A240797 Total number of occurrences of the pattern 1=2 in all preferential arrangements (or ordered partitions) of n elements.

0, 1, 9, 78, 750, 8115, 98343, 1324204, 19650060, 318926745, 5623615965, 107093749818, 2191142272410, 47944109702671, 1117341011896515, 27633982917342360, 722929036749464280, 19946727355457792853, 578926427416920550233, 17632301590672398115270

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 n! (these are the permutations on n elements).

Examples

			The 13 preferential arrangements on 3 points and the number of times the pattern 1=2 occurs are:
1<2<3, 0
1<3<2, 0
2<1<3, 0
2<3<1, 0
3<1<2, 0
3<2<1, 0
1=2<3, 1
1=3<2, 1
2=3<1, 1
1<2=3, 1
2<1=3, 1
3<1=2, 1
1=2=3, 3,
for a total of a(3) = 9.
		

Links

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

Crossrefs

Cf. A000670, A240763, A240796-A240800.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, [1, 0], add((p-> p+
      [0, p[1]*j*(j-1)/2])(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]]*j*(j - 1)/2}][b[n - j]]*Binomial[n, j], {j, 1, n}]]; a[n_] := b[n][[2]]; Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Feb 07 2017, after Alois P. Heinz *)

Formula

a(n) ~ n! * n / (4 * (log(2))^n). - Vaclav Kotesovec, May 03 2015

Extensions

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