On the Number of Permutations on n Objects with. greatest cycle length

Transcription

1 Ž. ADVANCES IN APPLIED MATHEMATICS 0, ARTICLE NO. AM O the Number of Permutatios o Obects with Greatest Cycle Legth k Solomo W. Golomb ad Peter Gaal Commuicatio Scieces Istitute, Uiersity of Souther Califoria, Los Ageles, Califoria Received May 3, 997; accepted July 5, 997 Let L permutatios o distict obects k, greatest cycle legth k 4. We observe each of the followig: For all, For k, For 3 k, L,, L, Ž.!, Ý Lk,! For 4 k 3,! Lk,. k k Ý /! k Lk,. k ž k k ž Ý /! 3k k ½ Lk, k k 3k k k $5.00 Copyright 998 by Academic Press All rights of reproductio i ay form reserved. Ý 3k 3k Ý Ý 5. Ž k.ž kl. 98 l

2 For 5 k 4, PERMUTATIONS ON OBJECTS 99 k 3k 3k! ½ k, Ý Ý Ý l L k k! Ž k.ž kl. etc. For k, 4k 4kl 4k Ý Ý Ý 3! k kl km Ž.Ž.Ž. l m 3k 4k 4k Ý Ý Ý k k! Ž k.ž kl. l k miž k, k. ž / 4k Ý 5, 3k k 4k! Lk, Ý Ý L t, k.!k Ž k.! The expected legth of the logest cycle i a radom permutatio o obects is give by E!Ý kkl k,, where for k the cotributio of the kth term is Ž!. k!k. Similarly, for 3 k, the cotributio is k Ý k ŽŽ k.., etc. The rel- ative expected legth of the logest cycle of a radom permutatio o obects is E, which has a limit Ž Golomb s costat. as. The total cotributio to of all k, k, is Ž % as. Similarly, the total cotributio of all k, 3 k, to is , or 7.643%, as, etc. Figure shows L i triagular form for all, 4 k, ad all k,. Ž The th row sum is!.. We cosider the symmetric group S of all! permutatios o obects. As a coveiet coceptual model, we regard each elemet of S as a deck of cards Ž the cards umbered from to. i a particular permuted order, so that we have! decks, each cotaiig the same cards, but each deck havig the cards i a differet order. For each deck, we partitio the cards ito cycles as follows: We deal out cards, startig from the top of the deck, i order,util the card umbered appears. This completes the first cycle. If there are cards still remaiig i the deck, there is a lowest-umbered remaiig card, bearig the umber a, with a. We cotiue dealig cards util the card umbered a ap- t

3 00 GOLOMB AND GAAL FIG.. The values of L k, for 4 ad k.

4 PERMUTATIONS ON OBJECTS 0 pears, ad this completes the secod cycle. If ay cards still remai i the deck, we deal these util the lowest remaiig umber a 3, with a a3, appears, which completes the third cycle. We cotiue this process util all the cards are dealt. ŽFor example, i a deck of 8 cards, if the umbers i order were ,thecycles would be Ž 634.Ž.Ž 85.Ž 7... Sice the umber occurs equally ofte i each positio of the deck, the first cycle has a legth uiformly distributed from to, with a expected legth of Ž.. If a permutatio i S cotais a cycle of legth k, where k, this is ecessarily the logest cycle i, sice the umber of remaiig elemets i is k k. The umber of permutatios i S cotaiig a cycle of legth k, k, is exactly!k, so this is also the umber of permutatios i S for which the greatest cycle legth is k. To obtai the umber!k, we observe that the k elemets o the cycle of legth k ca be chose i ž/ ways, ad k permuted i Ž k.! ways to form a cycle of legth k; the the remaiig k elemets ca occur i ay of Ž k.! permuted orders. Thus, the umber of permutatios i S for which the logest cycle has legth kisž/ Ž k!k!!k.. Ž. k We defie Lk, to be the umber of elemets i S for which the greatest cycle legth is k. Clearly this requires Ý k L! k, sice every elemet of S is couted i oe ad oly oe of the L k, s. We have ust show that L!k for k. If, however, k, k, there are two reasos!k overestimates L k,.if k, there may be a cycle of legth greater tha k i the permutatio. If k, eve if there is o cycle of legth greater tha k, the cycle of legth k may ot be uique. For example, if k, there could be two cycles, each of legth. The umber of ways this ca happe is ž /ž / k k Ž. 4 Ž. k! k!k, ad sice these cases have bee couted k k twice i!k Ž k.!k, they should be subtracted oce, to obtai / Ž k.! Ž k.! Ž k.! Lk,k, for all k. k k k ž k Other cases where k is oly slightly less tha are also relatively easy to cout. For 3 k, the correctios to!k for the umber of

