More Statistics on Permutation Pairs �Jean-Marc FedouLaboratoire Bordelais de Recherches en InformatiqueUniversit�e Bordeaux I33405 Talence, FranceDon Rawlings yMathematics DepartmentCalifornia Polytehnic State UniversitySan Luis Obispo, Ca. 93407Submitted: June 30, 1994; Accepted: October 21, 1994AbstractTwo inversion formulas for enumerating words in the free monoid by�-adjacencies are applied in counting pairs of permutations by vari-ous statistics. The generating functions obtained involve re�nementsof bibasic Bessel functions. We further extend the results to �nitesequences of permutations.�This work is partially supported by EC grant CHRX-CT93-0400 and PRC Maths-InfoyFinancial support provided by LaBRI, Universit�e Bordeaux I0

the electronic journal of combinatorics 1 (1994), #R11 11 IntroductionThe study of statistics on permutation pairs was initiated by Carlitz, Scoville,and Vaughan [4]. Stanley [18] q-extended their work to �nite sequences ofpermutations. In [6], we exploited the recursive technique of Carlitz et. al.to obtain some additional re�nements. We also discussed numerous relateddistributions. Our purpose here is to further extend the study of statistics on�nite permutation sequences. Our method is based on the theory of inversionpresented in [7]. For clarity, we primarily focus on permutation pairs.Let Sn denote the symmetric group on f1; 2; : : : ; ng. For a permutation� = �(1)�(2) � � ��(n) 2 Sn, the descent and rise sets are de�ned asDes� = fi : 1 � i � n� 1; �(i) > �(i+ 1)g ;Ris� = fi : 1 � i � n� 1; �(i) < �(i+ 1)g :These sets are of course complementary relative to f1; 2; : : : ; n � 1g. Thedescent and rise numbers of � are respectively de�ned to be the cardinalitiesof Des� and Ris�, that is,des � = jDes�j and ris� = jRis�j :Furthermore, letmaj� = Xk2Des� k ; comaj� = Xk2Des�(n� k) ;rin� = Xk2Ris� k ; corin � = Xk2Ris�(n � k) :The statistics in the �rst column were originally referred to as the greater andlesser indices by Major MacMahon [16]. Many authors have since adoptedthe term major index for the former. Being the sum of the rise indices, wewill refer to rin� as the rise index. The statistics of the second columnwill respectively be called the comajor and corise indices. Since Des � andRis� are complementary, des� + ris� = n � 1, maj� + rin� = �n2�, andcomaj� + corin� = �n2�.We also consider the number of common iddescents of a permutation pair;for (�; �) in the cartesian product S2n = Sn � Sn, letiddes(�; �) = jDes��1 \ Des��1jwhere ��1 denotes the inverse of �.

the electronic journal of combinatorics 1 (1994), #R11 2Our initial results involve the �rst two terms of a sequence fJ (i;j)� g��0 ofre�ned bibasic Bessel functions. For a positive integer n, let(a; q)n = (1 � a)(1� aq) � � � (1 � aqn�1) :By convention, (a; q)0 = 1. The q-binomial coe�cient (or Gaussian polyno-mial ) is de�ned to be " nk #q = (q; q)n(q; q)k(q; q)n�kwhen 0 � k � n and to be 0 when k > n. The function J (i;j)� is de�ned asJ (i;j)� (z; q; p) = Xn�0(�1)nq(n+�2 ) " i+ 1n+ � #q " j + nn #p zn+� :A few properties of fJ (i;j)� g��0 are worth immediate remark. First, J (i�1;j)0is a Hadamard product of the two series appearing in the q-binomial theoremand q-binomial series ([1, p. 36]):q-Binomial Theorem (t; q)i = Pn�0(�1)nq(n2) " in #q tn :q-Binomial Series For jqj; jtj < 1, 1=(t; q)j+1 = Pn�0 " j + nn #q tn :Also note that J (i�1;j)0 (z; q; 0) = (z; q)i.Second, for jqj; jpj < 1, special cases of the functionJ�(z; q; p) = limi;j!1 J (i;j)� (z; q; p) = Xn�0 (�1)nq(n+�2 )zn+�(q; q)n+�(p; p)narise in a variety of other contexts. As demonstrated by Delest and Fedou [5],the coe�cient of qmzn in the expansion of the quotient J1(zq; q; q)=J0(zq; q; q)is equal to the number of skew Ferrers' diagrams (also known as parallelo-gram polyominoes) having n columns and area m. Also, J0(�z; q; 0) is thesecond q-analogue of the exponential function [11, p. 9], often denoted byEq(z). Finally, omitting q(n+�2 ) and replacing zn+� in the series J�(z; q; q) by

