A RULE OF THUMB FOR RIFFLE SHUFFLING

Transcription

1 A RULE OF THUMB FOR RIFFLE SHUFFLING SAMI ASSAF, PERSI DIACONIS, AND K. SOUNDARARAJAN Abstract. We study how may riffle shuffles are required to mix cards if oly certai features of the deck are of iterest, e.g. suits disregarded or oly the colors of iterest. For a wide variety of features, the umber of shuffles drops from 3 log 2 2 to log 2. We derive closed formulae ad a asymptotic rule of thumb formula which is remarkably accurate.. Itroductio This paper studies the mixig properties of the Gilbert-Shao-Reeds model for riffle shufflig cards. Iformally, the deck is cut ito two piles by the biomial distributio, ad the cards are riffled together accordig to the rule: if the left packet has A cards ad the right has B cards, drop the ext card from the left packet with probability A/A + B ad from the right packet with probability B/A + B. Cotiue util all cards have bee dropped. This defies a measure, deoted Q 2 σ, o the symmetric group S. Repeated shuffles are defied by covolutio powers Q k 2 σ = Q 2 τq k 2 στ. τ S The uiform distributio is Uσ = /!. There are several otios of the distace betwee Q k 2 ad U: the total variatio distace 2 Q k 2 U TV = max A S Q k 2 A UA = 2 ad the separatio ad l metrics 3 SEPk = max σ Q k 2 σ Uσ σ S Q k, l k = max σ 2 σ Uσ, Q k 2 σ Uσ. I widely cited works, Aldous [2] ad Bayer-Diacois [3] show that 3 2 log 2 + c shuffles are ecessary ad sufficiet to make the total variatio distace small, while 2log 2 + c shuffles are ecessary ad sufficiet to make separatio ad l small. The distaces i 2 ad 3 look at all aspects of a permutatio. I may card games, oly some aspects of the permutatio matter. For example, i Black-Jack, suits are irrelevat; i Baccarat, suits are irrelevat ad all s ad picture cards are equivalet; ESP card guessig experimets use a Zeer deck of 25 cards with each of 5 symbols repeated five times. It is atural to ask how may shuffles are required i these situatios. These questios are studied by Coger ad Viswaath [8, 9] who derive remarkable umerical procedures givig useful aswers for cases of practical iterest. Their work is reviewed at the ed of this itroductio. I this paper, we develop formulae ad asymptotics for a deck of cards with D cards labelled, D 2 cards labelled 2,..., D m cards labelled m. Most of the results are proved from the deck startig i order, i.e. with s o top through m s at the bottom. I Sectio 5, we show that iitial order ca chage the coclusios. Date: October 5, 28.

2 2 ASSAF, DIACONIS, AND SOUNDARARAJAN I Sectio 2, we begi with D = ad D 2 =. The trasitio matrix for this case has iterestig properties, rivalig the Amazig Matrix i [2]. We show that log 2 + c shuffles are ecessary ad sufficiet for covergece i ay of our metrics. Sectio 3 studies D = R, D 2 = B, with, for example, R = B = 26 modelig the red-black patter for a stadard 52 card deck. We derive a simple formula for Q k 2 w for ay patter w ad use this to agai show that log 2 + c steps are ecessary ad sufficiet for covergece to uiformity. We fid this surprisig as followig a sigle card ivolves a state space of size, reds ad blacks ivolves a state space of size /2, ad yet the same umber of shuffles are eeded. I Sectio 4, we treat the geeral case, derivig a formula which ca be used for some limited calculatios. We also reprove a result of Coger-Viswaath determiig where the maximum for SEP ad l are achieved. A mai result is a uified formula, our rule of thumb: Theorem.. Cosider a deck of cards with D i cards of type i, i m with D i d 3, = D + + D m. The the separatio distace after k shuffles is 2 km m m + η j j +m + +m j 2 k, where η is a real umber satisfyig 2 m η +. 3d 22 k m + 2 j= This result does ot deped o the idividual details of the D i ad shows that the same umber of shuffles are ecessary ad sufficiet for a variety of questios. For umerical approximatio, we set η = ad simply compute the sigle sum. The boud o η gives explicit error estimates. We demostrate that the rule of thumb is accurate for both sigle card ad red-black problems studied i earlier sectios. This also agrees with the extesive simulatio results of Coger-Viswaath ad allows results where exact computatios ad simulatio seem out of reach. Some umerical results are summarized below. Table. Rule of Thumb for the separatio distace for k shuffles of 52 cards. k Bayer-Diacois blackjack redblack Remarks o Table. The first row gives exact results from the Bayer-Diacois formula for the full permutatio group. The other umbers are from the rule of thumb. Roughly, the sigle card or redblack umbers suggest that half the usual umber of shuffles suffice. The Black-Jack equivaletly Baccarat umbers suggest a savigs of two or three shuffles, ad the suit umbers lie i betwee. The fial row is the rule of thumb for the Zeer deck with 25 cards, 5 cards for each of 5 suits. I a appedix, we show that the processes studied below are quotiet walks with respect to Youg subgroups of S. We show how represetatio theory ca be used to derive results for features of the radom traspositio radom walk.

