A011968 Apply (1+Shift) to Bell numbers.

1, 2, 3, 7, 20, 67, 255, 1080, 5017, 25287, 137122, 794545, 4892167, 31858034, 218543759, 1573857867, 11863100692, 93345011951, 764941675963, 6514819011216, 57556900440429, 526593974392123, 4981585554604074, 48658721593531669, 490110875149889635

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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 Gérard, 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, Jul 17 2008

Examples

			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.
		

References

• Olivier Gérard and Karol Penson, A budget of set partitions statistics, in preparation, unpublished as of Sep 22 2011

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..500 n = 0..200 from Vincenzo Librandi.
• Cohn, Martin; Even, Shimon; Menger, Karl, Jr.; Hooper, Philip K.; On the Number of Partitionings of a Set of n Distinct Objects, Amer. Math. Monthly 69 (1962), no. 8, 782--785. MR1531841.
• Cohn, Martin; Even, Shimon; Menger, Karl, Jr.; Hooper, Philip K.; On the Number of Partitionings of a Set of n Distinct Objects, Amer. Math. Monthly 69 (1962), no. 8, 782--785. MR1531841. [Annotated scanned copy]
• Augustine O. Munagi, Extended set partitions with successions, European J. Combin. 29(5) (2008), 1298--1308.
• Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.

Crossrefs

Cf. A000110, A000296, A011969, A011970.

A diagonal of A011971 and A106436. - N. J. A. Sloane, Jul 31 2012

Cf. A000569, A240936, A321750.

Programs

Maple

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

Python

# requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
from itertools import accumulate
A011968_list, blist, b = [1,2], [1], 1
for _ in range(10**2):
    blist = list(accumulate([b]+blist))
    A011968_list.append(b+blist[-1])
    b = blist[-1] # Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 20 2014

Formula

For n >= 1, a(n+1) = exp(-1)*Sum_{k>=0} ((k+1)/k!)*k^n. - Benoit Cloitre, Mar 09 2008

For n >= 1, a(n) = Bell(n) + Bell(n-1). - Augustine O. Munagi, Jul 17 2008

G.f.: G(0) where G(k) = 1 - 2*x*(k+1)/((2*k+1)*(2*x*k-1) - x*(2*k+1)*(2*k+3)*(2*x*k-1)/(x*(2*k+3) - 2*(k+1)*(2*x*k+x-1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 19 2012

G.f.: 1 + x*E(0) where E(k) = 1 + 1/(1-x*k-x)/(1-x/(x+1/E(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 20 2013

G.f.: 1 + Sum_{k>=0} ( 1+1/(1-x-x*k) )*x^(k+1)/Product_{i=0..k} (1-x*i). - Sergei N. Gladkovskii, Jan 20 2013

a(n) ~ Bell(n) * (1 + LambertW(n)/n). - Vaclav Kotesovec, Jul 28 2021