A240800 Total number of occurrences of the pattern 1<2<3 in all preferential arrangements (or ordered partitions) of n elements.

0, 0, 1, 28, 570, 10700, 200235, 3857672, 77620788, 1641549000, 36576771165, 859032716740, 21251178078702, 553095031003060, 15122143306215855, 433634860865610320, 13020228528050054760, 408687299328542444880, 13389274565474007735009, 457150279686453405468780

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 A226316.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..120

Crossrefs

Cf. A000670, A240763, A240796-A240800, A226316.

Programs

Maple

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

Mathematica

b[n_, t_, h_] := b[n, t, h] = If[n == 0, {1, 0}, Sum[Function[{p}, p + {0, p[[1]]*j*h/6}][b[n - j, t + j, h + j*t]]*Binomial[n, j], {j, 1, n}]]; a[n_] := b[n, 0, 0][[2]]; Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Jun 08 2015, after Alois P. Heinz *)

Formula

a(n) ~ n! * n^3 / (72 * (log(2))^(n+1)). - Vaclav Kotesovec, May 03 2015

Extensions

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