Solutions for HW4. x k n+1. k! n(n + 1) (n + k 1) =.

Transcription

1 Exercse 13 (a Proe Soutos for HW4 (1 + x 1 + x (1 + x 2 + x (1 + x + x 2 + by ducto o M(Sν x S x ν(x Souto: Frst ote that sce the mutsets o {x 1 } are determed by ν(x 1 the set of mutsets o {x 1 } s bjecto wth N So 1 + x 1 + x x 1 x ν(x 1 1 x ν(x N M({x 1 }ν M({x 1 }ν x {x 1 } so our detty hods for 1 Now fx ad assume for S {x 1 x } we hae (1 + x 1 + x (1 + x 2 + x (1 + x + x 2 + The wth S S {x +1 } M(Sν x S (1 + x 1 + x (1 + x 2 + x (1 + x +1 + x IHOP (1 + x +1 + x N M(Sν x S x ν(x M(S ν x S x ν(x So our detty hods for a by ducto (b Show agebracay that ( ( 1 ( + 1 M(Sν x S x +1 x ν(x x ν(x Souto: ( ( 1 1! ( 1 ( ( 1 ( ( + 1! ( + 1 ( + 1 (c (a Wrte the geeratg fucto (both seres ad cosed form for the umber of wea compostos of wth parts [Ht: Ths shoud oo somethg e the geeratg fucto for mutsets] Souto: Sce the umber of wea compostos of wth parts s ( (( the geeratg fucto for wea compostos of wth parts s ( + 1 x (1 x 1 N

2 (b Wrte the geeratg fucto (both seres ad cosed form for the umber of (ot wea compostos of wth parts Souto: Sce the umber of compostos of wth parts s ( 1 1 the geeratg fucto for compostos of wth parts s ( 1 1 ( 1 x x +1 x(1 + x (c Wrte the geeratg fucto for the umber of wea compostos of wth parts a ess tha j Souto: Let κ( j be the umber of wea compostos of wth parts a ess tha j Start wth (1 + x 1 + x x j 1 1 (1 + x 2 + x x j 1 2 (1 + x + x xj 1 x α 1 1 xα N α 1 + +α 1 α j 1 The eauatg at x 1 x x we get N κ( j x N α 1 + +α 1 α j 1 x (1 + x + x x j 1 (1 x j (1 x (d Item ge a geeratg fucto proof that the umber of wea compostos of to parts wth each part ess tha j s ( ( + r 1 ( 1 s r s rs N r+sj Souto: expad Cotug from the preous part use the geerazed boma theorem to κ( j x (1 x j (1 x N ( s N ( s N ( s N ( ( ( x j s s r N ( ( ( 1 s x sj s r N ( ( ( 1 s x sj s r N ( ( x r r ( r ( 1 r x ( + r 1 r x

3 sce ( r ( 1 r ( +r 1 r Now usg the mutpcato rue for seres we hae that the coeffcet of x o the ast e s ( ( + r 1 ( 1 s s r rs N r+sj thus prog the desred resut Exercse 14 (a Derg mutoma coeffcets agebracay Let α (α 1 α be a composto of Use the formua (!!(! to compute ( α 1 α otg that you ca frst choose the α 1 tems from the α 2 from α 1 the α 3 from (α 1 + α 2 ad so o Souto: By frst choosg the α 1 tems from the α 2 from α 1 the α 3 from (α 1 +α 2 ad so o product rue says that ( α 1 α ( ( ( ( α1 (α1 + α 2 (α1 + α 1 α 1 α 2 α 3! ( α 1! ( (α 1 + α 2! α 1!( α 1! α 2!( (α 1 + α 2! α 3!( (α 1 + α 2 + α 3! ( (α α 1! α!( (α α!! α 1! α!!! α 1! α! sce (α α (b Mutoma theorem Foowg our proof of the boma theorem show that ( (x 1 + x x x α 1 1 α 1 α xα 2 2 xα (α 1 α N α 1 + +α [Ht: Reca that the ey computato for the boma theorem was that for S {x (1 x ( } x T + x x S(1 ( x ( so that (1 + x x ( T S x ( T T S x ( T T S The former we had to proe by ducto o Now fx ad et S {x (j 1 1 j } (so that there are dstct arabes assocated to each x ad wa through a smar proof] Souto: Frst cosder the case where 1 so that ( 1 x α 1 1 α 1 α xα 2 2 xα x 1 x 1x 1 x +1 x (α 1 α N α 1 + +α 1 Now assume (x 1 + x x 1 1 (α 1 α N α 1 + +α 1 α ( 1 α 1 α x 1 x α 1 1 xα 2 2 xα

