Absorption Processes: Models forq-Identities

ADVANCES IN APPLIED MATHEMATICS ARTICLE NO.

18,133]148 Ž1997.

AM960504

Absorption Processes: Models for q-Identities Don Rawlings Mathematics Department, California Polytechnic State Uni¨ ersity, San Luis Obispo, California 93407

Several extensions of Blomqvist’s absorption process are presented. Inherent in some of the associated distributions is a method for establishing q-identities ranging from properties of Gaussian polynomials to product expansions of basic hypergeometric series to extensions of results on Mahonian statistics. One process links the comajor index to Russian roulette. Also given are examples involving the Rogers]Ramanujan identities that demonstrate how q-expressions may be modeled with absorption processes. Q 1997 Academic Press

1. INTRODUCTION Blomqvist’s w4x absorption process may be modified so as to provide a common experimental setting for a wide range of q-identities. The first innovation considered herein is the variation of particle size. A sequence of j dots connected by a line will be referred to as a j-particle. The j-absorption process consists of sequentially propelling jparticles into a chamber l cells long according to the following Bernoulli trials scheme.

To begin, a first j-particle P is placed in the leftmost j cells, one dot per cell. A coin with probability q - 1 of landing tails up is then tossed until Ži. a heads occurs or Žii. P occupies the rightmost j cells and a tails occurs. In case Ži., P is advanced one cell to the right each time tails appears and comes to rest when the heads occurs. In case Žii., P is removed from the chamber. Successive j-particles are similarly propelled into the chamber as 133 0196-8858r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

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if occupied cells had been remo¨ ed. When j exceeds the number of empty cells, subsequent j-particles are partially inserted and immediately removed without a single coin toss. A j-particle that comes to rest in the chamber is deemed an absorption. If not absorbed, a j-particle is said to have escaped. To illustrate, suppose that three 2-particles are propelled into a chamber 7 cells long. If the three Bernoulli sequences are TH, H, and TT, then the final outcome is

The tally of absorptions is 2 and one 2-particle escaped Žnot shown.. The 1-absorption process is the original one of Blomqvist Žalso see w6, 10, 21x.. The probability generating functions associated with the j-absorption process may be expressed in terms of basic hypergeometric series. For an integer n G 1, the q-shifted factorial of a is defined as

Ž a; q . n s Ž 1 y a. Ž 1 y aq . ??? Ž 1 y aq ny 1 . . By convention, Ž a; q . 0 s 1. For integers n, k G 0, the q-binomial coefficient Žalso known as the Gaussian polynomial . is defined by n k

q

s

Ž q n ; qy1 . k Ž q; q . k

.

The basic hypergeometric series r f s with r upper and s lower parameters is r fs

a1 , . . . , a r ; q, z b1 , . . . , bs s

Ž a1 ; q . k Ž a2 ; q . k ??? Ž ar ; q . k k k Ž y1. q Ž 2 . ??? Ž bs ; q . k

Ý kG0 Ž q; q . k Ž b 1 ; q . k

ž

1q syr

z n,

//

where Ž 2k . s k Ž k y 1.r2. In section 2, the following results will be verified. THEOREM 1. Gi¨ en that n G 1 j-particles are propelled into a chamber of length l G 1, let Ab denote the number of absorptions. If 0 - q - 1 and l G jn or l ' y1 mod j, then the probability of k absorptions is Pj, l , n Ab s k 4 s q Ž nyk .Ž lyjkyjq1. Ž q lyjq1 ; qyj . k

n k

qj

.

Ž 1.

MODELS FOR

q-IDENTITIES

135

Moreo¨ er, when l G jn, the probability generating function of Ab is Pj, l , n Ž z . s Ž q lyjq1 ; qyj . n z n 1 f 1

qyjn

q

; q j, lyjnq1

q lq1 z

.

Ž 2.

THEOREM 2 ŽNegative q-binomial distribution.. For a chamber of length l G j and a positi¨ e integer k F lrj, let N denote the number of j-particles required to achie¨ e k absorptions. If 0 - q - 1, then the probability of the nth j-particle being the kth one absorbed is Q j, l , k  N s n4 s q Ž nyk .Ž lyjkq1. Ž q lyjq1 ; qyj . k

ny1 nyk

qj

.

Ž 3.

Moreo¨ er, the probability generating function of N is jk Q j, l , k Ž z . s Ž q lyjq1 ; qyj . k z k 1 f 0 q ; q j , zq lyjkq1 . y

Ž 4.

