Left-to-right maxima in words and multiset permutations

Transcription

1 Left-to-right axia in words and ultiset perutations Ay N. Myers Saint Joseph s University Philadelphia, PA <ayers@sju.edu> Herbert S. Wilf University of Pennsylvania Philadelphia, PA <wilf@ath.upenn.edu> March 23, 2007 Abstract We extend classical theores of Rényi by finding the distributions of the nubers of both weak and strong left-to-right axia a.k.a. outstanding eleents in words over a given alphabet and in perutations of a given ultiset. 1 Introduction Given a sequence w = w 1 w 2... w n of ebers of a totally ordered set, we say j is a strongly outstanding eleent of w if whenever i < j we have w i < w j. In this case we call w j a strongly outstanding value. We say j is a weakly outstanding eleent if w i w j whenever i < j, and call w j a weakly outstanding value. In this paper we will explore the contexts in which w is a perutation, a ultiset perutation, or a word over soe finite alphabet. A faous theore of Rényi [12] see also [1] states that the nuber of perutations of [k] with r strongly outstanding eleents [ ] is equal to the nuber of such perutations with r k cycles, the latter being given by, the unsigned Stirling nuber of the first kind. r In this paper we investigate additional properties of the outstanding eleents and values of perutations, and extend the to ultiset perutations and words on [k] = {1, 2,..., k}. An interesting sidelight to our results is that we obtain a proof of Gauss s celebrated 2 F 1 evaluation by coparing two fors of one of our generating functions, in section 6 below. 1

2 2 Suary of results 2.1 Multiset perutations For perutations 1 we have the following results. Theore 1 Let M = {1 a 1, 2 a 2,..., k a k } be a ultiset with N = a1 + a a k, and let fm, r denote the nuber of perutations of M that contain exactly r strongly outstanding eleents. Then F M x = def This result is also found in [4]. r fm, rx r = N 1!a kx a 1!a 2!... a k! P M x = a kx N k a i x. 1 N a 1 + a a i Corollary 1 The generating function for the probability that a randoly selected perutation of M has exactly r strongly outstanding eleents is k 1 1 +, 2 a i x N a 1 + a a i and the average nuber of strongly outstanding eleents aong perutations of M is k P M1 a i =. 3 a i + a i a k Theore 2 The generating function for the nuber of perutations of M that contain exactly t weakly outstanding eleents is given by where G M x = φ N,a1 xφ N a1,a 2 x... φ N a1 a k 2,a k 1 xx a k 4 φ N,a x = a N 1 x. a =0 Corollary 2 The average nuber of weakly outstanding eleents aong perutations of M is k a i a i+1 + a i a k Corollary 3 Let A = ax i {a i }. The aount by which the average nuber of weakly outstanding eleents exceeds the average nuber of strongly outstanding eleents is AA 1. π Note that for perutations, all outstanding eleents and values are strongly outstanding. 2

3 2.2 Words The next theore involves Stirling nubers of the second kind, denoted by { n }. This is defined as the nuber of ways to partition a set of n eleents into nonepty subsets. Theore 3 The nuber of n-letter words over an alphabet of k letters which have exactly r strongly outstanding eleents is given by fn, k, r = [ ]{ } k n, 6 r and the average nuber of strongly outstanding eleents aong such words is H k = k + o1 n. Theore 4 The generating function for the nuber gn, k, t of n-letter words over an alphabet of k letters which have exactly t weakly outstanding eleents is given by G k n, x = def k 1 k 1 gn, k, tx t = 1 k 1 t x + t n t t t=0 x + t 1 k 1 and the average nuber of weakly outstanding eleents aong these words is n n k 1 k + H k 1 + O. k, 7 Next we introduce the notion of a teplate for words on [k]. A perutation of 5 or ore letters atches the teplate Y N Y Y, for exaple, if 1, 4, and 5 are outstanding eleents, 2 is not an outstanding eleent, and 3 is unconstrained. For exaple the perutation atches this teplate. In this case we think of the letters of the teplate Y, N, * as representing yes, no, and unconstrained, respectively. We generalize the Y and N constraints to S, W, S, and O; where S indicates a strongly outstanding eleent, W a weakly outstanding eleent, S indicates the absence of a strongly outstanding, and O the absence of an outstanding eleent. We provide an algorith for producing the generating function for the nuber of words that atch a given teplate. When the word is a perutation, we have: Theore 5 Let τ be a given teplate of length at ost n, and let τ j denote the letter that appears in position j, counting fro the left, of the teplate τ. Since every eleent of a perutation is either strongly outstanding or not outstanding, the letters of τ are chosen 3

