A109092 Number of hierarchical orderings for n labeled elements with 2 possible classes A and B for levels l>=2. Labeled analog of A104460.

1, 6, 53, 619, 8972, 155067, 3109269, 70893872, 1810283331, 51151579619, 1583934062306, 53322541667501, 1938521128765093, 75673000809822670, 3156390306304019025, 140076451219218605087, 6589244960448222899044, 327461842184597424792623, 17141751726301435708168665

Offset

1,2

Links

Alois P. Heinz, Table of n, a(n) for n = 1..140

Robert Gill, The number of elements in a generalized partition semilattice, Discrete mathematics 186.1-3 (1998): 125-134. See Example 2.

Norihiro Nakashima, Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.

Formula

G.f.: exp(-(exp(z)-1)/(-3+2*exp(z))).

Example

Let | denote a separator among different hierarchies of the hierarchical ordering. Let : denote a separator between levels in a hierarchy.
Furthermore, let a[1], a[2],... denote labeled elements.
An element a[i] will be written as a[i,A] if it falls into class A and as a[i,B] if it falls into class B. Note that at level l=1 no classes appear.
Then a(2) = 6 because a[1]a[2], a[1]|a[2], a[1]:a[2,A], a[2]:a[1,A], a[1]:a[2,B], a[2]:a[1,B].

Maple

with(combstruct): A109092 := [T, {T=Set(Sequence(S, card>=1)), S=Sequence(U, card>=1), U=Set(Z, card>=1)}, labeled]; seq(count(A109092, size=j), j=1..20);

Mathematica

With[{nn=20}, CoefficientList[Series[Exp[-(Exp[x]-1)/(-3+2Exp[x])], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Sep 16 2016 *)

Crossrefs

Cf. A075729, A104460.

Sequence in context: A276365 A185148 A243921 * A068416 A360231 A360175

Adjacent sequences: A109089 A109090 A109091 * A109093 A109094 A109095

Keyword

nonn

Author

Thomas Wieder, Jun 18 2005

Status

approved