Combinatorial analysis
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This document discusses mathematical properties related to combinatorial analysis. It covers basic numbers like binomial coefficients, multinomial coefficients, and Stirling numbers. It also covers partitions and number theoretic functions. For each topic, it provides definitions, generating functions, closed forms, relations including recurrences, and asymptotic values or special cases. It includes 9 tables providing values for various combinatorial sequences and functions for reference.
![24. Combinatorial Analysis
Mathematical Properties
In ench sub-section of this chnpter we use a special nncl easily recognizable symbol, and
a fised format wliicli etnplinsizes tlic use and yet t h n t s-mbol must be easy to write. We have
methods or" cstcnding the nccotnpnriyinp tables. settled on a script capital 3 without any certainty
Thc format follows this forin : that we 11ave settled tliis question permanently.
1. Definitions We feel that tlie subscript-superscript notation
A. Combinntorinl emplinsizes tlie generating friiictioris (wliicli are
B. Generating functions p6wers of mutunlly inverse functions) from which
C. Closed form most of the important relations flow.
11. Relations
24.1. Basic Numbers
A. Recurrences
B. Checks in comput.ing 24.1.1 Binomial Coefficients
C. Basic use in numerical analysis 1. Definitions
111. Asymptotic and Special Values
In general tlie notations used ore stsndnrd.
This includes the difference operator A defined on
A. (z)
is the number of yays of choosing m
objects from a collection of n distinct objects
functions of t by Af(z)=f(r+I)-f(;t), An+y(r) without regard to order.
=A(A.J(z)), the Kronecker deltu the Rieninriri 13. Generating furictioiis
zeta function {(s) and the grentest common
divisor symbol (m, n), The range of the summands n=O,l,. . .
for n summation sign without limits is explained
to the right of the formula.
The notations which are not stnndnrd are those
for tlie multinomials whicli nre arbitrary sliort-
hand for use in this chapter, and those for the C. Closed form
Stirling numbers which have never been stand-
ardized. A short table of various notations for
these numbers follows :
n> m
Notations for the Stirling Numbers n(n-1). . . (n-m+l) -
-
Reference First Kind Second Kind m!
This chapter 'S
.! %!-) 11. Relations
124.21 Fort S:-' yy *
}
(24.71 Jordan A
S
: e: * A. Recurrences
(24.101 bIoser and Wyman S.: 0::
124.91 Milne-Thomson (
:I :)B!% (:) B:Z n>m>l
(24.151 Riordan d n , m) S(n, m)
(24.11 Carlitz
[24.3] Could
Miksa
(-1)n-"Sl(n-l,n-m)
S(n-m+1, n)
S2(m, n-m)
mSn
=( :)+( :)+. . . +rim) m
n>
(Unpublished B. Checks
tables)
124.171 Gupta u(n, m)
r+sL n
We feel that a capital S is natural for Stirling
numbers of the first kind; it is infrequently used
for other notation in this contest.. But once it r 2 ir+l
is used we have difficulty finding a suitable
symbol for Stirling numbers of tlie second kind.
The numbers are sufficiently important to warrant
"D>
(:')=(-
mo
('''I)
in.,
. . . (Inodp) pa prime
a22 *Rev pnge XI.](https://image.slidesharecdn.com/combinatorialanalysis-130220222738-phpapp01/85/Combinatorial-analysis-2-320.jpg)
![COMBINATORIAL ANALYSIS 823
where
m m
5 (-om
m=O (;)f(z-m)
n = E n,$, m = C mkpk 2Q n-k-1
)
k-0 k-0
p>mk, n k
=f:=O
k (-1)S-f 8-k ~kj(z--s) e
n
<
C. Numerical analysis
I 111. Special Values
2"(2n-1)(2n-3) . . . 3-1
("n">= n!
24.1.2 Multinomial Coefficients
I. DefinitiOM
A. (n; nl, %, . . ., n,) is the number of ways of putting n=nl+nz+. . . +nm different objects
into m different boxes with nk in the k-th box, k=I, 2, . . ., m.
(n; all %, . . ., a,,)* is the number of permutations of n=a1+2%+. . . +nu, symbols composed
of a k cycles of length k for k=1, 2, . . ., n.
