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 /= 3 remains unsolved. It is the purpose of this paper to provide the complete solution of this classical problemOffice of Naval Research
Contract No. NR-042-286http://archive.org/details/uniquemaximumprope00ble

Bleick, Willard Evan

Wang, Peter C. C.

Calhoun, Institutional Archive of the Naval Postgraduate School

ILIFOKNIA UNPS-53BL77011NAVAL POSTGRADUATE SCHOOLMonterey, CaliforniaUNIQUE MAXIMUM PROPERTY OF THESTIRLING NUMBERS OF THE SECONDbyKINDW. E. Bleick and Peter C. C. Wang25 January 1977Approved for public release; distribution unlimited.Prepared for:"-<r ice of Naval Research (Dr. Bruce McDonald)ti sties and Probability BranchDE£8°^:NPS.53BL77011 in9 t0"' VA 22217NAVAL POSTGRADUATE SCHOOLMonterey, CaliforniaRear Admiral Isham Under J. r. BorstingSuperintendent ProvostABSTRACT:Letting f (n) and £(n) the first and last maximum of the graphS(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) Sn h(n)/log n with h(n) subject only to h(n)-*» as n->». This resultwas 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 withn-*° n-H»Harper's result [Ann. Math. Stat. 38(1968), 410-414], it is concluded thatS(n,k) have, asymptotically, a single maximum. Lieb [J. of Comb. Theory5(1968), 203-206] shows that Stirling numbers of the second kind possessthe 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 problemof establishing that S(n,k) possess a "unique" maximum for all n £ 3remains unsolved. It is the purpose of this paper to provide the completesolution of this classical problem.This task was supported by: Office of Naval ResearchContract No. NR-042-286NPS-53BL7701125 January 1977UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGE (When Data Entered)REPORT DOCUMENTATION PAGE READ INSTRUCTIONSBEFORE COMPLETING FORM1. REPORT NUMBERNPS-53BL770112. GOVT ACCESSION NC 3 RECIPIENT'S CATALOG NUMBER4. TITLE (and Subtitle)Unique Maximum Property of theStirling Numbers of the Second Kind5. TYPE OF REPORT 4 PERIOD COVEREDTechnical Report6. PERFORMING ORG. REPORT NUMBERi7. AUTHORSW. E. BleickPeter C. C. Wang8. CONTRACT OR GRANT NUMBERfs)NR-042-2869. PERFORMING ORGANIZATION NAME AND ADDRESSNaval Postgraduate SchoolMonterey, CA 9394010. PROGRAM ELEMENT, PROJECT. TASKAREA « WORK UNIT NUMBERS11. CONTROLLING OFFICE NAME AND ADDRESSOffice of Naval Research (Dr. Bruce McDonald)Statistics and Probability BranchArlington, VA 2221712. REPORT DAT;25 January ]97713. NUMBER OF PAGES1114. MONITORING AGENCY NAME ft ADDRESSf/f different from Controlling Office) 15. SECURITY CLASS, (of this report)UNCLASSIFIED15«. DECLASSIFI CATION/ DOWN GRADINGSCHEDULE16. 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 NOTES19. KEY WORDS (Continue on reverse aide It naceaaary and Identify by block number)Stirling number of the second kindUnique Maximum propertyHermite's formula for finite differences20. 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 graphS(n,k);k = 1, 2, ... , n, Kanold [J. Reine Angew. Math 230(1968), 211-212] showsthat, 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 byHarborth [J. Reine Angew. Math 230(1968), 213-214] to yieldlim 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 OBSOLETES/N 0102-014-6601 |UNCLASSIFIEDSECURITY 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 ofStrong Logarithmic Concavity, and thus are unimodal. Dobson [J. of Comb.Theory 5 (1968), 212-214 and 7 (1969), 116-121] shows