A RANDOM PERMUTATION MODEL ARISING IN CHEMISTRY

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1 J. Appl. Prob. 45, Prited i Eglad Applied Probability Trust 2008 A RANDOM PERMUTATION MODEL ARISING IN CHEMISTRY MARK BROWN, The City College of New York EROL A. PEKÖZ, Bosto Uiversity SHELDON M. ROSS, Uiversity of Souther Califoria Abstract We study a model arisig i chemistry where elemets umbered, 2,..., are radomly permuted ad if i is immediately to the left of i + the they become stuck together to form a cluster. The resultig clusters are the umbered ad cosidered as elemets, ad this process keeps repeatig util oly a sigle cluster is remaiig. I this article we study properties of the distributio of the umber of permutatios required. Keywords: Radom permutatio; hat-check problem 2000 Mathematics Subject Classificatio: Primary 60C05 Secodary 92E20. Itroductio For the classic hat-check problem first proposed i 708 by Motmort [2], the followig variatio appears i [6, p. 93]. Each member of a group of idividuals throws his or her hat i a pile. The hats are shuffled, each perso chooses a radom hat, ad the people who receive their ow hat depart. The the process repeats with the remaiig people util everybody has departed; let N be the umber of shuffles required. With X i represetig the total umber of people who have departed after shuffle umber i, it is easy to show that X i i is a martigale ad, thus, by the optioal samplig theorem we elegatly see that E[N]. Someoe gettig their ow hat ca also be thought of as correspodig to a cycle of legth oe i a radom permutatio. Properties of cycles of various legths i radom permutatios have bee studied extesively; see [] ad [3] for etry poits to this literature. A variatio of this problem was preseted i [5], where it was give as a model for a chemical bodig process. Below we discuss this variatio ad study its properties. We quote the followig descriptio of the chemistry applicatio from [5], where a recursive formula was give to umerically compute the mea. There are 0 molecules i some hierarchical order operatig i a system. A catalyst is added to the system ad a chemical reactio sets i. The molecules lie up. I the lie-up from left to right molecules i cosecutive icreasig hierarchical order bod together ad become oe. A ew hierarchical order sets amog the fused molecules. The catalyst is added agai Received 30 September 2008; revisio received 7 November Postal address: Departmet of Mathematics, The City College of New York, New York, NY , USA. address: cybergarf@aol.com Supported by the Natioal Security Agecy uder grat H Postal address: Departmet of Operatios ad Techology Maagemet, Bosto Uiversity, 595 Commowealth Aveue, Bosto, MA 0225, USA. address: pekoz@bu.edu Postal address: Departmet of Idustrial ad System Egieerig, Uiversity of Souther Califoria, Los Ageles, CA 90089, USA. address: smross@usc.edu 060

2 A radom permutatio model 06 to the system ad the whole process starts all over agai. The questio raised is how may times catalysts are expected to be added i order to get a sigle lump of all molecules. This variatio preseted i [5] ca be abstractly stated as follows. Suppose that we have elemets umbered, 2,...,. These elemets are radomly permuted, ad if i is immediately to the left of i + the i ad i + become stuck together to form possibly with other adjacetly umbered elemets a cluster. These clusters are the radomly permuted ad if a cluster edig with i immediately precedes oe startig with i + the those two clusters joi together to form a ew cluster. This cotiues util there is oly oe cluster, ad we are iterested i N, the umber of permutatios that are eeded. For istace, suppose that 7 ad that the first permutatio is 3, 4, 5,, 2, 7, 6, which results i the clusters {3, 4, 5}, {, 2}, {6}, ad {7}. If a radom permutatio of these four clusters gives the orderig {6}, {7}, {3, 4, 5}, {, 2} the the ew sets of clusters are {6, 7}, {3, 4, 5}, ad {, 2}. If a radom permutatio of these three clusters gives the orderig {3, 4, 5}, {6, 7}, {, 2} the the ew sets of clusters are {3, 4, 5, 6, 7} ad {, 2}. If a radom permutatio of these two clusters gives the orderig {, 2}, {3, 4, 5, 6, 7} the there is ow a sigle cluster {, 2, 3, 4, 5, 6, 7} ad N7 4. The radom variable N ca be aalyzed as a first passage time from state to state of a Markov chai whose state is the curret umber of clusters. Whe the state of this chai is i, we will desigate the clusters as,...,i, with beig the cluster whose elemets are smallest, 2 beig the cluster whose elemets are the ext smallest, ad so o. For istace, i the precedig 7 case, the state after the first trasitio is 4, with beig the cluster {, 2}, 2 beig the cluster {3, 4, 5}, 3 beig the cluster {6}, ad 4 beig the cluster {7}. With this covetio, the trasitios from state i are exactly the same as if the problem bega with the i elemets,,...,i. I Sectio 2 we compute the trasitio probabilities of this Markov chai ad use them to obtai some stochastic iequalities. I Sectio 3 we obtai upper ad lower bouds o E[N], as well as bouds o its distributio. I Sectio 4 we give results for a circular versio of the problem. 2. The trasitio probabilities With the above defiitios, let D be the decrease i the umber of clusters startig from state. The we have the followig propositio. Propositio. For 0 k<, PD k k + k! k+ i.

