05/11/2016. MATH 154 : Lecture 20
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1 05/11/2016 MATH 154 Lecture 20
2 m Elemets Showed ( mtl Last Time Itroductio to ( likely amiliar basic coutig priciples rom the poit o view o discrete uctios Cosidered iite sets M{ 1 } ad N{ 1 } ad Fu ( mm { M N } 1 Fu ( mm I? Cosidered Ijlm { t Fu ( mm / ijective } proved I Ijlm l/8mtsm( I i Special case whe m we have I Ijc ±! o Ij ( m bijective uctios N N aka permutatios Today Fuctios backwards ibres Quatitative Pigeohole Priciple Surjective uctios Stirlig umbers
3 ball Its ote useul to thik about uctios backwards as the ordered list o their ibres ( ll Cz ilm For example the uctio { 12 } } { 12 } deied by lll 2 l 211 l 3 I which may be visualized as the box coiguratio is the same thig as the ordered list ( { 23 } { I } This backwards descriptio o is very atural Forward ( m questios Backward ( questios Q Which box is ball x i? Q Which balls i boxy? A ix A ( y
4 The The The Propositio ( Quatitative Pigeohole Priciple Let e Fu ( mm has a ibre o cardiality at most LFI ad a ibre o cardiality at least F l Example Suppose m 12 ad 4 ; the ( Fi TFI 3 uctios e Fu ( 1241 just coiguratios o 12 balls i 4 boxes Pigeohole priciple says that there is a box cotaiig 3 balls or less (i ot each box would cotai at least 4 balls or a total o 16>12 ad a box cotaiig 3 balls or more ( i ot each box would cotai at most 2 balls or a total o 8<12
5 But Suppose Suppose Sice Proo Upper boud existece o ye N such that I ly It that that I iy I > Fr or all ye N The 1141 gl Cy I > F ith m IMI > IMI which is alse So there exists yet with Iiy / E Fi sice I ly / is a iteger we ca tighte this to Iiy 1 LFI Lower boud existece o ye N such that I cy 17 TFI that I cy / < Fr or all ye N The IMI y{ liyil < which is alse So there exists tighte this to Itty I Ft > y Fr m IMKIMI ye Y such that I y / 2 Fr Iiyil is a iteger we ca D
6 ormula Example I the case m+i the quatitave pigeohole priciple reduces to the statemet that ay uctio { 1 +1 } { 1 } has a ibre o size 1 I Llttl I or less ad a ibre o size 7Fl+ til 2 or more Deiitio A uctio t Fu ( mm is said to be surjective it each o its ibres is o empty Let Sur ( mm c Fu ( mm deote the set o surjectio s Thikig o t Fu ( mm o as a coiguratio m balls placed i boxes Clearlyțhis cotais at least oe ball surjectivity just meas that every box ca oly happe whe the umber o balls is at least as large as the umber o boxes so I Fu ( mm / Smz to be determied
7 As discussed ve we The ( each ball Coversely ly ibres uiquely is placed i determies a Fu ( ( ( l subsets o the exactly box oe (M vector every o t a every Mz uctio be ca the as ibres list o its ordered 1 } i M is uio l M{ domai viewed M whose elemets M N (2 their ad mm They ( 1 disjoit w (2 u u y tyz C disjoit subsets o M such { 12 that C m y } cy i Such vectors uctios all o whose i called M N Uortuately My weak ad ibres which all weak o there o is Mir Mir umm via My part set compositios empty YEN correspod o compositios ( M to o M M strict M part there Thus is a bijective correspodece betwee Surjective uctios set compositios (M M N M o beig M those as ie empty o explicit ormula or the umber o strict part set compositios o M p ly those
8 ball Example Here the 238 uctios i Fu( 32 viewed both as box coiguratios ad as 2 part set compositios o { 143} ( surjectio marked with * * 3 3 ( 221 ( { I 2 I 23z I ( { 12 } { 3} * 3 i ( { 3 } { 12 } } { 12 } * i 2 ( { 11 { 231 * 2 ( { 2 } { 13} * 2 ( { 131{ 21 * 2 ( {231 {
9 The part However there is a ormula or Sur( mi / i terms o Stirlig umbers Deiitio A uordered list { M set partitio o M Stirlig set umber ( or a M } o disjoit o empty sets whose uio is M is called umber o Stirlig umber o the secod kid part set compositios o M is deoted { m } ad called a Example ( { 12 } { 3 } ad ( { 3 } { 12} dieret 2 part set compositios o { 1431 whereas { { 12 } { 3 } } ad { { 31 { 12 } } the 2 same part partitio o { 1231
10 m Propositio lsurlm l/! { m } Proo As we have discussed the umber Sur( mm I be ca thought o as the umber o set part set compositios o M { 1 } Each such compositio is obtaied oe by choosig o the { F } part set o partitios M ad the choosig oe o the! ways o listig its blocks D This is a somewhat usatisyig ormula or / Sur ( mm I because we dot have a ormula or { F} However we ll see soo that the Stirlig umbers ca be calculated recursively
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