A240799 Total number of occurrences of the pattern 1=2<3 in all preferential arrangements (or ordered partitions) of n elements.

0, 0, 1, 20, 310, 4660, 72485, 1193080, 20938764, 392485560, 7850724915, 167242351100, 3785057708146, 90775554103052, 2301045251519105, 61499717442074800, 1729026306941190680, 51022485837639054768, 1577126907722325214959, 50967150013960792511700

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 A001710 (1,3,12,60,360,...).

Links

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

Crossrefs

Cf. A000670, A240763, A240796-A240800, A001710.

Programs

Maple

b:= proc(n, t) option remember; `if`(n=0, [1, 0], add((p-> p+
      [0, p[1]*j*(j-1)*t/6])(b(n-j, t+j))*binomial(n, j), j=1..n))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=1..25);  # Alois P. Heinz, Dec 08 2014

Mathematica

b[n_, t_] := b[n, t] = If[n==0, {1, 0}, Sum[Function[p, p+{0, p[[1]]*j*(j-1)*t/6}][b[n-j, t+j]]*Binomial[n, j], {j, 1, n}]]; a[n_] := b[n, 0][[2]]; Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Feb 28 2017, after Alois P. Heinz *)

Formula

a(n) ~ n! * n^2 / (24 * (log(2))^n). - Vaclav Kotesovec, May 03 2015

Extensions

a(8)-a(20) from Alois P. Heinz, Dec 08 2014