4 fro {Y, N, }. The probability that a perutation of at least n letters atches the teplate τ is j j. 8 j:τ j = N The corresponding result for words is: j:τ j = Y Theore 6 Let τ be a word on {S, W,, S, O}. Suppose F k, τ, x = k 1 fk, τxk is the ordinary generating function for fk, τ, the nuber of words over the alphabet [k] that atch the teplate τ. Consider the adjunction of one new sybol, A {S, W,, S, O}, at the right end of τ. The generating function, F k, τa, x can be obtained fro F k, τ, x by applying an operator Ω A, i.e., where 3 Notation F k, τa, x = Ω A F k, τ, x, Ω S F k, τ, x = xf k, τ, x/1 x, 9 Ω W F k, τ, x = F k, τ, x/1 x, 10 Ω F k, τ, x = x d F k, τ, x, 11 dx Ω S F k, τ, x = x d dx x F k, τ, x, and 12 1 x Ω O F k, τ, x = x d dx 1 F k, τ, x x In the following sections we will count perutations, ultiset perutations, and words that contain a given nuber of strongly or weakly outstanding eleents. Let fk, r denote the nuber of perutations of [k] = {1, 2,..., k} that contain exactly r strongly outstanding eleents. For a ultiset M = {1 a 1, 2 a 2,..., k a k }, let fm, r denote the nuber of perutations of M that contain exactly r strongly outstanding eleents. Finally let fn, k, r denote the nuber of n letter words on the alphabet [k] that contain exactly r strongly outstanding eleents. When counting weakly outstanding eleents, we use g in place of f and t in place of r. Using the above notation, we define the following generating functions for strongly outstanding eleents: F k x = r fk, rxr, F M x = r fm, rxr, and F k n, x = 4

5 r fn, k, rxr. We define analogous generating functions for weakly outstanding eleents and use G in place of F. 4 Strongly outstanding eleents of ultiset perutations In this section we establish Theore 1 and its corollaries. Given a ultiset M = {1 a 1, 2 a 2,..., k a k }, let N = j a j. We construct the perutations of M that have exactly r strongly outstanding eleents as follows. We have N slots into which we will put the N eleents of M to ake these perutations. Take the a 1 1 s that are available and place the in soe a 1 -subset of the N slots that are available. There are two cases now. If the set of slots that we chose for the 1 s did not include the first leftost slot, then we can fill in the reaining slots with any perutation of the ultiset M/1 a 1 that has exactly r strongly outstanding eleents. On the other hand, if we did place a 1 into the first slot, then after placing all of the 1 s, the reaining slots can be filled in with any perutation of the ultiset M/1 a 1 that has exactly r 1 strongly outstanding eleents. Recall that fm, r denotes the nuber of perutations of the ultiset M that have exactly r strongly outstanding eleents. The arguent in the preceding paragraph shows that When we define N fm, r = a 1 N 1 a 1 1 fm/1 a 1, r + F M x = r fm, rx r, N 1 a 1 1 fm/1 a 1, r 1. we have the recurrence F M x = = N N 1 N 1 + x F M/1 a 1 x a 1 a 1 1 a 1 1 N 1 N 1 + x F M/1 a 1 x a 1 1 a 1 This shows that the generating polynoial resolves into linear factors over the integers. 5

