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A032032 Number of ways to partition n labeled elements into sets of sizes of at least 2 and order the sets. 10
1, 0, 1, 1, 7, 21, 141, 743, 5699, 42241, 382153, 3586155, 38075247, 428102117, 5257446533, 68571316063, 959218642651, 14208251423433, 223310418094785, 3699854395380371, 64579372322979335, 1182959813115161773, 22708472725269799933, 455643187943171348103 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

With m = floor(n/2), a(n) is the number of ways to distribute n different toys to m numbered children such that each child receiving a toy gets at least two toys and, if child k gets no toys, then each child numbered higher than k also gets no toys. Furthermore, a(n)= row sums of triangle A200091 for n>=2. - Dennis P. Walsh, Apr 15 2013

Row sums of triangle A200091. - Dennis P. Walsh, Apr 15 2013

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..400

C. G. Bower, Transforms (2)

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 245

I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637, 2013 and J. Int. Seq. 17 (2014) #14.1.1

Robert A. Proctor, Let's Expand Rota's Twelvefold Way For Counting Partitions!, arXiv:math.CO/0606404, Jan 05, 2007

Index entries for related partition-counting sequences

FORMULA

"AIJ" (ordered, indistinct, labeled) transform of 0, 1, 1, 1...

E.g.f.: 1/(2+x-exp(x)).

a(n) = n! * sum(k=1..n, sum(j=0..k, C(k,j) *stirling2(n-k+j,j) *j!/(n-k+j)! *(-1)^(k-j))); a(0)=1. - Vladimir Kruchinin, Feb 01 2011

a(n) ~ n! / ((r-1)*(r-2)^(n+1)), where r = -LambertW(-1,-exp(-2)) = 3.14619322062... - Vaclav Kotesovec, Oct 08 2013

EXAMPLE

For n=5, a(5)=21 since there are 21 toy distributions satisfying the conditions above. Denoting a distribution by |kid_1 toys|kid_2 toys|, we have the distributions

  |t1,t2,t3,t4,t5|_|, |t1,t2,t3|t4,t5|, |t1,t2,t4|t3,t5|, |t1,t2,t5|t3,t4|, |t1,t3,t4|t2,t5|, |t1,t3,t5|t2,t4|, |t1,t4,t5|t2,t3|, |t2,t3,t4|t1,t5|, |t2,t3,t5|t1,t4|, |t2,t4,t5|t1,t3|, |t3,t4,t5|t1,t2|, |t1,t2|t3,t4,t5|, |t1,t3|t2,t4,t5|, |t1,t4|t2,t3,t5|, |t1,t5|t2,t3,t4|, |t2,t3|t1,t4,t5|, |t2,t4|t1,t3,t5|, |t2,t5|t1,t3,t4|, |t3,t4|t1,t2,t5|, |t3,t5|t1,t2,t4|, and |t4,t5|,t1,t2,t3|. - Dennis P. Walsh, Apr 15 2013

MAPLE

spec := [ B, {B=Sequence(Set(Z, card>1))}, labeled ]; [seq(combstruct[count](spec, size=n), n=0..30)];

# second Maple program:

b:= proc(n) b(n):= `if`(n=0, 1, add(b(n-j)/j!, j=2..n)) end:

a:= n-> n!*b(n):

seq(a(n), n=0..25);  # Alois P. Heinz, Jul 29 2014

MATHEMATICA

a[n_] := n! * Sum[ Binomial[k, j] * StirlingS2[n-k+j, j]*j! / (n-k+j)! * (-1)^(k-j), {k, 1, n}, {j, 0, k}]; a[0] = 1; Table[a[n], {n, 0, 22}] (* Jean-Fran├žois Alcover, Sep 05 2012, from given formula *)

PROG

(PARI) x='x+O('x^66); Vec(serlaplace( 1/(2+x-exp(x)) ) ) \\ Joerg Arndt, Apr 16 2013

CROSSREFS

Cf. A102233, A232475.

Cf. column k=2 of A245732.

Sequence in context: A061961 A028248 A267609 * A084711 A183938 A060146

Adjacent sequences:  A032029 A032030 A032031 * A032033 A032034 A032035

KEYWORD

nonn

AUTHOR

Christian G. Bower

STATUS

approved

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Last modified October 20 21:31 EDT 2018. Contains 316404 sequences. (Running on oeis4.)