3 A RULE OF THUMB FOR RIFFLE SHUFFLING 3 Literature review of riffle shuffles. The basic shufflig model was itroduced by Gilbert ad Claude Shao i a upublished report [9]. The model was idepedetly itroduced ad studied by Jim Reeds i upublished work [2]. The first rigorous results are by Aldous [] who showed that asymptotically 3 2 log 2 shuffles are correct for total variatio. Separatio distace is itroduced i coectio with stoppig time argumets i Aldous ad Diacois [2]. They show that 2log 2 + c steps are ecessary ad sufficiet for separatio covergece. The cutoff pheomea is first oticed i this paper as well. A geeralizatio to a-shuffles is itroduced by Bayer-Diacois i [3]. Here the deck is cut ito a packets by a multiomial distributio, ad the cards are dropped from packets with probability proportioal to packet size. Lettig Q a σ deote this measure, they show 4 Q a Q b = Q ab. Thus it is eough to study a sigle a-shuffle. The mai result of their paper is the simple formula 5 Q a σ = +a r a, where r = rσ is the umber of risig sequeces i σ rσ = dσ + with d the umber of descets i σ. This allows simple closed form expressios for a variety of distaces. A umber of extesios ad variatios have sice developed. We will ot survey these here see [] for a thorough treatmet but metio that features of permutatios are show to achieve the correct limitig distributio i fewer shuffles. For example, 5 6 log 2 + c suffice for the logest icreasig subsequeces [7], log 2 for the descet structure [2], k arbitrarily slowly for the cycle structure [4] ad a sigle shuffle suffices for the logest cycle [4]. A recet additio is the work of Che ad Saloff-Coste [6] studyig radom combiatios of a-shuffles for radomly varyig a. Mark Coger ad D. Viswaath study the same type of problems as we do. I [8], they lay out the basic problems, develop a formalism for calculatios ivolvig descet polyomials a geeralizatio of Euleria polyomials, ad use these to derive a closed formula for the chace of a give arragemet after a a-shuffle for decks labelled {,2,...,h,x }. This icludes both our sigle card case ad the full deck case. They show that the probability of a arragemet is 6 a +h a m=r m r + h h a m l a m l, with r the umber of cards labelled c, c h, that are ot preceded by a card labelled c ad l the umber of cards labeled x that precede the card labeled h. This elegat expressio ca be aalyzed asymptotically usig the aalytic techiques of Sectios 2-5 below. Their mai results pertai to red-black decks where they derive equivalece relatios o cofiguratios that have the same probability. They poit out that startig with the reds o top or reds alteratig with blacks ca lead to differet coclusios. I [9], the authors use their earlier work o descet polyomials to develop a fasciatig Mote Carlo procedure for approximatig the total variatio distace. Our exact ad asymptotic calculatios overlap theirs i may places, ad i every case we fid their umbers spot o. This leads us to accept their estimates for problems of deck hads at bridge where we have ot foud a way to do exact calculatios. The results derived here add to the result of Coger-Viswaath i the followig ways. First, we preset some ew formulae e.g. the trasitio matrix for sigle card mixig or the red-black formula which allow exact computatios. Secod, we derive asymptotic approximatios for a variety of cases. There are o such computatios i previous work. Third, we have made sese of this sea of formulae ad approximatios through our rule of thumb.

4 4 ASSAF, DIACONIS, AND SOUNDARARAJAN Fially, we metio the broad extesios of riffle shufflig to radom walks o hyperplae arragemets due to Bidigare, Halo ad Rockmore see [] for a survey. The process iduced by observig which chamber of a sub-arragemet cotais the preset state of the origial walk is still Markov. Oe might try to solve the problems of rates of covergece for selected features for ay of these extesios. 2. Followig a sigle card Suppose oe otices that the ace of spades is o the bottom of a deck of cards. How may shuffles does it take util this oe card is close to uiformly distributed o {,2,...,}? As show i a appedix, uder repeated shuffles a sigle card moves accordig to a Markov chai. We begi by writig dow the trasitio matrix. Propositio 2.. Let P a i,j be the chace that the card at positio i is moved to positio j after a a-shuffle. The for i,j, we have P a i,j = a u j j a k r a k j r k i r a k + j i r, r i r k= r=l where the ier sum is from l = max,i + j + to u = mii,j. Proof. We calculate Q a j,i, the chace that a iverse a-shuffle brigs the card at positio j to positio i. For this to occur, the card at positio j may be labelled by k, k a. The r cards above this card may be labelled from to k. All will appear before the card at positio j i j r ways. The remaiig cards above must labelled from k + to a. Here r mij,i. Also if m cards below positio j are labelled from to k, the m + r = i,m < j ad so r i + j +. Fially, i r cards below positio j must be labelled from to k i j i r ways, ad the remaiig cards must be labelled from k + to a. For example, the trasitio matrices for = 2,3 are give below. a + a a + 2a + 2a2 a 2a 2a a a + 6a 2 2a 2 2a a 2 a 2a 2a 2 a + 2a + Two other special cases to ote are the extreme cases whe i = or i =, which are give by P a,j = a a a k j a k + j, P a,j = a a k j k j. k= These sigle card trasitio matrices are studied by Ciucu [7] who gives a closed form for all whe a = 2: 2 2 i + 2 i if i = j, P 2 i,j = j 2 i if i > j, j+ P 2 i+, j+ if i < j. These matrices share may properties of the amazig matrix developed by Holte [2]. The followig Propositio is essetially due to Ciucu [7]. Propositio 2.2. The trasitio matrices followig a sigle card have the followig properties: they are cross-symmetric, i.e. P a i,j = P a i +, j + ; 2 P a P b = P ab ; 3 the eigevalues are,/a,/a 2,...,/a ; 4 the right eige vectors are idepedet of a ad have the simple form: V m i = i i m i + i+m m i for /a m, m. k=

5 A RULE OF THUMB FOR RIFFLE SHUFFLING 5 Proof. The cross-symmetry follows from Propositio 2., ad the multiplicative property 2 follows from the shufflig iterpretatio ad equatio 4. Property implies that the eige structure is quite costraied; see [23]. Properties 3 ad 4 follow from results of Cuicu [7]. Remark 2.3. We ote that Holte s matrix arose from studyig the carries process of ordiary additio. Diacois ad Fulma [2] show that it is also the trasitio matrix for the umber of descets i repeated a-shuffles. We have ot bee able to fid a closer coectio betwee the two matrices. From Propositio 2. we obtai the followig Corollary, which also follows as a special case of Theorem 2.2 i [8]. Corollary 2.4. Cosider a deck of cards with the ace of spades startig at the bottom. The the chace that the ace of spades is at positio j from the top after a a-shuffle is 7 Q a j = P a,j = a a k j k j. From the explicit formula, we are able to give exact umerical calculatios ad sharp asymptotics for ay of the distaces to uiformity. The results below show that log 2 + c shuffles are ecessary ad sufficiet for both separatio ad total variatio ad there is a cutoff for these. This is surprisig sice, o the full permutatio group, separatio requires 2log 2 +c steps whereas total variatio requires 3 2 log 2 + c. Of course, for ay specific, these asymptotic results are just idicative. k= Table 2. Distace to uiformity for a deck of 52 distict cards T V SEP l Table 3. Distace to uiformity for a sigle card startig at the bottom of a 52 card deck T V SEP l Remarks o Table 3. We use Propositio 2. to give exact results whe = 52. For compariso, Table 2 gives exact results for the full deck usig [3]. Tables 3 ad 4 show that it takes about half as may or fewer shuffles to achieve a give degree of mixig for a card at the bottom of the deck. For example, the widely cited 7 shuffles for total variatio drops this distace to.334 for the full orderig, but this requires oly 4 shuffles to achieve a similar degree of radomess for a sigle card at the bottom, ad oly 2 for a sigle card startig i the middle. Similar statemets hold for the separatio ad l metrics.