Theorem 1 for j s 1 is due to Blomqvist w4x and Theorem 2 for j s 1 was discovered by Dunkl w6x. Other modifications of Blomqvist’s absorption process are considered in Sections 5 through 8. Beyond their intrinsic appeal, such distributions afford a method for establishing q-identities. To illustrate, Theorems 1 and 2 are used in Section 3 to give probabilistic proofs of known properties of Gaussian polynomials and of known product expansions for 1 f 0 and 1 f 1. In Section 9, distributions on an absorption ring are used to verify and extend two of MacMahon’s w14x fundamental results concerning the inversion number and the comajor index. One absorption ring distribution incidentally reveals some information about Russian roulette. Absorption processes also serve to model a variety of q-expressions. Included in Sections 5 and 6 are processes for the Rogers]Ramanujan identities. Although not of primary interest here, two asymptotic distributions involving q-analogs of the Bessel function J 0 and the cosine function are derived in Section 4 from Theorem 1. These are referred to respectively as the q-Bessel and Cauchy distributions, the former being discovered by Griffin w9x.

2. PROOF OF THEOREMS 1 AND 2 To prove Ž1., fix n G 1 and suppose that k of the n j-particles are absorbed. Note that, after the Ž i y 1.st absorption occurs where 1 F i F Ž k q 1., the probability associated with the next absorption is Ž1 y q lyjiq1 ..

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Also, the probability of an escape occurring between the Ž i y 1.st and ith absorptions is q lyjiq1 s q lyjkyjq1 q jŽ kq1yi .. As the Ž n y k . escapes may be distributed in any manner whatsoever between the absorptions, it follows that Pj, l , n Ab s k 4 k

s

Ł Ž 1 y q lyjiq1 .

q Ž nyk .Ž lyjkyjq1.

Žqj.

Ý

sŽ m .

,

0Fm 1Fm 2F ??? Fm nykFk

is1

where sŽ m. s m1 q m 2 q ??? qm nyk . Since the sum on the above right is equal to w nk x q j Žsee Theorem 3.1 in w1x., the proof of Ž1. is complete. To verify Ž2., first use the identity w nk x q s Ž q n ; qy1 . nykrŽ q; q . nyk and the k

fact that Ž a; qy1 . k s Ž ay1 ; q . k Žya. k q y Ž 2 . to rewrite formula Ž1. as Pj, l , n Ab s k 4 s q Ž nyk .Ž lyjkyjq1. s

Ž q lyjq1 ; qyj . n Ž q jn ; qyj . nyk Ž q lyjnq1 ; q j . ny k Ž q j ; q j . nyk

Ž q lyjq1 ; qyj . n Ž qyj n ; q j . nyk nyk Ž lq1.Ž nyk .qjŽ ny k . 2 . q Ž y1. Ž q lyjnq1 ; q j . ny k Ž q j ; q j . nyk

Then, n

Pj, l , n Ž z . s

Ý Pj, l , n Absk 4 z k ks0

sŽq

lyjq1

sŽq

lyjq1

yj

;q

yj

;q

ny k

Ž qyjn ; q j . ny k Ž y1. nyk q jŽ 2 . Ý j j lyjnq1 ; q j . nyk ks0 Ž q ; q . ny k Ž q n

.n .n

z

n

z

n

1f1

qyjn q lyjnq1

j

;q ,

q lq1 z

q lq1

nyk

ž / z

.

Formula Ž3. also follows from Ž1.. For a fixed positive integer k F lrj, the probability of the nth j-particle being the kth absorption is clearly Q j, l , k  N s n4 s Pj, l , ny1 Ab s k y 1 4 Ž 1 y q lyjkq1 . s q Ž nyk .Ž lyjkq1. Ž q lyjq1 ; qyj . ky 1 s q Ž nyk .Ž lyjkq1. Ž q lyjq1 ; qyj . k

ny1 ky1

ny1 nyk

qj

qj

Ž 1 y q lyjkq1 .

.

The generating function in Ž4. may then be easily derived from Ž3..

q-IDENTITIES

MODELS FOR

137

3. THE PROBABILISTIC APPROACH TO q-IDENTITIES Absorption distributions may be used to translate facts of probability into q-identities. First, let us consider Blomqvist’s distribution P1, l, n . Suppose k of Ž r q s . 1-particles are absorbed. Evidently, n of the first r 1-particles and Ž k y n . of the remaining s 1-particles are absorbed where 0 F n F k. Thus, k

P1, l , rqs Ab s k 4 s

Ý P1 , l , r  Ab s n 4 P1, ly n , s Ab s k y ¨ 4 .

