Exponential Generating Functions - J. T. Butler
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1 Epoetal Geeatg Fuctos - J. T. Butle Epoetal Geeatg Fuctos Geeatg fuctos fo pemutatos. Defto: a +a +a a + s the oday geeatg fucto fo the sequece of teges (a, a, a 2, a, ). Ep. Ge. Fuc.- J. T. Butle Ep. Ge. Fuc.- J. T. Butle 2 Obsevato: Oday geeatg fuctos ae ofte used fo combatos. Defto: a a a a K!! 2!! s the epoetal geeatg fucto fo the sequece of teges (a, a, a 2, a, ). Ep. Ge. Fuc.- J. T. Butle Ep. Ge. Fuc.- J. T. Butle 4 Obsevato: Epoetal geeatg fuctos ae ofte used fo pemutatos. Eample: K= e! 2!! s the epoetal geeatg fucto fo pemutatos of oe, oe, two, thee, detcal objects. Sce all objects ae the same, thee s oly oe pemutato. Ep. Ge. Fuc.- J. T. Butle 5 Ep. Ge. Fuc.- J. T. Butle 6
2 Epoetal Geeatg Fuctos - J. T. Butle Cosde the poduct of two geeatg fuctos as follows. G( ) = A( ) B( ), 2 whee A( ) = B( ) = ( ).! 2! Ep. Ge. Fuc.- J. T. Butle 7 Cosde the cotbuto to of G () fom sgle tem fom A() ad a sgle tem fom B(). p q ( p + q)! =!! p! q! ( p + q)! p q p+ q Numbe of ways to pemute p + q objects wth p of the fst kd ad q of the secod kd. Ep. Ge. Fuc.- J. T. Butle 8 F I K J F + + I 2 K J F I + + K J G( ) = + +!! 2!!!!! 2! 2 + ( + ) + ( 2 + 2) + ( 2 + 6) + K! 2!! a b aa ab aaa aab bb ba bbb aba baa bba bab abb Ep. Ge. Fuc.- J. T. Butle 9 K Eample: How may -dgt umbes ae geeated fom {,, 2,, 4}? Ep. Ge. Fuc.- J. T. Butle U V W = 2 The epoetal geeatg fucto s F I K J F I K J F K K K! 2!! 2!! 2! = e e e e e = e 5 The 5 tem s 5! Thus, thee ae 5 umbes. Ep. Ge. Fuc.- J. T. Butle F F K! 2!! 2! 2 I 2 K J. K I K J I K J Eample: How may -dgt umbes ae geeated fom {,, 2,, 4} whch the total umbe of s ad s s eve? Ep. Ge. Fuc.- J. T. Butle 2 U V W = 2 2
3 Epoetal Geeatg Fuctos - J. T. Butle Thee ae two ways to have a eve umbe of s ad s. eve s ad eve s 2. odd s ad odd s. eve s ad eve s F 2 4 I K J F I 2 4 K J F I K K K 2 K J!!!!!! 5 = + ( e e ) ( e + e ) e = e + e + e () Ep. Ge. Fuc.- J. T. Butle Ep. Ge. Fuc.- J. T. Butle 4 2. odd s ad odd s F I 2 K J F I K J F I K K K 2 K J!!!!!! 5 = ( e e ) ( e e ) e = e e + e Ep. Ge. Fuc.- J. T. Butle 5 (2) Addg () ad (2) yelds 5 e + e 2 F F 2 = K!!! 2! 2 2 I 2 K J + The coeffcet of Ep. Ge. Fuc.- J. T. Butle 6 c h s! 2 ( 5 + ) K I K J Dstbutg dstct objects to odstct cells The epoetal geeatg fucto fo the umbe of ways to dstbute dstct objects to dstct cells whee o cell s empty s 2 F I K ( e ) 2!! KJ = = F H G I K J = ( ) = F H G I K J = = F = H G I K J =! ( ) e ( ) =! ( ) ( ) ( ) Ep. Ge. Fuc.- J. T. Butle 7 Ep. Ge. Fuc.- J. T. Butle 8