a similar result ina stronger form. However, the classical problem of establishing thatS(n,k) possesses a "unique" maximum for all n>3 remains unsolved. It isthe purpose of this paper to provide the complete solution of thisclassical problem.UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGEfWhen Date Entered)I. IntroductionThe Stirling numbers of the second kind S(n,k) have come intorenewed salience, primarily due to the fact that S(n,k) is the numberof partitions of an n-set into k disjoint nonempty subsets andS(n,k) is the number of distinct fields defined on a finite samplespace with n elementary events to which each field contains exactlyk "£*"2 events [1]. Letting f(n) and £(n) the first and last maximaAof the graph S(n,k);k = 1, 2, ... , n, Kanold [2] shows that, forsufficiently large n, n/log n < f(n) < i(n) 2 n h(n)/log n withh(n) subject only to h(n)-*» as n-*». This result was subsequentlyimproved to yieldlim f (n)n log n = lim £(n)n log n = 1,by Harborth [3]. Together with Harper's result [4], it is concludedthat S(n,k) have, asymptotically, a single maximum. Earlier Lieb[5] shows that Stirling numbers of the second kind possess the propertyof Strong Logarithmic Concavity, and thus are unimodal. Dobson[6, 7] shows a similar result in a stronger form. However, theclassical problem of establishing that S(n,k) possess ajl "unique"maximum for all n ^ 3 remains unsolved. It is the purpose of thispaper 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)zn'k= Tf (l-jz)"1,n=k j=lconvergent for |z| < k , as a generating function for the Stirlingnumbers S(n,k) of the second kind. The reciprocal transformationz = w converts (1) to the Laurent series(2) £ S(n,k)w"n = 7f (w-j)"1 ,n=k j=lconvergent for |w| > k. The coefficient in the series (2) may beexpressed 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 integrandFrom (3) it follows that(4) S(n,k-1) - S(n,k) = Ar ( . vl^"1^ n2iri 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) = wn- (k+1) wn_1 .-2-But by [9;p.l0] we find that (4) can also be represented by a formula ofHermite as the repeated definite integral(6) S(n,k-l)-9(n,k)-J1 dt^ j'ldtj.... J^f (k_1) (u^dt^where u=l+t..+t9+. .+t,_.. .We imagine that t. , t„, .., t,_. constitute a set ofrectangular Cartesian coordinates and impose an orthogonal transformation ofcoordinates to u.,u„, •.,u» _ . We then perform the integration of (6) overthe variables u„,u_, ..,u, - . Because of the structure of the subspace ortho-gonal to u1we find that (6) becomes(7) S(n,k-1)-S(n,k) = Jkf(k_1) (u^gCu^di^-irv" if(k' l) (v+5)g((i)d5l-vwhere 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) (u1) = (n-l)![nu1n"k+1-(n-k+l)(k+l)u1n"k ]/(n-k+l)l.On substituting (8) in (7) we find that the even part of the integrand isproportional 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 valueswhich 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 forn=k=2.We have established the following Theorem: The Stirling numbers of thesecond kind S(n,k) possess a "unique" maximum for n>3.-3-References[1] Wang, Peter C. C, "Enumeration of Distinct Fields Over aFinite Probability Space", Notices of Amer. Math. Soc.16(1969)294.[2] Kanold , H. J., "Uber line asymptotische Abschatzung beiStirlingschen Zahlen 2. Art", J. Reine Angew. Math.230(1968), 211-212.[3] Harborth, H. , "Uber das Maximum bei Stirlingschen Zahlen2. 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 Functionfor Stirling Numbers", J. of Comb. Theory 5(1968), 203-206.