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

2,2

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)

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

Formula

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

Example

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}.

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 *)

Prog

(PARI) a(n)=sum(j=2, n--, (j+1)*stirling(n, j, 2)) \\ Charles R Greathouse IV, Jul 06 2011
(Magma) [Bell(n) -2: n in [3..30]]; // G. C. Greubel, Sep 13 2019
(Sage) [bell_number(n)-2 for n in (3..30)] # G. C. Greubel, Sep 13 2019
(GAP) List([3..30], n-> Bell(n)-2); # G. C. Greubel, Sep 13 2019

Crossrefs

Cf. A000110, A008826.

Sequence in context: A259338 A196907 A116427 * A026529 A363881 A357870

Adjacent sequences: A008824 A008825 A008826 * A008828 A008829 A008830

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Jan 02 2004

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

Status

approved