## Lengyel's Constant

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

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 is a Stirling Number of the Second Kind. 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)

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. Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/lngy/lngy.html

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 Unpublished manuscript, 1990.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.

Plouffe, S. The Lengyel Constant.'' http://www.lacim.uqam.ca/piDATA/lengyel.txt.