(n; al, %, . . ., an)' is the number of ways of partitioning a set of n=a1+2az+. . . +na. dif-
ferent objects into ak subsets containing k objects for k=1, 2, . . ., n.
B. Generating functions
(zl+%+. . . +z,)"=Z(n; nl, 121, . . ., nm)z;%P . . . z3 summed over a+-+. . . +nm=n
(2 P)
k-1
m
=m!
OD t"
n-m 12.
Z(n;al,a2, . . ., a ) z 1 ; .
n*;zr. .:
z-
summed over a1+2%+ . . . +m,,=n
C. Closed forms
(n; nl, Q, . . ., n,) =n!/w!n,! . . . n,! nl+%+. . .+nm=n
(n; all az, . . ., a,,) *=n!/lal@!2"Iag! . . . n".cr,,! a1+2%+ . . .+m,=n
{{ {
(n;al,az, . . ., ~ , ) ' = n ! / ( l ! ) ~ l a ~ ! ( 2 ! . ~ ! (n!)"a,!
). . al+%+. . . +m,=n
11. Relatiom
A. Recurrence
m
(n+m;nl+l,nz+l,. . . , n m + l ) = C (n+m-l;nl+l, . . .,nk-l+l,nk,nk+l+l, . . . ,n,,,+l)
k-1
B. Checks
all ni 1
* Z(n;nl,%, . . .,n,,,)= summed over nl -]-nz+ . . . +nm=n
Z(n;al,%, . . ., ~,,)*=(-1)"-?3~~' summed over a1+2%+ . . . +m,,=n and al+az+. . . +an=m
Z(n;a,,az, . . ., a,)'= SP)
C. Numerical analysis (FaA di Bruno's formula)
d" n
= C f("'(g(z))W; al,G, . . , an)' g'b) 1'1 g"(z) 1'2 . g'"'(z) 1
dz"g b ) )
-f(
m-0
. . . 'I
summed over al+2%+ . . . +nun=n and al+az+. . . +an=m.](https://image.slidesharecdn.com/combinatorialanalysis-130220222738-phpapp01/85/Combinatorial-analysis-3-320.jpg)
![COMBINATORIAL ANALYSIS 825
B. Generating function
B. Checks
n
(-1)"-=m! s =
p 1 where
m-0
((z))=z-[z]-& if z is not an integer
=O if x is an integer
11. Relationa
A. Recurrence
p(n)= & (-l)k-lp (-)
n? p(0)=1
1-
s 1 In
5
=I ol(k)p(n-k)
n k-1
B. Check
111. Asymptotics and Special Velum
* lim m-" sAm)=(mt)-l
n-m 111. Aapptoties
1 *d%&
P(4- -
4nd3e
24.2.2 Partitions Into Distinct Parte
I. Definitiono
A. q(n) is the number of decompositions of n
into distinct integer summands without regard to
order. E.g., 5=1+4=2+3 so that q(5)=3.
B. Generating function
m m OD
24.2. Partitions qq(n)z"= II (l+x")=
n- n-1
n-1n 11
z<
1
24.2.1 Unrestricted Partitions C. Closed form
I. Definitions
A. p(n) is the number of decompositions of n
into integer summands without regard to order. where Jo(x) is the Bessel function of order 0 and
E.g.,5=1+4=2+3=1+1+3=1+2+2=1+1+ was defined in part IC. of the previous
1+2=1+1+1+1+1 so that p(5)=7. subsection.
*See page 11.](https://image.slidesharecdn.com/combinatorialanalysis-130220222738-phpapp01/85/Combinatorial-analysis-5-320.jpg)
![COMBINATORIAL ANALYSIB 827
1 "
24.3.3 Divisor Functions
I. Definitions
-
n2 m=l
u1(m)=-+O
12
u2 CY>
-
A. uk(n) the sum of the k-th powers of the
is 24.3.4 Primitive Roots
divisors of n. Often udn) denoted by d(n),
is and
U l b )b y 4%).