[6] Dobson, A. J.,, "A Note on Stirling Numbers of the SecondKind", J. of Comb. Theory, 5(1968), 212-214.[7] Dobson, A. J, and Rennie, B. C. , "On Stirling Numbers of theSecond 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 LISTCopiesStatistics and ProbabilityprogramOffice of Naval ResearchAttn: Dr. B. J. McDonaldArlington, Virgainia 22217Director, Naval ResearchLaboratoryAttn; Library, Code 2029(ONRL)Washington, D. C. 20390Defense Documentation CenterCameron StationAlexandria, Virginia 22314Defense Logistics StudiesInformation ExchangeArmy Logistics Management CenterAttn: Arnold HixonFort Lee, Virginia 23801Technical Information DivisionNaval Research LaboratoryWashington, D. C. 20390Office of Naval ResearchNew York Area Office207 West 24th StreetAttn: Dr. Jack LadermanNew York, New YorkDirectorOffice of Naval ResearchBranch Office495 Summer StreetAttn: Dr. A. L. PowellBoston, Massachusetts 02210DirectorOffice of Naval ResearchBranch Office536 South Clark StreetAttn: Dr. A. R. DaweChicago, Illinois 6060512CopiesOffice of Naval ResearchBranch Office536 South Clark StreetAttn: Dr. Robert BuchalChicago, Illinois 60605DirectorOffice of Naval ResearchBranch Office1030 East Green StreetAttn: Dr. A. R. LauferPasadena, California 91101Office of -Naval ResearchBranch Office1030 East Green StreetAttn: Dr. Richard LauPasadena, California 91101Office of Naval ResearchSan Francisco Area Office50 Fell StreetSan Francisco, California 94102Technical LibraryNaval Ordanance StationIndian Head, Maryland 20640Naval Ship Engineering CenterPhiladelphiaDivision Technical LibraryPhiladelphia, Pennsylvania 19112Bureau of Naval PersonnelDepartment of the NavyTechnical LibraryWashington, D. C. 20370Library, Code 0212Naval Postgraduate SchoolMonterey, California 93940LibraryNaval Electronics Laboratory CenterSan Diego, California 92152-5-Copies CopiesNaval Undersea CenterTechnical LibrarySan Diego, California 92132Applied Mathematics LaboratoryNaval Ships Research andDevelopment CenterAttn: Mr. Gene H. GleissnerWashington, D. C. 20007 1Office of .Chief of NavalOperations (Op 964)Ballston Tower No. 2Attn: Mr. A. S. RhodesArlington, Virginia 22203 1Naval Ships Systems CommandShips 0311National Center No. 3Attn: Miss B. S. OrleansArlington, Virginia 20360 1University of ChicagoDepartment of StatisticsAttn: Prof. W. KruskalChicago, Illinois 60637 1Stanford UniversityDepartment of Operations ResearchAttn: Prof. G. J. LiebermanStanford, California 94305 1Florida State UniversityDepartment of StatisticsAttn: Prof. I. R. SavageTallahassee, Florida 32306 1Florida State UniversityDepartment of StatisticsAttn: Prof. R. A. BradleyTallahassee, Florida 32306 1Princeton UniversityDepartment of StatisticsAttn: Prof. J. W. TukeyPrinceton, New Jersey 08540 1Princeton UniversityDepartment of StatisticsAttn: Prof. G. S. WatsonPrinceton, New Jersey 05840Stanford UniversityDepartment of StatisticsAttn: Prof. T. W. AndersonStanford, California 94305University of CaliforniaDepartment of StatisticsAttn: Prof. P. J. BickelBerkeley, California 94720University of WashingtonDepartment of MathematicsAttn: Prof. Z. W. BirnbaumSeattle, Washington 98105Harvard UniversityDepartment of StatisticsAttn: Prof. W. G. CochranCambridge, Massachusetts 02139Columbia UniversityDepartment of Civil Engineeringand Engineering MechanicsAttn: Prof. C. DermanNew York, New York 