¨ s0

Replacing the probabilities above by the right-hand side of Ž1. and canceling like terms gives the q-analog of the Chu]Vandermonde identity w1, p. 37x: k

rqs k

q

s

Ý q Ž ry n .Ž ky n .

ns0

r n

q

s kyn

. q

As an application of Dunkl’s distribution Q1, l, k , fix k G 1 and let Ak Ž n. denote the event that the nth 1-particle is the kth absorption. For Ž k y 1. F m - n, the conditional probability of Aky 1Ž m. occurring given that A k Ž n. occurs may be computed with the aid of Ž3.: Prob  A ky 1 Ž m . < A k Ž n . 4 s s

Prob  Aky 1 Ž m . n Ak Ž n . 4 Prob  AkŽ n. 4 Q1, l , ky1 N s m4 Q1, lykq1, 1 N s n y m4 Q1, l , k  N s n4

s q my kq1

my1 mykq1

ny1 nyk

q

y1

. q

1  Ž .< Ž .4 Combining the observation that Ý ny ms ky1 Prob A ky1 m A k n s 1 with the above result implies that

ny1

q my kq1

Ý msky1

my1 mykq1

q

s

ny1 nyk

. q

A few cosmetic changes then reveal a standard property w1, p. 37x of the Gaussian polynomials: r

Ý qn

¨ s0

nqk n

q

s

rqkq1 kq1

. q

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Theorems 1 and 2 also imply product expansions for 1 f 1 and 1 f 0 . From the trivial observation that 1 s Pj, l, nŽ1., we have qyj n

1f1

q

lyjnq1

; q j , q lq1 s

1

Žq

lyjq1

; qyj . n

,

Ž 5.

where 0 - q - 1. Similarly, Ž4. implies 1 q jk ; q j , q lyjkq1 s . lyjq1 y ; qyj . k Žq

1f0

More general results for 1 f 1 and 1 f 0 may be found in w8, p. 236x. Evident in Dunkl’s w6x remark that the identity 1 s Q1, l, j Ž1. ‘‘expresses the known ŽHeine. sum of the 1 f 0 series,’’ the probabilistic method provides an experimental setting for establising q-identities. Further examples are given in w18x and in sections 7, 8, and 9. A probabilistic derivation of a formula for Ramanujan’s 1 c 1 sum was given by Kadell w11x. For later reference, two cases of the expansion for 1 f 1 are singled out. Setting Ž j, l . s Ž1, n. and letting n ª ` in Ž5. yields the identity qk

Ý kG0

2

Ž q; q.

2 k

s

1

, Ł i iG1 Ž 1 y q .

Ž 6.

which has been attributed to Euler w1, p. 20x. Selecting Ž j, l . s Ž2, 2 n. and then allowing n ª ` in Ž5. leads to an identity due to Cauchy w1, p. 20x: q2k

yk

2

1

sŁ . Ý 2 iy1 . iG1 Ž 1 y q kG0 Ž q; q . 2 k

Ž 7.

4. THE q-BESSEL AND CAUCHY DISTRIBUTIONS Before further modifying Blomqvist’s absorption process, a pause is taken to extract two asymptotic distributions from Theorem 1. For the j-absorption process, let Es denote the number of escapes. For Ž j, l . s Ž1, n., define B Es s k 4 to be the probability of having k escapes as n ª `. From Ž1., B Es s k 4 s lim P1, n , n Ab s n y k 4 s lim q k nª`

s

qk

Ž q n ; qy1 . nyk

2

nª`

2

Ž q; q. k

n

lim

Ł

nª` iskq1

Ž1 y q i . s

qk

n k

2

Ž q; q .

Ł Ž1 y q i . .

2 k iG1

q

MODELS FOR

q-IDENTITIES

139

Making use of the definition J 0, q Ž z . s

Ý Ž y1. kG0

qk

k

2

2

Ž q; q . k

2k

Ž zr2.

given by Jackson w8, p. 25x for one of the q-analogs of the Bessel function J 0 Ž z . and also in view of Ž6., it follows that 2

2

B Es s k 4 s

q k r Ž q; q . k J 0, q Ž 2'y 1 .

.

Ž 8.

Note that B Es s 04 s 1rJ0, q Ž2'y 1 .. Implicit in Blomqvist w4x, Ž8. was rediscovered by Griffin w9x and used to prove Ž6.. The connection with J 0 perhaps justifies referring to B Es s ?4 as the q-Bessel distribution. For the second distribution, let

Ž y1. q 2 k yk z 2 k . Ž q; q . 2 k k

cos q z s

Ý kG0

2

Note that cos q Ž1 y q . z ª cos z as q ª 1y. Other q-cosine functions have been considered by Jackson and Hahn w8, p. 23x. From Ž7., we have cos q Ž 'y 1 . s

1

. Ł 2 iy1 . iG1 Ž 1 y q

For the case Ž j, l . s Ž2, 2 n. of the j-absorption process, let C Es s k 4 be the asymptotic probability of having k escapes. By Ž1., C  Es s k 4 s lim P2, 2 n , n Ab s n y k 4 s nª`

q2k

yk

2

rŽ q ; q . 2 k

cos q Ž 'y 1 .