I. Definitions
B. Generating functions The integers not exceeding and relatively prime
to a fixed integer n form a group; the group is
cyclic if and only if n=2,4 or n is of the form pk or
2pkwhere p is an odd prime. Then g is a primitive
root of n if it generates that group; i.e., if g, g2, . . .,
g+'(")distinct modulo n. There are cp(cp(n))
are
C. Closed form primitive roots of n.
11. Relations
A. Recurrences. If g is a primitive root of a
prime p and gp-l$ l(mod p2) g is a primitive
then
root of pk for all k. If gp-' = 1(mod p2) then g+p
is a primitive root of pkfor all k.
If g is a primitive root of pk then either g or
g+pk,whichever is odd, is a primitive root of 2p'.
B. Checks. If g is a primitive root of n then gk
is a primitive root of n if and pnly if (k, cp(n),
= 1,
and each primitive root of n is of this form.
References
Texts [24.13] H . Rademacher, On the partition function, Proc.
London Math. SOC. 241-254 (1937).
43,
[24.1] L. Carlitr, Note on Norlunds polynomial B$),
[24.14] H. Rademacher and A. Whiteman, Theorems on
Proc. Amer. Math. SOC. 452-455 (1960).
11,
Dedekind sums, Amer. J. Math. 63, 377-407
[24.2] T. Fort, Finite differences (Clarendon Press, (1941).
Oxford, England, 1948).
[24.15] J. Riordan, An introduction to combinatorial
[24.3] H. W. Gould, Stirling number representation analysis (John Wiley & Sons, Inc., New York,
problems, Proc. Amer. Math. SOC.11, 447-451 N.Y., 1958).
(1960).
[24.16] J. V. Uspensky and M. A. Heaslet, Elementary
[24.4] G. H. Hardy, Ramanujan (Chelsea Publishing Co., number theory (McGraw-Hill Book Co., Inc.,
New York, N.Y., 1959). New York, N.Y., 1939).
[24.5] G. H. Hardy and E. M. Wright, An introduction
to the theory of numbers, 4th ed. (Clarendon Tables
Press, Oxford, England, 1960). [24.17] British Association for the Advancement of Science,
[24.6] L. K. Hua, On the number of partitions of a num- Mathematical Tables, vol. VIII, Number-divisor
ber into unequal parts, Trans. Amer. Math. SOC. tables (Cambridge Univ. Press, Cambridge,
51, 194-201 (1942). England, 1940). n S l 0 ' .
[24.7] C. Jordan, Calculus of finite differences, 2d ed. [24.18] H. Gupta, Tables of distributions, Res. Bull. East
(Chelsea Publishing Co., New York, N.Y., Panjab Univ. 13-44 (1950); 750 (1951).
1960). [24.19] H. Gupta, A table of partitions, Proc. London
[24.8] K. Knopp, Theory and application of infinite Math. SOC.39, 142-149 (1935) and 11. 42,
series (Blackie and Son, Ltd., London, England, 546-549 (1937). p(n), n=1(1)300; p(n), n=301
1951). (1)600.
[24.9] L. M. Milne-Thomson, The calculus of finite [24.20] G. KavBn, Factor tables (Macmillan and Co., Ltd.,
differences (Macmillan and Co., Ltd., London, London, England, 1937). n 1256,000.
England, 1951). [24.21] D. N. Lehmer, List of prime numbers from 1 to
[24.10] L. Moser and M. Wyman, Stirling numbers of the 10,006,721, Carnegie Institution of Washington,
second kind, Duke Math. J. 25, 29-43 (1958). Publication No. 165, Washington, D.C. (1914).
[24.11] L. Moser and M. Wyman, Asymptotic develop- [24.22] Royal Society Mathematical Tables, vol. 3, Table
ment of the Stirling numbers of the first kind, of binomial coefficients (Cambridge Univ. Press,
J. London Math. SOC. 133-146 (1958).
33, Cambridge, England, 1954). (:).for r <*n< 100.
i24.121 H. H. Ostmann, Additive Zahlentheorie, vol. I [24.23] G. N. Watson, Two tables of partitions, Proc.
(Springer-Verlag, Berlin, Germany, 1956). London Math. SOC. 550-556 (1937).