10027Columbia UniversityDepartment of MathematicsAttn: Prof. H. RobinsNew York, New York 10027New York UniversityInstitute of Mathematical ScienceAttn: W. M. HirschNew York, New York 10453University of North CarolinaDepartment of StatisticsAttn: Prof. W. L. SmithChapel Hill, North Carolina 27514-6-Copies CopiesUniversity of North CarolinaDepartment of Statistics' ttn: Prof. M. R. Leadbetterchapel Hill, North Carolina 27514 1,Jurdue Universitydepartment of Statistics\ttn: Prof. H. Rubin:afayette, Indiana 47907 1University of CaliforniaSan DiegoDepartment of MathematicsP. 0. Box 109Attn: Prof. M. RosenblattLa Jolla, California 92038 1New York UniversityDepartment of Industrial Engineeringand Operations ResearchAttn: Prof. J. H. K. KaoBronx, New York 10453 1University of WisconsinDepartment of StatisticsAttn: Prof. G. E. P. BoxMadison, Wisconsin 53706 1State University of New YorkI nairman, Department of StatisticsAttn: Prof. E. ParzenBuffalo, New York 14214 1University of CaliforniaOperations Research CenterAttn: Prof. R. E. BarlowBerkeley, California 94720 1Yale UniversityDepartment of StatisticsAttn: Prof. F. J. AnscombeNew Haven, Connecticut 1Purdue UniversityDepartment of StatisticsAttn: Prof. S. S. GuptaLafayette, Indiana 47907 1Cornell UniversityDepartment of Operations ResearchAttn: Prof. R. E. BechhoferIthaca, New York 14850Stanford UniversityDepartment of MathematicsAttn: Prof. S. KarlinStanford, California 94305Southern Methodist UniversityDepartment of StatisticsAttn: Prof. D. B. OwenDallas, Texas 75222University of GeorgiaDepartment of StatisticsAttn: Prof. R. E. BargmannAthens, Georgia 30601Daniel H. Wagner, AssociatesStation Square OneAttn: Dr. L. D. StonePaoli, Pennsylvania 19301Daniel H. Wagner, AssociatesStation Square OneAttn: Dr. B. BelkinPaoli, Pennsylvania 19301Stanford UniversityDepartment of Operations ResearchAttn: Prof. A. F. VeinottStanford, California 94305Stanford UniversityDepartment of Operations ResearchAttn: Prof. D. L. IglehartStanford, California 94305George Washington UniversityDepartment of StatisticsAttn: Prof. Herbert SolomonWashington, D. C. 20006University of North CarolinaDepartment of StatisticsAttn: Prof. C. R. BakerChapel Hill, North Carolina 27514-7-Copies CopiesClemson UniversityDepartment of MathematicsAttn: Prof. K. T. WalleniusClemson, South Carolina 29631 1University of CaliforniaDepartment of StatisticsAttn: Charles E. AntoniakBerkeley, California 94720 1Clarkson College of TechnologyDivision of ResearchAttn: Prof. M. ArozullahPotsdam, New York 13676 1University of Southern CaliforniaElectrical Sciences DivisionAttn: Prof. W. C. LindseyLos Angeles, California 90007 1Case Western Reserve UniversityDepartment of Mathematics andStatisticsAttn: Prof. S. ZacksCleveland, Ohio 44106 1University of FloridaDepartment of Electrical EngineeringAttn: Prof. D. G. ChildersGainesville, Florida 32601 1Stanford UniversityDepartment of StatisticsAttn: Prof. H. ChernoffStanford, California 94305 1Naval Research LaboratoryElectronics Division(Code 5267)Attn: Mr. Walton BishopWashington, D. C. 20390 1Commandant of the Marine Corps(Code AX)Attn: Dr. A. L. SlafkoskyScientific AdvisorWashington, D. C. 20380 1Program in LogisticsThe George Washington UniversityAttn: Dr. W. H. Marlow707 22nd Street, N. W.Washington, D. C. 20037Mississippi Test FacilityEarth Resources Laboratory(Code GA)Attn: Mr. Sidney L. WhitleyBay St. Louis, Mississippi 39520Naval