6 6 ASSAF, DIACONIS, AND SOUNDARARAJAN Table 4. Distace to uiformity for a sigle card startig at the middle of a 52 card deck T V SEP l For asymptotic results, we first derive a approximatio to separatio. Sice separatio is a upper boud for total variatio, this gives a upper boud for total variatio. Fially, we derive a matchig lower boud for total variatio. Propositio 2.5. After a a-shuffle, the probability that the bottom card is at positio i satisfies α i+ a α Q ai a α i α, where for brevity we have set α = /a. I particular, the separatio distace satisfies a α α SEPa α a α. Proof. Sice k/k a/a for all < k a we fid that 8 α i Q a Q a i α i Q a. Therefore so that = i Q a i Q a α i = Q a aα α, i= α Q a a α a α. Sice Q a = Q a + /a it follows that Q a a α. Usig 8 the desired upper boud for Q a i follows. Similarly, = Q a i Q a α i = Q a α α, i i= so that Q a a α. Sice Q a = Q a /a it follows that Q a α a α, ad from 8 the desired lower boud for Q a i follows. From 2 ad the above estimates we obtai our bouds o SEPa. α If a = 2 log 2 +c = 2 c, the our result shows that the SEPa is approximately 2 c e 2 c e 2 c, ad for large c this is 2 c. The fit to the data i Table 5 is excellet: for example after te shuffles of a fifty-two card deck we have 2 c = which is very early the observed separatio distace of.25.

7 A RULE OF THUMB FOR RIFFLE SHUFFLING 7 Remark 2.6. Propositio 2.5 gives a local limit for the probability that the origial bottom card is at positio j from the bottom. Whe the umber of shuffles is log 2 + c, the desity of this with respect to the uiform measure is asymptotically zce j/2c, with z a ormalizig costat zc = /2 c e j/2c. The result is uiform i j for c fixed, large. Propositio 2.7. Cosider a deck of cards with the ace of spades at the bottom. With α = /a, the total variatio distace for the mixig of the ace of spades after a a-shuffle is at most α + α aα2 α a α + log/α log α α +, ad at least α α a α α α + a log/α log α α. Proof. Let Q a i deote the probability that the ace of spades is at positio i from the top after a a shuffle. Note that Q a i is mootoe icreasig i i, ad let i be such that Q a i < / Q a i +. From Propositio 2.5 we fid that i satisfies 9 so that log a α i + a α < α i a α, α a α i log /α log α α + From Propositio 2.5 we have that the desired total variatio is ai Q i i i α i+ a α = i α i + α αi, ad also i i i i Usig 9 ad we obtai the Propositio. Q ai i α i α αi. Remark 2.8. After log 2 + c shuffles, that is whe a = 2 c, Propositio 2.7 shows that the total variatio distace is approximately with C = 2 c C log Ce /C + C loge/c e /C. Thus whe c is large ad egative, the total variatio is close to, ad whe c is large ad positive, the total variatio is close to. Thus total variatio ad separatio coverge at the same rate. This is a asymptotic result ad, for example, Table 3 supports this. Remark 2.9. From Propositio 4., the l distace is achieved for cofiguratios with the ace of spades back o the bottom. Propositio 2.5 gives a formula for this ad the argumets for Propositios 2.5 ad 2.7 show that log 2 + c shuffles are ecessary ad sufficiet for covergece i l. Remark 2.. Similar, but more demadig, calculatios show that if the ace of spades starts at positio i, ad maxi/, i/ A > for some fixed positive A, the 2 log 2 shuffles suffice for covergece i ay of the metrics. We omit further details.

8 8 ASSAF, DIACONIS, AND SOUNDARARAJAN 3. A red-black deck We focus ow o riffle shuffles of a deck cosistig of R red cards ad B black cards. The purpose of this sectio is to give a explicit descriptio of a-shuffles of the deck with iitial cofiguratio of red atop blacks. I Bayer-Diacois [3], the formula describig whe a a-shuffle of distict cards results i a particular permutatio has the simple form a a + r where r is the umber of risig sequeces i the permutatio. The aalysis for the red-black deck is markedly differet. Oe idicatio of this comes by oticig how likely the reverse deck is to occur. I the case of permutatios, the reverse deck has risig sequeces, ad so the Bayer-Diacois formula dictates that this cofiguratio caot occur uless a. However, i the red-black case, the reverse deck blacks atop reds may occur after a sigle 2-shuffle o matter the deck size. Theorem 3.. Cosider a deck with R red cards o top of B black cards. The probability that a a-shuffle will result i the deck cofiguratio w is Q a w = a R a R+B k R j k j a k bj a k + B bj k= j= where bj = b w j is the umber of black cards above the jth red card i the deck w. Proof. The geeral formula for the probability of w resultig from a a-shuffle is give by R + B 2 a probw A, A,...,A a A + +A a=r+b where the sum is over all o-egative compositios A = A,A 2,...,A a of R + B, i.e A i ad A +A 2 + +A a = R+B, ad probw A deotes the probability that w results from successively droppig cards from the piles A i. We break the sum ito the followig two cases: either there exists a iteger k such that A + A A k = R or ot. Cosider the case whe the sum of the first k piles is exactly R. The, the result of the subsequet riffle shuffle is equally likely to be ay of the R+B R possible deck cofiguratios. That is to say, give such a cut A, probw A = / R+B R for every w. Therefore the cotributio to Qa w from all such cuts is give by 3 A + +A a=r+b k s.t. A + +A k =R = = a a R+B a R+B R R + B A,...,A a k= A k = A k+ + +A a=b A + +A k =R A k a R a R+B a k B k= A k = R A k R+B = R, R R Ak A k k R A k = A,...,A k B A k+,...,a a a a R+B a k B k R k R. The choice to let k be the first idex such that A + + A k = R is ecessary i order to avoid over coutig compositios with may s. This choice seemigly breaks the symmetry betwee R ad B i the fial formulatio. However, the symmetric versio may be obtaied by takig k to be the last idex such that A + + A k = R. Fially, ote that sice B, we may i fact take the sum over k to rage from to a. Now cosider the alterative case whe there exists a pile ecessarily uique cotaiig both red ad black cards. The assumptio o A amouts to the existece of itegers k,x,y, with k=