the electronic journal of combinatorics 1 (1994), #R11 3(z=2)2n+� gives one of the sequences of q-Bessel (or basic Bessel) functionsoriginally studied by Jackson [15] and further explored by Ismail [14]. Thus,J (i;j)� (z; q; p) is indeed a re�ned bibasic Bessel function.Several theorems could be used to demonstrate our method. Our �rst ob-jective will be on determining the distribution of (iddes; des; comaj; ris; corin)over unrestricted pairs in S2n and over restricted pairs (�; �) 2 S2n with�(1) = n. In sections 3 through 7, we proveTheorem 1 The generating functions for the sequencesAn(t; x; y; q; p) = X(�;�)2S2n tiddes(�;�)xdes�qcomaj�yris�pcorin� ;A1n(t; x; y; q; p) = Xf(�;�)2S2n : �(1)=ng tiddes(�;�)xdes�qcomaj�yris�pcorin� (1)are Xn�0 An(t; x; y; q; p)zn(x; q)n+1(y; p)n+1 = Xi;j�0xiyj 1� tJ (i;j)0 (z(1� t); q; p)� t ; (2)Xn�0 A1n+1(t; x; y; q; p)zn+1(x; q)n+2(y; p)n+1 = Xi;j�0xiyj J (i;j)1 (z(1 � t); q; p)J (i;j)0 (z(1� t); q; p)� t : (3)For comparison with previously obtained results on permutations and per-mutation pairs, a number of corollaries are presented in the next section.Other closely related �ve-variate distributions on permutation pairs areconsidered in section 8. Speci�cally, we give the generating functions forthe distribution of (iddes; des; comaj; ris; corin) over pairs (�; �) 2 S2n satisfy-ing �(1) = n and for the distributions of (iddes; des; comaj;des; comaj) and(iddes; ris; corin; ris; corin) for unrestricted and restricted pairs in S2n. Thecorresponding re�ned bibasic Bessel functions are variations on J (i;j)� . Insection 9, we give two theorems for �nite sequences of permutations whichcontain the ones for permutation pairs as special cases.2 Selected CorollariesMultiplying (2) and (3) through by (1� x)(1� y) and then taking the limitas x; y! 1� leads respectively to the following corollaries:

the electronic journal of combinatorics 1 (1994), #R11 4Corollary 1 The distribution of (iddes; comaj; corin) on S2n is generated byXn�0 An(t; 1; 1; q; p)zn(q; q)n(p; p)n = 1 � tJ0(z(1� t); q; p)� t :Corollary 2 The distribution of (iddes; comaj; corin) on pairs (�; �) 2 S2n+1satisfying the condition �(1) = n+ 1 is generated byXn�0 A1n+1(t; 1; 1; q; p)zn+1(q; q)n+1(p; p)n = J1(z(1 � t); q; p)J0(z(1� t); q; p)� t :These corollaries are equivalent to special cases of Theorems 2 and 4 in [6].Several equivalent distributions are discussed in section 4 of [6]. Corollary 1is essentially due to Stanley [18].Further replacing z by z(1�q)(1�p) in Corollary 1 and letting q; p! 1�gives the initial result on permutation pairs due to Carlitz et al. [4]:Corollary 3 The distribution of iddes over S2n is generated byXn�0 An(t; 1; 1; 1; 1)znn!n! = 1� tPn�0(�1)nzn=n!n! � t :By appropriately selecting the values of various parameters, it is alsopossible to obtain generating functions for the analogues of the Eulerianpolynomials of Carlitz [2, 3] and of Stanley [18] respectively de�ned byCn(y; p) = X�2Sn yris�prin� and Sn(t; q) = X�2Sn tdes�qinv�where inv� denotes the number of inversions of �, that is, the number of pairs(i; j) such that 1 � i < j � n and �(i) > �(j). The bijective techniques ofFoata [8] may be easily adapted to show thatCn(y; p) = X�2Sn yris�pcorin� and Sn(t; q) = X�2Sn tides�qcomaj�where ides� = des��1. When x = 0, the only pairs contributing non-zeroweight in (1) are of the form (12 : : : n; �). Thus, An(1; 0; y; 0; p) = Cn(y; p).Similarly, An(t; 1; 0; q; 0) = Sn(t; q). We therefore have the following imme-diate corollaries of (2):

