A240796 Total number of occurrences of the pattern 1<2 in all preferential arrangements (or ordered partitions) of n elements.

0, 1, 15, 186, 2330, 31065, 447405, 6979588, 117745668, 2141106795, 41810587775, 873474855726, 19451904450654, 460209050303821, 11531197020389025, 305122289460210120, 8503747639606509128, 249020038061419770783, 7645072502094118876755, 245564189847880300238290

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 is 2^(n-1).

The total number of occurrences of the pattern 1<2 in all permutations on n elements is (n-1)*(n-1)! (cf. A010027, A001563).

Examples

			The 13 preferential arrangements on 3 points and the number of times the pattern 1<2 occurs are:
1<2<3, 3
1<3<2, 2
2<1<3, 2
2<3<1, 1
3<1<2, 1
3<2<1, 0
1=2<3, 2
1=3<2, 1
2=3<1, 0
1<2=3, 2
2<1=3, 1
3<1=2, 0
1=2=3, 0,
for a total of a(3) = 15.
		

Links

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

Crossrefs

Cf. A000670, A240763, A240796-A240800, A010027, A001563, A264082, A381299.

Programs

Maple

b:= proc(n, t) option remember; `if`(n=0, [1, 0], add((p-> p+
      [0, p[1]*j*t/2])(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*t/2}][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, Jun 08 2015, after Alois P. Heinz *)

Formula

a(n) ~ n! * n^2 / (8 * (log(2))^(n+1)). - Vaclav Kotesovec, May 03 2015

a(n) = Sum_{k=0..binomial(n,2)} k * A381299(n,k). - Alois P. Heinz, Feb 22 2025

Extensions

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