6 Indeed we get the explicit for N 1 N 1 N a1 1 N a1 1 F M x = + x + x... a 1 a 1 1 a 2 a 2 1 k N a1 a 2 a i 1 1 N a1 a 2 a i 1 1 = + x a i a i 1 = N 1!a k 1 kx a i x 1 + a 1!a 2!... a k! N a 1 + a a i This gives us Theore 1. Its corollaries follow by obvious calculations. We note that this result was previously obtained by Cléent, Flajolet, and Vallée [4] in the context of binary search trees. Since the generating polynoial has real zeros only, the probabilities {p r M} are uniodal and log concave. By Darroch s Theore [5], the value of r for which p r M is axiu differs fro P M 1 of 3 above by at ost 1. 5 Weakly outstanding eleents of ultiset perutations Next we prove Theore 2. Consider gm, t, the nuber of perutations of M = {1 a 1, 2 a 2,..., k a k } that have exactly t weakly outstanding eleents, and G M x = t gm, txt. To find gm, t, suppose the perutation begins with a block of exactly 0 1 s. Since the value that follows the last 1 in the block is not available for a 1, there reain N 1 slots into which the reaining 1 s can be put, in N 1 a 1 ways. Once all of the 1 s have been placed, if the reaining perutation of the ultiset M/1 a 1 has exactly t weakly outstanding eleents then the whole thing will have t weakly outstanding eleents. Hence we have gm, t = N 1 gm/1 a 1, t, a 0 1 if M/1 a 1 is nonepty, whereas if M = 1 a 1 only, then gm, t = δ t,a1. If we ultiply by x t and su on t we find that { } N 1 G M x = x G M/1 a 1 x, a 1 0 6

7 except that if M = 1 a 1 only, then G M x = x a 1. So if we write then we have φ N,a x = a N 1 x, a =0 G M x = φ N,a1 xφ N a1,a 2 x... φ N a1 a k 2,a k 1 xx a k. 14 This gives 4. The two sus φ N,a 1 = n a and φ N,a 1 = n a /n a+1 are eleentary, and they iply that φ N,a 1/φ N,a1 = 1/n a+1. Then logarithic differentiation of 14 and evaluation at x = 1 shows that the average nuber of weakly outstanding eleents in perutations of M is k a i a i+1 + a i a k Let A = ax i {a i }. If we copare 15 and 3 we find that the aount by which the average nuber of weakly outstanding eleents exceeds the average nuber of strongly outstanding eleents is k a i a i+1 + a i a k + 1 a i a i + a i a k = k AA 1 a i a i 1 a i+1 + a i a k + 1a i + a i a k k 1 k i + 1 π2 AA This estiate is best possible when A = 1, i.e., when every eleent occurs just once. 6 Strongly outstanding eleents of words Next we investigate fn, k, r, the nuber of n-letter words over an alphabet of k letters that have exactly r strongly outstanding eleents. In so doing we will prove Theore 3. Note first that the nuber of strongly outstanding eleents of such a word depends only on the perutation of the distinct letters appearing in the word that is achieved by the first 7

8 appearances of each of those letters, because a value j can be strongly outstanding in a word w only if it is the first i.e., leftost occurrence of j in w. Hence associated with each n-letter word w over an alphabet of k letters which has exactly r strongly outstanding eleents there is a triple S, P, σ consisting of 1. a subset S [k], which is the set of all of the distinct letters that actually appear in w, and 2. a partition P of the set [n] into = S classes, naely the ith class of P consists of the set of positions in the word w that contain the ith letter of S, and 3. a perutation σ S, = S, which is the sequence of first appearances in w of each of the k letters that occur in w. σ will have r strongly outstanding eleents. Conversely, if we are given such a triple S, P, σ, we uniquely construct an n-letter word w over [k] with exactly r strongly outstanding eleents as follows. First arrange the classes of the partition P in ascending order of their sallest eleents. Then perute the set S according to the perutation σ, yielding a list S. In all of the positions of w that are described by the first class of the partition P i.e., the class in which the letter 1 lives we put the first letter of S, etc., to obtain the required word w. Thus the nuber of words that we are counting is equal to the nuber of these triples, viz. k [ r ]{ n }. 16 fn, k, r = It is noteworthy that three flavors of Pascal-triangle-like nubers occur in this forula. Let ρn, k denote the average nuber of strongly outstanding eleents aong the n- letter words that can be fored fro an alphabet of k letters. To find ρn, k for large n, we have first that fn, k, r = { } k n [ ] = { } k n! r r r { } k n! k n n k, for large n, where we have used the facts that { n } n /! and [ ] r r =!. Siilarly, rfn, k, r = { } k n [ ] r = { } k n!h r r r { } k n!h k n H n k H k, 8

