Exponential Generating Functions

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Curtis Riley
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1 Epotl Grtg Fuctos COS 3 Dscrt Mthmtcs Epotl Grtg Fuctos (,,, ) : squc of rl umbrs Epotl Grtg fucto of ths squc s th powr srs ( )! 3 Ordry Grtg Fuctos (,,, ) : squc of rl umbrs Ordry Grtg Fucto of ths squc s th powr srs ( ) Epotl grtg fucto mpls Wht s th grtg fucto for th squc (,,,, )? !!! 3! Wht s th grtg fucto for th squc (,,,8, )? 3 ( ) ( ) !!! 3!

3 Ordry Grtg Fuctos (,,, ) : squc of rl umbrs Ordry Grtg Fucto of ths squc s th powr srs (

2 Oprtos o potl grtg fuctos Addto ( + b, + b, ) hs grtg fucto ( ) + b( ) Multplcto by fd rl umbr ( α, α, ) hs grtg fucto α( ) Shftg th squc to th rght (,,,, ) hs grtg fucto ( ) Shftg to th lft ( ) + hs grtg fucto (,, ) 5 Dffrtto d (,,3 3 ) hs grtg fucto ( ) (or '( ) ) d Itgrto (,,, ) hs grtg fucto f( t) dt 3 Multplcto of grtg fuctos ( )( b ) ( c ) c b 7 Substtutg α for (, α, α ) o ( α) hs grtg fuct Substtut for (,,,,,, ) hs grtg fucto ( ) Dffrtto ( ) s th potl grtg fucto fo r (,,, ) ( )! d ( ) d! ( )! d ( ) s th potl grtg fucto for (,, 3, ) d Dffrtto s quvlt to shftg th squc to th lft 6 8

f( t) dt 3 Multplcto of grtg fuctos ( )( b ) ( c ) c b 7 Substtutg α for (, α, α ) o ( α) hs grtg fuct Substtut for (,,,,,, ) hs grtg fucto ( )

3 Itgrto ( ) s th potl grtg fucto fo r (,,, ) tdt () ( )!! tdt +! ( + ) ( + )! tdt () s th potl grtg fucto f or (,,,, ) + Ordry Epotl Implctos of product rul C ( ) AB ( ) ( ) c b c b Usful for coutg wth dstgushbl objcts Itgrto s quvlt to shftg th squc to th rght 9 Usful for coutg wth ordrd objcts Ordry Epotl Multplcto ( )( b ) ( c ) c b b c!!! c b!! ( )!! c b!( )! b Itrprtto of Multplcto: Product Rul Gv rrgmts of typ A d typ B, df rrgmts of typ C for lbld objcts s follows: Dvd th group of lbld objcts to two groups, th Frst group d th Sco d group; rrg th Frst group by rrgmt of typ A d th S cod group by rrgmt of typ B. b c : umbr of rrgmts of typ A for objcts : umbr of rrgmts of typ B for objcts : umbr of rrgmts of typ C for objcts

9 Usful for coutg wth ordrd objcts Ordry Epotl Multplcto ( )(

4 Itrprtto of Multplcto: Product Rul b c b c : umbr of rrgmts of typ A for popl : umbr of rrgmts of typ B for popl : umbr of rrgmts of typ C for popl c b : potl grtg fucto A( ) : potl grtg fucto B( ) : potl grtg fucto C( ) D ( ) : potl grtg fucto for rrgmts of typ D wth ctly groups D ( ) A( ) D ( ) D ( ) A( ) A ( ) C ( ) AB ( ) ( ) 3 5 A( ) : potl grtg fucto for rrgmts of typ A : o mpty group llowd A( ) : potl grtg fucto for rrgmts of typ A : o mpty group llowd Df rrgmts of typ D for lbld objcts s follows: Df rrgmts of typ E for lbld objcts s follows: Dvd th group of lbld objcts to groups, th Frst group, Sco dgroup,, th group (,,, ) rrg ch group by rrgmt of typ A. Dvd th group of lbld objcts to groups, d rrg ch group by rrgmt of typ A (th groups r ot umbr d ). D) ( : potl grtg fucto for rrgmts of typ D E) ( : potl grtg fucto for rrgmts of typ E 6

A( ) : potl grtg fucto for rrgmts of typ A : o mpty group llowd Df rrgmts of typ D for lbld objcts s follows: Df rrgmts of typ E for lbld objcts s follows: Dvd th group of lbld objcts to groups, th

5 E ( ) : potl grtg fucto for rrgmts of typ E wth ctly groups A( ) E ( )! E ( ) E ( ) A( )! A( ) Empl How my squcs of lttrs c b formd from A, B, d C such tht th umbr of A's s odd d th umbr of B's s odd? 3 + rqurd EGF 3 + ( ) coffct of! 3 + ( ) rqurd umbr 7 9 Empl How my squcs of lttrs c b formd from A, B, d C such tht th umbr of A's s odd d th umbr of B's s odd? EGF for A's odd! EGF for B's EGF for C' s! rqurd EGF trm Empl How my wys c popl b rrgd to prs, (th prs r ot umbrd)? A ( ) : potl grtg fucto for sgl pr, for A ( ) E ( ) : potl grtg fucto for rrgg popl to prs E ( ) / /! ( /)!! ( /)!

3 + ( ) rqurd umbr 7 9 Empl How my squcs of lttrs c b formd from A, B, d C such tht th umbr of A's s odd d th umbr of B's s odd? EGF for A's odd!

6 Drgmts (or Htchc ldy rvstd) d : umbr of prmuttos o objcts wthout fd pot D ( ) : potl grtg fucto for umbr of drgmts A prmutto o [ ] c b costructd by pcg subst K of [ ], costructg drgmt of K d fg th lmts of [ ]- K. Evry prmutto of [] rss ctly oc ths wy.! EGF for ll prmuttos! EGF for prmuttos wth ll lmts fd! D ( ) Drgmts D ( ) D ( ) ( )! d ( ) coffct of!! ( ) d!!

drgmt of K d fg th lmts of [ ]- K. Evry prmutto of [] rss ctly oc ths wy.

Sequences and Series

Sequences and Series Secto 9. Sequeces d Seres You c thk of sequece s fucto whose dom s the set of postve tegers. f ( ), f (), f (),... f ( ),... Defto of Sequece A fte sequece s fucto whose dom s the set of postve tegers.