A102232 -id:A102232 - cp's OEIS frontend

A102233 Number of preferential arrangements of n labeled elements when at least k=3 elements per rank are required.

1, 0, 0, 1, 1, 1, 21, 71, 183, 2101, 13513, 64285, 629949, 5762615, 41992107, 427215283, 4789958371, 47283346849, 540921904725, 6980052633257, 85901272312905, 1129338979629643, 16398293425501375, 238339738265039119, 3588600147767147775, 58124879519314730741

Offset: 0

Thomas Wieder, Jan 01 2005

nonn

The labeled case for at least k=2 elements per rank is given by A032032 = Partition n labeled elements into sets of sizes of at least 2 and order the sets. The unlabeled case for at least k=3 elements per rank is given by A000930 = A Lamé sequence of higher order. The unlabeled case for at least k=2 elements per rank is given by A000045 = Fibonacci numbers.
With m = floor(n/3), 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 three 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 A200092 for n>=3. - Dennis P. Walsh, Apr 15 2013
Row sums of triangle A200092. - Dennis P. Walsh, Apr 15 2013

			Let 1,2,3,4,5,6 denote six labeled elements. Let | denote a separation between two ranks. E.g., if elements 1,2 and 3 are on rank (also called level) one and elements 3,4 and 5 are on rank two, then we have the ranking 123|456.
For n=9 we have a(9)=2101 rankings. The order within a rank does not count. Six examples are: 123|456|789; 123456789; 12345|6789; 129|345678; 1235|46789; 789|123456.

Alois P. Heinz, Table of n, a(n) for n = 0..400
Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013 and J. Int. Seq. 17 (2014) #14.1.1.

Cf. A000640, A102232, A032032, A232475.

Cf. column k=3 of A245732.

seq (n! *coeff (series (1- (z^2-2*exp(z)+2+2*z) /(4-2*exp(z)+2*z+z^2), z=0, n+1), z, n), n=0..30);
with(combstruct): SeqSetL := [S, {S=Sequence(U), U=Set(Z, card >= 3)}, labeled]: seq(count(SeqSetL, size=j), j=0..23); # Zerinvary Lajos, Oct 19 2006
# third Maple program:
b:= proc(n) b(n):= `if`(n=0, 1, add(b(n-j)/j!, j=3..n)) end:
a:= n-> n!*b(n):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 29 2014

CoefficientList[Series[1-(x^2-2*E^x+2+2*x)/(4-2*E^x+2*x+x^2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 29 2013 *)
b[n_] := b[n] = If[n==0, 1, Sum[b[n-j]/j!, {j, 3, n}]]; a[n_] := n!*b[n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 31 2016, after Alois P. Heinz *)

z='z+O('z^66); Vec(serlaplace( 1-(z^2-2*exp(z)+2+2*z) / (4-2*exp(z)+2*z+z^2) ) ) \\ Joerg Arndt, Apr 16 2013

E.g.f.: 1-(z^2-2*exp(z)+2+2*z)/(4-2*exp(z)+2*z+z^2).
a(n) = n! * sum(m=1..n, sum(k=0..m, k!*(-1)^(m-k) *binomial(m,k) *sum(i=0..n-m, stirling2(i+k,k) *binomial(m-k,n-m-i) *2^(-n+m+i) /(i+k)!))); a(0)=1. - Vladimir Kruchinin, Feb 01 2011
a(n) ~ 2*n!/((2+r^2)*r^(n+1)), where r = 1.56811999239... is the root of the equation 4+2*r+r^2 = 2*exp(r). - Vaclav Kotesovec, Sep 29 2013
a(0) = 1; a(n) = Sum_{k=3..n} binomial(n,k) * a(n-k). - Ilya Gutkovskiy, Feb 09 2020
E.g.f.: 1/(2 + x + x^2/2 - exp(x)). - Christian Sievers, Oct 27 2024

a(0) changed to 1 at the suggestion of Zerinvary Lajos, Oct 26 2006

A084416 Triangle read by rows: T(n,k) = Sum_{i=k..n} i!*Stirling2(n,i), n >= 1, 1 <= k <= n.