9 where we have used the additional fact that r r[ r ] =!H. If we divide these last two equations we find that li ρn, k = H k = n k. The proof of Theore 3 is coplete. Fro eq. 16 we can use the standard generating functions for the two kinds of Stirling nubers to show that k fn, k, ry r t n r = 1 + y. 1 jt r,n But fro 16 we also have fn, k, ry r t n r = [ ] k l y r { } n t n l r t l r,n l,r n = [ ] k l y r t l l r t 1 t1 2t... 1 lt l,r k l 1 y l t + j t l 1 t1 2t... 1 lt = l = l j=0 l k l j=1 j=1 y + j 1t. 1 jt Coparison of these two evaluations shows that we have found and proved the following identity: l k y + j 1t k = 1 + y. 17 l 1 jt 1 jt l j=1 j=1 But Gauss had done it earlier, since it is the evaluation of his well known [ ] k y/t 2F 1 1 1/t 1. 7 A calculus of teplates 7.1 Teplates and perutations In this section we prove Theores 5 and 6. We begin by establishing equation 8, which appeared first in [13]. 9

10 This result is a generalization of a theore of R. V. Kadison [9], who discovered the case where the teplate is NN NY, and proved it by the sieve ethod. To ake this paper self-contained, we include a proof of 8. Our proof is by induction on n. Suppose it has been proved that, for all teplates τ of at ost n 1 letters, the nuber of perutations of n 1 letters that atch τ is correctly given by 8, and let τ be soe teplate of n letters. If in fact the length of τ is < n then the forula 8 gives the sae result as it did when applied to n 1-perutations, which is the correct probability. In the case where the length of τ is n and the rightost letter of τ is Y, every atching perutation σ ust have σn = n. Hence the nuber of atching perutations is n 1!p n 1 τ, where τ consists of the first n 1 letters of τ, which is equal to n!p n τ, proving the result in this case. In the last case, where the length of τ is n and the rightost letter of τ is N, the probability of a perutation atch ust be p n 1 τ 1 1/n, since this case and the preceding one are exhaustive of the possibilities and the preceding one had a probability of p n 1 τ /n. But this agrees with the forula 8 for this case, copleting the proof of the theore. 7.2 Teplates on words Next we include results for words that are analogous to those we found for perutations. A preview of the kind of results that we will get is the following. Suppose F k, τ, x = k 1 fk, τxk, where fk, τ denotes the nuber of words over the alphabet [k] that atch the teplate τ. Then consider the adjunction of one new sybol, let s call it A, at the right end of τ. Then we will show that the new generating function, F k, τa, x can be obtained fro F k, τ, x by applying a certain operator Ω A. That is, F k, τa, x = Ω A F k, τ, x. The operator Ω A will depend only on the letter A that is being adjoined to the teplate τ. Hence, to find the generating function for a coplete teplate τ, we begin with F k,, x = 1/1 x, and we read the teplate τ fro left to right. Corresponding to each letter in τ we apply the appropriate operator Ω. When we have finished scanning the entire teplate the result will be the desired generating function for τ. The letters that we will allow in a teplate τ = τ 1 τ 2... τ l are {S, W, O,, S, O}. Their eanings are that if w = w 1... w l is a word of length l over the alphabet [k] then for w to atch the teplate it ust be that w i is 1. a strongly outstanding value whenever τ i = S, or 10

