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%I #34 Dec 28 2023 12:48:59
%S 1,1,3,7,17,38,87,191,421,911,1963,4186,8885,18724,39284,82005,170521,
%T 353214,729290,1501184,3081869,6311404,12896983,26301515,53541702,
%U 108815626,220824295,447524559,905850001,1831526719
%N Number of hierarchical ordered partitions of partitions.
%C 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).
%C 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.
%C 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.
%C With the generating function f := z -> 1/(mul(1-z^i/mul(1-z^j,j=1..i), i=1..25)); we get the asymptotic expansion using the command equivalent (f(z),z,n);
%C 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.
%C In short, a(n) is the number of finite sequences of integer partitions with weakly decreasing lengths and total sum n. The case of twice-partitions is A358831. A version choosing compositions is A218482. The strictly decreasing case is A358836. For ordered set partitions we have A005651. For weakly decreasing bigomega see A358335. - _Gus Wiseman_, Dec 05 2022
%H Seiichi Manyama, <a href="/A141199/b141199.txt">Table of n, a(n) for n = 0..1000</a>
%H Thomas Wieder, <a href="http://www.m-hikari.com/ams/ams-password-2009/ams-password53-56-2009/wiederAMS53-56-2009.pdf">The number of certain rankings and hierarchies formed from labeled or unlabeled elements and sets</a>, Applied Mathematical Sciences, vol. 3, 2009, no. 55, 2707 - 2724. [_Thomas Wieder_, Nov 14 2009]
%F G.f.: 1/Product_{i>=1} (1-x^i/Product_{j=1..i} (1-x^j)). - _Vladeta Jovovic_, Jul 16 2008
%e n=1:
%e [1]
%e -------------------------
%e n=2:
%e [1, 1],
%e [1, "|", 1],
%e [2]]
%e -------------------------
%e n=3:
%e [1, 2],
%e [1, "|", 1, "|", 1],
%e [1, 1, 1],
%e [3],
%e [2, "|", 1],
%e [1, 1, "|", 1],
%e [1, "|", 2]
%e -------------------------
%e n=4:
%e [1, 1, 1, "|", 1],
%e [1, 1, "|", 1, 1],
%e [2, 2],
%e [1, 3],
%e [1, 1, 1, 1],
%e [1, 1, 2],
%e [4],
%e [1, "|", 1, "|", 1, "|", 1],
%e [1, 2, "|", 1],
%e [1, 1, "|", 2],
%e [1, 1, "|", 1, "|", 1],
%e [2, "|", 1, "|", 1],
%e [1, "|", 2, "|", 1],
%e [1, "|", 1, "|", 2],
%e [1, "|", 3],
%e [3, "|", 1],
%e [2, "|", 2].
%p A Maple program to generate these "hierarchically ordered partitions of partitions" is available on request.
%p 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).
%o (PARI) my(N=40, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k/prod(j=1, k, 1-x^j))) \\ _Seiichi Manyama_, Jan 18 2022
%Y Cf. A055887, A083355, A140585.
%Y Cf. A000041, A000219, A001970, A005651, A063834, A129519, A218482, A358335, A358831, A358836, A358908.
%K nonn
%O 0,3
%A _Thomas Wieder_, Jun 13 2008, Jun 29 2008, Jul 28 2008
%E More terms from _Vladeta Jovovic_, Jul 16 2008
%E a(0)=1 prepended by _Seiichi Manyama_, Jan 18 2022
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