A375625 -id:A375625 - cp's OEIS frontend

A375623 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.

0, 0, 0, 2, 12, 30, 72, 132, 240, 380, 600, 870, 1260, 1722, 2532, 3080

Offset: 0

W. Edwin Clark, Aug 21 2024

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.

Cf. A047928, A126972, A375625.

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:

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

a(11)-a(13) from Hugo Pfoertner, Aug 23 2024
a(14) from Markus Sigg, Aug 25 2024
a(15) from Hugo Pfoertner, Sep 04 2024

cp's OEIS Frontend

A375623 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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