4 for a fxed Note that sce ( α 1 α ges the umber of ways of parttog [] to abeed sets of sze α 1 α respectey by tracg whch set goes to we hae that ( α 1 α ( 1 1 α 1 α 1 α 1 α +1 α (ths s the mutoma coeffcet geerazato of the detty ( ( 1 ( 1 + sce ( ( So ( ( (x 1 + x x IHOP 1 x x α 1 1 α 1 α xα 2 2 xα by ( 1 1 (α 1 α N α 1 + +α 1 (α 1 α N α 1 + +α 1 (α 1 α N α 1 + +α (α 1 α N α 1 + +α (α 1 α N α 1 + +α 1 ( 1 α 1 α ( 1 1 ( 1 ( + 1 α 1 α x (x α 1 1 xα 2 2 xα x α 1 1 xα 1 1 xα +1 x α xα 1 α 1 α 1 α 1 α +1 α α 1 α x α 1 1 xα 2 2 xα x α 1 1 xα 2 2 xα (c Lattce paths Proposto 121 EC1 says the foowg Let (a 1 a d N d ad et e deote the th ut coordate ector Z d The umber of attce paths Z d from the ( org ( to wth steps {e 1 e d } s ge by the mutoma coeffcet a1 + +a d a 1 a d ( Chec ths proposto for d 2 wth the pot (2 3 Souto: The attce paths from ( to (2 3 wth steps S {(1 ( 1} are ( of whch there are 1 (2+3! 2!3! ( ( Chec ths proposto for d 3 wth the pot (1 1 2 Souto: The attce paths from ( to (1 1 2 wth steps S {(1 ( 1 ( 1} are

5 of whch there are 12 (1+1+2! 1!1!2! ( ( Proe ths theorem (spe out the boo s proof wth more detas Souto: Let be a attce path from to (a 1 a d wth eemetary steps where a a d The cosder the sequece determed by the path ( (e 1 e 2 e S M where M s the mutset o S {e 1 e d } wth weght ν(e a sequece (e 1 e 2 e S M determes the path e 1 e 1 + e 2 e e Smary each from to e 1 + +e (a 1 a d So the attce paths from to (a 1 a d wth eemetary steps are bjecto wth permutatos S M of whch there are ( a 1 + +a d a 1 a d (d Iteger parttos A (teger partto λ (λ 1 λ of s a composto of satsfyg λ 1 λ 2 λ > We draw parttos as boxes ped up ad to the eft to a corer wth λ 1 boxes the frst row λ 2 boxes the secod row ad so o For exampe λ ( s a partto of 11 λ (5 4 3 s a partto of 12 λ (5 s a partto of 5 ad λ s a partto of The sx parttos to ft a 2 2 square are (1 (2 (1 1 (2 1 ad (2 2 Use attce paths to cout the umber of teger parttos fttg to a m rectage Souto: Note that by tag a partto sde a m rectage ad oerayg a m grd wth the org ( at the south-west corer we hae that tracg the south-east wa of the partto tracg the wa of the grd whe approprate determes a attce path from ( to ( m wth eemetary steps Vce ersa ay attce path from ( to ( m wth eemetary steps determes the partto whose th part eds at the m + 1st up-step For exampe ad Thus parttos sde a m rectage are bjecto wth attce paths from ( to ( m wth eemetary steps of whch there are ( +m m

6 Exercse 15 (a For each of the foowg ge exampes for sma aues of The express the foowg umbers terms of the Fboacc umbers ( Exampe: The umber of subsets S of the set [] {1 2 } such that S cotas o two cosecute tegers Aswer: Let a be the umber of good subsets of [] Note that a 1 { {1}} 2 ad a 2 { {1} {2}} 3 Now dde S to 2 cases: ether t cotas or t does t Sce eery good subset wthout s aso a good subset of [ 1] ad ce ersa the umber of good subsets wthout s a 1 Smary S S {} s a bjecto betwee good subsets of [] cotag ad good subsets of [ 2] the umber of good subsets of [] cotag s a 2 So a a 1 + a 2 wth a 1 2 f 3 a 2 3 f 4 Ths s the same recurrece that determes f but shfted so that a f +2 So there are f +2 good subsets of [] NOTE: For may of these ths s the strategy you wat Mae a recurrece reato that oos e the Fboacc recurrece ad shft appropratey For at east oe you wat to use a preous part ( The umber of compostos of to parts greater tha 1 Souto: Let S { compostos of to parts greater tha 1 } A { compostos of to parts greater tha 1 whose ast part s 2 } ad B { compostos of to parts greater tha 1 whose ast part s ote 2 } so that a S s the aue we wsh to eumerate ad S A B so that a A + B Note that a a 1 ad a 2 {(2} 1 The fucto from A whch taes (α 1 α 2 to (α 1 α s a fucto from A to S 2 sce α α + 2 ad α > 1 Its erse appedg a good composto of 2 by 2 s we-defed so A s bjecto wth S 2 Smary the fucto from B whch taes (α 1 α to (α 1 α 1 s a fucto from B to S 1 sce (α α 1 1 ad α > 1 for 1 1 ad α > 2 so α 1 > 1 Its erse addg 1 to the ast part s we-defed so B s bjecto wth S 1 Thus a S A + B S 2 + S 1 a 2 + a 1 Sce a 1 f ad a 2 1 f 1 we hae a f +1 ( The umber of compostos of to parts equa to 1 or 2 Souto: Let S { compostos of to to parts equa to 1 or 2 } A { compostos S whose ast part s 1 } ad B { compostos S whose ast part s 2 }

