A271271 Number of set partitions of [n] such that at least one pair of consecutive blocks (b,b+1) exists having no pair of consecutive numbers (i,i+1) with i member of b and i+1 member of b+1.

0, 0, 0, 0, 1, 9, 58, 341, 1983, 11776, 72345, 462173, 3075894, 21330762, 154050330, 1157493707, 9037925277, 73244123107, 615295131046, 5351329029624, 48126530239366, 447043890866154, 4284293705043796, 42317095568379559, 430355360965092107, 4501973706497500364

Offset: 0

Author

Alois P. Heinz, Apr 03 2016

Keywords

• nonn

Examples

			a(4) = 1: 13|2|4.
a(5) = 9: 124|3|5, 134|2|5, 135|2|4, 13|25|4, 13|2|45, 13|2|4|5, 14|23|5, 14|2|3|5, 1|24|3|5.
		

Links

• Wikipedia, Partition of a set

Crossrefs

Cf. A000110, A185982, A271270, A271273, A272065.

Programs

Maple

b:= proc(n, i, m, l) option remember; `if`(n=0,
      `if`(l=[] or {l[]}={1}, 1, 0), add(b(n-1, j, max(m, j),
      `if`(j=m+1, `if`(j=i+1, [l[],1], [l[],0]),
      `if`(j=i+1, subsop(j=1, l), l))), j=1..m+1))
    end:
a:= n-> combinat[bell](n)-b(n, 0$2, []):
seq(a(n), n=0..18);

Mathematica

b[n_, i_, m_, l_] := b[n, i, m, l] = If[n == 0, If[Union[l, {1}] == {1}, 1, 0], Sum[b[n-1, j, Max[m, j], If[j == m+1, Join[l, If[j == i+1, {1}, {0}] ], If[j == i+1, ReplacePart[l, j -> 1], l]]], {j, 1, m+1}]]; a[n_] := BellB[n] - b[n, 0, 0, {}]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Feb 02 2017, translated from Maple *)

Formula

a(n) = A000110(n) - A271270(n).