9 A RULE OF THUMB FOR RIFFLE SHUFFLING 9 k a, x R, y B, such that A + + A k = R x, A k = x + y, ad A k+ + + A a = B y. Give such a cut A, probw A = r x,y w/ R+B R x,x+y,b y, where rx,y w deotes the umber of risig subsequeces cosistig of x red cards followed by y black cards. The resultig cotributio to Q a w from all such cuts is give by R + B 4 a R+B probw A A,...,A a A + +A a=r+b k s.t. A + +A k <R ad A k+ + +A a<b = = a R+B a R+B a R k= x= y= a R k= x= y= B r x,y w A + +A k =R x A k+ + +A a=b y B r x,y wk R x a k B y. R x A,...,A k B y A k+,...,a a For the fial equatio to make sese, we adopt the covetio that =. Let bj deote the umber of black cards above the jth red card i w. We may cout risig subsequeces of w by the last red card used i the subsequece, givig the equatio 5 r x,y w = R j= j x B bj To see this, ote that the first biomial coefficiet couts the umber choices of x red cards before the jth red card, ad the secod biomial coefficiet couts the umber of choices for y black cards after the jth red card. Isertig this ito the x ad y summatios of 4 gives 6 a R+B a R k= x= y= = a R+B = a R+B B r x,y wk R x a k B y a R k= j= a k= j= R j k x x= R x y. B B bj a k B y y y= R k R j k j a k bj a k + B bj a k B bj. The probability Q a w is obtaied by addig the expressios i 3 ad 6. Sice a R a R k R j k R j k j a k B = k R a k B k k= j= = a k= we obtai the desired expressio. k= j= k R B /kr a a k /k = a k B k R k R, Give 5, Q a gives a completely explicit descriptio of a-shuffles, though this is difficult to evaluate for a arbitrary w. However, there are two special deck cofiguratios for which Q a simplifies icely, amely reds atop blacks where r x,y w = R B x y ad blacks atop reds where r x,y w =. By Propositio 4., the formulae below ca be used to give exact calculatios for separatio ad l. k=

10 ASSAF, DIACONIS, AND SOUNDARARAJAN Corollary 3.2. The probability of a a-shuffle resultig i the origial deck cofiguratio of reds atop blacks is a k R a R+B k R a k + B. k= The probability a a-shuffle resultig i the reverse deck cofiguratio of blacks atop reds is a a R+B a k B k R k R. k= Aother special case to cosider is trackig the positio of a sigle card startig at the bottom of the deck. For this case, takig B = ad R = i we recover Corollary 2.4. Note that if istead we cosider a sigle red card, i.e. R = ad B =, startig at the top, the the distributio is the same. More precisely, let Q a i deote the chace that, say, the 2 of hearts is at positio i from the top of the deck after a a-shuffle. The it is easy to verify that Q a i = Q a i +, which is just a special case of the cross-symmetry i Propositio 2.2. Fially, cosider the case of a sigle 2-shuffle for a arbitrary red-black deck. I this case, the left had summad of reduces to a sigle term evaluatig to. For the right had summad, ote that k = forces x = R, ad k = a forces y = B. Corollary 3.3. The probability of a 2-shuffle resultig i a deck cofiguratio w is 7 Q 2 w = 2 R+B 2 hw + 2 tw, where hw deotes the umber of red cards precedig the first black card i w, ad tw deotes the umber of black cards followig the fial red card of w. Equatio 7 ca be used to give a simple formula for the total variatio after a sigle 2-shuffle of a deck with red cards ad black cards. Here ote that ay two cofiguratios with the same umber of red cards o top ad black cards o bottom has the same likelihood of occurrece, so we get 8 Q 2 U TV = i= j= 2 i + 2 j i + j i + Usig this formula, the total variatio after a sigle 2-shuffle of a deck with 26 red ad 26 black cards is.579, which agrees with the umerical approximatios of Coger ad Viswaath i [8]. We do ot see how to compute total variatio effectively after more shuffles. Asymptotic results for the separatio distace for red-black cofiguratios appear i the followig sectio. 4. Approach to uiformity i separatio for geeral decks I this sectio we work with geeral decks cotaiig D i cards labelled i, i m. The followig lemma shows that the separatio distace is always achieved by reversig the iitial deck cofiguratio. Note this is equivalet to Theorem 2. from [8]. Propositio 4.. Let D be a deck as above. After a a-shuffle of the deck with s o top dow to m s o bottom, the most likely deck cofiguratio is this iitial deck ad the least likely cofiguratio is the reverse deck w with m s o top dow to s o the bottom. I particular, the separatio distace is achieved for w.

11 A RULE OF THUMB FOR RIFFLE SHUFFLING Proof. Note first that the iitial cofiguratio ca result from ay possible cut of the deck ito a piles. Moreover, from ay give cut of the deck, the idetity is at least as likely to occur as ay other cofiguratio. The first assertio ow follows. The oly cuts of the iitial deck which may result i w are those cotaiig o pile with distict letters. However, for all such cuts, each rearragemet of the deck is equally likely to occur. Therefore w miimizes Q a w ad so maximizes Q a w/u. The explicit formula for Q a w give i Corollary 3.2 facilitates exact computatios of SEPa for decks of practical iterest. Similarly, we ca compute Q a w for a arbitrary deck with D i i s, i =,...,m. Theorem 4.2. Cosider a deck with cards ad D i cards labeled i, i =,...,m. The the separatio distace after a a-shuffle of the sorted deck s followed by 2 s, etc is give by SEPa = m a a k m Dm kj k j D j k j k j D j. D... D m =k < <k m <a Proof. From Propositio 4., w may oly result from cuts with o pile cotaiig distict cards ad ay such cut is equally like to result i ay deck. Therefore Q a w is give by Q a w = a, A,...,A a D,...,D m A + +A a= A refies D where A refies D meas there exist idices k,...,k m such that A + + A k = D ad, for i = 2,...,m, A ki A ki = D i. Just as i the proof of Theorem 3. we may take the k i s to be miimal so that the expressio for Q a w simplifies to give 9 Q a w = a =k <k < <k m <a The result ow follows from Propositio 4.. m a k m Dm j= j= kj k j D j k j k j D j. Table 5. Separatio distace for k shuffles of 52 cards. k Bayer-Diacois blackjack A redblack Remarks o Table 5. We calculate SEP after repeated 2-shuffles for various decks usig Theorem 4.2: blackjack 9 raks, say A , with 4 cards each ad aother rak, say, with 6 cards; 4 distict suits, say clubs, diamods, hearts ad spades, of 3 cards each; A the ace of spades ad 5 other cards; redblack a two color deck with 26 red ad 26 black cards; ad a deck with 5 cards i each of 5 suits. The missig etries i Table 5 highlight the limitatios of exact calculatios usig Theorem 4.2.

