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24. Combinat orial AnalysisMathematical Properties
In ench sub-section of this chnpter we usea fised format wliicli
etnplinsizes tlic use a n dmethods or" cstcnding the nccotnpnriyinp
tables.T hc format follows this forin :1. D e f i n i t i o n s
A . Combinn orinlB . Generat ing functionsC . Closed formA .
RecurrencesB. Checks in comput. ingC . Basic use in numerical
analysisIn general tlie notations used ore stsndnrd.T his includes
the difference oper at or A defined onfunctions of t by Af(z)= f(r+
I)-f(;t), An+ y(r)
= A(A.J (z)), t he K ron ecker deltu the Rieninririzeta function
{ ( s ) and the grentest commondivisor sym bol (m, n ) , T he range
of th e summ andsfor n summation sign without limits is explainedto
the right of the formula.T he notat ions which are not s tnndnrd
are th osefor tlie multinomials whicli nre arbitrary sliort-hand
for use in this chapter , and those for theStir l ing numbers which
have never been stand-ardized. A shor t table of various notations
forthese numbers follows :
11 . R e l a t i o n s
111. A s y m p t o t i c a n d S p e c ia l V a lu e s
N o t a t i o n s for t h e S t i r l i n g NumbersR e fer en ce
F i r s t K i n d S e c o n d K i n d
This chapter S %!-)124.21 Fort S:-' y y *(24.71 Jordan A e
*(24.101 bIoser and Wyman S.: 0: :(24.151 Riordan d n , m) S ( n ,
m)
124.91 Milne-Thomson (: I :) B!% (:) B : Z(24.11 Carlitz} ( - 1
) n - " S l ( n - l , n - m ) S2 (m, n - m )[24.3]
Could(UnpublishedMiksa S ( n -m + 1 , n) m S ntables)
a special nncl easily recognizable symbol, andyet t h n t s\
-mbolmust be easy to write . We havesett led on a script capital 3
without any cer ta intyt h a t we 11ave sett led tliis quest ion
permanently.We feel th at tlie subscript-super script notat
ionemplinsizes tlie generating friiictioris (wliicli arep6wers of
mutunlly inverse functions) from whichmost of the impor t ant r
elations flow.
24.1 . Ba s ic N um be r s24.1.1 Bin om ial Coeff ic ient s
1. D e f i n i t i o n sA . ( z)s the nu mb er of \ yays of
choosing mobjects from a collection of n dist inct objectswithout
regard to order .13 . G ener at ing fur ict ioiis
n = O , l , . .
C. Closed form
n > mn ( n - 1 ) . . . ( n -m + l)m !- -11 . R e l a t i o n
s
A. Recurrencesn > m > l
=( :)+( :::)+. . . + r im )> mB. Checks
124.171 Gupta u( n , m )We feel that a capital S is natural for
Stirlingnumbers of the first kind; it is infrequently usedfor other
notat ion in th is cont est . . Bu t once i tis used we have
difficulty finding a suitablesymbol for Stir l ing numbers of tlie
second kind.T he num bers are suffic iently import ant to warr
ant
a22 *Rev p n g e XI .
r + s L nr 2 r + l
(:')= ("D> ( ' ' ' I ) . . . ( Inodp) pa pr imemo in.,
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826 COMBINATORIAL ANUYBIS11. Relations
A. Recurrences
= O otherwise
B. Checkf - r( -l) ' q( n -( 3k' ~kf ) ) =lf n=O< ;tkktk
1.primes
B. Gener ating functions2 p( n ) n -s=l/ r ( s ) ~ ' s >
Ia-111. Relations
A. Recurrencep( m n ) =p( m ) p( n )f ( m ,n ) =l
= O if (m , n ) > lB. Check
cc( d ) = bC. Numerical analysisg( n ) pf(d)or a ll n f and only
ifn f ( 4g ( d) g( n / d)or all n
g( n )11 f(d) for alln f and only if
g( z ) =E j ( z / n )or all z >O i f and only ifdln f (n )= n
g( n / d) r ( "or all ndl na-1
j ( x > g ( n ) g( z/ n )or all z >oa-1
g( z) 2 (m ) or all z >O if and o n l y ifn-1 m= 5( n ) g(
7=)or all z>on-1m a -and if I f ( m nz)= c( n )f(nz) I
converges.
The cyclotomic polynomial of order n ietu-1 n-1 n-1
I I ($- l ) r ( n / d )dl n
111. A e y m p t o t i c s
24.3.2 T h e Euler T o t i e n t F u n c t i on1.
Definitions
A . p( n )s the number of integers not exceedingB. Gener ating
functions
and relat ively prime to n .
C. Closed form
over dist inct primes p dividing n .11. Relations
A. Recurrence( P ( m4= d m )&)B . Checks
ar(" ) 1 (mod n)111. Asympto t ics
aa>2
14"- u1(m)= -+ O -2 m =l24.3.4 P r i m i t i v e R o o t s
I . DefinitionsT he int egers not exceeding and relatively pr
ime
to a fixed integer n form a group; the group iscyclic if an d o
n l y if n = 2 , 4 or n is of th e form pkor2pkwhere p is an odd
prime. Then g is a primitiveroot of n if i t genera tes t hat group
; i.e., if g, g2, . . .,g+ ' ( ")re dist inct modulo n . There are
cp(cp(n)pr imitive root s of n .11. R elat ions
A . Recurrences. If g is a primitive root of aprime p and gp-l$
l (mod p2) hen g is a primitiveroot of pk for all k. If gp-' =
1(mod p2) hen g + pis a pr imitive root of pk or all k .If g is a
primitive root of p k then either g org+ p k ,whichever is odd, is
a pr imitive root of 2p'B. Checks. If g is a p rimitive root of n
then gkis a p rimitive root of n if and pnly if (k, cp( n) ,1,and
each p rimitive root of n is of this form.
ReferencesTexts
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