A008827 Number of proper partitions of a set of n labeled elements.

0, 3, 13, 50, 201, 875, 4138, 21145, 115973, 678568, 4213595, 27644435, 190899320, 1382958543, 10480142145, 82864869802, 682076806157, 5832742205055, 51724158235370, 474869816156749, 4506715738447321, 44152005855084344, 445958869294805287, 4638590332229999351

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Previous name: Coefficients from fractional iteration of exp(x) - 1.

From Harry Richman, Mar 18 2023: (Start)

A "proper partition" of a set is a set partition in which there is more than one part, and there is some part which has more than one element.

a(n) is the number of chains of length 2 from the top element to the bottom element in the partition lattice on n labeled objects.

(End)

Examples

			For n = 3 there are a(3) = 3 proper partitions of {1,2,3}, which can be represented {12|3}, {13|2}, {23|1}.
For n = 4 there are a(4) = 13 proper partitions of {1,2,3,4}, which can be represented {123|4}, {124|3}, {134|2}, {234|1}, {12|34}, {13|24}, {14|23}, {12|3|4}, {13|2|4}, {14|2|3}, {23|1|4}, {24|1|3}, {34|1|2}.
		

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 148.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 2..200
• Ivo Rosenberg, The number of maximal closed classes in the set of functions over a finite domain, J. Combinatorial Theory Ser. A 14 (1973), 1-7. See Table I (it is not certain that this is the same sequence. - _N. J. A. Sloane_, Jun 25 2015)
• Ivo Rosenberg and N. J. A. Sloane, Correspondence, 1971

Crossrefs

Cf. A000110, A008826.

Programs

GAP

List([3..30], n-> Bell(n)-2); # G. C. Greubel, Sep 13 2019

Magma

[Bell(n) -2: n in [3..30]]; // G. C. Greubel, Sep 13 2019

Maple

seq(combinat[bell](n)-2, n=2..31); # Zerinvary Lajos, Sep 29 2006

Mathematica

Table[BellB[n] - 2, {n, 3, 30}] (* Vladimir Joseph Stephan Orlovsky, Jul 06 2011 *)

PARI

a(n)=sum(j=2,n--,(j+1)*stirling(n,j,2)) \\ Charles R Greathouse IV, Jul 06 2011

Sage

[bell_number(n)-2 for n in (3..30)] # G. C. Greubel, Sep 13 2019

Formula

a(n) = A000110(n) - 2.

Extensions

More terms from Vladeta Jovovic, Jan 02 2004

Name changed and a(2)=0 prepended by Harry Richman, Mar 18 2023