A129118 For each permutation p of {1,2,...,n} define minabsjump(p) = min(|p(i) - i|, 1<=i<=n); a(n) is the sum of minabsjumps of all p.

0, 1, 2, 10, 48, 295, 2068, 16654, 150382, 1508500, 16631696, 199966907, 2603709640, 36501212971, 548150650582, 8779185528284, 149376644391508, 2690852138104504, 51161190374132154, 1023850096381041159, 21512688329462044264

Offset

1,3

Links

Table of n, a(n) for n=1..21.

Formula

a(n) = Sum_{k=1..floor(n/2)} k * A299789(n,k). - Alois P. Heinz, Jan 21 2019

Maple

n:=8; with(combinat); P:=permute(n); ct:= 0; for j to factorial(n) do ct:= ct+min(seq(abs(P[j][i]-i), i=1..n)) end do: ct; # yields a(n) for the specified n - Emeric Deutsch, Aug 24 2007
# second Maple program:
b:= proc(s) option remember; (n-> `if`(n=1, x^(s[]-1), add((p->
      add(coeff(p, x, i)*x^min(i, abs(n-j)), i=0..degree(p)))(
        b(s minus {j})), j=s)))(nops(s))
    end:
a:= n-> (p-> add(coeff(p, x, i)*i, i=1..n-1))(b({$1..n})):
seq(a(n), n=1..15);  # Alois P. Heinz, Jan 21 2019

Mathematica

b[s_] := b[s] = Function[n, If[n == 1, x^(s - 1), Sum[Function[p, Sum[ SeriesCoefficient[p, {x, 0, i}]*x^Min[i, Abs[n - j]], {i, 0, Exponent[p, x]}]][b[s ~Complement~ {j}]], {j, s}]]][Length[s]] // Expand;
a[n_] := a[n] = If[n == 1, 0, Function[p, Sum[SeriesCoefficient[p, {x, 0, i}]*i, {i, 1, n - 1}]][b[Range[n]][[1]]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 12}] (* Jean-François Alcover, May 21 2020, after 2nd Maple program *)

Prog

(C++) #include <iostream> #include <vector> #include <algorithm> using namespace std; inline int mabsj(const vector<int> & s) { const int n = s.size() ; int mi =n ; for(int i=0; i<n; i++) { const int thisdiff = abs(s[i]-i-1) ; if ( thisdiff < mi ) mi=thisdiff ; } return mi ; } int main(int argc, char *argv[]) { for(int n=1 ;; n++) { vector<int> s; for(int i=1; i<=n; i++) s.push_back(i) ; unsigned long long resul=0 ; do { resul += mabsj(s) ; } while( next_permutation(s.begin(), s.end()) ) ; cout << n << " " << resul << endl ; } return 0 ; } /* R. J. Mathar, Nov 01 2007 */

Crossrefs

Cf. A130153, A018927, A299789.

Sequence in context: A200540 A166922 A302557 * A037256 A270664 A330543

Adjacent sequences: A129115 A129116 A129117 * A129119 A129120 A129121

Keyword

more,nonn

Author

Emeric Deutsch, Vladeta Jovovic, Jun 07 2007

Extensions

One more term from Emeric Deutsch, Aug 24 2007

a(11)-a(13) from R. J. Mathar, Nov 01 2007

a(14) from Donovan Johnson, Sep 24 2010

a(15)-a(21) from Alois P. Heinz, Jan 21 2019

Status

approved