A075729 Number of different hierarchical orderings that can be formed from n labeled elements: these are divided into groups and the elements in each group are then arranged in a "preferential arrangement" or "weak order" as in A000670.

1, 1, 4, 23, 173, 1602, 17575, 222497, 3188806, 50988405, 899222457, 17329515172, 362164300173, 8155216185781, 196789115887252, 5064722539020379, 138457553073641465, 4006059432756066914, 122284085809137076203, 3926775294104305483621, 132313462760902116605534

Offset: 0

Author

Thomas Wieder and N. J. A. Sloane, Oct 06 2002

Keywords

• nonn
• nice

Comments

If all individuals form a single society ("uniparate society"), then the number of different hierarchies for that single society is equal to the ordered Bell number Bell_ordered(n) (A000670).

Represent a labeled pre-order (quasi-order, topology, A000798) as a directed graph. a(n) is the number of such digraphs in which the underlying graph of each component is complete. a(3)=23 because there are 29 such digraphs but o->o<-o and o<-o->o are not counted. Each has 3 labelings. 29 - 6 = 23. - Geoffrey Critzer, Jul 30 2014

Examples

			a(3) = 23: Let the n = 3 individuals be named 1, 2 and 3. Let a pair of parentheses () indicate a society and let square brackets [] denote a set of disparate societies. Finally, let the ranks be ordered from left to right and separated by a colon, e.g., (1,2:3) is a society with individual 3 on top and individuals 1 and 2 on the same bottom rank.
Then the hierarchical ordering for n = 3 is composed of the following sets: [(1),(2),(3)], [(1,2)(3)], [(3,2)(1)], [(3,1)(2)], [(1:2)(3)], [(3:2)(1)], [(1:3)(2)], [(2:1)(3)], [(2:3)(1)], [(3:1)(2)], [(3:2:1)], [(1:3:2)], [(2:1:3)], [(1:2:3)], [(3:1:2)], [(2:3:1)], [(1,3:2)], [(3,2:1)], [(2,1:3)], [(3:1,2)], [(1:2,3)], [(2:3,1)], [(1,2,3)].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..419 (first 101 terms from T. D. Noe)
• José A. Adell and Sithembele Nkonkobe, A unified generalization of Touchard and Fubini polynomial extensions, Integers (2023) 23, #A80.
• INRIA, Algolib: The Algorithms Project's Library
• P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 571.
• Robert Gill, The number of elements in a generalized partition semilattice, Discrete mathematics 186.1-3 (1998): 125-134. See Example 2.
• Norihiro Nakashima and Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.
• K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, Hierarchical Dobinski-type relations via substitution and the moment problem [J. Phys. A 37 (2004), 3475-3487]
• N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.
• Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
• Thomas Wieder, The number of certain rankings and hierarchies formed from labeled or unlabeled elements and sets, Applied Mathematical Sciences, vol. 3, 2009, no. 55, 2707 - 2724. - _Thomas Wieder_, Nov 14 2009
• Index entries for sequences related to parenthesizing

Crossrefs

Cf. A000670, A075744. See A075900 for the unlabeled case.

Programs

Maple

A075729 := n->n!*exp(1/4/ln(2)-3/4)/2/sqrt(Pi)/(2*ln(2))^(1/4)*exp(-n*ln(ln(2)))*exp(sqrt(2*n/ln(2)))*n^(-3/4);
with(combstruct); SetSeqSetL := [T, {T=Set(S), S=Sequence(U,card >= 1), U=Set(Z,card >=1)},labeled]; seq(count(SetSeqSetL,size=j),j=1..12);
# alternative Maple program:
b:= proc(n) option remember: `if`(n<2, 1,
      (2*n-1)*b(n-1) -(n-1)*(n-2)*b(n-2))
    end:
a:= n-> add(b(k)*Stirling2(n,k), k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, May 22 2018

Mathematica

Range[0, 20]!CoefficientList[Series[E^(1/(2 - E^x) - 1), {x, 0, 20}], x] (* Robert G. Wilson v, Jul 13 2004 *)
Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; Fubini[0, 1] = 1; a[0] = 1; a[n_] := a[n] = (n-1)! Sum[a[n-k] Fubini[k, 1]/((n-k)! (k-1)!), {k, 1, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 31 2016 *)
Table[Sum[BellY[n, k, PolyLog[-Range[n], 1/2]/2], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)
With[{nn=20},CoefficientList[Series[Exp[1/(2-Exp[x])-1],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 26 2022 *)

Maxima

a(n):= sum(sum(stirling2(n,k)*k!*binomial(k-1,m-1), k,m,n)/m!, m,1,n) /* Vladimir Kruchinin, Aug 10 2010 */

Formula

E.g.f.: exp(f(x)-1) where f(x) = 1/(2-exp(x)) = e.g.f. for A000670.

STIRLINGi transform of A000262.

a(n) = (n-1)! * Sum_k=1^n a(n-k)*b(k)/((n-k)!*(k-1)!); a(n) = a(n) + C(n-1, k-1)*a(n-k)*b(k) (where b(n) = A000670(n)). - Thomas Wieder, Dec 31 2002

a(n) = (Sum_{j=1..n} m(j))*(n!*Product_{j=1..n} B(j)^m(j))/(Product_{j=1..n} (m(j))!*(j!)^m(j)), where the sum is over all (m(1),m(2),...,m(n)) such that Sum_{j=1..n} (j*m(j)) = n. - Thomas Wieder, May 18 2003

a(n) is asymptotic to exp(1/(4*log(2))-3/4) /(2*sqrt(Pi*sqrt(2*log(2)))) *n!*exp(-log(log(2))*n)*exp(sqrt(2*n /log(2))) /n^(3/4). Calculated using the Maple package "algolib", using the command "equivalent(exp(1/(2-exp(x))-1), x, n);". - Thomas Wieder, Nov 12 2002

a(n) = Sum_{k=0..n} A079641(n,k)*A000110(k). - Vladeta Jovovic, Sep 25 2006

a(n) = sum(sum(stirling2(n,k)*k!*C(k-1,m-1), k=m..n)/m!, m=1..n). - Vladimir Kruchinin, Aug 10 2010