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 A140585 Total number of all hierarchical orderings for all set partitions of n. 9
 1, 4, 20, 129, 1012, 9341, 99213, 1191392, 15958404, 235939211, 3817327362, 67103292438, 1273789853650, 25973844914959, 566329335460917, 13150556885604115, 324045146807055210, 8446201774570017379, 232198473069120178475, 6715304449424099384968 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 1..420 (first 160 terms from Alois P. Heinz) 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] FORMULA Stirling transform of A005651 = Sum of multinomial coefficients: a(n) = Sum_{i=1..n} S2(n,k) A005651(k). E.g.f.: 1/Product_{k>=1} (1 - (exp(x)-1)^k/k!). - Thomas Wieder, Sep 04 2008 a(n) ~ c * n! / (log(2))^n, where c = 1/(2*log(2)) * Product_{k>=2} 1/(1-1/k!) = A247551 / (2*log(2)) = 1.82463230250447246267598544320244231645906135137... . - Vaclav Kotesovec, Sep 04 2014, updated Jan 21 2017 EXAMPLE We are considering all set partitions of the n-set {1,2,3,...,n}. For each such set partition we examine all possible hierarchical arrangements of the subsets. A hierarchy is a distribution of elements (sets in the present case) onto levels. A distribution onto levels is "hierarchical" if a level l+1 contains less or equal elements than the level l. Thus for n=4 the arrangement {1,2}:{3}{4} is not allowed. Let the colon ":" separate two consecutive levels l and l+1. n=2 --> 1+3=4 {1,2} {1}{2} {1}:{2} {2}:{1} n=3 --> 1+9+10=20 1*1 3*3=9 1*10 {1,2,3} {1,2}{3} {1}{2}{3} {1,3}{2} {2,3}{1} {1}{2}:{3} {3}{1}:{2} {1,2}:{3} {2}{3}:{1} {1,3}:{2} {2,3}:{1} {1}:{2}:{3} {3}:{1}:{2} {3}:{1,2} {2}:{3}:{1} {2}:{1,3} {1}:{3}:{2} {1}:{2,3} {2}:{1}:{3} {3}:{2}:{1} n=4 --> 1+12+9+60+47=129 1*1 4*3=12 3*3=9 6*10=60 1*47 {1,2,3,4} {1,2,3}{4} {1,2}{3,4} {1,2}{3}{4} {1}{2}{3}{4} {1,2,4}{3} {1,3}{2,4} {1,2}{3}:{4} {1,3,4}{2} {1,4}{2,3} {1,2}{4}:{3} {1}{2}:{3}:{4} {2,3,4}{1} {1}{2}:{3,4} {1}{3}:{2}:{4} {1,2}:{3,4} {1,2}:{3}:{4} {1}{4}:{2}:{3} {1,2,3}:{4} {1,3}:{2,4} {1,2}:{4}:{3} {1}{2}:{4}:{3} {1,2,4}:{3} {1,4}:{2,3} {1}:{2}:{3,4} {1}{3}:{4}:{2} {1,3,4}:{2} {3,4}:{1,2} {2}:{1}:{3,4} {1}{4}:{3}:{2} {2,3,4}:{1} {2,4}:{1,3} {1}:{3,4}:{2} {2,3}:{1,4} {2}:{3,4}:{1} {2}{3}:{1}:{4} {4}:{1,2,3} {2}{4}:{1}:{3} {3}:{1,2,4} likewise for: {2}{3}:{4}:{1} {2}:{1,3,4} {3,4}{1}{2} {2}{4}:{3}:{1} {1}:{2,3,4} {2,4}{1}{3} {2,3}{1}{4} {3}{4}:{1}:{2} {1,4}{2}{3} {3}{4}:{2}:{1} {1,3}{2}{4} {1}{2}:{3}{4} {1}{3}:{2}{4} {1}{4}:{2}{3} {2}{3}:{1}{4} {2}{4}:{1}{3} {3}{4}:{1}{2} {2}{3}{4}:{1} {1}{3}{4}:{2} {1}{2}{4}:{3} {1}{2}{3}:{4} {1}:{2}:{3}:{4} +23 permutations MAPLE A140585 := proc(n::integer) local k, Result; Result:=0: for k from 1 to n do Result:=Result+stirling2(n, k)*A005651(k); end do; lprint(Result); end proc; E.g.f.: series(1/mul(1-(exp(x)-1)^k/k!, k=1..10), x=0, 10). # Thomas Wieder, Sep 04 2008 # second Maple program: with(numtheory): b:= proc(k) option remember; add(d/d!^(k/d), d=divisors(k)) end: c:= proc(n) option remember; `if`(n=0, 1, add((n-1)!/ (n-k)!* b(k)* c(n-k), k=1..n)) end: a:= n-> add(Stirling2(n, k) *c(k), k=1..n): seq(a(n), n=1..30); # Alois P. Heinz, Oct 10 2008 MATHEMATICA Table[n!*SeriesCoefficient[1/Product[(1-(E^x-1)^k/k!), {k, 1, n}], {x, 0, n}], {n, 1, 20}] (* Vaclav Kotesovec, Sep 03 2014 *) CROSSREFS Cf. A005651, A075729, A034691, A083355, A109186, A000670. Sequence in context: A126674 A196557 A082032 * A132436 A307006 A208735 Adjacent sequences:  A140582 A140583 A140584 * A140586 A140587 A140588 KEYWORD nonn AUTHOR Thomas Wieder, May 17 2008 EXTENSIONS More terms from Alois P. Heinz, Oct 10 2008 STATUS approved

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Last modified January 17 20:36 EST 2020. Contains 330987 sequences. (Running on oeis4.)