Unique maximum property of the Stirling numbers of the second kind
By: by W.E. Bleick and Peter C.C. Wang
Contributors: Wang, Peter C. C., 1937-; Naval Postgraduate School (U.S.)
Publisher: Naval Postgraduate School
Published: 1977
Place of publication: Monterey, California
Language: eng
Subjects: Asymptotic expansions
Description: Letting f(n) and (n) be the first and last maxim of the graph S(n,k); k = 1, 2, ... , n, Kanold [J. Reine Angew. Math 230 (1968), 211-212] shows that, for sufficiently large n, n/log n < f(n) </= (n) </= n h(n)/log n with h(n) subject only to h(n) [to infinity] as n [to infinity]. This result was subsequently improved by Harborth {J. Reine Angew. Math 230 (1968), 213-214} to yield lim f(n)n¹log n = lim (n)n¹log n = 1.  Together with Harper's result , it is concluded that S(n,k) have, asymptotically, a single maximum.  Lieb [J. of Comb. Theory 5 (1968), 203-206] shows that Stirling numbers of the second kind possess the property of Strong Logarithmic Concavity, and thus are unimodal.  Dobsn [J. of Comb. Theory 5 (1968). 212-214 and 7 (1969), 116-121] shows a similar result in a stronger form.  However, the classical problem of establishing that S(n,k) possesses a "unique" maximum for all n >/= 3 remains unsolved.  It is the purpose of this paper to provide the complete solution of this classical problem.
Notes: Title from cover.

"Prepared for: Office of Naval Research Statisitics and Probability Branch"--Cover.

"25 January 1977"--Cover.

"NPS-53BL77011"--Cover.

Author(s) key words: Stirling number of the second kind, unique maximum property, Hermit's formula for finite difference.

Includes bibliographical references (p. 4).

"Approved for public release; distribution unlimited"--Cover.

Technical report; 1977.

kmc/kmc 9/9/09.