4 Epoetal Geeatg Fuctos - J. T. Butle Thus, the umbe of ways to place dstct objects to dstct cells wth o cell left empty s F ( ) H G I ( )! (, ) K J = S = Thus, the umbe of ways to place dstct objects to odstct cells s S(, )! = ( ) ( ) = F H G I K J Ep. Ge. Fuc.- J. T. Butle 9 Ep. Ge. Fuc.- J. T. Butle 2 Note: S(,) = S(,) = S(,) = S( -, -) + S( -, ) fo < < The secod equato ca be see as follows. Select oe object α, ad place t by tself. The, dstbute the emag objects the emag cells. Ths s couted by S (, ). Ep. Ge. Fuc.- J. T. Butle 2 Ep. Ge. Fuc.- J. T. Butle 22 Othewse, dstbute all objects ecept α cells, ad the place α oe of the cells. Ths s couted by S(, ). I effect, thee ae ow dstct cells because they cota dstct objects, whch ca be chose ways. Ep. Ge. Fuc.- J. T. Butle 2 Stlg umbe tagle B() Bell umbes Ep. Ge. Fuc.- J. T. Butle 24 4
5 Epoetal Geeatg Fuctos - J. T. Butle B() epesets the Bell umbes, amed afte Ec Temple Bell, who was the fst to aalyze them detal. He ded 96. I the 92 s ad 9 s, he wote scece fcto ude the ame Joh Tae. B() s the umbe of ways to place dstct objects to odstct cells, whee cells ca cota ay umbe of objects cludg. Ep. Ge. Fuc.- J. T. Butle 25 Ep. Ge. Fuc.- J. T. Butle 26 The fst te Bell umbes B() ,47 Bell umbes ca be descbed by B(, ) = B(, ) + B(, ), B( ) = B(, ) = B(, ) B(, ) = Ep. Ge. Fuc.- J. T. Butle 27 Ep. Ge. Fuc.- J. T. Butle 28 Bell tagle Sum B() Ep. Ge. Fuc.- J. T. Butle 29 Iteestg popetes `. The sum of the hozotal ow s the secod dagoal. 2. If the sum of the ow s added to the umbe at the ed of the ow, the et Bell umbe s obtaed.. Evey thd Bell umbe s eve. Ep. Ge. Fuc.- J. T. Butle 5
6 Epoetal Geeatg Fuctos - J. T. Butle Bell umbes. Cout the umbe of ways a umbe wth dstct pme factos ca be factoed. Fo eample, has dstct pme factos, 2,, ad 5. The umbe of ways to facto (2 5, 5 6,, 2 5, ad ) s 5, whch s B(). 2. Note that a way to hyme 5 les coespods to dstbutg 5 dstct objects (les of a poem) to odstct cells (each le a cell hymes wth all othe les that cell). Ep. Ge. Fuc.- J. T. Butle Ep. Ge. Fuc.- J. T. Butle 2 Couts the umbe of ways to hyme les of poety. Fo eample, B (5) appeas the Tale of the Gej, a famous Japaese book wtte by Lady Muasak, who lved fom about 978 to. Evey chapte of ths 54-chapte book, ecept the fst ad the last beg wth oe of the 52 ways to hyme fve les of poety. Hozotal les jo les that hyme. The 52 ways to hyme fve les of poety s show o the et two pages. Ep. Ge. Fuc.- J. T. Butle Ep. Ge. Fuc.- J. T. Butle 4 Lmeck Ep. Ge. Fuc.- J. T. Butle 5 Ep. Ge. Fuc.- J. T. Butle 6 6
7 Epoetal Geeatg Fuctos - J. T. Butle Ep. Ge. Fuc.- J. T. Butle 7 7
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