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I LIFOKNIA U NPS-53BL77011 NAVAL POSTGRADUATE SCHOOL Monterey, California UNIQUE MAXIMUM PROPERTY OF THE STIRLING NUMBERS OF THE SECOND by KIND W. E. Bleick and Peter C. C. Wang 25 January 1977 Approved for public release; distribution unlimited. Prepared for: " -<r ice of Naval Research (Dr. Bruce McDonald) ti sties and Probability Branch D E £8°^:NPS.53BL77011 in 9 t0 "' VA 22217 NAVAL POSTGRADUATE SCHOOL Monterey, California Rear Admiral Isham Under J. r. Borsting Superintendent Provost ABSTRACT: Letting f (n) and £(n) the first and last maximum 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) S n h(n)/log n with h(n) subject only to h(n)-*» as n->». This result was subsequently improved by Harborth [j. Reine Angew. Math 230(1968), 213-214] to yield lim f (n)n~ log n ■ lim *(n)n~ log n - 1. Together with n-*° n-H» Harper's result [Ann. Math. Stat. 38(1968), 410-414], 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. Dobson [J. of Comb. Theory 5(1968), 212-214 and Vol. 7(1969), 116-121] shows a similar result in a stronger form. However, the classical problem of establishing that S(n,k) possess 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. This task was supported by: Office of Naval Research Contract No. NR-042-286 NPS-53BL77011 25 January 1977 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) REPORT DOCUMENTATION PAGE READ INSTRUCTIONS BEFORE COMPLETING FORM 1. REPORT NUMBER NPS-53BL77011 2. GOVT ACCESSION NC 3 RECIPIENT'S CATALOG NUMBER 4. TITLE (and Subtitle) Unique Maximum Property of the Stirling Numbers of the Second Kind 5. TYPE OF REPORT 4 PERIOD COVERED Technical Report 6. PERFORMING ORG. REPORT NUMBER i 7. AUTHORS W. E. Bleick Peter C. C. Wang 8. CONTRACT OR GRANT NUMBERfs) NR-042-286 9. PERFORMING ORGANIZATION NAME AND ADDRESS Naval Postgraduate School Monterey, CA 93940 10. PROGRAM ELEMENT, PROJECT. TASK AREA « WORK UNIT NUMBERS 11. CONTROLLING OFFICE NAME AND ADDRESS Office of Naval Research (Dr. Bruce McDonald) Statistics and Probability Branch Arlington, VA 22217 12. REPORT DAT; 25 January ]977 13. NUMBER OF PAGES 11 14. MONITORING AGENCY NAME ft ADDRESSf/f different from Controlling Office) 15. SECURITY CLASS, (of this report) UNCLASSIFIED 15«. DECLASSIFI CATION/ DOWN GRADING SCHEDULE 16. DISTRIBUTION STATEMENT (of this Report) Approved for public release; distribution unlimited. 17. DISTRIBUTION STATEMENT (of the abatract entered In Block 20, If different from Report) 18. SUPPLEMENTARY NOTES 19. KEY WORDS (Continue on reverse aide It naceaaary and Identify by block number) Stirling number of the second kind Unique Maximum property Hermite's formula for finite differences 20. ABSTRACT (Continue on reverae aide It naceaaary and Identity by block number) Letting f(n) and ^( n ) A tne first and last maxim*£aUof 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)-*» as n-*°. This result was subsequently improved by Harborth [J. Reine Angew. Math 230(1968), 213-214] to yield lim f(n)n log n = lim £(n)n log n = 1. Together with Harper's result [Ann. n-x» n-*°° DD 1 JAN 73 1473 EDITION OF 1 NOV 65 IS OBSOLETE S/N 0102-014-6601 | UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (Whan Data Entered) UNCLASSIFIED ,L(_UK|TY CLASSIFICATION OF THIS PAGEC>W>en Dmta Entered) Math. Stat. 38 (1967), 410-414], it Is concluded that S(n,k) have, asymp- totically, a single maximum. Lieb [J. of Comb. Theory 5 (]968), 203-206] shows that Stirling numbers of the second kind possess the property of Strong Logarithmic Concavity, and thus are unimodal. Dobson [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. UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGEfWhen Date Entered) I. Introduction The Stirling numbers of the second kind S(n,k) have come into renewed salience, primarily due to the fact that S(n,k) is the number of partitions of an n-set into k disjoint nonempty subsets and S(n,k) is the number of distinct fields defined on a finite sample space with n elementary events to which each field contains exactly k "£*" 2 events [1]. Letting f(n) and £(n) the first and last maxima A of the graph S(n,k);k = 1, 2, ... , n, Kanold [2] shows