Let denote the partition lattice of the set . The maximum element of is

(1)

and the minimum element is

(2)

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

(3)

where and is a Stirling number of the second kind. The first few values of for , 2, ... are then 1, 1, 4, 32, 436, 9012, 262760, ... (Sloane's A005121).

Lengyel (1984) proved that the quotient

(4)

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

(5)

(Sloane's A086053).

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 Unpublished manuscript, 1990.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."

CITE THIS AS:

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