11 2. a weakly outstanding value whenever τ i = W, or 3. unrestricted whenever τ i =, or 4. not a strongly outstanding value whenever τ i = S, or 5. neither a weakly nor a strongly outstanding value whenever τ i = O. Let s consider what happens to the count of atching words when we adjoin one of these letters to a teplate whose counting function is known. Let fk, τ denote the nuber of words of length l = lengthτ, on the alphabet [k] that atch the teplate τ. 1. If w is one of the words counted by fk, τs, and if we delete its last letter, we obtain one of the words that is counted by fi, τ for soe 1 i < k, and consequently k 1 fk, τs = fi, τ. If F k, τ, x = k 1 fk, τxk, then we have F k, τs, x = x F k, τ, x. 1 x Thus we have found the operator Ω S, and it is defined by Ω S F x = xf x/1 x. This is equation Siilarly, if w is one of the words counted by fk, τw, and if we delete its last letter, we obtain one of the words that is counted by fi, τ for soe 1 i k, and consequently k fk, τw = fi, τ. If F k, τ, x = k 1 fk, τxk, then we have F k, τw, x = F k, τ, x 1 x. Thus we have found the operator Ω W, and it is defined by Ω W F x = F x/1 x. This is

12 Since the arguent in each case is easy and siilar to the above we will siply list the reaining three operators, equations 11, 12, and 13, as follows: Ω F x = x d dx F x Ω S F x = x d dx x F x 1 x Ω O F x = x d dx 1 F x 1 x The successive applications of these operators can be started with the generating function for the epty teplate, F, x = 1 1 x. Thus to find the generating function for soe given teplate τ, we begin with the function 1/1 x, and then we read one letter at a tie fro τ, fro left to right, and apply the appropriate one of the five operators that are defined above. As an exaple, how any words of length 3 over the alphabet [k] atch the teplate τ = S S? This is the coefficient of x k in 1 F S S, x = Ω S Ω Ω S 1 x = x d dx = x + 3x2 1 x 4 x 1 x x d x 1 dx 1 x 1 x The required nuber of words over a k letter alphabet that atch the teplate τ is the coefficient of x k in the above. Since these generating functions will always be of the for P t/1 x r, with P a polynoial, we note for ready reference that [x k ] { j a } jx j 1 x r = j r + k j 1 a j. r 1 In the exaple above we have r = 4, a 1 = 1, a 2 = 3, so k + 2 k + 1 fk, S S =

13 8 Weakly outstanding eleents of words Finally we prove Theore 4. Recall that gn, k, t is the nuber of n-letter words over the alphabet [k] which have exactly t weakly outstanding eleents. Consider just those words w that contain exactly 1 s, < n. If w begins with a block of exactly l 1 s, 0 l then by deleting all of the 1 s in w we find that the reaining word is one with n letters over an alphabet of k 1 letters and it has exactly t l weakly outstanding eleents. Thus we have the recurrence gn, k, t = n 1 =0 n l 1 l When we set G k n, x = t gn, k, txt, we find that G k n, x = n 1 =0 l=0 k n l 1 l gn, k 1, t l + δ t,n. x l G k 1 n, x + x n. n, k 1; G 0 n, x = 0 18 To discuss this recurrence, let = x be the usual forward difference operator on x, i.e., x fx = fx + 1 fx. Then fro the recurrence above we discover that x 1 G 1 n, x = x n ; G 2 n, x = x x n x 1; G 3 n, x = 2 xx n. 2 This leads to the conjecture that G k n, x = k 1 x = t=0 { } x 1 x n k 1 k 1 k 1 1 k 1 t x + t n t x + t 1 k 1 To prove this it would suffice to show that the function G k n, x above satisfies the recurrence 18. If we substitute the conjectured for of G k into the right side of 18 we find that the sus over l and can easily be done, and the identity to be proved now reads as k 2 k 2 1 k t t t=0 x + t 1 k 2 k 1 k 1 = 1 k 1 t x + t n t t=0 13. x + t t + 1 x + t + 1n x n + x n x + t 1 k 1.

