Combiatorially Thiig SIMUW 2008: July 4 25 Jeifer J Qui jjqui@uwashigtoedu Philosophy We wat to costruct our mathematical uderstadig To this ed, our goal is to situate our problems i cocrete coutig cotexts Most mathematicias appreciate clever combiatorial proofs But faced with a idetity, how ca you create oe? This course will provide you with some useful combiatorial iterpretatios, lots of examples, ad the challege of fidig your ow combiatorial proofs Throughout the ext two wees, your matra should be to eep it simple
Cotets Basic Tools 3 Some Combiatorial Iterpretatios 3 2 Coutig Techique : As a Questio ad Aswer Two Ways 6 3 Coutig Techique 2: Descriptio-Ivolutio-Exceptio (DIE) 7 2 Biomial Idetities 8 2 Worig Together 8 22 O Your Ow 9 23 What s the Parity of? 0 3 Fiboacci Idetities 3 3 Tilig with Squares ad Domioes 3 32 Worig Together 4 33 O Your Ow 5 34 Combiatorial Proof of Biet s Formula 6 4 Geeralizatios Lucas, Giboacci, ad Liear Recurreces 8 4 Tilig i Circles, Tilig with Weights 8 42 Worig Together 9 43 O Your Ow 9 44 Biet Revisited 20 5 Cotiued Fractios 2 5 Itroductios ad Exploratios 2 52 O Your Ow 22 53 Primes of form 4m + 23 6 Alteratig Sums 24 6 Worig Together 24 62 O Your Ow 26 63 Exted ad Geeralize by Playig with Parameters 26 7 Determiats 27 7 The Big Formula 27 72 Worig Together 28 73 O Your Ow 29 74 Vadermode Determiat 30 2
Basic Tools Some Combiatorial Iterpretatios First, we eed combiatorial iterpretatios for the objects occurrig i our idetities While there are may possible iterpretatios, oly oe is preseted for each mathematical object tryig, of course, to eep it simple I have icluded at least oe method to compute each object for completeess though we will rarely rely o computatio! factorial Combiatorial: The ways to arrage umbers, 2, 3,, i a lie Computatioal:! ( ) ( 2) 2 ( ) biomial coefficiet, choose Combiatorial: The ways to select a subset cotaiig elemets from the set [] {, 2, 3,, }! Computatioal:!( )! (( )) multichoose Combiatorial: The ways to cast votes for elemets from the set [] {, 2, 3,, } Computatioal: () + [ ] (usiged) Stirlig umber of the first id Combiatorial: The ways to arrage people aroud idetical (oempty) circular tables [ ] { [ ] 0 Computatioal: Recursively 0 0 0 ad ( )! For 2, [ ] [ ] + ( + ) [ ] 3
{ } Stirlig umber of the secod id Combiatorial: The ways to distribute people ito idetical (oempty) rooms { } { { } 0 Computatioal: Recursively 0 0 0 ad For 2, { } f the th Fiboacci umber { } + { } Combiatorial: The ways to tile a board usig squares ad 2 domioes Computatioal: f 0, f, ad for 2 f f + f 2 or f φ + 5 + φ where φ + 5 2 WARNING: You might be used to the Fiboacci umbers defied i the more traditioal way F 0 0, F, ad for 2 F F + F 2 L the th Lucas umber Combiatorial: The ways to tile a circular board usig squares ad 2 domioes Computatioal: L 0 2, L, ad for 2 L L + L 2 G the th Giboacci umber or L φ + φ Combiatorial: The ways to tile a board usig squares ad 2 domioes where the first tile is distiguished There are G choices for a leadig square ad G 0 choices for a leadig domio Computatioal: G 0 ad G are give ad for 2 G G + G 2 or G αφ + β φ where α (G + G 0 /φ)/ 5 ad β (φg 0 G )/ 5 4
D the th Deragemet umber Combiatorial: The ways to arrage, 2,, i a lie so that o umber lies i its atural positio ( Computatioal: D!! + 2! 3! + + ) ( )! C the th Catala umber Combiatorial: The umber of lattice paths from (0, 0) to (, ) usig right ad up edges ad stayig below the lie y x Computatioal: C 2 + [a 0,a,,a ] the fiite cotiued fractio a 0 + a + a 2 + + a p q Combiatorial: Numerator: The ways to tile a + board usig squares ad domioes where cell i ca cotai half a domio or as may as a i squares, 0 i Deomiator: The ways to tile a board usig squares ad domioes where cell i ca cotai half a domio or as may as a i squares, i Computatioal: Attac with algebra to ratioalize the complex fractio det(a) the determiat of the matrix A {a ij } Combiatorial: The siged sum of oitersectig -routes i a directed graph with origis, destiatios, ad a ij directed paths from origi i to destiatio j Computatioal: det(a) sig(σ)a σ() a 2σ(2) a σ() σ S Example: det 2 5 5 8 2 0 2 8 2+2 2 0+5 5 0 8 5 5 2 2 2 0 5