42,](https://image.slidesharecdn.com/combinatorialanalysis-130220222738-phpapp01/85/Combinatorial-analysis-7-320.jpg)
Combinatorial analysis
- 1. 24. Combinatorial Analysis 'M E. K. GOLDBERG. . NEWMAN,' HAYNSWORTH~ Contents page Mathematical Properties . . . . . . . . . . . . . . . . . . . . 822 24.1. Basic Numbers . . . . . . . . . . . . . . . . . . . . 822 24.1.1 Binomial Coefficients . . . . . . . . . . . . . . 822 24.1.2 Multinomial Coefficients . . . . . . . . . . . . . 823 24.1.3 Stirling Numbers of the First Kind . . . . . . . . 824 24.1.4 Stirling Numbers of the Second Kind . . . . . . . 824 24.2. Partitions . . . . . . . . . . . . . . . . . . . . . . 825 24.2.1 Unrestricted Partitions . . . . . . . . . . . . . . 825 24.2.2 Partitions Into Distinct Parts . . . . . . . . . . . 825 24.3. Number Theoretic Functions . . . . . . . . . . . . . . 826 24.3.1 The Mobius Function . . . . . . . . . . . . . . 826 24.3.2 The Euler Function . . . . . . . . . . . . . . . 826 24.3.3 Divisor Functions . . . . . . . . . . . . . . . . 827 24.3.4 Primitive Roots . . . . . . . . . . . . . . . . . 827 References . . . . . . . . . . . . . . . . . . . . . . . . . . 827 Table 24.1. Binomial Coefficients (;) . . . . . . . . . . . . . . 828 n 1 5 0 .m 1 2 5 Table 24.2. Multinomials (Including a List of Partitions) . . . . . . 831 nllO Table 24.3. Stirling Numbers of the First Kind Si"') . . . . . . . . . 833 n525 Table 24.4. Stirling Numbers of the Second Kind si"') . . . . . . . 835 n125 Table 24.5. Number of Partitions and Partitions Into Distinct Parts . . 836 P b ) . n )n1500 d . Table 24.6. Arithmetic Functions . . . . . . . . . . . . . . . . 840 d n )4 n ) ui(n>. . . n51000 Table 24.7. Factorizations . . . . . . . . . . . . . . . . . . . . 844 n<10000 Table 24.8. Primitive Roots. Factorization of p- 1 . . . . . . . . . 864 n<10000 Table 24.9. Primes . . . . . . . . . . . . . . . . . . . . . . . 870 pll06 1. 2 National Bureau of Standards. 3 National Bureau of Standards. (Presently. Auburn Univenrity.) 821
- 2. 24. Combinatorial Analysis Mathematical Properties In ench sub-section of this chnpter we use a special nncl easily recognizable symbol, and a fised format wliicli etnplinsizes tlic use and yet t h n t s-mbol must be easy to write. We have methods or" cstcnding the nccotnpnriyinp tables. settled on a script capital 3 without any certainty Thc format follows this forin : that we 11ave settled tliis question permanently. 1. Definitions We feel that tlie subscript-superscript notation A. Combinntorinl emplinsizes tlie generating friiictioris (wliicli are B. Generating functions p6wers of mutunlly inverse functions) from which C. Closed form most of the important relations flow. 11. Relations 24.1. Basic Numbers A. Recurrences B. Checks in comput.ing 24.1.1 Binomial Coefficients C. Basic use in numerical analysis 1. Definitions 111. Asymptotic and Special Values In general tlie notations used ore stsndnrd. This includes the difference operator A defined on A. (z) is the number of yays of choosing m objects from a collection of n distinct objects functions of t by Af(z)=f(r+I)-f(;t), An+y(r) without regard to order. =A(A.J(z)), the Kronecker deltu the Rieninriri 13. Generating furictioiis zeta function {(s) and the grentest common divisor symbol (m, n), The range of the summands n=O,l,. . . for n summation sign without limits is explained to the right of the formula. The notations which are not stnndnrd are those for tlie multinomials whicli nre arbitrary sliort- hand for use in this chapter, and those for the C. Closed form Stirling numbers which have never been stand- ardized. A short table of various notations for these numbers follows : n> m Notations for the Stirling Numbers n(n-1). . . (n-m+l) - - Reference First Kind Second Kind m! This chapter 'S .! %!-) 11. Relations 124.21 Fort S:-' yy * } (24.71 Jordan A S : e: * A. Recurrences (24.101 bIoser and Wyman S.: 0:: 124.91 Milne-Thomson ( :I :)B!% (:) B:Z n>m>l (24.151 Riordan d n , m) S(n, m) (24.11 Carlitz [24.3] Could Miksa (-1)n-"Sl(n-l,n-m) S(n-m+1, n) S2(m, n-m) mSn =( :)+( :)+. . . +rim) m n> (Unpublished B. Checks tables) 124.171 Gupta u(n, m) r+sL n We feel that a capital S is natural for Stirling numbers of the first kind; it is infrequently used for other notation in this contest.. But once it r 2 ir+l is used we have difficulty finding a suitable symbol for Stirling numbers of tlie second kind. The numbers are sufficiently important to warrant "D> (:')=(- mo ('''I) in., . . . (Inodp) pa prime a22 *Rev pnge XI.