7 so that a S s the aue we wsh to eumerate ad S A B so that a A + B Note that a ad a 1 {(1} 1 The fucto from A whch taes (α 1 α 1 to (α 1 α s a fucto from A to S 1 sce (α α + 1 ad α 1 or 2 Its erse appedg a good composto of 1 by 1 s we-defed so A s bjecto wth S 1 Smary the fucto from B whch taes (α 1 α 2 to (α 1 α s a fucto from B to S 2 sce (α α + 2 ad α 1 or 2 Its erse appedg a good composto of 2 by 2 s we-defed so B s bjecto wth S 2 Thus a S A + B S 1 + S 2 a 1 + a 2 Sce a f ad a 1 1 f 1 we hae a f ( The umber of compostos of to odd parts Souto: Let S { compostos of to to odd parts } A { compostos S whose ast part s 1 } ad B { compostos S whose ast part s ot 1 } so that a S s the aue we wsh to eumerate ad S A B so that a A + B Note that a ad a 1 {(1} 1 The fucto from A whch taes (α 1 α 1 to (α 1 α s a fucto from A to S 1 sce (α α + 1 ad α s odd for a Its erse appedg a good composto of 1 by 1 s we-defed so A s bjecto wth S 1 Smary the fucto from B whch taes (α 1 α to (α 1 α 2 s a fucto from B to S 2 sce (α α + 2 ad α s odd ad α 3 ad odd so that α 2 1 ad odd Its erse addg 2 to the ast part s we-defed so B s bjecto wth S 2 Thus a S A + B S 1 + S 2 a 1 + a 2 Sce a f ad a 1 1 f 1 we hae a f ( The umber of sequeces (ε 1 ε 2 ε of s ad 1s such that ε 1 ε 2 ε 3 ε 4 Souto: Let S { good sequeces } A { sequeces S whose aue s } ad B { compostos S whose aue s 1 } so that a S s the aue we wsh to eumerate ad S A B so that a A + B Note that a 1 {( (1} 2 ad a 2 {( ( 1 (1 1} 3 We cosder two cases:

8 ( odd so that the ast equaty s If ε A the ε 1 ε whch puts o restrcto o ε 1 or ay preous ε So good sequeces ε 1 ε 2 ε 3 ε 1 are bjecto wth good sequeces ε 1 ε 2 ε 3 ε 1 e A s bjecto wth S 1 If ε B the ε 1 ε 1 so ε 1 1 Howeer ε 2 ε 1 1 puts o restrcto o ε 2 or ay preous ε So good sequeces ε 1 ε 2 ε 3 ε are bjecto wth good sequeces ε 1 ε 2 ε 3 ε 2 e B s bjecto wth S 2 ( ee so that the ast equaty s Ths foows exacty as the preous case except ow A s bjecto wth S 2 ad B s bjecto wth S 1 Ether way a S A + B S 1 + S 2 a 1 + a 2 Sce a 1 2 f 3 ad a 2 3 f 4 we hae a f +2 ( The umber of sequeces (T 1 T 2 T of subsets T of [] such that T 1 T 2 T 3 T 4 Souto: For each [] assg a sequece ε ( (ε ( 1 ε( { by ε ( f / T j j 1 f T j Note that ad T j T j+1 f ad oy f ε ( j T j T j+1 f ad oy f ε ( j ε ( j+1 for a [] ε ( j+1 for a [] So by spag oer [] we get a sequece ε (ε (1 ε ( of sequeces ε ( (ε ( 1 ε of 1 s ad s satsfyg ε ( 1 ε ( 2 ε ( 3 ε ( 4 Smary ay such a sequece of good sequeces ges a sequece of subset T of [] such that T 1 T 2 T 3 T 4 Sce the sequeces correspodg to each are depedet of each other ad the umber of good sequeces ε ( s f +2 by the preous part we hae the umber of good sequeces of subsets s ge by {ε (1 } {ε ( } f +2 f +2 f +2

