A104525 The number of hierarchical orderings among the parts of the integer partitions of the integer n.

1, 4, 12, 40, 123, 395, 1227, 3851, 11944, 37032, 114144, 351040, 1075316, 3285398, 10007731, 30409157, 92169561, 278738219, 841132013, 2533138770, 7614144053, 22845435104, 68427663680, 204623945617, 610951554377, 1821438443615, 5422608839874, 16121857331124

Offset

1,2

Comments

Euler transform of A055887 = number of ordered partitions of partitions.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..750

N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.

Thomas Wieder, Comments on A104525

Example

Let * denote an element, let : denote separator among different levels within a hierarchy, let | denote a separator between different hierarchies. Furthermore, the braces {} indicate a frame. For n=3 one has a(3) = 12 because:
{*:**}, {*:*}:{*}, {*}:{**}, {*:*:*}, {*}:{*}:{*}, {**}|{*}, {*}|{*:*}, {*}|{*}|{*}, {**}:{*}, {*}:{*:*}, {*}:{*}|{*}, {***}.

Maple

We can use combstruct to actually construct the structures A104525(n). %1 := Sequence(Set(Set(Z))).
with(combinat): with (numtheory): b:= proc(n) local k; option remember; `if`(n=0, 1, add (numbpart(k) * b(n-k), k=1..n)) end: a:= proc(n) option remember; `if` (n=0, 1, add (add (d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n) end: seq (a(n), n=1..30); # Alois P. Heinz, Feb 02 2009

Mathematica

max = 30; A055887 = CoefficientList[1/(2 - 1/QPochhammer[x, x]) + O[x]^(max + 1), x] ; s = 1/Product[(1 - x^n)^A055887[[n + 1]], {n, 1, max}] + O[x]^max; CoefficientList[s, x] // Rest (* Jean-François Alcover, Jan 10 2016 *)

Crossrefs

Cf. A034691, A034899, A055887, A104460, A104500, A109186.

Sequence in context: A056274 A335806 A058353 * A126986 A341990 A090576

Adjacent sequences: A104522 A104523 A104524 * A104526 A104527 A104528

Keyword

nonn

Author

Thomas Wieder, Mar 12 2005. Definition revised Mar 28 2009

Extensions

More terms from Alois P. Heinz, Feb 02 2009

Status

approved