0,2

Starting with n=3 (a(3)=15), number of set partitions of n+2 with at least one singleton and the smallest element in any singleton is exactly n-2. The maximum number of singletons is therefore 5. Alternatively, starting with n=3, number of set partitions of n+2 with at least one singleton and the largest element in any singleton is exactly 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 3 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>2, a(n) is the number of partitions of V(n+1) into blocks of nonconsecutive integers. - Augustine O. Munagi (amunagi@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.

If n>2, then bell(n)+3*bell(n-1)+3*bell(n-2)+bell(n-3). - Augustine O. Munagi (amunagi@yahoo.com), Jul 17 2008

a(3)=15 because the set {1,3,5,7} has 15 different partitions which are necessarilty into blocks of nonconsecutive integers.

with(combinat): 1, 4, 8, seq(`if`(n>2, bell(n)+3*bell(n-1)+3*bell(n-2)+bell(n-3), NULL), n=3..22); - Augustine O. Munagi (amunagi@yahoo.com), Jul 17 2008

Cf. A000110.

Cf. A011968, A011969.

Adjacent sequences: A011967 A011968 A011969 this_sequence A011971 A011972 A011973

Sequence in context: A027961 A018921 A103536 this_sequence A111988 A110652 A059373

nonn

njas