.

Note that C Es s 04 s 1rcos q Ž'y 1 .. In view of Ž7., C Es s ?4 . may be reasonably referred to as Cauchy’s distribution. A ‘‘modular’’ analog of C Es s ?4 is given in Theorem 5 of Section 8. Bernoulli schemes for the reciprocals of a q-analog of the number e and of a q-analog of the Riemann-zeta function are presented in w17, 18x. Although not done so here, they may be easily reformulated in the context of the absorption ring considered in Section 7.

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5. REBOUND ABSORPTION PROCESSES: MODELING q-SERIES In a relatively straightforward manner, absorption processes may be used to model many q-series and q-products. In this section and the next, processes are described for both sides of the identity qk

2

1

sŁ Ý 5iy4 .Ž 1 y q 5iy1 . iG1 Ž 1 y q kG0 Ž q ; q . k

Ž 9.

of Rogers and Ramanunjan w1, p. 104x. The rebound absorption process involves sequentially propelling 1-particles into a chamber, as in Section 1, with the following modifications. Upon reaching the rightmost cell, the ‘‘active’’ 1-particle P immediately reverses direction. If P returns to the leftmost empty cell and a tails is tossed, it is then removed. Furthermore, a new cell is adjoined to the right of the chamber in the event that P ever occupies the rightmost cell during the course of its run. For instance, suppose two 1-particles are sent into an empty chamber of length 3. If the sequences TH and TTT are tossed, then the result is

where one 1-particle escapes and the chamber length is increased to 4. Let Ts denote the number of tails required of a newly inserted 1-particle to reach the rightmost cell Žthat is, Ts is the number of empty cells minus 1.. The asymptotic probability of Ts being equal to k, denoted by R Ts s k 4 , may be computed using the theory of Markov chains. Propelling the nth 1-particle is to be viewed as the nth step in the chain. Considering the number of tails required to reach the rightmost cell as the state, let pk, m be the transition probability of moving from state k to state m in one step. Since pk, m s 0 whenever < k y m < ) 1, the chain is a birth-and-death process. Note that pk, ky1 s Ž1 y q k . and pk, kq1 s q 2 kq1. From the theory of Markov chains Že.g., see w5, p. 283x., it follows that p 0, 1 p1, 2 ??? pky1 , krp1, 0 p 2, 1 ??? pk , ky1 R Ts s k 4 s Ý kG0 p0, 1 p1, 2 ??? pky1, krp1, 0 p2, 1 ??? pk , ky1 2

s

q k rŽ q; q. k 2

Ý kG0 q k r Ž q; q . k 2

.

By Ž9., R Ts s k 4 s Ž q k rŽ q; q . k . P iG 1Ž1 y q 5iy4 .Ž1 y q 5iy1 ..

MODELS FOR

q-IDENTITIES

141

A minor alteration of the rebound absorption process leads to a measure involving the identity q kŽ kq1.

1

sŁ , Ý 5iy3 .Ž 1 y q 5iy2 . iG1 Ž 1 y q kG0 Ž q; q . k also of Rogers and Ramanujan w1, p. 104x: To wit, a direction change in the rightmost cell requires a tails Ži.e., rebound is not immediate as time is allotted for ‘‘turning around’’.. The modified chain is still a birth-and-death process. The relevant transition probabilities are pUk, ky1 s Ž1 y q k . and pUk, kq1 s q 2 kq2 . Letting S Ts s k 4 be the stationary probability of k tails being required to reach the rightmost cell, we see that S  Ts s k 4 s

s

q kŽ kq1.r Ž q; q . k

Ý kG0 q kŽ kq1.r Ž q; q . k q kŽ kq1.

Ž q; q . k

Ł Ž 1 y q 5iy3 .Ž 1 y q 5iy2 . .

iG1

The Rogers]Ramanujan identities have appeared before in a probabilistic context, namely, the hard hexagon model developed by Baxter w2x. Similar absorption models may be given for the Heine]Euler distributions first considered by Benkherouf and Bather w3x Žin connection with oil exploration. and further investigated by Kemp w12, 13x.