that, for sufficiently large n, n/log n < f(n) < i(n) 2 n h(n)/log n with h(n) subject only to h(n)-*» as n-*». This result was subsequently improved to yield lim f (n)n log n = lim £(n)n log n = 1, by Harborth [3]. Together with Harper's result [4], it is concluded that S(n,k) have, asymptotically, a single maximum. Earlier Lieb [5] shows that Stirling numbers of the second kind possess the property of Strong Logarithmic Concavity, and thus are unimodal. Dobson [6, 7] shows a similar result in a stronger form. However, the classical problem of establishing that S(n,k) possess ajl "unique" maximum for all n ^ 3 remains unsolved. It is the purpose of this paper to provide the complete solution of this classical problem. -1- II. Unique Maximum Property of S(n,k). In Riordan [8; p. 43] has given the Taylor series (1) £ S(n,k)z n ' k = Tf (l-jz)" 1 , n=k j=l convergent for |z| < k , as a generating function for the Stirling numbers S(n,k) of the second kind. The reciprocal transformation z = w converts (1) to the Laurent series (2) £ S(n,k)w" n = 7f (w-j)" 1 , n=k j=l convergent for |w| > k. The coefficient in the series (2) may be expressed as the contour integral (3) S(n,k) ■= 2ir± J (w-1) (w-2) . . . (w-k) where the contour C encloses the singular points of the integrand From (3) it follows that (4) S(n,k-1) - S(n,k) = Ar ( . vl^" 1 ^ n 2iri j (w-1) (w-2) ... (w-k) In Milne-Thomson [9;p.ll] we find that (4) is the divided difference [123.. k] of order k-1 of the polynomial (5) f (w) = w n - (k+1) w n_1 . -2- But by [9;p.l0] we find that (4) can also be represented by a formula of Hermite as the repeated definite integral (6) S(n,k-l)-9(n,k) -J 1 dt^ j'ldtj.... J^f (k_1) (u^dt^ where u=l+t..+t 9 +. .+t, _.. . We imagine that t. , t„, . . , t, _. constitute a set of rectangular Cartesian coordinates and impose an orthogonal transformation of coordinates to u.,u„, • . ,u» _ . We then perform the integration of (6) over the variables u„,u_, . . ,u, - . Because of the structure of the subspace ortho- gonal to u 1 we find that (6) becomes (7) S(n,k-1)-S(n,k) = J k f (k_1) (u^gCu^di^ - i r v " i f (k ' l) (v+5)g((i)d5 l-v where v=(l+k)/2=u -5 and g is a positive even function of £ independent of n. By differentiation of (5) we obtain (8) f (k_1) (u 1 ) = (n-l)![nu 1 n " k+1 -(n-k+l)(k+l)u 1 n " k ]/(n-k+l)l. On substituting (8) in (7) we find that the even part of the integrand is proportional to (9) g(f){nC[(v+5) n " k -(v-^ n " k ]-v(n-2k+2)[(v+ 5 ) n - k +(v- 5 ) n - k ]}. Since we are interested in finding more than one pair of n and k values which make (7) vanish, and since g(?) is independent of n, we see that (9) must be identically zero for all £. But (9) vanishes identically only for n=k=2. We have established the following Theorem: The Stirling numbers of the second kind S(n,k) possess a "unique" maximum for n>3. -3- References [1] Wang, Peter C. C, "Enumeration of Distinct Fields Over a Finite Probability Space", Notices of Amer. Math. Soc. 16(1969)294. [2] Kanold , H. J., "Uber line asymptotische Abschatzung bei Stirlingschen Zahlen 2. Art", J. Reine Angew. Math. 230(1968), 211-212. [3] Harborth, H. , "Uber das Maximum bei Stirlingschen Zahlen 2. Art", J. Reine Angew. Math. 230(1968), 213-214. [4] Harper, L. H. , "Stirling Behavior is Asymptotically Normal", Ann. Math. Stat. 38(1967), 410-414. [5] Lieb, E. H. , "Concavity Properties and a Generating Function for Stirling Numbers", J. of Comb. Theory 5(1968), 203-206. [6] Dobson, A. J.,, "A Note on Stirling Numbers of the Second Kind", J. of Comb. Theory, 5(1968), 212-214. [7] Dobson, A. J, and Rennie, B. C. , "On Stirling Numbers of the Second Kind", J. of Combinatorial Theory, 7(1969), 116-121. [8] Riordan, John, An Introduction tu Combinatorial Analysis , John Wiley & Sons, Inc* » 1958. [9] Milne- Thorns on, L. M. , The Calculus of Finite Differences , MacMillan and Co., Ltd., London, 1933. -4- DISTRIBUTION LIST Copies Statistics and Probability program Office of Naval Research Attn: Dr. B. J. McDonald Arlington, Virgainia 22217 Director, Naval Research Laboratory Attn; Library, Code 2029 (ONRL) Washington, D. 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