Lengyel's Constant

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

Let $L$ denote the partition lattice of the Set $\{1, 2, \ldots, n\}$. The Maximum element of $L$ is

$$M=\{\{1, 2, \ldots, n\}\}$$ (1)

and the Minimum element is

$$m=\{\{1\}, \{2\}, \ldots, \{n\}\}.$$ (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 $s(n,k)$ is a Stirling Number of the Second Kind. Lengyel (1984) proved that the Quotient

$$r(n)={Z_n\over (n!)^2(2\ln 2)^{-n} n^{1-(\ln 2)/3}}$$ (4)

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

$$\Lambda\equiv \lim_{n\to\infty} r(n) = 1.0986858055\ldots.$$ (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 Unpublished manuscript, 1990.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.