A088814 -id:A088814 - cp's OEIS frontend

A084357 Number of sets of sets of lists.

1, 1, 4, 23, 171, 1552, 16583, 203443, 2813660, 43258011, 731183365, 13466814110, 268270250977, 5744515120489, 131525839441428, 3205279987587275, 82812074976214547, 2260364854328771548, 64979726427408468055, 1961976154991285214707, 62065551492895731512852

Offset: 0

N. J. A. Sloane, Jun 22 2003

nonn

In the book by Flajolet and Sedgewick on page 139 incorrectly gives a(5) = 1542. - Vaclav Kotesovec, Jul 11 2020

T. S. Motzkin, Sorting numbers ...: for a link to an annotated scanned version of this paper see A000262.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.

T. D. Noe, Table of n, a(n) for n = 0..100
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 139.
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]
K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, Hierarchical Dobinski-type relations via substitution and the moment problem
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.

Row sums of A079005 and row sums of A088814.

Cf. A000110, A008297, A271703.

with(combstruct); SetSetSeqL := [T, {T=Set(S), S=Set(U,card >= 1), U=Sequence(Z,card >=1)},labeled]; [seq(count(%,size=j),j=1..12)];

a[n_] = Sum[ n!/k!*Binomial[n-1, k-1]*BellB[k], {k, 0, n}]; a[0] = 1; Array[a, 20, 0]
(* Jean-François Alcover, Jun 22 2011, after Vladeta Jovovic *)

E.g.f.: exp(exp(x/(1-x))-1). Lah transform of Bell numbers: Sum_{k=0..n} n!/k!*binomial(n-1, k-1)*Bell(k). - Vladeta Jovovic, Sep 28 2003

A256892 Triangular array read by rows, the matrix product of the unsigned Lah numbers and the Stirling set numbers, T(n,k) for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 13, 9, 1, 0, 73, 79, 18, 1, 0, 501, 755, 265, 30, 1, 0, 4051, 7981, 3840, 665, 45, 1, 0, 37633, 93135, 57631, 13580, 1400, 63, 1, 0, 394353, 1192591, 911582, 274141, 38290, 2618, 84, 1, 0, 4596553, 16645431, 15285313, 5633922, 999831, 92358, 4494, 108, 1

Offset: 0

Peter Luschny, Apr 12 2015

Also the Bell transform of A000262(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016

			Triangle starts:
1;
0,    1;
0,    3,    1;
0,   13,    9,    1;
0,   73,   79,   18,   1;
0,  501,  755,  265,  30,  1;
0, 4051, 7981, 3840, 665, 45, 1;

See also A088814 and A088729 for variants based on an (1,1)-offset of the number triangles. See A131222 for the product Lah * Stirling-cycle.
A079640 is an unsigned matrix inverse reduced to an (1,1)-offset.
Cf. A000262, A045943, A084357, A088729, A088814.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> simplify(hypergeom([-n, -n-1], [], 1)), 9); # Peter Luschny, Jan 29 2016

BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, HypergeometricPFQ[{-n, -n-1}, {}, 1]], rows = 12];
Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

def Lah(n, k):
    if n == k: return 1
    if k<0 or  k>n: return 0
    return (k*n*gamma(n)^2)/(gamma(k+1)^2*gamma(n-k+1))
matrix(ZZ, 8, Lah) * matrix(ZZ, 8, stirling_number2)  # as a square matrix

T(n+1,1) = A000262(n).
T(n+1,n) = A045943(n).
Row sums are A084357.

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A084357 Number of sets of sets of lists.

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