Postgraduate SchoolDepartment of Operations Research andAdministrative SciencesAttn: Prof. P. A. W. LewisMonterey, California 93940Southern Methodist UniversityDepartment of StatisticsAttn: Prof. W. R. SchucanyDallas, Texas 75222Webb Institute of Naval ArchitectureAttn: Prof. 0. J. KarstCrescent Beach RoadGlen Cove, New York 11543University of MissouriDepartment of StatisticsAttn: Prof. W. A. Thompson, Jr.Columbia, Missouri 65201Rice UniversityDepartment of Mathematical SciencesAttn: Prof. J. R. ThompsonHouston, Texas 77001University of CaliforniaSystem Science DepartmentAttn: Prof K. YaoLos Angeles, California 90024Naval Postgraduate SchoolDepartment of MathematicsAttn: P. C. C. WangMonterey, California 93940Naval Postgraduate SchoolDepartment of MathematicsAttn: Prof. Willard Bleick2016•8-Copies CopiesRaytheon CompanySubmarine Signal DivisionAttn: Dr. W. S. Liggett, Jr.Portsmouth, Rhode Island 02971 1Systems Control, Inc.Attn: Dr. L. P. Seidman260 Sheridan AvenuePalo Alto, California 44306 1University of CaliforniaDepartment of Information andComputer ScienceAttn: Prof. E. MasryLa Jolla, California 92037 1University of CaliforniaSchool of EngineeringAttn: Prof. N. J. BershadIrvine, California 92664 1University of CaliforniaSchool of Engineering andApplied ScienceAttn: Prof. I. RubinLos Angeles, California 90024 1Virginia Polytechnic InstituteDepartment of StatisticsAttn: Prof. C. KramerBlacksburg, Virginia 24061 1New York UniversityDepartment of Electrical EngineeringAttn: Prof. I. YagodaBronx, New York 10453 1University of RochesterDepar>:ient of StatisticsAttn: Prof. J. KeilsonRochester, New York 14627 1University of MichiganDepartment of Industrial EngineeringAttn: Prof. R. L. DisneyAnn Arbor, Michigan 48104 1Cornell UniversityDepartment of Computer ScienceAttn: Prof. J. E. HopcroftIthaca, New York 14850 1Smithsonian InstitutionAstrophysical ObservatoryAttn: Dr. C. A. LundquistCambridge, Massachusetts 02138 1Naval Postgraduate SchoolDepartment of Operations Researchand Administrative SciencesAttn: Prof. J. D. EsaryMonterey, California 93940 1Polytechnic Institute of BrooklynDepartment of Electrical EngineeringAttn: Prof. M. L. ShoomanBrooklyn, New York 11201 1Union CollegeInstitute of Industrial AdministrationAttn: Prof. L. A. AroianSchenectady, New York 12308 1Ultrasystems, Inc.Attn: Dr. D. C. Dorrough500 Newport Center DriveNewport Beach, California 92660 1University of New MexicoDepartment of Mathematics and StatisticsAttn: Prof. W. J. ZimmerAlbuquerque, New Mexico 87106 1Princeton UniversityDepartment of StatisticsAttn: Prof. G. SimonPrinceton, New Jersey 08540 1Naval Ordnance Systems Command, NORD 035Attn: Mr. 0. SeidmanRoom 6E08, National Center #2Arlington, Virginia 20360 1Naval Coastal Systems LaboratoryCode P761Attn: Mr. C. M. BennettPanama City, Florida 32401 1Food and Drug AdministrationStatistics and Information Science DivisionHealth Protection BranchAttn: Dr. A. Petrasovits, Head,Survey Design and Quality Control355 River Road, 4th FloorVanier, Ontario, Canada 1-9-DUDLEY KNOX LIBRARY • RESEARCH REPORTS5 6853 01071052 8

Unique maximum property of the Stirling numbers of the second kind

https://archive.org/download/uniquemaximumprope00blei/uniquemaximumprope00blei.pdf

Unique maximum property of the Stirling numbers of the second kind

Abstract

Similar works

Full text

Available Versions

Calhoun, Institutional Archive of the Naval Postgraduate School