I
LIFOKNIA U
NPS-53BL77011
NAVAL POSTGRADUATE SCHOOL
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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
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Unique Maximum Property of the
Stirling Numbers of the Second Kind
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W. E. Bleick
Peter C. C. Wang
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NR-042-286
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Naval Postgraduate School
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Office of Naval Research (Dr. Bruce McDonald)
Statistics and Probability Branch
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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
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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-
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