A271788 Number of set partitions of [n] having exactly one pair (m,m+1) such that m is in some block b and m+1 is in block b+1.

0, 1, 3, 7, 16, 39, 105, 314, 1035, 3723, 14494, 60670, 271544, 1293147, 6523495, 34724247, 194357190, 1140402612, 6995760364, 44760085240, 298054873358, 2061644525813, 14787185811993, 109804829195145, 842928183558160, 6680572760715182, 54595535222727960

Offset: 1

Author

Alois P. Heinz, Apr 14 2016

Keywords

• nonn

Examples

			a(2) = 1: 1|2.
a(3) = 3: 12|3, 13|2, 1|23.
a(4) = 7: 123|4, 124|3, 12|34, 134|2, 13|2|4, 14|23, 1|234.
a(5) = 16: 1234|5, 1235|4, 123|45, 1245|3, 124|3|5, 125|34, 12|345, 1345|2, 134|2|5, 135|2|4, 13|25|4, 13|2|45, 145|23, 14|23|5, 15|234, 1|2345.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..500
• Wikipedia, Partition of a set

Crossrefs

Column k=1 of A185982.

Programs

Maple

b:= proc(n, i, m, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
       add(`if`(j=i+1 and k=0, 0, b(n-1, j, max(m, j), k-
      `if`(j=i+1, 1, 0))), j=1..m+1))
    end:
a:= n-> b(n, 1, 0, 1):
seq(a(n), n=1..30);

Mathematica

b[n_, i_, m_, k_] := b[n, i, m, k] = If[n == 0, If[k == 0, 1, 0], Sum[If[j == i + 1 && k == 0, 0, b[n - 1, j, Max[m, j], k - If[j == i + 1, 1, 0]]], {j, 1, m + 1}]];
a[n_] :=  b[n, 1, 0, 1];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, May 27 2018, translated from Maple *)