12 2 ASSAF, DIACONIS, AND SOUNDARARAJAN Propositio 4. may be used with the Coger-Viswaath formula i 6 to give a simple expressio for separatio after a a-shuffle for a deck of size h + with cards labelled,2,...,h ad cards labelled x: SEPa = + h + a +h a k=h k a k. h Now we derive a basic asymptotic tool, Propositio 4.3, which allows asymptotic approximatios for geeral decks. As motivatio, cosider agai the case of oe card mixig, i.e. begi with cards with the ace of spaces at the bottom of the iitial deck. How may shuffles are required to radomize the ace of spades? Recall from Corollary 2.4 that the chace that the ace of spades is at positio i from the top after a a-shuffle is give by Q a i = a a k i k i, with the covetio =. Therefore from Propositio 4., we have 2 SEPa = Q a = a a k. k= Exact calculatios whe = 52 are give i Table 5. Propositio 4.3. Let a be a positive real umber, ad let r ad s be atural umbers with r, s 2. Let ξ be a real umber i [,]. The Sa,ξ;r,s := a r+s k + ξ r a k ξ s where θ is a real umber i [,]. k a ξ k= r!s! = a r + s +! + θ r!s! 6a r + s! r +, s Proof. Put fx = x r x s for x [,] ad fx = otherwise. The sum that we wish to evaluate is 2 ˆfalelξ, k Z fk + ξ/a = a l Z by the Poisso summatio formula. Here, we write ex = e 2πix ad ˆfy = fxe xydx deotes the Fourier trasform. Now ote that 22 ˆf = x r x s r!s! dx = r + s +!. Further ˆfy = x r x s e 2πixy dx = 2πiy f xe 2πixy dx = 2πiy 2 f xe 2πixy dx, upo itegratig by parts twice, ad sice r, s 2 we have f = f = f = f =. Therefore ˆfy 4π 2 y 2 f x dx. Now r f x = x s 2x r r x s x x 2 + s x 2 x r x s,

13 A RULE OF THUMB FOR RIFFLE SHUFFLING 3 ad so f x dx = r x s 2x r x s dx + x 2 r + 2. s r!s! r + s! r x 2 + s x 2 x r x s dx Combiig the above estimates with 2 ad 22 we coclude that our sum equals r!s! a r + s +! + θ r!s! 2π 2 a r + s! r + s l 2 for some θ [,]. Sice l= l 2 = π 2 /6 the Propositio follows. Now suppose we have red cards ad black cards, so 2 cards altogether, with the red cards startig o top. I this case, the uiform distributio Uw = U = / 2. Agai we use Propositio 4. this time with Corollary 3.2 to give a formula for the separatio distace, 2 2 a 23 SEPa = Q a w = a 2 a k k k For exact computatios whe 2 = 52, see Table 5. We ow use Propositio 4.3 to calculate asymptotic expressios for this separatio distace. Corollary 4.4. For 2 cards startig with red cards o top, we have, with α = /a k= l Z l = SEPa = a 2 + α2+ + 2θ 3a 2 α2, for some real umber θ [,]. I particular, for large with a = 2 log 2 2+c, SEPa = 2 c e 2 c + O. a Proof. Note that a a 2 a k k k = k= a a k k + ξ dξ a 2 k= a = a 2 a + ξ k + ξ k + ξ dξ. k= Usig Propositio 4.3 we see that the ier sum over k above equals a + ξ 2!! + a + ξ 2 2θ 2! 6 Usig these observatios i 23 we obtai that SEPa = a + ξ With a little calculus the Corollary follows. a!! 2 2! 2dξ θ a a + ξ a. 2 2dξ.

14 4 ASSAF, DIACONIS, AND SOUNDARARAJAN The approximatio a 2 24 a k k k a2+ a k= which is the basis of our Corollary above is more accurate tha suggested by the simple error bouds that we have give. For example, whe = 26 ad a = 6, the actual separatio distace give i Table 5 differs from the approximatio of the Corollary by about 7 2. Put differetly, ote that the LHS ad the RHS of 24 are both polyomials i a of degree 2, ad i fact the coefficiets of both polyomials match for all degrees betwee ad 2. Before movig to geeral decks, we establish a geeralizatio of Propositio 4.3. Propositio 4.5. Let m 2 ad a be atural umbers, let ξ,..., ξ m be real umbers i [,]. Let r,..., r m be atural umbers all at least r 2. Let S m a;ξ,r = a + ξ r a m + ξ m rm. a,...,a m a +...+a m=a The r! r m! S m a;ξ,r r r m + m! a + ξ ξ m r +...+r m+m m m j a + ξ ξ m r +...+r m+m 2j r! r m!. j 3r r r m + m 2j! j= Proof. We establish this by iductio o m. The case m = 2 follows from Propositio 4.3, takig there a to be what we would ow call a + ξ + ξ 2. Let ow m 3 ad suppose the result has bee established for m variables. Now 25 S m a;ξ,r = a+ξ ξ m a = a r S m a a ; ξ, r with ξ = ξ 2,...,ξ m ad r = r 2,...,r m, ad iterpretig the terms with a a as beig. Usig the iductio hypothesis we have that S m a a ; ξ, r 2! r m! r r r m + m 2! a a + ξ ξ m r r m+m 2 26 m 2 m 2 j a a + ξ ξ m r r m+m 2 2j r 2! r m!. j 3r r r m + m 2 2j! j= Note that the above estimate is valid eve if a + ξ ξ m a a sice the RHS is larger tha the mai term that is beig subtracted i the LHS. We use this estimate i 25, ad the ivoke Propositio 4.3 to hadle each of the m ew sums that arise. Thus, the cotributio of the mai term i 26 is, for some θ, r! r m! r r m + m! + a + ξ ξ m r +...+r m+m θ 3r r! r m! a + ξ ξ m r+...+rm+m 3, r r m + m 3!

15 A RULE OF THUMB FOR RIFFLE SHUFFLING 5 while the j-th term o the RHS of 26 cotributes m 2 j a + ξ ξ m r +...+r m+m 2j r! r m! j 3r r r m + m 2j! + 3r a + ξ ξ m r +...+r m 2j 2 r r m + m 2j 2! Usig these i 26 ad 25, ad usig the triagle iequality, ad that m j = m 2 j + m 2 j we obtai the Propositio. Cosider ow a geeral deck of cards with D s followed by D 2 2 s ad so o edig with D m m s. Recall that the separatio is maximum for the reverse cofiguratio of the deck, ad that probability is give i Theorem 4.2. We ow use Propositio 4.5 to fid asymptotics for that separatio distace. The followig is our rule of thumb. Theorem 4.6. Cosider a deck of cards of m-types as above. Suppose that D i d 3 for all i m. The the separatio distace is a m m m + η j j +m, + + m j a where η is a real umber satisfyig 2 m η +. 3d 2a m + 2 j=. Proof. Recall the expressio for the separatio distace give i Theorem 4.2. To evaluate this, we require a uderstadig of a +...+a m=a a j = a Dm m m j= a D j j a j D j a +...+a m=a a j a Dm m m We ow ivoke Propositio 4.5. Thus the above equals for some θ m a m + ξ ξ m D j!! j= +θ m j= m j 3d 2 j= D j a j + ξ j D j dξ j. j a m + ξ ξ m 2j 2j! We may simplify the above as { 2 m } D! D m! + θ + 3d 2a m + 2!... a m + + ξ ξ m dξ dξ m, dξ dξ m.

