A271270 Number of set partitions of [n] such that for each pair of consecutive blocks (b,b+1) at least one pair of consecutive numbers (i,i+1) exists with i member of b and i+1 member of b+1.

1, 1, 2, 5, 14, 43, 145, 536, 2157, 9371, 43630, 216397, 1137703, 6313675, 36848992, 225464838, 1442216870, 9620746697, 66781675113, 481413175433, 3597627996006, 27825925290597, 222422033403527, 1834910286704787, 15603508329713182, 136616625732498989

Offset: 0

Author

Alois P. Heinz, Apr 03 2016

Keywords

• nonn

Examples

			A000110(4) - a(4) = 15 - 14 = 1: 13|2|4.
A000110(5) - a(5) = 52 - 43 = 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, A271271, A271272, A272064.

Programs

Maple

b:= proc(n, i, m, l) option remember; `if`(n=0,
      `if`({l[], 1}={1}, 1, 0), add(b(n-1, j, max(m, j),
      `if`(j=m+1, [l[], `if`(j=i+1, 1, 0)],
      `if`(j=i+1, subsop(j=1, l), l))), j=1..m+1))
    end:
a:= 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_] := b[n, 0, 0, {}]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Jan 30 2017, translated from Maple *)

Formula

a(n) = A000110(n) - A271271(n).