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A005121
Number of ultradissimilarity relations on an n-set.
49
1, 1, 4, 32, 436, 9012, 262760, 10270696, 518277560, 32795928016, 2542945605432, 237106822506952, 26173354092593696, 3375693096567983232, 502995942483693043200, 85750135569136650473360, 16583651916595710735271248, 3611157196483089769387182064, 879518067472225603327860638128
OFFSET
1,3
COMMENTS
First column in A154960. - Mats Granvik, Jan 18 2009
Number of chains from minimum to maximum in the lattice of set partitions of {1, ..., n} ordered by refinement. - Gus Wiseman, Jul 22 2018
REFERENCES
L. Babai and T. Lengyel, A convergence criterion for recurrent sequences with application to the partition lattice, Analysis 12 (1992), 109-119.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 316-321.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
D. Barsky and J.-P. Bézivin, p-adic Properties of Lengyel's Numbers, Journal of Integer Sequences, 17 (2014), #14.7.3.
P. J. Cameron, Some treelike objects, Quart. J. Math. Oxford, 38 (1987), 155-183. See p. 170 - N. J. A. Sloane, Apr 18 2014
Emily H. Dickey and Noah A. Rosenberg, Labeled histories and maximally probable labeled topologies with multifurcation, arXiv:2511.16799 [q-bio.PE], 2025. See pp. 9, 12.
Steven R. Finch, Lengyel's Constant [Broken link]
Steven R. Finch, Lengyel's Constant [From the Wayback machine]
T. Lengyel, On a recurrence involving Stirling numbers, Europ. J. Combin., 5 (1984), 313-321.
T. Lengyel, On some 2-adic properties of a recurrence involving Stirling numbers, p-Adic Numbers Ultrametric Anal. Appl. 4, No. 3, 179-186 (2012).
F. Murtagh, Counting dendrograms: a survey, Discrete Appl. Math., 7 (1984), 191-199.
M. Schader, Hierarchical analysis: classification with ordinal object dissimilarities, Metrika, 27 (1980), 127-132. [Annotated scanned copy]
M. Schader, Letter to N. J. A. Sloane, Aug 25 1981.
Eric Weisstein's World of Mathematics, Lengyel's Constant
FORMULA
a(n) = Sum_{i=1..n-1} N_i(n), where N_k(m) = Sum_{j=k..m-1} Stirling2(m, j)*N_{k-1}(j), m=3..n, k=2..m-1; N_1(2)=N_1(3)=...=N_1(n)=1.
a(n) = Sum_{k=1..n-1} Stirling2(n, k)*a(k) [Lengyel]. - Vladeta Jovovic, Apr 16 2003
E.g.f. satisfies Z(z) = 1/2 * (Z(exp(z)-1) - z). [Lengyel]
Asymptotic growth: a(n) ~ C_L*(n!)^2*(2log(2))^(-n)*n^(-1-1/3*log(2)) (Babai and Lengyel), with C_L = 1.0986858055... = A086053 [Flajolet and Salvy].
Sum_{k>=1} a(k-1)/fallfac(n,k) = -1/n^2 + 2*Sum_{k>=1} a(k-1)/n^k, with the falling factorials fallfac(n,k) = Product_{j=0..k-1}(n-j). - Vaclav Kotesovec, Aug 04 2015
EXAMPLE
From Gus Wiseman, Jul 22 2018: (Start)
The (3) = 4 chains from minimum to maximum in the lattice of set partitions of {1,2,3}:
{{1},{2},{3}} < {{1,2,3}}
{{1},{2},{3}} < {{1},{2,3}} < {{1,2,3}}
{{1},{2},{3}} < {{2},{1,3}} < {{1,2,3}}
{{1},{2},{3}} < {{3},{1,2}} < {{1,2,3}}
(End)
MATHEMATICA
a[1] = 1; a[n_] := a[n] = Sum[StirlingS2[n, k]*a[k], {k, 1, n-1}]; Array[a, 19]
(* Jean-François Alcover, Jun 24 2011, after Vladeta Jovovic *)
PROG
(PARI) {a(n) = local(A); if( n<1, 0, for(k=1, n, A = truncate(A) + x*O(x^k); A = x - A + subst(A, x, exp(x + x*O(x^k)) - 1)); n! * polcoeff(A, n))} /* Michael Somos, Sep 22 2007 */
CROSSREFS
KEYWORD
nonn,nice,easy
EXTENSIONS
More terms from Vladeta Jovovic, Apr 16 2003
STATUS
approved