0,2

Starting with n=2 (a(2)=5), number of set partitions of n+2 with at least one singleton and the smallest element in any singleton is exactly n-1. The maximum number of singletons is therefore 4. Alternatively, starting with n=2, number of set partitions of n+2 with at least one singleton and the largest element in any singleton is exactly 4. E.g. a(3)=10 counts the following set partitions of [5]: {1345, 2}, {13, 2, 45}, {145, 2, 3}, {134, 2, 5}, {15, 2, 34}, {135, 2, 4}, {14, 2, 35}, {13, 2, 4, 5}, {14, 2, 3, 5}, {15, 2, 3, 4} - Olivier GERARD (olivier.gerard(AT)gmail.com), Oct 29 2007

Let V(N)={v(1),v(2),...,v(N)} denote an ordered set of increasing positive integers containing 2 pairs of adjacent elements that differ by at least 2, that is, v(i),v(i+1) with v(i+1)-v(i)>1. Then for n>1, a(n) is the number of partitions of V(n+1) into blocks of nonconsecutive integers. - Augustine O. Munagi (amunagi(AT)yahoo.com), Jul 17 2008

Olivier Gerard and Karol Penson, A budget of set partitions statistics, in preparation.

A. O. Munagi, Extended set partitions with successions, European J. Combin. 29(5) (2008), 1298--1308.

For n>=1, a(n+2)= exp(-1)*sum(k>=0,(k+1)^2/k!*k^n) - Benoit Cloitre (abmt(AT)wanadoo.fr), Mar 09 2008

If n>1, then a(n)=bell(n)+2*bell(n-1)+bell(n-2) - Augustine O. Munagi (amunagi(AT)yahoo.com), Jul 17 2008

a(3)=10 because the set {1,3,5,6} has 10 different partitions into blocks of nonconsecutive integers: 15/36, 16/35, 135/6, 136/5, 1/35/6, 1/36/5, 13/5/6, 15/3/6, 16/3/5, 1/3/5/6.

with(combinat): 1, 3, seq(`if`(n>1, bell(n)+2*bell(n-1)+bell(n-2), NULL), n=2..22); - Augustine O. Munagi (amunagi(AT)yahoo.com), Jul 17 2008

Cf. A000110.

Cf. A011968.

Sequence in context: A132332 A002039 A007695 this_sequence A003187 A100885 A003186

Adjacent sequences: A011966 A011967 A011968 this_sequence A011970 A011971 A011972

nonn

N. J. A. Sloane (njas(AT)research.att.com).