A272065 Number of set partitions of [n] such that at least one pair of consecutive blocks (b,b+1) exists having not exactly one pair of consecutive numbers (i,i+1) with i member of b and i+1 member of b+1.

0, 0, 0, 0, 2, 17, 101, 545, 2935, 16351, 95335, 583373, 3745903, 25208633, 177505205, 1305468285, 10009943248, 79880835800, 662319435622, 5696570446421, 50749156111271, 467630493212126, 4451067568592918, 43709810099960739, 442331477265626019

Offset

0,5

Links

Table of n, a(n) for n=0..24.

Wikipedia, Partition of a set

Formula

a(n) = A000110(n) - A272064(n).

Example

a(4) = 2: 13|24, 13|2|4.
a(5) = 17: 124|35, 124|3|5, 134|25, 134|2|5, 135|24, 13|245, 13|24|5, 135|2|4, 13|25|4, 13|2|45, 13|2|4|5, 14|235, 14|23|5, 14|25|3, 14|2|3|5, 1|24|35, 1|24|3|5.

Maple

b:= proc(n, i, m, l) option remember; `if`(n=0,
      `if`({l[], 1}={1}, 1, 0), add(`if`(j<m+1 and
           j=i+1 and l[j]=1, 0, 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-> 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[Append[l, 1]] == {1}, 1, 0], Sum[If[j < m+1 && j == i+1 && l[[j]] == 1, 0, b[n-1, j, Max[m, j], If[j == m+1, Append[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 03 2017, translated from Maple *)

Crossrefs

Cf. A000110, A185982, A271271, A272064.

Sequence in context: A023260 A174365 A119363 * A129977 A213787 A105652

Adjacent sequences: A272062 A272063 A272064 * A272066 A272067 A272068

Keyword

nonn

Author

Alois P. Heinz, Apr 19 2016

Status

approved