A212397 Denominator of the average number of move operations required by an insertion sort of n (distinct) elements.

1, 1, 2, 6, 6, 30, 5, 35, 140, 1260, 1260, 13860, 13860, 180180, 180180, 180180, 360360, 6126120, 2042040, 38798760, 7759752, 2586584, 2586584, 59491432, 178474296, 4461857400, 4461857400, 40156716600, 40156716600, 1164544781400, 1164544781400

Offset: 0

Author

Alois P. Heinz, May 14 2012

Keywords

• nonn
• frac

Comments

The average number of move operations is 1/n! times the number of move operations required to sort all permutations of [n] (A212395), assuming that each permutation is equiprobable.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Wikipedia, Insertion sort
• Index entries for sequences related to sorting

Crossrefs

Numerators are in A212396.

Cf. A000142, A212395.

Programs

Maple

b:= proc(n) option remember;
      `if`(n=0, 0, b(n-1)*n + (n-1)! * (n-1)*(n+4)/2)
    end:
a:= n-> denom(b(n)/n!):
seq(a(n), n=0..30);

Mathematica

a[n_] := Denominator[n (n + 7)/4 - 2 HarmonicNumber[n]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 29 2018, from 2nd formula in A212396 *)

Formula

a(n) = denominator of A212395(n)/A000142(n).