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A059849
Number of pairs of partitions of {1,2,...,n} whose meet is the partition {{1}, {2}, ..., {n}}.
23
1, 1, 3, 15, 113, 1153, 15125, 245829, 4815403, 111308699, 2985997351, 91712874487, 3189130896077, 124366296990757, 5395176819674205, 258547307299130037, 13603419571939001827, 781604484498111072195, 48806254671145521802863, 3298007680091577596528415
OFFSET
0,3
P. J. Cameron, D. A. Gewurz and F. Merola, Product action, Discrete Math., 308 (2008), 386-394.
E. R. Canfield, Meet and join in the partition lattice, Electronic Journal of Combinatorics, 8 (2001) R15.
B. Pittel, Where the typical set partitions meet and join, Electronic Journal of Combinatorics, 7 (2000) R5.
Frank Simon, Algebraic Methods for Computing the Reliability of Networks, Dissertation, Doctor Rerum Naturalium (Dr. rer. nat.), Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden, 2012. - N. J. A. Sloane, Jan 04 2013
FORMULA
E.g.f. M(x) satisfies the equation M(exp(x)-1) = sum_{n >= 0)} (B_n)^2 * x^n/n!, where B_n is the n-th Bell number (A000110).
E.g.f.: Sum_{n>=0} exp( (1+x)^n - 2 ) / n!. - Paul D. Hanna, Jul 24 2018
a(n) = Sum_{k=0..n} Stirling1(n, k)*Bell(k)^2. - Vladeta Jovovic, Oct 01 2003
EXAMPLE
a(2) = 3 because there are two partitions of {1,2} and of the four possible pairs, only the pair ( {{1,2}}, {{1,2}} ) fails to have meet equal to {{1},{2}}.
MATHEMATICA
a[n_] := Sum[StirlingS1[n, k]*BellB[k]^2, {k, 0, n}]; Array[a, 20] (* Robert G. Wilson v, Jul 24 2018 *)
PROG
(PARI) /* From Vladeta Jovovic's formula: */
{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}
{a(n)=sum(k=0, n, Stirling1(n, k)*Bell(k)^2)}
CROSSREFS
Cf. Bell numbers A000110. Also A007311 and Stirling numbers of the second kind, A000225.
Sequence in context: A056053 A295758 A343707 * A123853 A357794 A335531
KEYWORD
nonn
AUTHOR
E. R. Canfield (erc(AT)cs.uga.edu), Feb 26 2001
EXTENSIONS
More terms from Vladeta Jovovic, Mar 04 2001
STATUS
approved

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Last modified September 20 13:29 EDT 2024. Contains 376072 sequences. (Running on oeis4.)