A059849 Number of pairs of partitions of {1,2,...,n} whose meet is the partition {{1}, {2}, ..., {n}}.
1, 1, 3, 15, 113, 1153, 15125, 245829, 4815403, 111308699, 2985997351, 91712874487, 3189130896077, 124366296990757, 5395176819674205, 258547307299130037, 13603419571939001827, 781604484498111072195, 48806254671145521802863, 3298007680091577596528415
Offset: 0
Keywords
Examples
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}}.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..325
- 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
Programs
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Mathematica
a[n_] := Sum[StirlingS1[n, k]*BellB[k]^2, {k, 0, n}]; Array[a, 20] (* Robert G. Wilson v, Jul 24 2018 *) -
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)}
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
Extensions
More terms from Vladeta Jovovic, Mar 04 2001