6. ALTERNATELY SIZED PARTICLES: MODELING q-PRODUCTS The q-product in Ž9. arises from a natural variation of the j-absorption process: Particles of different sizes are to be alternately sent into the chamber. Beginning with a 2-particle, suppose that n 2-particles and n 3-particles are alternately propelled into a chamber of length l as in Section 1. Let R l, 2 n Ab s k 4 denote the probability that k absorptions occur. For l s 5n, the probability of all 2 n particles being absorbed is clearly n

R 5n , 2 n Ab s 2 n4 s R 5n , 2 n Es s 0 4 s

Ł Ž1 y q 5iy4 .Ž 1 y q 5iy1 . .

is1

Thus lim n ª` R 5n, 2 n Es s 04 is the reciprocal of the right-hand side of Ž9.. The distribution R l, 2 n Ab s ?4 may be recursively determined. We first extend the definition of R l, m Ab s ?4 to include odd m. Beginning with a

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DON RAWLINGS

3-particle, suppose that n 3-particles and Ž n y 1. 2-particles are alternately propelled into the chamber. We let R l, 2 ny1 Ab s k 4 be the probability of having k absorptions. In regards to R l, 2 n Ab s k 4 , the first 2-particle is either absorbed or escapes. Thus, R l , 2 n Ab s k 4 s Ž 1 y q ly1 . R ly2, 2 ny1 Ab s k y 1 4 q q ly1 R l , 2 ny1 Ab s k 4 holds for l G 2 and n G 1. Similarly, R l , 2 ny1 Ab s k 4 s Ž 1 y q ly2 . R ly3 , 2 ny2  Ab s k y 1 4 q q ly2 R l , 2 ny2  Ab s k 4 is true for l G 3 and n G 1. The initial conditions are R l, 0 Ab s 04 s 1, R l, m Ab - 04 s 0, R 2, 2 ny1 Ab s 04 s q ny 1, R 2, 2 ny1 Ab s 14 s Ž1 y q ny 1 ., and R l, m Ab s k 4 s 0 whenever k ) min lr2, m4 . 7. THE ABSORPTION RING: MAHONIAN DISTRIBUTIONS AND RUSSIAN ROULETTE To further illustrate the probabilistic approach to q-identities, two processes from w16x are recast here in the context of an absorption ring. Generalizations of the two are sketched in Section 8. These processes are then used in Section 9 to prove and extend MacMahon’s w14x results on the inversion number and the comajor index. To set the stage, let n1 , n 2 , . . . , n l denote a sequence of non-negative integers. Put n s n1 q n 2 q ??? q n l . The set of functions mapping  1, 2, . . . , n4 to  1, 2, . . . , l 4 that take on the value i exactly n i times will be denoted by M w1n1 2 n 2 ??? l n l x. Such a function f will be expressed as a list of its range elements: f s f Ž1. f Ž2. ??? f Ž n.. The descent set of f g M w1n1 2 n 2 ??? l n l x, denoted by Des f, consists of the indices k, 1 F k F Ž n y 1., such that f Ž k . ) f Ž k q 1.. The in¨ ersion number and comajor index of f are inv f s <  Ž k , m . : k - m, f Ž k . ) f Ž m . 4 <

and

comaj f s

Ý Ž n y k. . kgDes f

For example, the function f s 2 2 1 2 1 g M w12 2 3 x has Des f s  2, 44 , inv f s 5, and comaj f s 4. Also needed is the q-multinomial coefficent: n n1 n 2 ??? n l

q

s

Ž q; q . n . Ž q; q . n1 Ž q; q . n 2 ??? Ž q ; q . n l

MODELS FOR

q-IDENTITIES

143

An absorption ring of type ª n s Ž n1 , n 2 , . . . , n l . is constructed by first stretching a chamber having l cells, consecutively numbered from 1 to l, into a ring so that the 1st and lth cells are joined at their outer edges Žmaking escape impossible.. Cell i is then partitioned into n i subcells, thereby expanding its absorption capacity to n i 1-particles. The in¨ ersion number process of type ª n s Ž n1 , n 2 , . . . , n l . consists of sequentially propelling n s Ž n1 q n 2 q ??? qn l . 1-particles from the first empty subcell. For each 1-particle, a coin with probability q - 1 of landing tails up is tossed until heads occurs. The active 1-particle advances to the next empty subcell each time tails occurs and comes to rest Žis absorbed. when heads appears. For expedience, the Bernoulli sequences of the n 1-particles will be strung together into a single sequence. As a partial illustration, suppose that the two 1-particles are accordingly propelled into an absorption ring of type ª n s Ž3, 1, 2.. If the string TTTHTTTTTTH occurs, then the result is

The outcome of the inversion number process of type ª n s Ž n1 , n 2 , . . . , n l . may be partially encoded by a function f g M w1n1 2 n 2 ??? l n l x; just set f Ž i . equal to the number of the cell Žcontaining the subcell . in which the ith 1-particle comes to rest. For instance, if l s 3 and ª n s Ž2, 1, 2., then the function associated with the sequence HTTHTHTTHH is f s 1 3 2 1 3. Let mbs f denote the Bernoulli sequence of minimal length that generates f. In general, inv f is equal to the number of tails occurring in mbs f. The following result was established in w16x. As the proof given therein contains a minor flaw, a corrected version is inlcuded below. For permutations Žª n s Ž1, 1, . . . , 1.., Ž10. was first obtained by Rawlings and Treadway w20x.