16 6 ASSAF, DIACONIS, AND SOUNDARARAJAN ad evaluatig the itegrals above this is { + θ + The Theorem follows. 2 m } m D! D m! 3d 2a m + 2! j= j m j a j m+. Remark 4.7. For simplicity we have restricted ourselves to the case whe each pile has at least three cards. With more effort we could exted the aalysis to iclude doubleto piles. The case of some sigleto piles eeds some modificatios to our formula, but this variat ca also be worked out. Remark 4.8. From Theorem 4.6 oe ca show that for a geeral decks as above, oe eeds a of size about m before the separatio distace becomes small. We ote that whe a is of size about m, the quatity η appearig i Theorem 4.6 is of size about /md 2, so that the estimates furished above represet a true asymptotic uless both m ad d happe to be small. I other words, whe we either have may piles, or a small umber of thick piles, Theorem 4.6 gives a good asymptotic. Remark 4.9. While asymptotic, Theorem 4.6 is astoishigly accurate for decks of practical iterest. For example, comparig exact calculatios i Table 5 with approximatios usig this rule of thumb i Table shows that after oly 3 shuffles, the umbers agree to the give precisio. Moreover, the simplicity of the formula i Theorem 4.6 allows much further computatios tha are possible usig the formula i Theorem 4.2. We ow give a heuristic for why our rule of thumb is umerically so accurate; this was hited at previously i our remark followig Corollary 4.4. Let k be a iteger, ad defie f k z = r k z r, r= with the covetio that =. Thus f z = / z, f z = z/ z 2, ad i geeral f k z = A k z/ z k+ where A k z deotes the k-th Euleria polyomial. The sum over a,..., a m appearig i our proof of Theorem 4.6 is simply the coefficiet of z a i the geeratig fuctio z m f D z f Dm z. Our rule of thumb may be iterpreted as sayig that 27 z m f D z f Dm z D! D m! + m! zm f +m z. To explai the sese i which 27 holds, ote that f k z exteds meromorphically to the complex plae, ad it has a pole of order k+ at z =. Moreover it is easy to see that f k z k!/ z k+ has a pole of order at most k at z =. Therefore, the LHS ad RHS of 27 have poles of order + at z =, ad their leadig order cotributios match. Therefore the differece betwee the RHS ad LHS of 27 has a pole of order at most at z =. But i fact, this differece ca have a pole of order at most d at z =, ad thus the approximatio i 27 is tighter tha what may be expected a priori. To obtai our result o the order of the pole, we record that oe ca show f k z = k! z z k+ log z k+ + ζ k + O z. 5. Comparig 2-shuffles with differet startig patters Coger ad Viswaath ote that the iitial cofiguratio ca affect the speed of covergece to statioary. I this sectio, we ivestigate this for a deck with red ad black cards. Cosider first startig with reds o top. If the iitial cut is at the most likely value the the red-black

17 A RULE OF THUMB FOR RIFFLE SHUFFLING 7 patter is perfectly mixed after a sigle shuffle. More geerally, by Corollary 3.3, the chace of the deck w resultig from a sigle 2-shuffle of a deck with red cards atop black cards is give by Q 2 w = hw + 2 tw. Cosider ext the result of 2-shuffles o the alteratig deck red-black-red-black-. As motivatio, we recall a popular card trick: Begi with a deck of 2 cards arraged alterately red, black, red, black, etc. The deck may be cut ay umber of times. Have the deck tured face up ad cut with cuts completed util oe of the cuts results i the two piles havig cards of opposite color uppermost. At this poit, ask oe of the participats to riffle shuffle the two piles together. The resultig arragemet has the top two cards cotaiig oe red ad oe black, the ext two cards cotaiig oe red ad oe black, ad so o throughout the deck. This trick is called the Gilbreath Priciple after its ivetor, the mathematicia Norma Gilbreath. It is developed, with may variatios, i Chapter 4 of [8]. From the trick we see that begiig with a alteratig deck severely limits the possibilities. Which start mixes faster? The followig developmets both explai the trick ad give a useful formula for aalysis. Lemma 5.. The umber of deck patters resultig from a cut with a odd umber of cards i both piles followed by a riffle shuffle is 2. Similarly, the umber of deck patters resultig from a cut with both piles eve followed by a riffle shuffle is 2. Proof. For the case of a odd cut, the last two cards after the riffle shuffle must be a red ad a black card. No matter what piles these two cards fell from, the ext two cards must also cosist of oe red ad oe black card. Cotiuig o, the possible resultig decks are exactly those where the ith ad i + st cards have differet colors for i =, 3,..., 2. The umber of such decks is exactly 2, sice each of the order of each of the pairs is idepedet. For a eve cut, we proceed by iductio otig that the case whe =,2,3 are easily solved by ispectio. I this case, the oly resultig decks will ecessarily begi with a red card ad ed with a black card. The umber of decks begiig with two red cards or edig with two black cards is determied by the previous case sice removig the top or bottom card from each pile results i piles with a odd umber of cards, givig 2 possibilities. However, we must discout the over couted case of decks begiig with two red cards ad edig with two black cards, ad, by iductio sice the piles are agai both eve, there are 2 3 such decks. Fially, the remaiig case must be decks begiig ad edig with a red card followed by a black card. I this case, agai, the piles remai eve ad by iductio the umber of such decks is 2 3. Therefore the total cout for cuts with both piles eve is = 2. The proof of the lemma shows exactly why the card trick is a success: to have differet colors o the top of the two piles, the cut must have bee odd. Therefore the first two cards dropped cosist of oe red ad oe black, ad the ext two cards dropped cosist of oe red ad oe black, ad so o. Also from the lemma, we see that the oly deck that ca result from either a odd cut or a eve cut is the idetity. Propositio 5.2. The chace of a 2-shuffle of the alteratig deck resultig i a deck cofiguratio w is give by if w = w Q 2 w = if w O \ w, 2 if w E \ w, otherwise,

