Generating Function for Partitions with Parts in A.P

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Geetig Fuctio fo Ptitio wi Pt i AP Hum Reddy K # K Jkmm * # Detmet of Memtic Hidu Coege Gutu 50 AP Idi * Detmet of Memtic 8 Mi AECS Lyout B BLOCK Sigd Bgoe 5604 Idi Abtct: I i e we deive e geetig fuctio fo e umbe of i ove G titio whe e t e i AP we o obti fomu fo e umbe of e met t of titio of Key Wod: Ptitio -titio Ove Ptitio Gtitio Geetig fuctio Sub : Cifictio :P8 Eemety Theoy of Ptitio I INTRODUCTION S yvecotee d Loveoy [] iitited e tudy of ovetitio of Hum ReddyK [] deived fomu fo e met t of ovetitio of G titio wee itoduced by GVRKSg[] I i e we itoduce oveg titio d oveg titio of d obti geetig fuctio i i cotext k A titio of i G titio if met t e of e fom k N If0 G titioof i G titio of wi excty t A oveg titio of i G titio of fit (euivety e fi) occuece of t i ove ied u to oveg titio of by G d it cdiity by G i i which time ucceivey We deote e et of we o obti e geetig fuctio fo G t II GENERATING FUNCTION FOR NUMBER OF PARTITIONS WITH PARTS IN AP Hum ReddyK [ ] etbihed t e geetig fuctio fo e umbe of titio of ito t wi ditict t occuig wi fixed feuecie f f f f i f f f f f f f f f f ; f f f f f f f f We exted i eut foow Theoem : The geetig fuctio fo e umbe of titio of ito t wi ditict t occuig wi fixed feuecie f f f f d ee t beog to e et f d f f d f f f d f f f d f f f f ; f f f d f f f d f d f d S k d dk N i d e i AP f Poof: Ay titio of umbe ito t c be witte i e fom f f fi f f f d eeet occuece of i i fi time A ime exme i whee wwwitemi Pge 4

Recig by ( ) d ( ) d d d i fit t d ee e two oibiitie fo e ecod t by d o eve it uteed d d We c eie ece Thu give e feuecie f f f f ytemtic chge of t of oibe titio ito t of which e ditict d occu wi ecified feuecie c be ited i y d geetig fuctio fo ech coum c be defied Let f f be give feuecie Sttig wi e titio ( ) d ( ) d d d we coide e obem whe e fit ce oy i teed Rece i e fit t wi (feuecie f ) by yied two titio The ext t coud be d o d Thee choice Fo exme if we choe d d f ou titio 5+5+++ give two titio 7+7+++ d 7+7+5+5+5 We ge em i two ow foow: 5+5+++ 7+7+++ d 7+7+5+5+5 We eet i oce ow ecig 7 i e ecod ow by 9 to get e id ow 9+9+++ 9+9+5+5+5 d 9+9+7+7+7 Succeive ictio of i oce yied e foowig titio i e ow i i e gee ce 5 5 5 Whe e get t i wi e feuecy f e ow i d d d d d d d d d d d d d d d d d d d d d o o d e t eemet i i ow i d d d d d We eet i oce d obti tigu y i which e fit t of e titio i e fit coum icee by d wi feuecy f e fit d ecod t icee by d wi feuecie f f d o o f d f d f d f d f d 0 Aocite Let wwwitemi Pge 5

k d f k d fo e eemet i e Thi yied e euied geetig fuctio i e foowig fom The geetig fuctio fo ech coum i k coum k k d f f k d f k d d f k d f f df k0 0 k0 0 k0 The euied geetig fuctio i which covege whe 0 k k f f d 0 d f d f d f f I e gee ce e geetig fuctio i f f d f f f d fd f d f d f d f d f whee Hece e geetig fuctio fo e umbe of titio of ito t wi ditict t occuig wi fixed feuecie f f f f d ee t beog to e et S d d N i f d f f d f f f d f f f d f f f f ; : S f f f d f f f d f f d f d Note: If we ut d b i e bove we get fomu which i deived by Hum ReddyK i [ ] Poof: Sice e geetig fuctio fo e umbe of titio of ito t wi ditict t occuig wi fixed feuecie f f f f i f f f f f f f f f f ; f f f f f f f f The geetig fuctio fo e umbe of titio of ito t fo ie combitio of f f f uch t f f f i f f f f f f f f f f f f f f f f f So e geetig fuctio fo e umbe of ovetitio of ito t wwwitemi Pge 6

d Exme (i) The f (ii) f f f f f f f f f f f f f The e oibiitie e f o f f (iii) The e oibiitie e f o f f o f f o f f f III GENERATING FUNCTION FOR G t I i ectio we ooe fomu fo fidig e umbe of met t of titio of Theoem: If wwwitemi Pge 7

k k k k d Poof: Let G t t t k t k t k be y G titio Fo fixed t ee ce ie Ce:Let By ubtctig k t If k wi ditict t m k e k fom ech t of k Hece t i k Coeodig to i ee e umbe of t ech eu to Ce : Omit k ' oveg titio i t k k we get t t titio of time k t wi t ovetitio of Thu e umbe of k i e et k t The t k ovetitio i k k t k k fom t t ce e t i t G titio of oveg titio Thu e umbe of Ce : time of t t k t k wi ditict t e et t beig k of t wi et t k - ditict t d ech ³ k t We kow t e tot hvig excty t met Coeodig to i ee e oveg titio hvig moe t met t ech beig k i k k G f t t t The t of e G titio e eu d ech t i of e fom N of e fom oveg titio of The umbe of G titio of wi eu t ech beig k Sice e umbe of uch divio of i e umbe of k Thi Ptitio h k i eu to oveg titio of i k i d The umbe of devie of Fom ce () () d () it foow t e umbe of which occu t time i oveg titio of wi met t k t k k k k d G f t t t wwwitemi Pge 8

t t k k k k G f t t t k k d t t k k k k d t t t Fom [] e umbe of met t i oveg titio of i k k k k k d G t t t Iuttio k t k t k A iuttio we tke = 6 = d = The titio ude coidetio e ited beow wi tot idicted t e ed of e ow The tot umbe of udeied t i 7 detied beow 5 ovet 09 4 09 4 8 4 4 0 0 5 4 5 We ow y e fomu 5 4 4 5 4 4 5 7 5 6 6 6 6 7 Ackowedgemet: The uo e kfu to (Rtd) ofeo IRmbhd m fo hi vube uggetio d commet duig etio of i e REFERENCES [] Syve Cotee d Jeemy Love oy :Ove titio TAmeMSoc 56(4) 6-65 [] Humeddy K (00) A tudy of -titio ei ubmitted to Achy Ngu uiveity fo wd of hd i Memtic [] Seg GVRK: A tudy of M titio Thei ubmitted Achy Ngu uiveity fo wd of PhD i Memtic wwwitemi Pge 9

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