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DON RAWLINGS

THEOREM 3. For the in¨ ersion number process of type ª ns Ž n1 , n 2 , . . . , n l ., let F Ž i . denote the number of the cell in which the ith 1-particle is absorbed. For 0 F q F 1, the probability of f g M w1n1 2 n 2 ??? l n l x occurring is y1

n n1 n 2 . . . n l

inv f Iª nFsf4 sq

.

Ž 10 .

q

Proof. Clearly, Ž10. is true for l s 1. For l ) 1 and f g M w1n1 2 n 2 ??? l n l x, the first heads to appear in a Bernoulli sequence that generates f occurs either Ži. after the nth toss or Žii. on or before the nth toss. In case Ži., the process may be viewed as being restarted on the Ž n q 1.st toss. In case Žii., if n denotes the toss on which the first heads occurs, then n1 q n 2 q ??? qn f Ž1.y1 - n F n1 q n 2 q ??? qn f Ž1. .

Ž 11 .

Let g s f Ž2. f Ž3. ??? f Ž n. and say that f Ž1. s i. We then have n Iª n  F s f 4 s q Iª nFsf4 q

Ý q ny1 Ž 1 y q . In *  F s g 4 , ª

Ž 12 .

n

where ª n* s Ž n1 , . . . , n i y 1, . . . , n l . and the sum is over all n satisfying Ž11.. Since mbs f s TT . . . TH mbs g, where the sequence immediately preceding mbs g contains Ž n1 q n 2 q ??? qn iy1 . tails, it is clear that inv f s the number of tails in mbs f s n1 q n 2 q ??? qn iy1 q inv g . Theorem 3 then follows from Ž12. by induction. The second process considered in this section, referred to as the comajor index process of type ª n s Ž n1 , n 2 , . . . , n l ., proceeds as does the inversion number process with one exception: For 2 F i F n, the ith 1-particle is inserted into the first empty subcell measured from the beginning of the cell in which the Ž i y 1.st 1-particle has been absorbed. The proof of Ž10. may be modified to establish the first half of THEOREM 4. For the comajor index process of type ª n s Ž n1 , n 2 , . . . , n l ., let F Ž i . denote the number of the cell in which the ith 1-particle is absorbed. For 0 F q - 1, the probability of f g M w1n1 2 n 2 ??? l n l x occurring is comaj f Cª nFsf4 sq

n n1 n 2 . . . n l

y1

.

Ž 13 .

q

Moreo¨ er, if des f denotes the number of descents in f, then the probability generating function of des relati¨ e to the measure in Ž13. is n Cª n Ž z . s Ž z ; q . nq 1 n n ??? n 1 2 l

y1 q

l

Ý kG0

zk Ł is1

k q ni ni

. q

Ž 14 .

MODELS FOR

q-IDENTITIES

145

A key point for proving Ž13. and for Section 9 is that comaj f is equal to the number of tails in the shortest Bernoulli sequence that generates f relative to the comajor index process. Formula Ž14. follows from MacMahon’s w14, Vol. 2, p. 211x solution to the q-Simon Newcomb problem. For permutations, Ž13. was first deduced in other terms by Moritz and Williams w15x; the connection with the comajor index was made by Rawlings and Treadway w20x. As the comajor index process for ª n s Ž1, 1, . . . , 1. is clearly equivalent to a game of Russian roulette with n participants Ževen assuming the winner does not take another turn after the runner-up’s death., some macabre information may be extracted from Theorem 4. Let Sn s M w11 2 1 . . . n1 xŽi.e., the set of permutations of  1, 2, . . . , n4.. For ª n s Ž1, 1, . . . , 1., Ž13. is the probability that a given ‘‘order of death’’ occurs; letting w n x s Ž1 q q .Ž1 q comaj a w x  4 q q q 2 . ??? Ž1 q q q ??? qq ny 1 ., we have Cª r n q ! for n Fsa sq all a g Sn . Although computationally impractical for large n, the probability of participant k winning at Russian roulette is therefore Prob  k wins 4 s

1

q comaj a , w nx q! Ý a

where the sum is over all a g Sn satisfying a Ž n. s k. From Ž14., a measure of the courage needed to win may be computed. As des a is equal to the minimum number of times that the comajor index process X Ž . must ‘‘wrap around the ring’’ in order to generate a , Cª n 1 gives the minimum number of times the winner may expect ot pull the trigger. Incidentally, the undertaker’s waiting time is just the negative binomial distribution of order Ž n y 1..