Original entry on oeis.org

1, 3, 2, 13, 12, 6, 75, 74, 60, 24, 541, 540, 510, 360, 120, 4683, 4682, 4620, 4080, 2520, 720, 47293, 47292, 47166, 45360, 36960, 20160, 5040, 545835, 545834, 545580, 539784, 498960, 372960, 181440, 40320, 7087261, 7087260, 7086750, 7068600, 6882120, 6048000, 4142880, 1814400, 362880

Offset: 1

N. J. A. Sloane, Jun 24 2003

Interpolates between A000670 and factorials.
From Thomas Scheuerle, Apr 25 2022: (Start)
Number of preferential arrangements of n labeled elements when at least k ranks are required.
This sequence starts for k and n with offset 1. If it would start with k = 0, we would observe in column k = 0 an exact copy of column k = 1 with a preceding one at n = 0, k = 0. The difference between 0 ranks and one rank (all in the same rank) is only for n = 0 where k = 0 allows zero-filled ranks as an valid arrangement, too. (End)

			Triangle begins with T(n,k):
   k=   1,   2,   3,   4,   5
  n=1   1
  n=2   3,   2
  n=3  13,  12,   6
  n=4  75,  74,  60,  24
  n=5 541, 540, 510, 360, 120
...
From _Thomas Scheuerle_, Apr 25 2022: (Start)
If we would add n = 0, k = 0 to the data of this sequence:
   k=   0,   1,   2,
  n=0   1
  n=1   1,   1
  n=2   3,   3,   2
...
T(n, 3) with 3 preceding zeros is: 0,0,0,6,60,510,4620,...
This sequence has the e.g.f.: (e^x-1)^3/(2-e^x).
.
13 arrangements for n = 3 and k = 1 (one rank required):
1,2,3  1,2|3  2,3|1  1,3|2  1|2,3  2|1,3  3|1,2  1|2|3  1|3|2  2|1|3  2|3|1  3|1|2  3|2|1
12 arrangements for n = 3 and k = 2 (two ranks required):
1,2|3  2,3|1  1,3|2  1|2,3  2|1,3  3|1,2  1|2|3  1|3|2  2|1|3  2|3|1  3|1|2  3|2|1
6 arrangements for n = 3 and k = 3 (three ranks required):
1|2|3  1|3|2  2|1|3  2|3|1  3|1|2  3|2|1
. (End)

Mirror image of array in A084417.
Cf. A005649, A069321 (row sums).
A000670(n) (column k = 1), A052875(n) (column k = 2), A102232(n) (column k = 3).
Cf. A000142, A008277.

T := (n,k)->sum(i!*Stirling2(n,i),i=k..n): seq(seq(T(n,k),k=1..n),n=1..10);

row(n) = vector(n, k, sum(i=k, n, i!*stirling(n, i, 2))); \\ Michel Marcus, Apr 20 2022

E.g.f. for m-th column: (exp(x)-1)^m/(2-exp(x)). - Vladeta Jovovic, Sep 14 2003
T(n, k) = Sum_{m = k..n} A090582(n + 1, m + 1).
From Thomas Scheuerle, Apr 25 2022: (Start)
Sum_{k = 0..n} T(n, k) = A005649(n). Column k = 0 is not part of data.
Sum_{k = 1..n} T(n, k) = A069321(n).
T(n, 0) = A000670(n). Column k = 0 is not part of data.
T(n, 1) = A000670(n), for n > 0.
T(n, 2) = A052875(n).
T(n, 3) = A102232(n).
T(n, n) = n! = A000142. (End)

More terms from Emeric Deutsch, May 11 2004

More terms from Michel Marcus, Apr 20 2022

cp's OEIS Frontend

A102233 Number of preferential arrangements of n labeled elements when at least k=3 elements per rank are required.

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