5 0 GOLOMB AND GAAL permutatios o obects with greatest cycle legth k are:!k for the possibility that two cycles of legth k are tied for logest;!kk Ž. for the possibility that there is a loger cycle, of legth k, for each, k. For 3 k, there are at most two cycles with legth k. Hece: Ž k.! Lk,k ½ 5, k k ad! k Lk, Ý, k. k ½ k k 3 5 Because there may be a tie for logest cycle i a particular permutatio, the expressios i curly brackets do ot represet the probability that a cycle of legth k is logest, or the probability that it is uiquely logest, but rather is the weightig factor required to cout the logest cycle oly oce per permutatio, eve whe there is a tie for logest. For this reaso also, as we go to values of k where Ž r. k r with icreasig r, the usual iclusioexclusio rules of coutig do ot apply, but must be specially modified for the possibility that as may as r cycles of legth k may be tied for logest. Specifically, we fid: Ž 3k.! k Lk,3k ž Ý, all k, k k 3k k ½ / 5 ½ / k ž / Ý k k k5 Ž 3k.! L k,3k ž ad more geerally, k k 3k, all k, k 3k 3k! ½ k, Ý Ý Ý l L k k Ž k.ž kl. for 4 k 3. 3k Ý 5, k k 3k

6 PERMUTATIONS ON OBJECTS 03 I the ext regio, with 5 k 4, we have k 3k 3k! ½ k, Ý Ý Ý l L k k! Ž k.ž kl. 4k 4kl 4k Ý Ý Ý 3! k kl km Ž.Ž.Ž. l m 3k 4k 4k Ý Ý Ý k k! Ž k.ž kl. l ž / 4k Ý 5. 3k k 4k For the specific case 5k, it is ecessary to modify oly the right-most term, 4k, replacig it with Ž 4k.Ž 5k.. The geeral patter, for Ž r. k r, is readily deduced from the cases r, 3, ad 4, illustrated above. Startig from the other directio, at k, we clearly have L,, for all, sice there is exactly oe permutatio which has each obect i a cycle by itself. To evaluate L,, we cout the permutatios of order i S. We fid the umber of such permutatios with -cycles for each,, to obtai Similarly, Ý ž /ž / ž /,! L, all. ž /ž / ž / L3, Ý L, 34, all 3.! The geeral result is easily show to be k Ž k.! 4 k k k Ý ž /ž / ž / k, k k! k L Ž. mi k, k Ý t L t, k k miž k, k.! Ý Ý L t, k, k,!k Ž k.! t

7 04 GOLOMB AND GAAL which expresses L recursively i terms of L where k k, k, k ad. Figure shows the umerical triagle for L for all,4, ad all k, k. The expected legth of the logest cycle, over the set of elemets i S,is give by E Ý kkl k,, ad the relatie expected legth of the logest cycle is E. This subect has bee discussed i 4, where it is show that lim exists, with ŽD. Kuth 5 has referred to as Golomb s costat.. It is iterestig that E is a decreasig sequece, while E Ž. is a icreasig sequece, both covergig to. ŽThe sequece E Ž. appears to coverge to cosiderably faster.. The proof that b E Ž. is a icreasig sequece, first give i, is as follows: I our deck-of-cards model, suppose that a Ž st. card is itroduced i a radom positio Žthere are equally likely positios. i a radom deck of cards. The probability would be E Ž. that it would lad i the previously logest cycle, thereby icreasig the legth of the logest cycle by, if logest cycles were always uique. Sice logest cycles are ot ecessarily uique for, the actual probability that the ew card will icrease the expected legth by exceeds E Ž. for all. Thus, E E E Ž. E ŽŽ.. E Ž. Ž. for, with strict iequality for. Dividig by, we fid: E E b b, for, with strict iequality for. It is equally easy to show that c E is a mootoe decreasig sequece for, as follows. Suppose we have a deck of cards Žwith the iduced cycle structure described earlier. ad we radomly discard oe of them. If the logest cycle was uique, the the probability that we have decreased its legth by oe is E. For, the logest cycle is ot always uique, ad discardig a card from a cycle which was tied for logest does ot decrease the legth of the logest cycle. Hece, the expected decrease i the legth of the logest cycle, for, is less tha E. Thus, for, E E E E Ž., ad dividig by yields E 4 Ž. E, i.e., c c for. Sice b is a mootoic icreasig sequece for ad is clearly bouded from above by, it has a limit by a well-kow theorem of Weierstrass. This is the proof, give i, that the limit exists. ŽClearly c 4 has the same limit, sice c b Ž E. ŽE Ž.. Ž. as.. I respose to the research problem, several closed-form expressios for were foud, all ivolvig defiite itegrals of o-elemetary fuctios Žsee 3 ad. 4. k,