14 If we replace the duy index of suation t by t 1 on the left side, the identity to be proved becoes k 1 k 2 x + t 2 x + t 1 1 k t 1 x + t n x n + x n t 1 k 2 t t=1 k 1 k 1 = 1 k 1 t x + t n t t=0 x + t 1 k 1 It is now trivial to check that for 1 t k 1, the coefficient of x + t n on the left side is equal to that coefficient on the right. If we cancel those ters and divide out a factor x n fro what reains, the identity to be proved becoes k 1 k 2 x + t 2 x + t 1 1 k t 1 t 1 k 2 t t=1. x = 1 k 1. k 1 The su that appears above is a special case of Gauss s original 2 F 1 [a, b; c 1] evaluation, and the proof of 7 is coplete. A straightforward calculation now shows that the average nuber of weakly outstanding eleents aong n-letter words over the alphabet [k] is = n n k 1 k + H k 1 + O. k 9 Related results In the literature, the outstanding eleents and values of sequences go by various naes: éléents salients Rényi, left-to-right axia, records, and others; and appear in several different contexts and results, for exaple: It is well known that the probability of obtaining at least r strongly outstanding eleents in a sequence X 1, X 2,..., X n of n independent, identically distributed, continuous rando variables approaches 1 as n. Glick s survey [7], for exaple, contains this result and those that follow in this paragraph and the next. Let Y i = 1 if i is a strongly outstanding eleent of such a sequence, and 0 otherwise. The expected value is E[Y i ] = 1/i and the variance is V [Y i ] = 1/i 1/i 2. The nuber of strongly outstanding eleents in a sequence of continuous iid rando variables is therefore R i = i Y i with expectation E[R i ] = n 1/i and variance V [R i ] = n 1/i n 1/i2. Note n 1/i lnn Euler s constant = , and n 1/i2 π 2 /6 = for n. Let N r denote the r th outstanding eleent in a sequence of n iid continuous rando variables. Then N 1 = 1, E[N r ] = for r 2, and E[N r+1 N r ] = as well. The 14

15 probability P [N 2 = i 2, N 3 = i 3,..., N r = i r ] = 1/i 2 1i i r 1i r for 1 < i 2 < i 3 < < i r. The probability that an iid[ sequence ] of n continuous rando variables has k exactly r strongly outstanding eleents is /r 1! for large n. r Chern and Hwang [3] consider the nuber f n,k of k consecutive records strongly outstanding eleents in a sequence of n iid continuous rando variables. They iprove upon known results for the liiting distribution of f n,2. In particular they show f n,k is asyptotically Poisson for k = 1, 2, and this is not the case for k 3. They give the probability generating function for f n,2, and observe that the distribution of f n,2 is identical to that for the nuber of fixed points j in a rando perutation of [n] for 1 j < n. They give a recurrence for the probability generating function for f n,k, and copute the ean and variance for this nuber. Are there siilar results for discrete distributions? Prodinger [11] considers left-to-right axia in both the strict strongly outstanding and loose weakly outstanding senses for geoetric rando variables. He finds the generating function for the probability that a sequence of n independent geoetric rando variables each with probability pq v 1 of taking the positive integer value v, where q = p 1 has k strict left-to-right axia. This probability is the coefficient of z n y k in F z, y = 1 + yzpqk 1. 1 z1 q k k 1 For loose left-to-right axia the analogous generating function is k 0 1 z1 q k 1 z + zq k 1 py. In this paper, Prodinger also finds the asyptotic expansions for both the expected nubers of strict and loose left-to-right axia, and the variances for these nubers. He does all of the above for unifor rando variables as well. Knopfacher and Prodinger [10] consider the value and position of the r th left-to-right axiu for n geoetric rando variables. The position is the r th strongly outstanding eleent, and the value is that taken by the rando variable in that position again the value v is taken with probability pq v 1, where q = 1 p. For the r th strong left-to-right axiu, p r 1, n log q 1 respectively. q and the asyptotic forulas for value and position are r and 1 p r 1! For the rth weak left-to-right axiu, the value and position are asyptotically rq p 1 r 1! functions. r 1. p log 1 n These results are obtained by first coputing the relevant generating q 15

