Comments on A104525 Thomas Wieder (thomas.wieder(AT)t-online.de), May 31 2008 A104525 gives the number of hierarchical orderings among the parts of the integer partitions of the integer n. In order to explain this structure we first consider the ordered partitions of partitions (opops) of the integer n, that is sequence A055887(n). Let the colon ":" be a position separator. We think of hierarchies where ":" separates two consecutive ranks. The semicolon ";" just separates two opops. For n=3 we have A055887(3)=8. These 8 opops are: {{1},{1},{1}}; {1,1,1}; {1}:{1}:{1}; {1,1}:{1}; {1}:{{1},{1}}; {{1},{1}}:{1}; {1}:{1,1}; {1},{1,1}. An equivalent (and may be more appealing) representation is: 1,1,1; 3; 1:1:1; 2:1; 1:{1,1}; {1,1}:1; 1:2; 1,2. We see that an opop can be interpreted as an hierarchical ordering. In an hierarchical ordering either unlabeled or labeled elements are distributed among labeled ranks. The nature of the elements in the case of an opop is particular, the elements are subsets of the n-set {1,1,...,1} composed of n unlabeled elements "1". In the second step, we introduce a second separator "|". Two parts (subsets) of a partition belong to two different opops if they are separated by "|". In this way we introduce multiple opopos for a single partition. For our example with n=3 we get the 12 structures {{1},{1},{1}}; {1},{1},{1}; {1,1,1}; {1}:{1}:{1}; {1},{1}:{1}; {1,1}:{1}; {1}:{{1},{1}}; {1},{{1},{1}}; {{1},{1}}:{1}; {1}:{1,1}; {1},{1,1}; {{1},{1,1}}. Or equivalently: 1,1,1; 1|1|1; 3; 1:1:1; 1|1:1; 2:1; 1:{1,1}; 1|{1,1}; {1,1}:1; 1:2; 1|2; 1,2. Thus the hierarchical ordering A104525(n) follows from the application of the two separators ":" and "|" on the parts on the integer partition of n. A104525 is therefore the partition analogon to sequence A034691. Whereas for A104525 the elements are parts ("groups"), for A034691 the elements may be thought as unlabeled atoms ("individuals"). The labeled analoga for this point of view are A109186 and A075729. The e.g.f. of A104525 follows from an Euler transform of the opop A055887. If a(n) denotes A055887(n), then (in Maple notation) e.g.f. = mul(1/(1-x^n)^a(n),n=1..inf) For definitions of transforms on integer sequences see: M. Bernstein & N. J. A. Sloane, Some canonical sequences of integers, Linear Algebra and its Applications, 226-228 (1995), 57-72. We can use combstruct to actually construct the structures A104525(n). The combstruct command is Test := [V,{V=Set(U),U=Sequence(S,card>=1),S=Set(T,card>=1),T=Set(Z,card>=1)}, unlabelled]; seq(count(Test,size=j),j=1..20); For n=3 the comstruct command "allstructs(Test,size=3);" returns [Set(Sequence(Set(Set(Z), Set(Z, Z)))), Set(Sequence(Set(Set(Z)), Set(Set(Z)), Set(Set(Z)))), Set(Sequence(Set(Set(Z), Set(Z)), Set(Set(Z)))), Set(%1, Sequence(Set(Set(Z, Z)))), Set(Sequence(Set(Set(Z), Set(Z), Set(Z)))), Set(%1, %1, %1), Set(Sequence(Set(Set(Z, Z)), Set(Set(Z)))), Set(%1, Sequence(Set(Set(Z), Set(Z)))), Set(%1, Sequence(Set(Set(Z)), Set(Set(Z)))), Set(Sequence(Set(Set(Z)), Set(Set(Z), Set(Z)))), Set(Sequence(Set(Set(Z, Z, Z)))), Set(Sequence(Set(Set(Z)), Set(Set(Z, Z))))] %1 := Sequence(Set(Set(Z))).