3 062 M. BROWN ET AL. Proof. Lettig A i be the evet that i immediately precedes i + i the radom permutatio, the D is the umber of evets A,...,A that occur. The, with S j PA i A ij, 0<i < <i j < the iclusio/exclusio idetity see [4, p. 06] gives j PD k S j j+k. k jk Now cosider PA i A ij. If we thik of a permutatio of elemets as havig degrees of freedom the, for each evet A i i the itersectio, oe degree of freedom i the permutatio is dropped. For istace, suppose that we wat PA 2 A 3 A 6. The, i order for these three evets to occur, 2, 3, ad 4 must be cosecutive values of the permutatio, as must be 6 ad 7. Because there are 5 other values, there are thus 3! such permutatios. Similarly, for the evet A 2 A 4 A 6 to occur, 2 ad 3 must be cosecutive values of the permutatio, as must be 4, 5 ad 6, 7. As there are 6 other values, there are 3! such permutatios. Cosequetly, for 0 <i < <i j <, j! PA i A ij.! As a result, jk j! S j j! which yields j PD k j+k j k j! k k i k + i k i k i k! k i k + k i k k! j, j! k i i! k 2 i k + + k k. k! k! Thus, the result follows oce we show that k k k + k! k k! + k k! + k+ k +! or, equivaletly, that k k k + k! k! + k+ k +!

4 A radom permutatio model 063 or k + k, k + k which is immediate. Remark. A recursive expressio for PD k, though ot i closed form, was give i [5]. From Propositio we immediately coclude that D coverges i distributio to a Poisso radom variable with mea. Corollary. We have lim PD k e /k!. We ow preset two results that will be used i the ext sectio. Recall from [6, p. 33] that a discrete radom variable X is said to be likelihood ratio smaller tha Y if PX k/ PY k is oicreasig i k. Corollary 2. With the above defiitios, D is likelihood ratio smaller tha a Poisso radom variable with mea. Proof. We eed to show that k! PD k is oicreasig i k. But, with B k k! PD k we have B k B k k+ k > 0, i i + k + 2 k+2 k + 2! which proves the result. Corollary 3. The state of the Markov chai after a trasitio from state, D, is likelihood ratio icreasig i. Proof. From Propositio, P D k k + k+ i. k! Cosequetly, P + D + k P D k + + k. As the precedig is icreasig i k, the result follows. 3. The radom variable N Let X i be the ith decrease i the umber of clusters, so that S k is the state of the Markov chai, startig i state, after k trasitios, k. k X i

5 064 M. BROWN ET AL. Propositio 2. We have PN > k e k k i. Proof. Let the Y i,i,...,k, be idepedet Poisso radom variables, each with mea. Now, because likelihood ratio is a stroger orderig tha stochastic order see Propositio 4.20 of [6], it follows by Corollary 2 that X i, coditioal o X,...,X i,is stochastically smaller tha a Poisso radom variable with mea. Cosequetly, the radom vector X,...,X k ca be geerated i such a maer that X i Y i for each i,...,k.but this implies that PN > k PX + +X k < PY + +Y k < We ow cosider bouds o E[N]. e k k i. Propositio 3. We have Proof. First ote that E[N] + i. E[D ] Pi immediately precedes i +. Because the Markov chai caot make a trasitio from a state ito a higher state ad E[D ] is odecreasig i, it follows from Propositio 5.23 of [6] that Propositio 4. We have Proof. To begi, ote that E[N] i2 E[N] + Z k E[D i ] + i. e e + e e 2 j. j2 k X i E[X i X,...,X i ], k, 2 is a zero-mea martigale. Hece, by the martigale stoppig theorem, E[Z N ]0. 3

6 A radom permutatio model 065 Now, because E[X i X,...,X i ] is the expected decrease from state S i, it follows from that E[X i X,...,X i ]E[X i S i ]E[D Si S i ]. S i Usig this, ad the fact that N X i, we obtai, from 2 ad 3, [ N E[N]+E S i ] 0. Now otatioally suppressig its depedece o the iitial state, let T j deote the amout of time that the Markov chai speds i state j, j >. The Hece, E[N] + j2 N S i j2 j E[T j ] + T j j. e e 2 j + j2 e e, where, for the iequality, we made use of the followig propositio. Propositio 5. We have E[T ] PD > 0 e e, E[T j ] PT j > 0 PD j > 0 e e 2. To prove Propositio 5, we will eed a series of lemmas. Lemma. Let W j, 2 j <, deote the state of the Markov chai from which the first trasitio to a state less tha or equal to j occurs. The, for r>j, PT j > 0 W j r PT j > 0 W j j + PD j+ D j+. Proof. Let Y r r D r. The, PT j > 0 W j r PD r r j D r r j PY r j Y r j PY r j j PY r i j PY r i/py r j. 4 But, for i j, it follows from Corollary 4 that PY r+ j PY r j PY r+ i PY r i