16 A nuber of additional properties of outstanding eleents of perutations are in Wilf [13]. Key [8] describes the asyptotic behavior of the nuber of records strongly outstanding eleents and weak records weakly outstanding eleents that occur in an iid sequence of integer valued rando variables. When the sequence w 1, w 2,..., w n under consideration is a rando perutation on n letters, then the expected nuber of strongly outstanding elents is the nth haronic nuber H n = The variance is H 2 2 n n = H n 2, where H n 2 = n 2 denotes the nth haronic nuber of the second order. Banderier, Mehlhorn, and Brier [2] show that the average nuber of left-to-right axia strongly outstanding eleents in a partial perutation is logpn+γ+2 1 p p p 2 p 2 O 1 n where γ = is Euler s constant. To obtain a partial perutation, we begin 2 with the sequence 1, 2,..., k and select each eleent with probability p; then take one of the pn! perutations of [pn] uniforily at rando and let it act on the selected eleents, while the nonselected eleents stay in place. Foata and Han [6] investigate the right-to-left lower records of signed perutations. A signed perutation is a word w = w 1 w 2... w n for which the letters w i are positive or negative integers, and w 1 w 2... w n is a perutation of {1, 2,..., n}. A lower record of such a word is a letter w i such that w i < w j for all j with i + 1 j n. The authors consider the signed subword obtained by reading the lower records of w fro left to right; and derive generating functions for signed perutations in ters of signed subwords, nubers of positive and negative letters in signed subwords, and other statistics. 1 + n 16

17 References [1] Louis Cotet, Advanced Cobinatorics, Reidel, [2] Cyril Banderier, Kurt Mehlhorn, and Rene Beier, Soothed analysis of three cobinatorial algoriths, Matheatical Foundations of Coputer Science 2003: 28th International Syposiu, Proceedings, Lecture Notes in Coputer Science 2747, Springer, 2003, [3] Hua-Huai Chern and Hsien-Kuei Hwang, Liit distribution of the nuber of consecutive records, Rando Structures Algoriths 26 no , [4] Julien Cléent, Philippe Flajolet, and Brigitte Vallée, The analysis of hybrid trie structures, Proceedings of the Ninth Annual ACM-SIAM Syposiu on Discrete Algoriths San Francisco, CA, 1998, , ACM, New York, [5] J. N. Darroch, On the distribution of the nuber of successes in independent trials, Ann. Math. Stat., , [6] Doinique Foata and Guo-Niu Han, Signed words and perutations II: The Euler- Mahonian polynoials, Electron. J. Cobin. 11 no /06, Research Paper 22, 18 pp. electronic. [7] Ned Glick, Breaking records and breaking boards, Aerican Matheatical Monthly, , [8] Eric S. Key, On the nuber of recordes in an iid discrete sequence, Journal of Theoretical Probability, 18 no , [9] Richard V. Kadison, Strategies in the secretary proble, Expositiones Matheaticae , [10] Arnold Knopfacher and Helut Prodinger, Cobinatorics of geoetrically distributed rando variables: Value and position of the r th left-to-right axiu. Discrete Math. 226 no , [11] Helut Prodinger, Cobinatorics of geoetrically distributed rando variables: Leftto-right axia. Discrete Matheatics, , [12] A. Rényi, Théorie des éléents saillants d une suite d observations, in Colloquiu Aarhus, 1962,

18 [13] Herbert S. Wilf, On the outstanding eleents of perutations, < wilf>,