2 Coutig Techique : As a Questio ad Aswer Two Ways We will use oe of two techiques to cout a idetity The first poses a questio ad the aswers it i two differet ways Oe aswer is the left side of the idetity; the other aswer is the right side Sice both aswers solve the same coutig questio, they must be equal Idetity For,!! Questio: The umber of ways to arrage, 2, 3,, except for Aswer : Aswer 2: Idetity 2 For, 0, () + Questio: How may ways ca we allocate votes to cadidates? Aswer : Aswer 2: 6
3 Coutig Techique 2: Descriptio-Ivolutio-Exceptio (DIE) The secod techique is to create two sets, cout their sizes, ad fid a correspodece betwee them The correspodece could be oe-to-oe, may-to-oe, almost oe-to-oe, or almost may-to-oe Idetity 3 For, 0, () + Descriptio: Set : Set 2: Ivolutio: Exceptio: Idetity 4 For 0, ) 0( ) ( 0 Descriptio: Set : Set 2: Ivolutio: Exceptio: What happes is we chage the upper idex of the summatio to somethig smaller tha? larger tha? 7
2 Biomial Idetities 2 Worig Together Let s use these techiques to prove some idetities Idetity 5 For 0!!( )! Idetity 6 The Biomial Theorem For 0, (x + y) x y 0 + 0 x y + x 2 y 2 + + 2 x 0 y Idetity 7 For 0 0 2 2 8
22 O Your Ow Idetity 8 For 0, (except 0), + The techique above ca be modified to prove: Idetity 9 For 0, 0, (except 0), Idetity 0 For, Idetity For, [ ] { } [ ] + ( ) { } + () () + [ { } ] (( )) Idetity 2 For, 0 2 Idetity 3 For, 2 0 Idetity 4 For q 0, q q 2 q ( q ) Idetity 5 For m, 0, m m j j j Idetity 6 For j 0, m ( m j )( m j ) + + Idetity 7 For oegative itegers, 2,,, let N i i 2 The i + i<j ( i 2 ) j + 3 2 i 4 N 2 9
23 What s the Parity of? Pascal s Triagle 2 3 3 4 6 4 5 0 0 5 6 5 20 5 6 7 2 35 35 2 7 8 28 56 70 56 28 8 9 36 84 26 26 84 36 9 0 45 20 20 252 20 20 45 0 55 65 330 462 462 330 65 55 Serpisi-lie Triagle 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Theorem For 0, the umber of odd itegers i the th row of Pascal s triagle is equal to 2 b where b the umber of s i the biary expasio of Questio How may odd umbers occur i the 76 th row of Pascal s triagle? LemmaThe parity of will be the same as the parity of the umber of palidromic sequeces with oes ad zeros Descriptio Biary sequeces with oes ad zeros Ivolutio Exceptio As: 76 (0000) 2 Eight odd umbers 0
Cout palidromic sequeces with oes ad zeros eve odd eve eve odd eve odd odd Cosequece If is eve ad is odd, the ( ) is eve, otherwise, has the same parity as /2 /2 where we roud /2 ad /2 dow to the earest iteger, if ecessary Examples Compute the parity: 76 76 76 5 36 2 Thi ( biary! ) (0000)2 (0000)2 (0000)2 (000) 2 (00000) 2 (00000) 2
Oly way to have a odd umber is if the s i the biary represetatio of are directly below s i biary represetatio of It tells us exactly which umbers produce odd biomial coefficiets: biary represetatio 0000 76 00000 72 00000 68 000000 64 00000 2 000000 8 000000 4 0000000 0 Extesio There a similar procedure to determie the remaider of whe divided by ay prime p? Lucas Theorem For ay prime p, we ca determie the remaider of whe divided by p from the base p expasios of ad If b t p t + b t b t + + b p + b 0 c t p t + c t b t + + c p + c 0 the ad bt bt c t c t b b0 c c 0 have the same remaider whe divided by p Example Calculate the remaider of 97 whe divided by 5 35 As: ( 97 35) ( 3 )( 4 2)( 2 0) 3 mod 5 2