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A141199 Number of hierarchical ordered partitions of partitions. 2
1, 3, 7, 17, 38, 87, 191, 421, 911, 1963, 4186, 8885, 18724, 39284, 82005, 170521, 353214, 729290, 1501184, 3081869, 6311404, 12896983, 26301515, 53541702, 108815626, 220824295, 447524559, 905850001, 1831526719 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Consider the "ordered partitions of partitions" as described in A055887. They are produced by introducing separators (a term used by J. Riordan) between the parts of a partition. If a partition has P parts, then it is possible to introduce 1, 2, ... P-1 separators. Let "|" denote such a separator. We just append 1,2,...,P-1 separators to each integer partition of n and subsequently form all permutation of the resulting list (which is composed of parts and separators).

There are some rules: If we do not append a separator, then we do not perform any permutation. Furthermore, we do not accept permutations which have a dangling separator in front of the integer parts or past them. E.g. the permutations [|,1,2,3] and [1,2,3,|] are forbidden. Furthermore, sequences of separators as "|,|" are forbidden.

Now we impose a further restriction on the permutations. Consider the elements between two separators. We call their number "occupation number". We just request that the occupation number of a ordered partition is monotonically decreasing (if we start from the left to the right of a permutation written in our notation). If we interpret a separator as a level, then we can speak of a hierarchy. E.g. we do not count [1,|,2,3,|,4] as a hierarchy, but we accept [1,2|,3,4] as a hierarchy. We thus speak of "hierarchically ordered partitions of partitions" for this sequence.

With the generating function f := z -> 1/(mul(1-z^i/mul(1-z^j,j=1..i), i=1..25)); we get the asymptotc expansion using the command equivalent (f(z),z,n);

The result is 3.788561346*exp(-n)^(-log(2)) + O(1/n*exp(-n)^(-log(2))). Let fas := n -> 3.788562346*exp(-n)^(-log(2)); then for n=60 we get fas(60)/A141199(60)= .4367915009e19/4344507472742893655 = 1.005387846.

LINKS

Table of n, a(n) for n=1..29.

Thomas Wieder, The number of certain rankings and hierarchies formed from labeled or unlabeled elements and sets, Applied Mathematical Sciences, vol. 3, 2009, no. 55, 2707 - 2724. [Thomas Wieder, Nov 14 2009]

FORMULA

G.f.: 1/Product(1-x^i/Product(1-x^j,j=1..i), i=1..infinity). - Vladeta Jovovic, Jul 16 2008

EXAMPLE

n=1:

[1]

-------------------------

n=2:

[1, 1],

[1, "|", 1],

[2]]

-------------------------

n=3:

[1, 2],

[1, "|", 1, "|", 1],

[1, 1, 1],

[3],

[2, "|", 1],

[1, 1, "|", 1],

[1, "|", 2]

-------------------------

n=4:

[1, 1, 1, "|", 1],

[1, 1, "|", 1, 1],

[2, 2],

[1, 3],

[1, 1, 1, 1],

[1, 1, 2],

[4],

[1, "|", 1, "|", 1, "|", 1],

[1, 2, "|", 1],

[1, 1, "|", 2],

[1, 1, "|", 1, "|", 1],

[2, "|", 1, "|", 1],

[1, "|", 2, "|", 1],

[1, "|", 1, "|", 2],

[1, "|", 3],

[3, "|", 1],

[2, "|", 2].

MAPLE

A Maple program to generate these "hierarchically ordered partitions of partitions" is available on request.

An asymptotic expansion can be found using the generating function given by Vladeta Jovovic. For that purpose we use the Maple program "equivalent" from Bruno Salvy (http://ago.inria.fr/libraries/libraries.html).

CROSSREFS

Cf. A055887, A083355, A140585.

Sequence in context: A026396 A176502 A319003 * A003478 A119587 A127984

Adjacent sequences:  A141196 A141197 A141198 * A141200 A141201 A141202

KEYWORD

nonn

AUTHOR

Thomas Wieder, Jun 13 2008, Jun 29 2008, Jul 28 2008

EXTENSIONS

More terms from Vladeta Jovovic, Jul 16 2008

STATUS

approved

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Last modified June 4 08:18 EDT 2020. Contains 334825 sequences. (Running on oeis4.)