- 3. COMBINATORIAL ANALYSIS 823 where m m 5 (-om m=O (;)f(z-m) n = E n,$, m = C mkpk 2Q n-k-1 ) k-0 k-0 p>mk, n k =f:=O k (-1)S-f 8-k ~kj(z--s) e n < C. Numerical analysis I 111. Special Values 2"(2n-1)(2n-3) . . . 3-1 ("n">= n! 24.1.2 Multinomial Coefficients I. DefinitiOM A. (n; nl, %, . . ., n,) is the number of ways of putting n=nl+nz+. . . +nm different objects into m different boxes with nk in the k-th box, k=I, 2, . . ., m. (n; all %, . . ., a,,)* is the number of permutations of n=a1+2%+. . . +nu, symbols composed of a k cycles of length k for k=1, 2, . . ., n. (n; al, %, . . ., an)' is the number of ways of partitioning a set of n=a1+2az+. . . +na. dif- ferent objects into ak subsets containing k objects for k=1, 2, . . ., n. B. Generating functions (zl+%+. . . +z,)"=Z(n; nl, 121, . . ., nm)z;%P . . . z3 summed over a+-+. . . +nm=n (2 P) k-1 m =m! OD t" n-m 12. Z(n;al,a2, . . ., a ) z 1 ; . n*;zr. .: z- summed over a1+2%+ . . . +m,,=n C. Closed forms (n; nl, Q, . . ., n,) =n!/w!n,! . . . n,! nl+%+. . .+nm=n (n; all az, . . ., a,,) *=n!/lal@!2"Iag! . . . n".cr,,! a1+2%+ . . .+m,=n {{ { (n;al,az, . . ., ~ , ) ' = n ! / ( l ! ) ~ l a ~ ! ( 2 ! . ~ ! (n!)"a,! ). . al+%+. . . +m,=n 11. Relatiom A. Recurrence m (n+m;nl+l,nz+l,. . . , n m + l ) = C (n+m-l;nl+l, . . .,nk-l+l,nk,nk+l+l, . . . ,n,,,+l) k-1 B. Checks all ni 1 * Z(n;nl,%, . . .,n,,,)= summed over nl -]-nz+ . . . +nm=n Z(n;al,%, . . ., ~,,)*=(-1)"-?3~~' summed over a1+2%+ . . . +m,,=n and al+az+. . . +an=m Z(n;a,,az, . . ., a,)'= SP) C. Numerical analysis (FaA di Bruno's formula) d" n = C f("'(g(z))W; al,G, . . , an)' g'b) 1'1 g"(z) 1'2 . g'"'(z) 1 dz"g b ) ) -f( m-0 . . . 'I summed over al+2%+ . . . +nun=n and al+az+. . . +an=m.
