s[s][s[s[s][k]]]]][k]][s],
⋯39⋯
,s[s[s[k[k[k[s][k[s[k[s[s][s[s[s][k]]]]][k]][s]]][k[k[s[k[s[s][s[s[s][k]]]]][k]][s]][k[s][k[s[k[s[s][s[s[s][k]]]]][k]][s]]]]][k[k[k[s[k[s[s][s[s[s][k]]]]][k]][s]][k[s][k[s[k[s[s][s[s[s][k]]]]][k]][s]]]][k[k[s][k[s[k[s[s][s[s[s][k]]]]][k]][s]]][k[k[s[k[s[s][s[s[s][k]]]]][k]][s]][k[s][k[s[k[s[s][s[s[s][k]]]]][k]][s]]]]]][s[s[s][k]][k[k[
⋯1⋯
][
⋯1⋯
]]][
⋯1⋯
]]]]]}
large output
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set size limit...
In[]:=
LeafCount/@%
Out[]=
{13,21,21,21,19,20,30,18,28,38,50,48,60,46,59,83,57,45,69,93,131,117,115,153,113,152,216,150,124,112,176,240,344,304,290,288,392,286,391,561,389}
In[]:=
ListLinePlot[%]
Out[]=