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
Keywords
Examples
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].
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.
Programs
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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);
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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 *)
Formula
G.f.: exp(-(exp(z)-1)/(-3+2*exp(z))).