18 8 ASSAF, DIACONIS, AND SOUNDARARAJAN where w is the iitial alteratig deck ad O respectively, E is the set of decks that ca result from rifflig together the two piles from cuttig the alteratig deck whe both piles have a odd respectively, eve umber of cards. Proof. Let w, u O. The the total umber of ways w ca result from ay odd cut is equal to the total umber of ways u ca result from ay odd cut. The same is true replacig O with E ad odd with eve. From the biomial idetity k = k k= k odd 2 = k k eve 2, k we must have both the right-had sums equal to 2 2. Therefore, by Lemma 5., the total umber of ways w ca result from a odd cut assumig it ca is 2 2 /2 = 2, ad, similarly, the total umber of ways w ca result from a eve cut assumig it ca is 2 2 /2 = 2. It follows from 28 that the separatio distace for a 2-shuffle is SEP2 = whe 3. Furthermore, sice 2 2, we ca compute the total variatio of a 2-shuffle to be 29 Q 2 U TV = , which goes to.5 expoetially fast as goes to ifiity. I cotrast, startig with reds above blacks, asymptotic aalysis of 8 shows that the total variatio teds to after a sigle shuffle whe is large. Thus a alteratig start leads to faster mixig. Ackowledgemets The authors thak Jaso Fulma for careful commets ad refereces. We also thak MSRI ad the participats of the combiatorial represetatio theory program where this work bega. Refereces [] D. Aldous. Radom walks o fiite groups ad rapidly mixig Markov chais. I Semiar o probability, XVII, volume 986 of Lecture Notes i Math., pages Spriger, Berli, 983. [2] D. Aldous ad P. Diacois. Shufflig cards ad stoppig times. Amer. Math. Mothly, 935: , 986. [3] D. Bayer ad P. Diacois. Trailig the dovetail shuffle to its lair. A. Appl. Probab., 22:294 33, 992. [4] S. Boyd, P. Diacois, P. Parrilo, ad L. Xiao. Symmetry aalysis of reversible Markov chais. Iteret Math., 2:3 7, 25. [5] T. Ceccherii-Silberstei, F. Scarabotti, ad F. Tolli. Harmoic aalysis o fiite groups, volume 8 of Cambridge Studies i Advaced Mathematics. Cambridge Uiversity Press, Cambridge, 28. Represetatio theory, Gelfad pairs ad Markov chais. [6] G.-Y. Che ad L. Saloff-Coste. The cutoff pheomeo for radomized riffle shuffles. Radom Structures Algorithms, 323: , 28. [7] M. Ciucu. No-feedback card guessig for dovetail shuffles. A. Appl. Probab., 84:25 269, 998. [8] M. Coger ad D. Viswaath. Riffle shuffles of decks with repeated cards. A. Probab., 342:84 89, 26. [9] M. Coger ad D. Viswaath. Normal approximatios for descets ad iversios of permutatios of multisets. J. Theoret. Probab., 22:39 325, 27. [] P. Diacois. Group represetatios i probability ad statistics. Istitute of Mathematical Statistics Lecture Notes Moograph Series,. Istitute of Mathematical Statistics, Hayward, CA, 988. [] P. Diacois. Mathematical developmets from the aalysis of riffle shufflig. I Groups, combiatorics & geometry Durham, 2, pages World Sci. Publ., River Edge, NJ, 23. [2] P. Diacois ad J. Fulma. Carries, shufflig ad a amazig matrix. preprit, 28. [3] P. Diacois ad S. P. Holmes. Radom walks o trees ad matchigs. Electro. J. Probab., 7:o. 6, 7 pp. electroic, 22. [4] P. Diacois, M. McGrath, ad J. Pitma. Riffle shuffles, cycles, ad descets. Combiatorica, 5: 29, 995. [5] P. Diacois ad M. Shahshahai. Geeratig a radom permutatio with radom traspositios. Z. Wahrsch. Verw. Gebiete, 572:59 79, 98.

19 A RULE OF THUMB FOR RIFFLE SHUFFLING 9 [6] A. Fässler ad E. Stiefel. Group theoretical methods ad their applicatios. Birkhäuser Bosto Ic., Bosto, MA, 992. Traslated from the Germa by Baoswa Dzug Wog. [7] J. Fulma. Applicatios of symmetric fuctios to cycle ad icreasig subsequece structure after shuffles. J. Algebraic Combi., 62:65 94, 22. [8] M. Garder. Marti Garder s New Mathematical Diversios from Scietific America. Simo & Schuster, New York, 966. [9] E. Gilbert. Theory of shufflig. Techical memoradum, Bell Laboratories, 955. [2] J. M. Holte. Carries, combiatorics, ad a amazig matrix. Amer. Math. Mothly, 42:38 49, 997. [2] J. Reeds. Theory of shufflig. Upublished mauscript, 976. [22] J.-P. Serre. Liear represetatios of fiite groups. Spriger-Verlag, New York, 977. Traslated from the secod Frech editio by Leoard L. Scott, Graduate Texts i Mathematics, Vol. 42. [23] J. R. Weaver. Cetrosymmetric cross-symmetric matrices, their basic properties, eigevalues, ad eigevectors. Amer. Math. Mothly, 92:7 77, 985. Appedix A. Radom walks o groups I this appedix, we reformulate shufflig i terms of radom walks o the symmetric group S, so that our ivestigatio of particular properties of a deck becomes the quotiet walk o Youg subgroups of S. Let G be a fiite group with Qg, g G Qg = a probability o G. The walk i may be called the left walk sice it cosists of repeatedly pickig elemets idepedetly with probability Q, say g,g 2,g 3,..., ad, startig at the idetity G, multiplyig o the left by g i. The geerates a radom walk o G, G, g, g 2 g, g 3 g 2 g,... By ispectio, the chace that the walk is at g after k steps is Q k g, where Q g = δ G,g. A algebraic method of focusig o aspects of the walk is to use the quotiet walk. Let H G be a subgroup of G, ad set X = G/H = {xh} to be the set of left cosets of H i G. The quotiet walk is derived from the walk above by simply reportig to which coset the curret positio of the walk belogs. The quotiet walk is a Markov chai o X with trasitio matrix give by 3 Kx,y = QyHx = h H Qyhx. Note that K is well-defied i.e. idepedet of the choice of coset represetatives ad that K is doubly stochastic. Thus the uiform distributio o X, Ux = H / G, is a statioary distributio for K. The chai K is reversible if ad oly if Q is symmetric i.e. Qg = Qg. Note that this is ot the case for riffle shuffles. While ituitively obvious, the followig shows the basic fact that powers of the matrix K correspod to covolvig ad takig cosets. Propositio A.. For Q a probability distributio o a fiite group G ad K as defied i 3, we have K l x,y = Q l yhx. Proof. The result is immediate from the defiitios for l =,. We prove the result for l = 2, the geeral case beig similar. Note that K 2 x,y = Kx,zKz,y = Qzh x Qyh 2 z. z z h,h 2 Settig h 2 = hh, otig that zh rus over G as z rus over X ad h over H, ad settig g = gx, we have K 2 x,y = h Qgx Qyhg = g h g Qg Qyhx g = Q2 yhx.