9 ( The sum α 1 α 2 α oer a 2 1 compostos α (α 1 α 2 α of [Ht: ths sum couts the umber of ways of sertg at most oe ertca bar each of the 1 spaces betwee stars a e of stars ad the crcg oe star each compartmet Now try repacg bars u-crced stars ad crced stars by 1 s 2 s ad 1 s respectey Use a preous part] Souto: Let S be the set of strog stars ad bars arragemets wth stars ad 1 bars together wth a choce of dstgushed star for each of the regos For exampe f 1 ad 3 eemets of S cude ad Sce strog star ad bar arragemets wth stars ad 1 bars are bjecto wth compostos of wth parts ad for each such arragemet correspodg to the composto (a 1 a there are a 1 a ways to choose the dstgushed stars (product rue we hae S α 1 α comps α of Now for each arragemet S assg the sequece of 1 s ad 2 s by repacg bars ucrced stars ad crced stars by 1 s 2 s ad 1 s respectey For exampe the sequeces correspodg to the arragemets aboe are ad respectey For a fxed there are 1 bars crced stars ad ucrced stars so the aues sum to ( ( whch s depedet of So we hae mapped S to compostos of 2 1 whose parts are a 1 s ad 2 s Now ets cosder the erse operato Sce 2 1 s odd ay composto of 2 1 whose parts are a 1 s ad 2 s w hae a odd umber of 1 s So for ay such composto repace a the 2 s by stars ad the repace each of the 1 s from eft to rght ateratg wth crced stars ad bars (the frst 1 becomes a crced star the secod 1 becomes a bar ad so o If there are r 2 s the there are s 2 1 2r 1 s; ad (s + 1/2 of those 1 s w become crced stars So the subsequet arragemet w cosst of r + (s + 1/2 r + (2 1 2r + 1/2 r + r ad (s 1/2 (2 1 2r 1/2 r 1 bars stars (crced ad ucrced [Note that sce oer a such sequeces r ca rage from to 1 the umber of bars ca rage from to 1 as desred] So ths erse from a compostos of 2 1 whose parts are 1 s ad 2 s s a we-defed map to S Therefore we hae a bjecto Thus by part (a( α 1 α S f 2 1 comps α of (b Cosder the detty f +1 (

10 ( Chec ths detty for 2 ad 3 Souto: 1 ( ( ( 1 1 f ( ( ( ( f ( Proe ths detty recursey by showg that t satsfes the Fboacc recurrece ad that t hods for the frst 2 aues Souto: Reca the detty that ( ( ( a a 1 a 1 + b b b 1 ad et F ( + 1 ( The 1 ( 2 1 ( 2 F ( + F ( ( 1 1 ( 1 + (settg ( 1 1 (( ( ( 1 ( + 1 ( sce ( But the ast e s just F ( + 1 So F ( satsfes the Fboacc recurrece ad F (2 f 2 ad F (3 f 3 Thus F ( f ( Proe ths detty combatoray Namey frst show combatoray that the umber of -subsets of [ 1] cotag o two cosecute tegers s ( ad the use (a( Souto: Let S be -subsets of [ 1] cotag o two cosecute tegers For A {a 1 < a 2 < < a } S cosder the map ϕ : {a 1 a 2 a } {a 1 a 2 1 a 3 2 a + 1} {b 1 b 2 b 3 b } e b a + 1 Sce A cotas o two cosecute tegers b +1 b a +1 ( (a + 1 a +1 a 1 > so ths map s a we-defed map of sets Moreoer b 1 a 1 1 ad b a ad b 1 < b < b so ϕ(a ( [ ] The erse sedg a set B {b 1 < b 2 < < b } ( [ ] to ϕ 1 : {b 1 b 2 b } {b 1 b b b + 1}

11 s we-defed wth mage S sce we hae dated the etres of B (gg o two cosecute eemets ad b So S s bjecto wth ( [ ] Thus by (a( ( 1 ( 1 S f 1+2 f +1 (c Note that EC1 Exampe 1112 to computes the geeratg fucto for the sequece (a N where a f +1 (so a 1 a 1 1 a 2 2 a 3 3 ad so o Repeat ths computato for (f Z> mag the approprate chages to accommodate the shft Souto: Just do t