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 A060639 Number of pairs of partitions of [n] whose join is the partition {{1,2,...,n}}. 23
 1, 1, 3, 15, 119, 1343, 19905, 369113, 8285261, 219627683, 6746244739, 236561380795, 9356173080985, 413251604702069, 20215438754502217, 1087524296159855603, 63950948621703499839, 4089003767746536828183, 282970817307108139386841, 21107742616278461624923449, 1690957890908364634072451893 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS It appears that a(n) = 2*A001188(n) - 1 for n > 0. This holds for the first 50 terms. - Charles R Greathouse IV, Mar 21 2012 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..325 E. R. Canfield, Meet and join in the partition lattice, Electronic Journal of Combinatorics, 8 (2001) R15. I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, et al., Enumeration of idempotents in diagram semigroups and algebras, arXiv preprint arXiv:1408.2021 [math.GR], 2014; Table 3. 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. FORMULA The e.g.f. J(x) satisfies the equation Sum_{n>=0} (B_n)^2 x^n/n! = exp(J(x)-1), where B_n is the n-th Bell number. a(0) = 1; a(n) = Bell(n)^2 - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Bell(n-k)^2 * k * a(k). - Ilya Gutkovskiy, Jan 17 2020 EXAMPLE J(2) = 3 because there are two partitions of {1,2} and of the four pairs of partitions, only the pair ( {{1},{2}}, {{1},{2}} ) fails to have join {{1,2}}. MATHEMATICA list[n_] := Module[{t}, t = Table[BellB[k-1]^2/(k-1)!, {k, 1, n+1}]; CoefficientList[1+Log[O[x]^(n+1)+Sum[t[[i]] x^(i-1), {i, 1, Length[t]}]], x]]; list[17] Range[0, 17]! (* Jean-François Alcover, Nov 03 2018, from PARI *) PROG (PARI) Bell(n)=round(suminf(k=0, k^n/k!)/exp(1)) list(n)=my(v=Vec(log(O(x^(n+1))+Polrev(vector(n+1, k, Bell(k-1)^2/(k-1)!))))); concat(1, vector(n, i, v[i]*i!)) \\ Charles R Greathouse IV, Mar 21 2012 CROSSREFS Bell numbers: A000110, Stirling numbers of the second kind: A000225, number of pairs whose meet equals {{1}, {2}, ..., {n}}: A059849. Cf. A001188, A318815. Sequence in context: A136654 A145161 A121422 * A068052 A068859 A006454 Adjacent sequences:  A060636 A060637 A060638 * A060640 A060641 A060642 KEYWORD nonn AUTHOR E. R. Canfield (erc(AT)cs.uga.edu), Apr 16 2001 EXTENSIONS More terms from Vladeta Jovovic, Apr 18 2001 STATUS approved

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Last modified June 15 04:47 EDT 2021. Contains 345043 sequences. (Running on oeis4.)