Lengyel's Constant

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


and the minimum element is


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


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


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


(OEIS A086053).

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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 Unpublished manuscript, 1990.Lengyel, T. "On a Recurrence Involving Stirling Numbers."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.

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