1,2

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 monotonously

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

Thomas Wieder, Home Page.

Thomas Wieder, (Old) Home Page.

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

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].

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).

Cf. A055887, A083355, A140585.

Adjacent sequences: A141196 A141197 A141198 this_sequence A141200 A141201 A141202

Sequence in context: A111210 A033489 A026396 this_sequence A003478 A119587 A127984

nonn

Thomas Wieder (thomas.wieder(AT)t-online.de), Jun 13 2008, Jun 29 2008, Jul 28 2008

More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Jul 16 2008