8. EXPERIMENTING WITH THE ABSORPTION RING The absorption ring provides the context for a range of experimentation. Some possibilities include sending in particles of different sizes, varying the initial placement of particles, and stipulating that each tails advances the active particle by c ) 1 cells Žwhich may be viewed as a variation on the problem of Josephus.. However, experimentation here will be limited to the introduction of 2-particles into an absorption ring. Some preliminaries are needed. Relative to a sequence i1 - i 2 - ??? i k of integers, Ž i k , i1 . and pairs of the form Ž i m , i mq1 . are said to be ring consecuti¨ e. A permutation g s g Ž1.g Ž2. ??? g Ž2 n. g S2 n is said to be pairwise ring consecuti¨ e if, for 1 F i F k, the pair Žg Ž2 k y 1.,g Ž2 k .. is ring consecutive relative to the sequence obtained by deleting g Ž1.,g Ž2., . . . , g Ž2 k y 2. from 1 - 2 - ??? - 2 n. The set of such permu-

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tations is denoted by P R C2 n . To illustrate, g s 2 3 8 1 7 4 5 6 g P R C8 . The pairwise in¨ ersion number of g g P R C2 n , denoted by pinv g , is defined to be the number of pairs Ž k, m. with 1 F k F n and 2 k F m F 2 n that satisfy the condition maxg Ž2 k y 1.,g Ž2 k .4 ) g Ž m.. Observe that pinv 2 3 8 1 7 4 5 6 s 9. As the first experiment ŽExp Y ., let us consider sequentially propelling n 2-particles into a ring of 2 n cells consecutively numbered from 1 to 2 n. A first 2-particle P is placed in cells 1 and 2, one dot per cell. A coin with probability of q - 1 of landing tails up is tossed until heads occurs. Particle P advances a cell each time tails occurs and comes to rest when heads appears. Subsequently, the remaining Ž n y 1. 2-particles are propelled around the ring as if any occupied cells had been removed. Each outcome of Exp Y corresponds in a natural way to a permutation g g P R C2 n : For 1 F i F n, let g Ž2 i y 1. and g Ž2 i ., respectively, be the cell numbers where the trailing and leading dots of the ith 2-particle come to rest. For instance, if the ring has 8 cells and the sequence HTTTTTHTHH is tossed, then g s 1 2 8 3 5 6 4 7. By observing that pinv g is equal to the number of tails in the shortest Bernoulli sequence generating g , a minor adaptation of the proof of Ž10. leads to the following modular analog of Cauchy’s distribution. THEOREM 5. For Exp Y, let G Ž2 i y 1. and G Ž2 i ., respecti¨ ely, denote the cell number in which the trailing and leading dots of the ith 2-particle come to rest. For 0 F q F 1, the probability of g g P R C2 n occurring is Y2 n G s g 4 s

q pinv g

w 2 x q w 4 x q ??? w 2 n x q

where, for a positi¨ e integer i, w i x q s 1 q q q ??? q q iy1. As a second experiment ŽExp Z ., let us repeat Exp Y with one modification: The ith 2-particle for 2 F i F n is to be inserted in the first two empty cells in front Žrelative to the orientation of motion. of where the Ž i y 1.st 2-particle has been absorbed. As comaj g equals the number of tails in the shortest Bernoulli sequence generating g , the proof of the following is routine. THEOREM 6. For Exp z, let G Ž2 i y 1. and G Ž2 i ., respecti¨ ely denote the cell numbers in which the trailing and leading dots of the ith 2-particle come to rest. For 0 F q - 1, the probability of a gi¨ en g g P R C2 n occurring is Z2 n G s g 4 s

q comaj g

w 2 x q w 4 x q ??? w 2 n x q

.

MODELS FOR

q-IDENTITIES

147

9. MORE EXAMPLES OF THE PROBABILISTIC METHOD  4  4 As the measures Iª n F s ? and Cª n F s ? of Section 7 trivially satisfy  4  4 Ý f Iª n F s f s 1 s Ý f Cª n F s f , Theorems 3 and 4 imply the advertised results of MacMahon w14x; namely,

Ý

fg M w1n1 2 n 2 ??? l n l x

n q inv f s n n ??? n 1 2 l

q

s

Ý

q comaj f .

fg M w1n1 2 n 2 ??? l n l x

In the permutation case ª n s Ž1, 1, . . . , 1., a bijection c :Sn ª S n satisfying inv a s comaj c Ž a .

