A159324 n! times the average number of comparisons required by an insertion sort of n (distinct) elements.

0, 0, 2, 16, 118, 926, 7956, 75132, 777456, 8771184, 107307360, 1416252960, 20068629120, 304002322560, 4903642679040, 83928856838400, 1519397749094400, 29010025797580800, 582647327132774400, 12280347845905305600, 271030782903552000000, 6251213902855219200000

Offset: 0

Author

Harmen Wassenaar (towr(AT)ai.rug.nl), Apr 10 2009

Keywords

• nonn

Examples

			For n=3, insertion sorting 123, 213, 213, 231, 312, 321 takes 3+3+3+2+3+2 = 4*3+2*2 = 16 comparisons.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..448
• Wikipedia, Insertion sort
• Index entries for sequences related to sorting

Crossrefs

Row sums of triangle A159323.

Cf. A000254, A001008, A002805, A138772, A212395, A285231.

Programs

Maple

a:= proc(n) option remember;
      `if`(n<2, 0, a(n-1)*n + (n-1)! * (n-1)*(n+2)/2)
    end:
seq(a(n), n=0..30); # Alois P. Heinz, May 14 2012
# second Maple program:
a:= proc(n) option remember; `if`(n<3, [0$2, 2][n+1],
      ((2*n^3-n^2-5*n+2)*a(n-1)-(n+2)*(n-1)^3*a(n-2))/((n-2)*(n+1)))
    end:
seq(a(n), n=0..30); # Alois P. Heinz, Dec 16 2016

Mathematica

a[n_] := n! ((n+1)(n+2)/4 - HarmonicNumber[n] - 1/2); Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 12 2017, after Gary Detlefs *)

Formula

a(n) = a(n-1)*(n) + n! *(n+1)/2 - (n-1)!.

a(n) = Sum_k A159323(n,k) = Sum_k A129178(n,k) * (n(n-1)/2 - k).

a(n) = n!/4 *(n^2+3*n-4*H(n)), where H(n) = Sum_{k=1..n} 1/k. - Gary Detlefs, Sep 02 2010

a(n) = A138772(n+1) - A000254(n). - Gary Detlefs, May 13 2012

a(n) = ((2*n^3-n^2-5*n+2)*a(n-1)-(n+2)*(n-1)^3*a(n-2))/((n-2)*(n+1)) for n>2. - Alois P. Heinz, Dec 16 2016

a(n) = 2 * A285231(n+1). - Alois P. Heinz, Apr 15 2017