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A375623
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Maximum value of F(p) = Sum (|i-j| - |p(i)-p(j)|)^2 where the sum is over all 1 <= i < j <= n, for all permutations p in the symmetric group S_n.
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0
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0, 0, 0, 2, 12, 30, 72, 132, 240, 380, 600, 870, 1260, 1722
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OFFSET
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0,4
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COMMENTS
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The function F was defined by Dan Asimov on the Mailing list Math-Fun on Aug. 18, 2024. It can be considered as a sort of entropy of a permutation p like the function Sum_{k=1..n} (p(k)-k)^2 in A126972.
The terms for even n seem to agree with A047928.
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LINKS
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MAPLE
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F := proc(S) local i, j, M;
M := 0;
for j from 1 to nops(S) do
for i from 1 to j-1 do
M := M + (abs(i - j) - abs(S[i] - S[j]))^2
od:
od: M end:
a := proc(n) local P, m, u, mm;
P := combinat:-permute(n);
m := 0;
for u in P do
mm := F(u);
if mm > m then m := mm fi;
od: m end:
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PROG
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(PARI) a375623(n) = my(m=0); forperm(n, p, m=max(m, sum(i=1, n, sum(j=1, i-1, (abs(i-j)-abs(p[i]-p[j]))^2)))); m \\ Hugo Pfoertner, Aug 22 2024
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CROSSREFS
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KEYWORD
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nonn,more,new
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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