7 066 M. BROWN ET AL. or, equivaletly, that PY r+ i PY r+ j PY r i PY r j. Thus, by 4, PT j > 0 W j r is odecreasig i r. Lemma 2. For all j 2, PD j+ D j+ e e. Proof. Let M k k i /. By Propositio we eed to show that M j+ j + 2M j+2 /j + e e. That is, we eed to show that, for all 3, M e e + M + 0. Case. Suppose that is eve ad that >2. The, M e e + M + [ M e e + M + e e +! e + e e e! > 0, e +! where we used the fact that M > e. Case 2. Suppose that is odd. I this case, M e e + M + M + M + + e e + e e e M +! e +! e e +! e + +! ] +! e e + +!, M +

8 A radom permutatio model 067 which will be oegative provided that e 2 +! e or, equivaletly, that +! ee, which is easily see to be true whe 3. This completes the proof of Lemma 2. We eed oe additioal lemma. Lemma 3. As, PD 0 e. Proof. By Propositio, PD 0 + M +, yieldig lim PD 0 e. To show that the covergece is mootoe, ote that + M M +2 + M M ! M !. Whe is odd, the precedig is clearly positive. Whe is eve, M + M / +!, ad, thus, we must show that or, equivaletly, that which follows sice, for eve, M +! +! M!, M M +!!. Proof of Propositio 5. Give that state j is etered, the time spet i that state will have a geometric distributio with parameter PD j > 0. Hece, E[T j ] PT j > 0 PD j > 0. Now, PT > 0, ad, by Lemma 3, PD > 0 e, which verifies the first part of Propositio 5. Also, for 2 j<, Lemmas ad 2 yield Hece, by Lemma 3, PT j > 0 PD j+ D j+ e E[T j ] e 2 e e 2, which completes the proof of Propositio 5. e e.

9 068 M. BROWN ET AL. Table. Lower boud Upper boud Corollary 4. We have e + e + e 2 e 2 l E[N] + l. 2 3 Proof. Let X be uiformly distributed betwee j 2 ad j + 2. The, j + /2 j+/2 [ ] l j /2 x dx E X E[X] j, j /2 where the iequality used Jese s iequality. Hece, j2 j l /2 2 l 3/2 3 ad the upper boud follows from Propositio 3. To obtai the lower boud, we use Propositio 4 alog with the iequality j + j+ l j j x dx j. Remarks.. Corollary 4 yields the results give i Table. 2. It follows from Corollary 3, usig a couplig argumet, that N is stochastically icreasig i., 4. The circular case Whereas we have previously assumed that at each stage the clusters are radomly arraged i a liear order, i this sectio we suppose that they are radomly arraged aroud a circle, agai with all possibilities beig equally likely. We suppose that if a cluster edig with i is immediately couterclockwise to a cluster begiig with i + the these clusters merge. Let N deote the umber of stages eeded util all elemets are i a sigle cluster, ad let D deote the decrease i the umber of clusters from state. Lemma 4. For 2, E[D ]vard. Proof. If B i is the evet that i is the couterclockwise eighbor of i + the D Bi.

10 A radom permutatio model 069 Now, ad, for i j, Hece, ad PB i 2!!, i,...,, PB i B j 3!!. E[D ] vard !! 2 Propositio 6. We have E[N ]. Proof. The proof is by iductio o. Because PN 2, it is true whe 2, ad so assume that E[Nk ]k for all k 2,...,. The, E[N D ] + E[N D D ], 5 yieldig E[N ] + E[N i] PD i which proves the result. + E[N ] PD 0 + E[N i] PD i + E[N ] PD 0 + i PD i + E[N ] PD 0 + PD 0 E[D ] + E[N ] PD 0 + PD 0, Remark. Propositio 6 could also have bee proved by usig a martigale stoppig argumet, as i the proof of Propositio 4. Propositio 7. For >2, varn.

11 070 M. BROWN ET AL. Proof. Let V varn. The proof is by iductio o. As it is true for 3, sice N 3 is geometric with parameter 2, assume it is true for all values betwee 2 ad. Now, varn D varn D D ad, from 5 ad Propositio 6, Hece, by the coditioal variace formula, E[N D ] D. V V ipd i + vard V PD 0 + V ipd i +. 6 Now, because PD 2 0 ad V 0, the iductio hypothesis yields Hece, from 6, V ipd i i PD i. which proves the result. V V PD 0 + PD 0 E[D ]+ V PD 0 + PD 0, Refereces [] Arratia, R. ad Tavaré, S The cycle structure of radom permutatios. A. Prob. 20, [2] De Motmort, P. R Essay d Aalyse sur le Jeux de Hazard. Quillau, Paris. [3] Diacois, P., Fulma, J. ad Guralick, R O fixed poits of permutatios. J. Algebraic Combi. 28, [4] Feller, W A Itroductio to Probability Theory ad Its Applicatios, Vol. I, 3rd ed. Joh Wiley, New York. [5] Rao, M. B. ad Kasala, S A discrete probability problem i chemical bodig. Preprit, Uiversite de Techologie de Compiege, Frace. [6] Ross, S. M. ad Peköz, E A Secod Course i Probability. ProbabilityBookstore.com, Bosto, MA.