the electronic journal of combinatorics 1 (1994), #R11 5Corollary 4 The distribution of (ris; corin) over Sn is generated byXn�0 Cn(y; p)zn(y; p)n+1 = Xj�0 yj1� [j + 1]p zwhere [j + 1]p = (1� pj)=(1� p).Corollary 5 (Stanley) The distribution of (des; inv) on Sn is generated byXn�0 Sn(t; q)zn(q; q)n = 1� tEq(�z(1� t))� twhere Eq(z) = J0(�z; q; 0).Another generating function for Cn(y; p) was given by Garsia [9].3 A key partition lemmaIn proving Theorem 1, we make repeated use of a result on partitions. Forlater purposes, we present this result in the language of the free monoid.Let A be an alphabet, that is, a non-empty set whose elements are referredto as letters. A �nite sequence (possibly empty) w = a1a2 : : : an of n lettersis said to be a word of length n. The length of w will be denoted by l(w).The empty word is signi�ed by 1. The set of all words that may be formedwith the letters from A along with the concatenation product is known asthe free monoid generated by A and is denoted by A�. We let An signify theset of words in A� of length n.To state the needed partition result, we select the alphabet N of non-negative integers and let Nr = f0; 1; 2; : : : ; rg. For w = a1a2 : : : an 2 Nn,set kwk = a1 + a2 + : : :+ an :For K � f1; 2; : : : ; n� 1g, a partition belonging to the setCnr (K) = f = 1 2 : : : n 2 Nnr : 1 � 2 � : : : � n; k < k+1 if k 2 Kghas at most n parts (each bounded by r) and is said to be compatible withK. We de�ne the index of a set K � f1; 2; : : : ; n� 1g to be

the electronic journal of combinatorics 1 (1994), #R11 6indK = Xk2K(n� k) :For � 2 Sn, note that ind(Des�) = comaj� and ind(Ris�) = corin�. Thekey partition result for the coming argumentation isLemma 1 For K � f1; 2; : : : ; n� 1g and r a non-negative integer,X 2Cnr (K) qk k = qindK " r � jKj+ nn #q :Proof. This is a trivial consequence of a well-known result in the theory ofpartitions. As may be referenced in [1, p. 33],X0��1��2�:::��n�s qk�k = " s+ nn #q (4)where � = �1�2 : : : �n. Suppose = 1 2 : : : n 2 Cnr (K). The bijection 1 2 : : : n ! �1�2 : : : �n de�ned by �j = ( j � jfi 2 K : i < jgj) satis�es theproperties that 0 � �1 � �2 � : : : � �n � (r � jKj) and k k = k�k+ indK.The desired result then follows from (4).4 Words by �-adjacenciesThe essence of our proof to Theorem 1 relies on two inversion theorems thatenumerate words in the free monoid by �-adjacencies. Let � be a binaryrelation on the alphabet A. A word w = a1a2 : : : an 2 An is said to have a�-adjacency in position k if ak�ak+1. The set of �-adjacencies and the numberof �-adjacencies of w = a1a2 : : : an are respectively denoted by�Adjw = fk : 1 � k � n� 1; ak�ak+1g and �adjw = j�Adjwj :An element of the set TA;� = fw = a1a2 : : : al(w) 2 A� : a1�a2� : : : �al(w)gis said to be a �-chain. We let T +A;� denote the set of �-chains of positivelength. In Z[t] << A >>, the algebra of formal power series on A� withcoe�cients from the ring of polynomials in t having integer coe�cients, thefollowing inversion formulas hold:

