0,2

Number of hierarchical orderings for n labeled elements (see A075729) when there are two kinds A and B of elements.

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

Thomas Wieder, Home Page.

a(n)=2^n*A000262(n) =2^n*n!*sum{k=0..n, C(n-1,k)/(k+1)!}. - Paul Barry (pbarry(AT)wit.ie), Apr 28 2007

With p(n) = the number of integer partitions of n, d(i) = the number of different parts of the i-th partition of n, m(i, j) = multiplicity of the j-th part of the i-th partition of n, sum_{i=1}^{p(n)} = sum over i and prod_{j=1}^{d(i)} = product over j one has: a(n)=sum_{i=1}^{p(n)} n!/(prod_{j=1}^{d(i)} m(i, j)!) * 2^(n)

Let "a_i" and "b_j" be elements situated in the classes A and B with _i and _j as labels. Let : be a separator among levels (ranks). Let | be a separator among groups. E.g. a_1:b_2|b_1 is a hierarchy composed of two groups which contain three elements in total.

a(2) = 12 from b_2:b_1, b_2:a_1, b_2|b_1, a_1:a_2, b_2:a_1, a_1|a_2, a_1:b_2, a_2:a_1, b_1:a_2, a_2:b_1, b_1|a_2, b_2:b_1.

SetSeqUnnL := [T, {T=Set(S, card>=1), S=Sequence(U, card>=1), U=Union(a, b), a=Atom, b=Atom}, labeled]; seq(count(SetSeqUnnL, size=j), j=1..20);

A104533 := proc(n::integer) local i, j, prttnlst, prttn, liste, ZahlVerschiedenerTeile, H, Mltplztt; Mltplztt:=vector[1000]; prttnlst:=partition(n); H := 0; for i from 1 to nops(prttnlst) do prttn := prttnlst[i]; liste := convert(prttn, multiset); ZahlVerschiedenerTeile := nops(liste); for j from 1 to ZahlVerschiedenerTeile do Mltplztt[j] := op(2, op(j, liste)); od; H := H + (n!/mul(Mltplztt[j]!, j=1..ZahlVerschiedenerTeile)) * 2^n; od; print(n, H); end proc;

Cf. A075729, A034691, A034899.

Equals 2^n * A000262(n).

Sequence in context: A152254 A157328 A061632 this_sequence A125031 A085867 A141133

Adjacent sequences: A104530 A104531 A104532 this_sequence A104534 A104535 A104536

nonn

Thomas Wieder (wieder.thomas(AT)t-online.de), Mar 13 2005

Edited by N. J. A. Sloane (njas(AT)research.att.com), May 06 2008, at the suggestion of Joerg Arndt