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Lengyel's Constant

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Let L denote the partition lattice of the set {1,2,...,n}. The maximum element of L is

M={{1,2,...,n}}
(1)

and the minimum element is

m={{1},{2},...,{n}}.
(2)

Let Z_n denote the number of chains of any length in L containing both M and m. Then Z_n satisfies the recurrence relation

Z_n=sum_(k=1)^(n-1)s(n,k)Z_k,
(3)

where Z_1=1 and s(n,k) is a Stirling number of the second kind. The first few values of Z_n for n=1, 2, ... are then 1, 1, 4, 32, 436, 9012, 262760, ... (OEIS A005121).

Lengyel (1984) proved that the quotient

r(n)=(Z_n(2ln2)^nn^(1+(ln2)/3))/((n!)^2)
(4)

is bounded between two constants as n->infty, and Flajolet and Salvy (1990) improved the result of Babai and Lengyel (1992) to show that

Lambda=lim_(n->infty)r(n)=1.0986858055...
(5)

(OEIS A086053).

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References

Babai, L. and Lengyel, T. "A Convergence Criterion for Recurrent Sequences with Application to the Partition Lattice." Analysis 12, 109-119, 1992.Finch, S. R. "Lengyel's Constant." §5.7 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 316-321, 2003.Flajolet, P. and Salvy, B. "Hierarchal Set Partitions and Analytic Iterates of the Exponential Function." Unpublished manuscript, 1990.Lengyel, T. "On a Recurrence Involving Stirling Numbers." Europ. J. Comb. 5, 313-321, 1984.Sloane, N. J. A. Sequences A005121/M3649 and A086053 in "The On-Line Encyclopedia of Integer Sequences."

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Lengyel's Constant

Cite this as:

Weisstein, Eric W. "Lengyel's Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LengyelsConstant.html

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