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A059849 Number of pairs of partitions of {1,2,...,n} whose meet is the partition {{1}, {2}, ..., {n}}.

%I A059849 #36 Jan 07 2025 02:00:31
%S A059849 1,1,3,15,113,1153,15125,245829,4815403,111308699,2985997351,
%T A059849 91712874487,3189130896077,124366296990757,5395176819674205,
%U A059849 258547307299130037,13603419571939001827,781604484498111072195,48806254671145521802863,3298007680091577596528415
%N A059849 Number of pairs of partitions of {1,2,...,n} whose meet is the partition {{1}, {2}, ..., {n}}.
%H A059849 Alois P. Heinz, <a href="/A059849/b059849.txt">Table of n, a(n) for n = 0..325</a>
%H A059849 P. J. Cameron, D. A. Gewurz and F. Merola, <a href="http://www.maths.qmw.ac.uk/~pjc/preprints/product.pdf">Product action</a>, Discrete Math., 308 (2008), 386-394.
%H A059849 E. R. Canfield, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v8i1r15">Meet and join in the partition lattice</a>, Electronic Journal of Combinatorics, 8 (2001) R15.
%H A059849 B. Pittel, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v7i1r5">Where the typical set partitions meet and join</a>, Electronic Journal of Combinatorics, 7 (2000) R5.
%H A059849 Frank Simon, <a href="https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa-101154">Algebraic Methods for Computing the Reliability of Networks</a>, Dissertation, Doctor Rerum Naturalium (Dr. rer. nat.), Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden, 2012. - _N. J. A. Sloane_, Jan 04 2013
%F A059849 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).
%F A059849 E.g.f.: Sum_{n>=0} exp( (1+x)^n - 2 ) / n!. - _Paul D. Hanna_, Jul 24 2018
%F A059849 a(n) = Sum_{k=0..n} Stirling1(n, k)*Bell(k)^2. - _Vladeta Jovovic_, Oct 01 2003
%e A059849 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}}.
%t A059849 a[n_] := Sum[StirlingS1[n, k]*BellB[k]^2, {k, 0, n}]; Array[a, 20] (* _Robert G. Wilson v_, Jul 24 2018 *)
%o A059849 (PARI) /* From Vladeta Jovovic's formula: */
%o A059849 {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
%o A059849 {Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}
%o A059849 {a(n)=sum(k=0, n, Stirling1(n, k)*Bell(k)^2)}
%Y A059849 Cf. Bell numbers A000110. Also A007311 and Stirling numbers of the second kind, A000225.
%K A059849 nonn
%O A059849 0,3
%A A059849 E. R. Canfield (erc(AT)cs.uga.edu), Feb 26 2001
%E A059849 More terms from _Vladeta Jovovic_, Mar 04 2001