the electronic journal of combinatorics 1 (1994), #R11 7Theorem 2 Words by �-adjacencies are generated byXw2A� t�adjww = 11 +Pw2T +A;� (�1)l(w)(1 � t)l(w)�1w :Theorem 3 For a non-empty set X � A, let A�X = fva 2 A� : a 2 Xg.Then, words ending in a letter from X by �-adjacencies are generated byXw2A�X t�adjww = �Pw2TA;�X (�1)l(w)(1� t)l(w)�1w1 +Pw2T +A;� (�1)l(w)(1� t)l(w)�1wwhere TA;�X = fva 2 TA;� : a 2 Xg and where the ratio is to be interpretedas the product of the reciprocal of its denominator (the left factor) with itsnumerator (the right factor).A number of related theories of inversion [12, 13, 18, 20, 21] have been devel-oped and applied to a wide range of combinatorial problems. Both Theorems2 and 3 may be readily deduced from the theory of inversion presented in [7].5 The role played by Theorems 2 and 3To see precisely how Theorems 2 and 3 intervene in the proof of Theorem 1,we �rst rewrite them asXw2A� t�adjww = 1� tPw2TA;� (�1)l(w)(1� t)l(w)w � t ; (5)Xw2A�X t�adjww = �Pw2TA;�X (�1)l(w)(1� t)l(w)wPw2TA;� (�1)l(w)(1 � t)l(w)w � t : (6)Next, let � be the binary relation on N�N consisting of the pairs ��ij�; � km��satisfying i > k and j � m; ij!� km! () i > k and j � m : (7)

the electronic journal of combinatorics 1 (1994), #R11 8Thus, the set of �-adjacencies for a biword �vw� = �a1a2:::anb1b2:::bn� 2 (N �N)n is�Adj vw! = fk : 1 � k � n� 1; ak > ak+1; bk � bk+1g :Moreover, �vw� = �a1a2:::anb1b2:::bn � 2 (Ni �Nj)n is a �-chain if and only ifi � a1 > a2 > : : : > an and j � b1 � b2 � : : : � bn: (8)As will be seen, the crucial step in establishing Theorem 1 is the evalua-tion of X(vw)2(Ni�Nj)�t�adj(vw)W vw! and X(vw)2(Ni�Nj)�Xit�adj(vw)W vw!where Xi denotes the set of biletters Ni � N0 = f�k0� : 0 � k � ig andwhere W is the homomorphism on (N �N)� obtained by multiplicativelyextending the weight W�ij� = qipjz de�ned on each �ij� 2 N �N . In view of(5) and (6), this can be accomplished by computing a sum of the formX(�1)l(vw)(1� t)l(vw)W vw! (9)twice; once summed over the set TNi�Nj ;� of �-chains in (Ni �Nj)� and oncesummed over the set TNi�Nj ;�Xi of �-chains ending in a biletter from Xi.By (8), expression (9) summed over TNi�Nj ;� is equal toXn�0(�1)n(1 � t)nzn Xi�a1>a2>:::>an�0 qkvk Xj�b1�b2�:::�bn�0 pkwkwhich, by Lemma 1, reduces toXn�0(�1)nq(n2) " i+ 1n #q " j + nn #p (1� t)nzn = J (i;j)0 (z(1� t); q; p) :Summarizing, we have established thatX(vw)2TNi�Nj;�(�1)l(vw)(1 � t)l(vw)W vw! = J (i;j)0 (z(1 � t); q; p) :