20 2 ASSAF, DIACONIS, AND SOUNDARARAJAN We may idetify permutatios i S with arragemets of a deck of cards by settig σi to be the label of the card at positio i from the top. Thus the permutatio is associated with four cards where 2 is o top, followed by, followed by 4, ad fially 3 is o the bottom. If we cosider the cards labelled,2,...,k to be red cards, ad the cards labelled k+,k+2,..., to be black cards, with all cards of the same color idistiguishable, the coset space X = S /S k S k is aturally associated with the k arragemets of red ad black ulabeled cards. Here, of course, we idetify a elemet of S k S k S as permutig the first k ad last k cards amog themselves. Similar costructios work for suits or values. Thus Propositio A. shows that the processes studied i the body of this paper are Markov chais. Appedix B. Shufflig by radom traspositios Let L 2 X = {f : X C} be the set of complex-valued fuctios o X with ier product defied by 3 f f 2 = f xf 2 x. X If K is symmetric, the real-valued fuctios may be used. The trasitio matrix K operates o L 2 via 32 Kfx = y x Kx, yfy. I the preset case, L 2 X = Id G H, the usual permutatio represetatio of G actig o left cosets X = G/H, with T g fx = fg x. By costructio, the actio of G commutes with K, i.e. 33 T g Kf = KT g f for all f L 2 X ad all g G. This implies that group represetatio theory ca be used to reduce the operator K or diagoalize K i the case whe K is symmetric. This classical topic is well developed i Fässler-Steifel [6] ad Boyd, et. al. [4]. Let Ĝ deote the set of irreducible represetatios of the fiite group G. For ρ Ĝ, the Fourier trasform of f L 2 G at ρ is defied by fρ = g G fgρg. As usual, Fourier trasform turs covolutio ito products, i.e. Q k ρ = Qρ k. Schur s lemma implies that the uiform distributio has zero trasform { if ρ is trivial, Ûρ = otherwise. The Fourier iversio theorem recostructs f from { fρ} by fg = dim ρ Tr fρρg. G ρ bg For backgroud, see Serre [22], Diacois [] or Ceccherii, et. al [5] where may applicatios are give.

21 A RULE OF THUMB FOR RIFFLE SHUFFLING 2 Suppose the iduced represetatio L 2 X decomposes ito irreducibles as 34 L 2 X = ρ bg V aρ ρ. The sice K commutes with G, K seds each of the spaces Vρ aρ ito itself. Further reductios may be possible if Q has suitable symmetries. The followig widely studied special case is relevat. Defiitio B.. The pair H G is a Gelfad pair if L 2 X is multiplicity free, i.e. all a ρ i 34 are either or. For example, whe k /2, S k S k S is a Gelfad pair with k 35 L 2 X = S i,i. i= Recall that the irreducible represetatios of S are idexed by partitios λ of. If S λ deotes the λth represetatio Specht modules, the sum i 35 rus over partitios ito two parts with the smaller part at most k. For further backgroud o Gelfad pairs, icludig examples ad applicatios, see [, 5]. Now we study a deck of red ad black cards after repeated radom traspositio shuffles. Recall that Diacois-Shahshahai [5] show that it takes 2log + c shuffles to mix distict cards. To be precise, the measure o S that drives the walks is Qσ = / if σ = id, 2/ 2 if σ = i,j, otherwise. Throughout the followig, all walks begi at the idetity permutatio, ad we use the covetio that πi is the label of the card at positio i. First, we follow the positio of the top card; i.e. the two of hearts is the oly red card followed by black cards. The trasitio matrix for this walk is give by 36 Pi,j = if i = j, 2 2 if i j. Note that this is symmetric, with Πi = / as the statioary distributio. Propositio B.2. For the trasitio matrix Pi,j above ad all l, we have 37 P l + 2 l if i = j, i,j = 2 l if i j. From this it follows that SEPl = 2 l ad P Π TV = 2 l. Proof. The results for the separatio ad total variatio distaces follow from 37 ad the defiitios. It is possible to give a direct combiatorial argumet for 37, but the followig represetatio theoretic argumet geeralizes readily to fid similar formula for j-tuples of cards.

22 22 ASSAF, DIACONIS, AND SOUNDARARAJAN The radom traspositio measure Q is costat o cojugacy classes of S ad so acts o each irreducible represetatio as a costat times the idetity. These costats are give explicitly by Diacois-Shahshahai [5], ivolvig characters ad dimesios of the represetatio. Cosider the operator Kσ,τ = Qτσ o the regular represetatio. The fuctio fσ = δ,σi / lies i the copies of the -dimesioal represetatio correspodig to the partitio,. The operator K acts o this space by multiplicatio by 2/. Thus P σ card labelled at positio i after l shuffles = Kl fσ = 2 l fσ = 2 l δ,σi. Here σ is the startig arragemet. Evaluatig the right-had side gives 37. Next we cosider the deck with N = 2 cards where the origial top cards are red ad the origial bottom cards are black. I this case, we thik of the the radom traspositio operator actig o the quotiet space S N /S S. For x,y S N /S S, the iduced Markov chai is 38 Kx, y = N 2 N + 2 if x y differ by a traspositio, N if x = y, 2 otherwise. This chai has uiform statioary distributio Πx = / N. The chai K is ivariat uder S N, i.e. Kx,y = Kσx,σy, so the distace to statioary does ot deped o the origial cofiguratio. As oted earlier, the pair S S, S N is a Gelfad pair, so 35 allows a easy determiatio of the eige values ad rate of covergece. Propositio B.3. For the Markov chai K o S N /S S, the eige values are β =, β j = N + N j 2 N 2 N j + j 2 3j,. Moreover, there is a uiversal costat A such j =,...,. The multiplicity of β j is m j = N j that if l = 4Nlog N + C, the K l Π Ae c/2. TV Proof. The operator K acts o L 2 S N /S S as the elemet of the group algebra N Id + 2 N 2 i,j. As show i [3], this elemet acts o the irreducibles S j,j as a costat times the idetity, with the costat beig β j ad the multiplicity beig the dimesio of S j,j. This proves the first part. The remaiig claims ca be proved followig the argumet i [3]: boud the total variatio distace by the L 2 orm, express this i terms of the eige values ad average over the startig state. This reduces the problem to boudig m j βj 2l. The lead term i this is j= i<j N 2 2l e c. N For l of the form 4Nlog N + c, the other terms are smaller ad sum i a reasoably stadard fashio. The terms are the same as i [3], so we suppress further details.