COS 341 Discrete Mathematics. Exponential Generating Functions

Transcription

1 COS 341 Dscrete Mathematcs Epoetal Geeratg Fuctos 1

2 Read the tetbook Trouble keepg pace? Optoal referece tet (Rose) has may more solved eercses ad practce problems Start early o homework assgmets Come to offce hours, make separate appotmets Lear from dscussos wth fellow studets Tutorg: Seors: See Dea Rchard Wllams (48 West College, 8-55) Juors: See Dea Frak Ordway (44 West College, ) Sophomores: See Drector of Studes your home college

3 Ordary Geeratg Fuctos ( a, a, a, ) : sequece of real umbers 1 Ordary Geeratg Fucto of ths sequece s the power seres a( ) = a = 3

4 Epoetal Geeratg Fuctos ( a, a, a, ) : sequece of real umbers 1 Epoetal Geeratg fucto sequece s the power seres a ( ) a = =! of ths 4

5 Epoetal geeratg fucto eamples What s the geeratg fucto for the sequec e (1,1,1,1, )? = 3 1 = = e! 1!! 3! What s the geeratg fucto for the sequec e (1,,4,8, )? = 3 ( ) ( ) = =! 1!! 3! e 5

6 Operatos o epoetal geeratg fuctos Addto ( a+ b, a1+ b1, ) has geeratg fucto a( ) + b( ) Multplcato by fed real umber ( αa, αa1, ) has geeratg fucto αa( ) Shftg the sequece to the rght (,, a, a, ) has geeratg fucto a( ) 1 Shftg to the left ( a, a 1, ) has geeratg fucto k 1 a ( ) a = k k+ 6

7 Substtutg α for ( a, αa, α a ) o a( α) 1 has geeratg fuct Substtute for ( a,,, a1,,, a ) has geeratg fucto a( ) 1 1 7

8 Dfferetato d ( a1, a,3 a3 ) has geeratg fucto a( ) (or a'( ) ) d Itegrato 1 1 a a1 3a has geeratg fucto f tdt (,,, ) ( ) Multplcato of geeratg fuctos ( )( ) ( a ) b = c = = = c = a b k= k k 8

9 Dfferetato a ( ) s the epoetal geeratg fucto fo r ( a, a, a, ) a ( ) a = =! d a a a ( ) = = d! ( 1)! = = d a ( ) s the epoetal geeratg fucto for ( a 1, a, a 3, ) d Dfferetato s equvalet to shftg the sequece to the left 9

10 Itegrato a ( ) s the epoetal geeratg fucto fo r ( a, a, a, ) atdt () a ( ) a = =! a = =! tdt + 1 a = =! ( + 1) 1 a = = ( + 1)! atdt () s the epoetal geeratg fucto f or (, a, a, a, ) Itegrato s equvalet to shftg the sequece to the rght 1

11 Multplcato Ordary ( )( ) ( a ) b = c = = = c = a b k= k k Epoetal a b c = = = =!!! c a b k k = k=!! k ( k)! c! = a b k!( k)! k= k k = a b k k= k k 11

12 Implcatos of product rule C ( ) = AB ( ) ( ) Ordary c = a b k= k k Useful for coutg wth dstgushable objects Epoetal c = a b k k k= k Useful for coutg wth ordered objects 1

13 Iterpretato of Multplcato: Product Rule Gve arragemets of type A ad type B, defe arragemets of type C for labeled objects as follows: Dvde the group of labeled objects to two groups, the Frst group ad the Seco d group; arrage the Frst group by a arragemet of type A ad the S ecod group by a arragemet of type B. a b c : umber of arragemets of type A for objects : umber of arragemets of type B for objects : umber of arragemets of type C for objects 13

14 Iterpretato of Multplcato: Product Rule a b c : umber of arragemets of type A for people : umber of arragemets of type B for people : umber of arragemets of type C for people c = a b k k k= k a b c : epoetal geeratg fucto A( ) : epoetal geeratg fucto B( ) : epoetal geeratg fucto C( ) C ( ) = AB ( ) ( ) 14

15 A( ) : epoetal geeratg fucto for arragemets of type A a = : o empty group allowed Defe arragemets of type D for labeled objects as follows: Dvde the group of labeled objects to k groups, the Frst group, Seco dgroup,, kthgroup ( k =,1,, ) arrage each group by a arragemet of type A. D) ( : epoetal geeratg fucto for arragemets of type D 15

16 Dk ( ) : epoetal geeratg fucto for arragemets of type D wth eactly k groups D ( ) = A( ) k k D ( ) = Dk ( ) k= = A( ) k k= 1 = 1 A ( ) 16

17 A( ) : epoetal geeratg fucto for arragemets of type A a = : o empty group allowed Defe arragemets of type E for labeled objects as follows: Dvde the group of labeled objects to k groups, ad arrage each group by a arragemet of type A (the groups are ot umbere d ). E) ( : epoetal geeratg fucto for arragemets of type E 17

18 Ek ( ) : epoetal geeratg fucto for arragemets of type E wth eactly k groups k A( ) Ek ( ) = k! E ( ) = Ek ( ) k= = k= ( ) = e A A( ) k! k 18

19 Eample How may ways ca people be arraged to pars, (the pars are ot umbered)? term A ( ) : epoetal geeratg fucto for a sgle par E ( ) = a = 1, a = for A ( ) = : epoetal geeratg fucto for arragg people to / pars E ( ) = 1 =! ( /)! e e e =! / ( /)! 19

20 Deragemets (or Hatcheck lady revsted) d : umber of permutatos o objects wthout a fed pot D ( ) : epoetal geeratg fucto for umber of deragemets A permutato o [ ] ca be costructed by pckg a subset K of [ ], costructg a deragemet of K ad fg the elemets of [ ]- K. Every permutato of [] arses eactly oce ths way. EGF for all permutatos! 1 = =! 1 = EGF for permutatos wth all elemets fed 1 1 = D ( ) e 1! = = = e

21 Deragemets 1 = D ( ) e 1 1 D ( ) = e 1 = ( 1)! = = k d ( 1) = coeffcet of =! k = k! k ( 1) d =! k! k= Dfferet proof Matousek 1., problem 17 1

22 Eample How may sequeces of letters ca be formed from A, B, ad C such that the umber of A's s odd ad the umber of B's s odd? EGF for A's EGF for B's EGF for C' s requred EGF = e = = odd! e e = = =! e = e = e e e e e e 3 + 4

23 Eample How may sequeces of letters ca be formed from A, B, ad C such that the umber of A's s odd ad the umber of B's s odd? requred EGF = coeffcet of = requred umber = e e e ( 1) +! ( 1) 4 3