the electronic journal of combinatorics 1 (1994), #R11 9Similarly,X(vw)2TNi�Nj;�Xi(�1)l(vw)(1� t)l(vw)W vw! = �J (i;j)1 (z(1� t); q; p) :The last two identities together with (5) and (6) implyX(vw)2(Ni�Nj)�t�adj(vw)W vw! = 1 � tJ (i;j)0 (z(1� t); q; p) � t ; (10)X(vw)2(Ni�Nj)�Xit�adj(vw)W vw! = J (i;j)1 (z(1� t); q; p)J (i;j)0 (z(1� t); q; p) � t : (11)6 Component bijectionsTo connect the left-hand sides of (10) and (11) with pairs of permutations,we have the following lemma.Lemma 2 For each n � 0, there is a bijection f � g from the setf �; �; �! : �; � 2 Sn; 2 Cni (Des�); � 2 Cnj (Ris�)gto the set (Ni �Nj)n such that, iff � g �; �; �! = f(�; )g(�; �)! = vw!;then k k = kvk, k�k = kwk, andk 2 Des��1\Des ��1 () k 2 �Adj vw! : (12)Moreover, if w = b1b2 : : : bn, we have�(1) = n whenever bn = 0 (13)

the electronic journal of combinatorics 1 (1994), #R11 10Proof. The bijection f � g is described in terms of two component bijectionsf and g. The map f sends elements from the set of pairsf(�; ) : � 2 Sn; 2 Cni (Des�)gto the set Nni by f(�; ) = ��1(1) ��1(2) : : : ��1(n) :The inverse of f is easily described: For v = a1a2 : : : an 2 Nni and s � 0,let Ps(v) = fr : ar = sg. Furthermore, let " Ps(v) denote the increasingword consisting of the integers from Ps(v) and " v be the non-decreasingrearrangement of v. Then,f�1(v) = (" P0(v) " P1(v) : : : " Pi(v); " v) :As an illustration, v = 30 3 0 2 2 3 2 N73 is mapped tof�1(3 0 3 0 2 2 3) = (" f2; 4g " ; " f5; 6g " f1; 3; 7g; " 3 0 3 0 2 2 3)= (2 4 5 6 1 3 7; 0 0 2 2 3 3 3) :Note that 0 0 2 2 3 3 3 2 C73(f4g). The bijection f�1 was previously used byGarsia and Gessel [10] and in [17] in the study of statistics on Sn.As a partial veri�cation of (12), suppose f(�; ) = a1a2 : : : an 2 Nni , thatis, ��1(k) = ak for 1 � k � n. >From the characterization of f�1 and fromthe observation that Des��1 consists of the integers k such that (k + 1)appears to the left of k in �, we have k 2 Des��1 if and only if ak > ak+1.Also note that k k = ka1a2 : : : ank.The bijection g is similarly de�ned fromf(�; �) : � 2 Sn; � 2 Cnj (Ris�)gto Nnj by setting g(�; �) = ���1(1)���1(2) : : : ���1(n) :For w = b1b2 : : : bn 2 Nnj , let # Ps(w) denote the decreasing word consistingof integers from the set Ps(w) = fr : br = sg. Then,g�1(w) = (# P0(w) # P1(w) : : : # Pj(w); " w) :The properties of g listed in Lemma 1 are easily veri�ed.

the electronic journal of combinatorics 1 (1994), #R11 117 Proof of Theorem 1Using the q-binomial series, the left-hand side of (2) expands asXn�0 zn Xi;j�0xiyj iXl=0 jXk=0An;l;k " i� l + nn #q " j � k + nn #pwhere An;l;k =X tiddes(�;�)qcomaj�pcorin�summed over pairs (�; �) 2 S2n satisfying des� = l and ris� = k. Combina-tion with Lemma 1 givesXn�0 An(t; x; y; q; p)zn(x; q)n+1(y; p)n+1 = Xi;j�0xiyj Xn�0 zn X(�;�)2S2n tiddes(�;�)X qk kpk�kwhere the last sum is over ( ; �) 2 Cni (Des�) � Cnj (Ris�). The bijection ofLemma 2 then impliesXn�0 An(t; x; y; q; p)zn(x; q)n+1(y; p)n+1 = Xi;j�0xiyj X(vw)2(Ni�Nj)�t�adj(vw)W vw! : (14)In view of (10), the proof of (2) is complete.To establish (3), begin by noting that (13) implies that f�g is a bijectionfromf �; �; �! : �; � 2 Sn; �(1) = n; 2 Cni (Des�); � 2 Cnj (Ris�); �1 = 0gto the set (Ni � Nj)n�1Xi where Xi = Ni � N0. Then, by steps similar tothose used in deriving (14), it may be shown thatXn�0 A1n+1(t; x; y; q; p)zn+1(x; q)n+2(y; p)n+1 = Xi;j�0 xiyj X(vw)2(Ni�Nj)�Xit�adj(vw)W vw! :Together with (11), this implies (3).

