Permutations appear scattered irregularly among all tuples. This notebook maps their positions to reveal a surprising structure. By visualizing deviations from a uniform equilibrium, we discover that the apparent chaos cancels out to a precise constant for every .
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Let poslist be any list of positive integers. Define the predicate
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permlistQ@poslist_:=Sort[poslist]==(Range@Length@poslist)
Now let n be a positive integer and consider the function
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tupls@n_:=Tuples[Range@n,n]
Note that tupls@n returns a list of length . For example,
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n
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tupls@2
Out[]=
{{1,1},{1,2},{2,1},{2,2}}
Note that permlistQ picks out all of the permutations:
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permlistQ/@%
Out[]=
{False,True,True,False}
This is true for any value of n .
The output of tupls@n has terms and must include all n! permutations of Range@n. It is the purpose of this work to exhibit a symmetry property obeyed by the positions of these permutations amongst the list of terms. For any positive integer n these positions are given by
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permlocs@n_:=Flatten@Position[permlistQ/@(tupls@n),True]
A plot shows a bit of a symmetry for n = 4:
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ListPlot@permlocs@4
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Here are some equally - spaced "equilibrium points"
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reflocs@n_:=Tablek,{k,n!}
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n!
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ListPlot@reflocs@4
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It makes sense to look at displacements from equilibrium :
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deltas@n_:=permlocs@n-reflocs@n
Now the symmetry becomes more obvious:
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ListPlot@deltas@4
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Define
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symmcheck@n_:=(deltas@n+Reverse@deltas@n)
Now the symmetry is so obvious that it becomes a constant:
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ListPlot@symmcheck@4
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To evaluate the constant define
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unioncheck@n_:=-Union@(deltas@n+Reverse@deltas@n)
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unioncheck@4
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29
3
The symmetry goes beyond n=4 and holds true for all values of n that have so far been evaluated. For example,
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unioncheck/@Range@7//Flatten
Out[]=
0,1,,,,,
7
2
29
3
601
24
319
5
116929
720
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Hidden symmetry in permutation space
by David Vasholz
Wolfram Community, STAFF PICKS, February 17, 2026
https://community.wolfram.com/groups/-/m/t/3641283
by David Vasholz
Wolfram Community, STAFF PICKS, February 17, 2026
https://community.wolfram.com/groups/-/m/t/3641283