- 4. 824 COMBINATORIAL ANALYSIS P, 1 0 ... 0 Pz Pl 2 ... P, Pz Pl ... ... =2(-1)"-mi(n; u,, &, . . ., u,)*PflP,"2.. . P, : ... 0 . .. n-1 Pn Pn-1 Pn-2 ... P, I. Definitions I Sin)I - (n- I)! (r+ln n)m-l/ -1) ! (m A. (-l)n-mSim) number of permutations is the for m=o(ln n) of n symbols which have exactly m cycles. B. Generating functions n z(x-1) . . . (x-n+l)=C S!,m)z" m-0 {ln (1+r)jm=m! m n-m a 2" Sim) bl < C. Closed form (see closed form for $3,"')) 11. Relations I A. Recurrences 24.1.4 Stirling Numbers of the Second Kind S;y),= Sj,m-l)_nS;m) n2.mrl I. Definitions (:) ~ i m ) =-m-r ~ c) S Lk i - ) A ) Smr nzmzr A. aim'isthe number of ways of partitioning a set of n elements into m non-empty subsets. B. Generating functions B. Checks n z"=C gpX(x-1) . . . (2-m+l) m -0 C. Numerical analysis I lm' z< - C. Closed form if convergent.
- 5. COMBINATORIAL ANALYSIS 825 B. Generating function B. Checks n (-1)"-=m! s = p 1 where m-0 ((z))=z-[z]-& if z is not an integer =O if x is an integer 11. Relationa A. Recurrence p(n)= & (-l)k-lp (-) n? p(0)=1 1- s 1 In 5 =I ol(k)p(n-k) n k-1 B. Check 111. Asymptotics and Special Velum * lim m-" sAm)=(mt)-l n-m 111. Aapptoties 1 *d%& P(4- - 4nd3e 24.2.2 Partitions Into Distinct Parte I. Definitiono A. q(n) is the number of decompositions of n into distinct integer summands without regard to order. E.g., 5=1+4=2+3 so that q(5)=3. B. Generating function m m OD 24.2. Partitions qq(n)z"= II (l+x")= n- n-1 n-1n 11 z< 1 24.2.1 Unrestricted Partitions C. Closed form I. Definitions A. p(n) is the number of decompositions of n into integer summands without regard to order. where Jo(x) is the Bessel function of order 0 and E.g.,5=1+4=2+3=1+1+3=1+2+2=1+1+ was defined in part IC. of the previous 1+2=1+1+1+1+1 so that p(5)=7. subsection. *See page 11.
- 6. 826 COMBINATORIAL ANUYBIS 11. Relations =2 g(z) f(m)for all z>O if and only if n-1 A. Recurrences m=5 n-1 for all r(n)g(7=) z>o and if m a 1 If(mnz) - =c If(nz) I converges. q(n) tu-1 n-1 n-1 =O otherwise The cyclotomic polynomial of order n ie II ($- l)r(n/d) dln B. Check 111. Aeymptotics f-r (-l)'q(n-(3k'~kf))=l if n =- O<;tkktk<n 2 =O otherwise. 111. Aspptotics 24.3. Number Theoretic Functions 24.3.2 The Euler Totient Function 24.3.1 The Mobius Function 1. Definitions I. Definition8 A. p(n)the number of integers not exceeding is A. p(n)=l if n=l and relatively prime to n. =(-l)& n is the product of k distinct if B. Generating functions primes =O if n is divisible by a square >1. aa>2 B. Generating functions 2p(n)n-s=l/r(s) a-1 ~'s>I 14<1 C. Closed form 11. Relations A. Recurrence over distinct primes p dividing n. p(mn)=p(m)p(n) if (m,n)=l 11. Relations =O if (m, n)>l A. Recurrence B. Check (m, n)=1 (P(m4 =dm)&) cc(d)=b B. Checks C. Numerical analysis g(n)=pf(d) for all n if and only if n =g f(4 r(d)g(n/d) n for all =11 g(n) f(d) for all n if and only if dln f(n) =n g(n/d)r(" n for all ar(") 1 (mod n) = (a, = 1 n) dln g(z)=Ej(z/n) z>O if and only if for all 111. Asymptotics a-1 j(x> =gp(n)g(z/n) z>o a-1 for all