the electronic journal of combinatorics 1 (1994), #R11 128 Other distributions on permutation pairsWith the aim of presenting theorems for �nite sequences of permutations, wegive the generating functions for some other �ve-variate distributions on S2n.We �rst consider (iddes; des, comaj, ris, corin) over pairs (�; �) 2 S2n with�(1) = n. LetB1n(t; x; y; q; p) = Xf(�;�)2S2n : �(1)=ng tiddes(�;�)xdes�qcomaj�yris�pcorin� :The sequence of re�ned bibasic Bessel functionsF (i;j)� (z; q; p) = Xn�0(�1)nq(n+�2 ) " in #q " j + n+ �n+ � #p zn+�plays the role of J (i;j)� . Actually, F (i+1;j)0 = J (i;j)0 . De�ne � as in (7). Sincef � g is a bijection from the setf �; �; �! : �; � 2 Sn; �(1) = n; 2 Cni (Des�); 1 = 0; � 2 Cnj (Ris�)gto the set ((Ninf0g) �Nj)n�1Yj where Yj = N0 � Nj, computations similarto those of sections 5 and 7 may be used to verifyTheorem 4 The sequence fB1n+1gn�0 is generated byXn�0 B1n+1(t; x; y; q; p)zn+1(x; q)n+1(y; p)n+2 = Xi;j�0 xiyj F (i;j)1 (z(1� t); q; p)F (i+1;j)0 (z(1 � t); q; p)� t :Next, we determine the distribution of (iddes; des, comaj, des, comaj)over unrestricted and restricted permutation pairs. De�ne Cn(t; x1; x2; q1; q2)and C1n(t; x1; x2; q1; q2) to beX(�;�) tiddes(�;�)xdes�1 qcomaj�1 xdes�2 qcomaj�2summed respectively over S2n and over pairs in S2n with �(1) = n. Theappropriate sequence of re�ned bibasic Bessel functions isG(i1;i2)� (z; q1; q2) = Xn�0(�1)nq(n+�2 )1 q(n+�2 )2 " i1 + 1n + � #q1 " i2n #q2 zn+� :

the electronic journal of combinatorics 1 (1994), #R11 13Let � be the binary relation on N �N consisting of the pairs ��ij�; � km�� suchthat i > k and j > m. The map f � f fromf �; �; �! : �; � 2 Sn; 2 Cni1(Des�); � 2 Cni2(Des �)gto the set (Ni1 �Ni2)n de�ned byf � f �; �; �! = f(�; )f(�; �)! = vw!is a bijection satisfying the properties that k k = kvk, k�k = kwk, andk 2 Des��1\Des��1 () k 2 �Adj vw! :Then, proceeding as in sections 5 and 7, we haveTheorem 5 The sequences fCngn�0 and fC1n+1gn�0 are respectively gener-ated byXn�0 Cn(t; x1; x2; q1; q2)zn(x1; q1)n+1(x2; q2)n+1 = Xi1 ;i2�0xi11 xi22 1 � tG(i1;i2+1)0 (z(1 � t); q1; q2)� t ;Xn�0 C1n+1(t; x1; x2; q1; q2)zn+1(x1; q1)n+2(x2; q2)n+1 = Xi1;i2�0 xi11 xi22 G(i1;i2)1 (z(1 � t); q1; q2)G(i1;i2+1)0 (z(1� t); q1; q2)� t :Finally, we consider the distribution of (iddes; ris, corin, ris, corin). De�neDn(t; y1; y2; p1; p2) and D1n(t; y1; y2; p1; p2) to beX(�;�) tiddes(�;�)yris�1 pcorin�1 yris�2 pcorin�2summed respectively over S2n and over pairs in S2n with �(1) = n. SetH(j1;j2)� (z; p1; p2) = Xn�0(�1)n " j1 + n+ �n+ � #p1 " j2 + nn #p2 zn+� :Take � to be the binary relation on N �N consisting of the pairs ��ij�; � km��such that i � k and j � m. Using the bijection g � g, we obtain