8 PERMUTATIONS ON OBJECTS 05 Perhaps the simplest of these, due to Natha Fie, is H LiŽt. e dt 0 Ž. x where Li x H0 dtl t is the itegral logarithm fuctio popularized by Gauss i his study of the distributio of prime umbers. Alteratively, as show by L. A. Shepp, e u t EiŽt. e e dt, where EiŽ t. du. H H u 0 t The cycles of legth k, for k, each cotribute to E, for a total cotributio of Ž. to E, ad the cotributio to the limit is , or % of Numerical calculatio shows that the total cotributio to of k with 3 k is , or % of, which leaves less tha.3% for all k 3. Of this amout, k with 4 k 3 cotributes or % of, leavig oly % as the total cotributio of all cycles of legth 4 to the relative expected legth of the logest cycle. For 6a, the total cotributio of k, 3 k, to E is give by k Ý Ý k k k3 ½ 5 3a a a Ý Ý Ý a, k a 4a ka which ca be used either umerically or aalytically to obtai the limitig value metioed above as the portio of which arises from this regio of k-values. The aalytic approach uses 3a a a a Ý Ý ž Ý k a 4a/ ka 3a dt a tdt a tdt a H H H 6a a t Ž a. t 4at ž / ž / ž / l a Ž a l. 4al a 6 a 6a 6a l l l 3 l as a. which agrees precisely with the umerical data for large.

9 06 GOLOMB AND GAAL Similar, though ever more complicated expressios correspod to each r-regio, Ž r. k r. I the collectio of the! decks of cards each, represetig the permutatios i S, there is a grad total of! cards, divided up ito cycles with legths from to. Amog the! cards, there are exactly! cards i cycles of legth k, for every k from to. Thus the total umber of cycles, amog all! permutatios, is! Ž 3 4., so the aerage Ž or expected. umber of cycles i a radom permutatio o obects is Ý Ž k. l ož. k as, where is the EulerMascheroi costat. ŽThus, i a radom permutatio o 0 3 or 0 obects, the expected umber of cycles is oly about 7.. From our card model, i a radom sequece of the umbers from to, this is also the expected umber of terms i the sequece which are smaller tha all subsequet terms. Several other radom processes have bee show to have distributios which mimic the distributio of cycle legths i a radom permutatio. Dickma 6 studied the sizes of the prime factors of a radom iteger N o the rage, x. Amog other thigs, he showed that if p is the expected largest prime factor of N, the lim Ž log p log N. x This may be stated as follows: The expected fractio of the digits of N which comes from its largest prime factor teds to the limit as N. Sice a ratio of logarithms is ivariat uder a chage of logarithmic base, it does ot matter whether the digits are i base, or base 0, or ay other base b. The aalysis of cycle legths i this paper, based o k ŽŽ r,r,. should correspod directly to the Ž Ž r. aalysis of the sizes of prime factors based o p N, N r,or equivaletly to log p Ž log N. Ž r., Ž log N. r. ŽHere log p correspods to k, ad log N to, i the aalogy with cycles of a radom permutatio.. Aother model for this distributio arises i the study of the degrees of the irreducible factors of a radom polyomial of degree over GFŽ q.. ŽIf d is the expected value of the highest degree of ay irreducible factor of a polyomial of degree over GFŽ q., the lim Ž d... Aother iterestig result Žsee,, or 3. is the followig: THEOREM. The probability of the logest cycle i a radom permutatio o obects Ž large. haig a relatie legth L aislž a. l a for a. I particular, the probability that the logest cycle has relative legth 0.5 is l 69.35%; ad it is a eve bet that the relative legth exceeds ' e

10 PERMUTATIONS ON OBJECTS 07 REFERENCES. S. W. Golomb, Cycles from Noliear Shift Registers, Jet Propulsio Laboratory Progress Report 0-389, August S. W. Golomb, Radom Permutatios, Research Problem No., Bull. Amer. Math. Soc. 70, No. 6 Ž Nov. 964., S. W. Golomb, Shift Register Sequeces, HoldeDay, Sa Fracisco, CA, S. W. Golomb, Shift Register Sequeces, rev. ed., Aegea Park Press, Lagua Hills, CA, D. E. Kuth, The Art of Computer Programmig, Vol., Semiumerical Algorithms, pp. 8 ad 454, AddisoWesley, Readig, MA, K. Dickma, O the frequecy of umbers cotaiig prime factors of a certai relative magitude, Ark. Mat. Astro. Fys. A, No. 0, Hafte Ž 930., 4.