A240798 Total number of occurrences of the pattern 1=2=3 in all preferential arrangements (or ordered partitions) of n elements.

0, 0, 1, 12, 130, 1500, 18935, 262248, 3972612, 65500200, 1169398065, 22494463860, 464072915878, 10225330604580, 239720548513355, 5959152063448080, 156592569864940040, 4337574220496785680, 126329273251232688069, 3859509516112803668220, 123426111134706786806890

Offset: 1

Author

N. J. A. Sloane, Apr 13 2014

Keywords

• nonn

Comments

There are A000670(n) preferential arrangements of n elements - see A000670, A240763.

The number that avoid the pattern 1=2=3 is given in A080599.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..420

Crossrefs

Cf. A000670, A240763, A240796-A240800, A080599.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, [1, 0], add((p-> p+
      [0, p[1]*binomial(j, 3)])(b(n-j))*binomial(n, j), j=1..n))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=1..25);  # Alois P. Heinz, Dec 08 2014

Mathematica

b[n_] := b[n] = If[n==0, {1, 0}, Sum[Function[p, p+{0, p[[1]]*Binomial[j, 3]} ][b[n-j]]*Binomial[n, j], {j, 1, n}]]; a[n_] := b[n][[2]]; Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Feb 28 2017, after Alois P. Heinz *)

Formula

a(n) ~ n! * n / (12 * (log(2))^(n-1)). - Vaclav Kotesovec, May 03 2015

Extensions

a(8)-a(21) from Alois P. Heinz, Dec 08 2014