A272064 Number of set partitions of [n] such that for each pair of consecutive blocks (b,b+1) exactly 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, 13, 35, 102, 332, 1205, 4796, 20640, 95197, 467694, 2435804, 13394117, 77490260, 470198899, 2984034004, 19757370537, 136171758636, 975002124101, 7239322944625, 55648169854405, 442195755123607, 3627392029179270, 30679238282421267, 267215329668444337

Offset

0,3

Links

Table of n, a(n) for n=0..26.

Wikipedia, Partition of a set

Formula

a(n) = A000110(n) - A272065(n).

Example

A000110(4) - a(4) = 15 - 13 = 2: 13|24, 13|2|4.
A000110(5) - a(5) = 52 - 35 = 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-> 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_] := b[n, 0, 0, {}]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Feb 03 2017, translated from Maple *)

Crossrefs

Cf. A000110, A185982, A271270, A272065.

Sequence in context: A258450 A131868 A339294 * A000747 A151259 A149853

Adjacent sequences: A272061 A272062 A272063 * A272065 A272066 A272067

Keyword

nonn

Author

Alois P. Heinz, Apr 19 2016

Status

approved