for all a g S n

may be easily extracted from the processes of Section 7. Just define c Ž a . to be the result of Ži. first encoding a g Sn as its Bernoulli sequence of minimal length mbs a under the rules of the inversion number process and Žii. then decoding mbs a according to the comajor index process. For instance,

a s 4 3 1 5 2 g S5 ª TTTHTTHHTHH ª 4 2 3 1 5 s c Ž a . g S5 . As may be verified, inv 4 3 1 5 2 s 6 s comaj 4 2 3 1 5. Foata w7x gave the first such bijection for general ª n s Ž n1 ,n 2 , . . . , n l .. Analogous information may be similarly obtained from the distributions of Section 8. In view of Theorems 5 and 6, we see that

Ý

q pinv g s w 2 x q w 4 x q ??? w 2 n x q s

gg P R C2 n

Ý

q comaj g .

gg P R C2 n

Also, for a g P R C2 n with minimal Bernoulli sequence B relative to Exp Y, let LŽ a . be the permutation in P R C2 n having B as its minimal sequence relative to Exp Z. The map L: P R C2 n ª P R C2 n is a bijection satisfying pinv a s comaj L Ž a .

for all a g P R C2 n .

As an example,

a s 3 4 2 5 8 1 6 7 ª TTHTHTTTHH ª 3 4 6 7 5 8 1 2 s L Ž a . and pinv 3 4 2 5 8 1 6 7 s 6 s comaj 3 4 6 7 5 8 1 2. Further consideration of statistics on ring consecutive permutations is given in w19x.

REFERENCES 1. G. E. Andrews, ‘‘The Theory of Partitions,’’ Addison]Wesley, Reading, MA, 1976. 2. R. J. Baxter, Rogers]Ramanujan identities in the hard hexagon model, J. Statist. Phys. 26Ž1981., 427]452.

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3. L. Benkerhouf and J. A. Bather, Oil exploration: Sequential decisions in the face of uncertainty, J. Appl. Probab. 25 Ž1988., 529]543. 4. N. Blomqvist, On an exhaustion process, Skandina¨ isk Aktuarietidskrift 35 Ž1952., 201]210. 5. K. L. Chung, ‘‘Elementary Probability Theory with Stochastic Processes,’’ SpringerVerlag, New York, 1979. 6. C. F. Dunkl, The absorption distribution and the q-binomial theorem, Comm. Statist. Theory Methods 10 Ž1981., 1915]1920. 7. D. Foata, On the Netto inversion number of a sequence, Proc. Amer. Math. Soc. 19 Ž1968., 236]240. 8. G. Gasper and M. Rahman, ‘‘Basic Hypergeometric Series,’’ Cambridge Univ. Press, Cambridge, UK, 1990. 9. K. Griffin, The q-derangement problem relative to the inversion number, Cal Poly State University, San Luis Obispo, California preprint Ž1996.. 10. N. L. Johnson, S. Kotz, and A. W. Kemp, ‘‘Univariate Discrete Distributions,’’ Wiley, New York, 1992. 11. K. W. J. Kadell, A probabilistic proof of Ramanujan’s 1 c 1 sum, SIAM J. Math. Anal. 6 Ž1987., 1539]1548. 12. A. W. Kemp, Steady-state Markov chain models for the Heine and Euler distributions, J. Appl. Probab. 21 Ž1992., 571]588. 13. A. W. Kemp, Heine]Euler extensions of the Poisson distribution, Comm. Statist. Theory Methods 21 Ž1992., 571]588. 14. P. A. MacMahon, ‘‘Combinatory Analysis,’’ Cambridge Univ. Press, LondonrNew York, 1915 Žreprinted by Chelsea, New York, 1960.. 15. R. H. Moritz and R. C. Williams, A coin-tossing problem and some related combinatorics, Math. Mag. 61 Ž1988., 24]29. 16. D. P. Rawlings, Bernoulli trials and permutation statistics, Internat. J. Math. Math. Sci. 15 Ž1992., 291]312. 17. D. P. Rawlings, Bernoulli trials and number theory, Amer. Math. Monthly 101 Ž1994., 948]952. 18. D. P. Rawlings, A probabilistic approach to some of Euler’s number theoretic identities, Trans. Amer. Math. Soc., to appear. 19. D. P. Rawlings, A generalized Mahonian Statistic on absorption ring mappings, J. Combin. Theory Ser. A, to appear. 20. D. P. Rawlings and J. A. Treadway, Bernoullli trials and Mahonian statistics: A tale of two q’s, Math. Mag. 67 Ž1994., 345]354. 21. S. Zacks and D. Goldfarb, Survival probabilities in crossing a field containing absorption points, Na¨ al Res. Logist. 13 Ž1966., 35]48.