Lengyel's Constant
Let 
denote the partition lattice of the set ![<span style=]()
{1,2,...,n}" src="http://mathworld.wolfram.com/images/equations/LengyelsConstant/Inline2.gif" style="height:15px; width:69px" />. The maximum element of 
is
![M=<span style=]() {{1,2,...,n}} " src="http://mathworld.wolfram.com/images/equations/LengyelsConstant/NumberedEquation1.gif" style="height:15px; width:109px" /> |
(1)
|
and the minimum element is
![m=<span style=]() {{1},{2},...,{n}}. " src="http://mathworld.wolfram.com/images/equations/LengyelsConstant/NumberedEquation2.gif" style="height:15px; width:130px" /> |
(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, ... (OEIS 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)
|
(OEIS A086053).
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 Unpublished manuscript, 1990.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."
