0,2

Number of set partitions of n+2 with at least one singleton and the smallest element in any singleton is exactly n. The maximum number of singletons is therefore 3. Alternatively, number of set partitions of n+2 with at least one singleton and the largest element in any singleton is exactly 3 (or n+2 if n+2<3). For example, a(3)=7 counts the following set partitions of [5]: {1245, 3}, {12, 3, 45}, {124, 3, 5}, {15, 24, 3}, {125, 3, 4}, {14, 25, 3}, {12, 3, 4, 5} - 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 a pair 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>0, 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+1)= exp(-1)*sum(k>=0,(k+1)/k!*k^n) - Benoit Cloitre (abmt(AT)wanadoo.fr), Mar 09 2008

If n>0, then a(n)=bell(n)+bell(n-1). - Augustine O. Munagi (amunagi(AT)yahoo.com), Jul 17 2008

a(3)=7 because the set {1,3,4,5} has 7 different partitions into blocks of nonconsecutive integers: 14/35, 135/4, 1/35/4, 13/4/5, 14/3/5, 15/3/4, 1/3/4/5.

with(combinat): seq(`if`(n>0, bell(n)+bell(n-1), 1), n=0..21); - Augustine O. Munagi (amunagi(AT)yahoo.com), Jul 17 2008

Cf. A000110.

Cf. A000296.

Sequence in context: A110498 A006073 A052402 this_sequence A080021 A032313 A032223

Adjacent sequences: A011965 A011966 A011967 this_sequence A011969 A011970 A011971

nonn

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