- 7. COMBINATORIAL ANALYSIB 827 1 " 24.3.3 Divisor Functions I. Definitions - n2 m=l u1(m)=-+O 12 u2 CY> - A. uk(n) the sum of the k-th powers of the is 24.3.4 Primitive Roots divisors of n. Often udn) denoted by d(n), is and U l b )b y 4%). I. Definitions B. Generating functions The integers not exceeding and relatively prime to a fixed integer n form a group; the group is cyclic if and only if n=2,4 or n is of the form pk or 2pkwhere p is an odd prime. Then g is a primitive root of n if it generates that group; i.e., if g, g2, . . ., g+'(")distinct modulo n. There are cp(cp(n)) are C. Closed form primitive roots of n. 11. Relations A. Recurrences. If g is a primitive root of a prime p and gp-l$ l(mod p2) g is a primitive then root of pk for all k. If gp-' = 1(mod p2) then g+p is a primitive root of pkfor all k. If g is a primitive root of pk then either g or g+pk,whichever is odd, is a primitive root of 2p'. B. Checks. If g is a primitive root of n then gk is a primitive root of n if and pnly if (k, cp(n), = 1, and each primitive root of n is of this form. References Texts [24.13] H . Rademacher, On the partition function, Proc. London Math. SOC. 241-254 (1937). 43, [24.1] L. Carlitr, Note on Norlunds polynomial B$), [24.14] H. Rademacher and A. Whiteman, Theorems on Proc. Amer. Math. SOC. 452-455 (1960). 11, Dedekind sums, Amer. J. Math. 63, 377-407 [24.2] T. Fort, Finite differences (Clarendon Press, (1941). Oxford, England, 1948). [24.15] J. Riordan, An introduction to combinatorial [24.3] H. W. Gould, Stirling number representation analysis (John Wiley & Sons, Inc., New York, problems, Proc. Amer. Math. SOC.11, 447-451 N.Y., 1958). (1960). [24.16] J. V. Uspensky and M. A. Heaslet, Elementary [24.4] G. H. Hardy, Ramanujan (Chelsea Publishing Co., number theory (McGraw-Hill Book Co., Inc., New York, N.Y., 1959). New York, N.Y., 1939). [24.5] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 4th ed. (Clarendon Tables Press, Oxford, England, 1960). [24.17] British Association for the Advancement of Science, [24.6] L. K. Hua, On the number of partitions of a num- Mathematical Tables, vol. VIII, Number-divisor ber into unequal parts, Trans. Amer. Math. SOC. tables (Cambridge Univ. Press, Cambridge, 51, 194-201 (1942). England, 1940). n S l 0 ' . [24.7] C. Jordan, Calculus of finite differences, 2d ed. [24.18] H. Gupta, Tables of distributions, Res. Bull. East (Chelsea Publishing Co., New York, N.Y., Panjab Univ. 13-44 (1950); 750 (1951). 1960). [24.19] H. Gupta, A table of partitions, Proc. London [24.8] K. Knopp, Theory and application of infinite Math. SOC.39, 142-149 (1935) and 11. 42, series (Blackie and Son, Ltd., London, England, 546-549 (1937). p(n), n=1(1)300; p(n), n=301 1951). (1)600. [24.9] L. M. Milne-Thomson, The calculus of finite [24.20] G. KavBn, Factor tables (Macmillan and Co., Ltd., differences (Macmillan and Co., Ltd., London, London, England, 1937). n 1256,000. England, 1951). [24.21] D. N. Lehmer, List of prime numbers from 1 to [24.10] L. Moser and M. Wyman, Stirling numbers of the 10,006,721, Carnegie Institution of Washington, second kind, Duke Math. J. 25, 29-43 (1958). Publication No. 165, Washington, D.C. (1914). [24.11] L. Moser and M. Wyman, Asymptotic develop- [24.22] Royal Society Mathematical Tables, vol. 3, Table ment of the Stirling numbers of the first kind, of binomial coefficients (Cambridge Univ. Press, J. London Math. SOC. 133-146 (1958). 33, Cambridge, England, 1954). (:).for r <*n< 100. i24.121 H. H. Ostmann, Additive Zahlentheorie, vol. I [24.23] G. N. Watson, Two tables of partitions, Proc. (Springer-Verlag, Berlin, Germany, 1956). London Math. SOC. 550-556 (1937). 42,