the electronic journal of combinatorics 1 (1994), #R11 14Theorem 6 The sequences fDngn�0 and fD1n+1gn�0 are respectively gener-ated byXn�0 Dn(t; y1; y2; p1; p2)zn(y1; p1)n+1(y2; p2)n+1 = Xj1;j2�0 yj11 yj22 1� tH(j1;j2)0 (z(1� t); p1; p2)� t ;Xn�0 D1n+1(t; y1; y2; p1; p2)zn+1(y1; p1)n+2(y2; p2)n+1 = Xj1;j2�0 yj11 yj22 H(j1;j2)1 (z(1� t); p1; p2)H(j1 ;j2)0 (z(1� t); p1; p2)� t :9 Permutation sequencesWe now consider distributions on �nite sequences of permutations. For inte-gers r; s � 0 not both zero, let i = (i1; i2; : : : ; ir) and j = (j1; j2; : : : ; js). Se-lect U � f1; 2; : : : ; rg and V � f1; 2; : : : ; sg. Further let i(U) = (i01; i02; : : : ; i0r)where i0l = il if l =2 U and i0l = il + 1 if l 2 U .The required multibasic extension of the previously appearing sequencesof re�ned bibasic Bessel functions is de�ned byK(i ; j)� (z;U; V ) = Xn�0(�1)nQ1Q2P1 P2 zn+�where Q1 = Yl=2U q(n+�2 )l " il + 1n+ � #ql ; Q2 = Yl2U q(n+�2 )l " iln #ql ;P1 = Ym=2V " jm + n+ �n+ � #pm ; P2 = Ym2V " jm + nn #pm :For r = s = 1 with U = ; and V = f1g, note that K(i ; j)� (z; ;; f1g) =J (i1 ; j1)� (z; q1; p1). For r = 2, s = 0, U = f2g, and V = ;, we haveK(i ; j)� (z; f2g; ;) = G(i1 ; i2)� (z; q1; q2). Similar choices of r, s, U , and V givethe other re�ned bibasic Bessel functions appearing in section 8.We de�ne the number of common iddescents of a sequence (�;�) =(�1; �2; : : : ; �r;�1; �2; : : : ; �s) 2 Srn � Ssn to beiddes(�;�) = j r\k=1Des��1k \ s\m=1Des��1m j :

the electronic journal of combinatorics 1 (1994), #R11 15Furthermore, setxdes� = xdes�11 xdes�22 : : : xdes�rr and yris� = yris�11 yris�22 : : : yris�ss :The symbols qcomaj� and pcorin� are to be similarly interpreted. Finally, letMn(t; U; V ) =X tiddes(�;�)xdes�qcomaj�yris�pcorin�where the sum is over all (�; �) 2 Srn � Ssn with �l(1) = n for l 2 U and�m(1) = n for m 2 V . Then, the map f (r) � g(s) consisting of r copies ofthe component bijection f and s copies of the component bijection g alongwith judicious use of the analysis of sections 5 and 7 imply our theorems onpermutation sequences:Theorem 7 The sequence fMn(t; ;; ;)gn�0 is generated byXn�0 Mn(t; ;; ;)zn(x;q)n+1(y;p)n+1 = Xi;j�0 xiyj (1� t)K(i(U) ; j)0 (z(1� t);U; V )� twhere (x;q)n = (x1; q1)n � � � (xr; qr)n and (y;p)n = (y1; p1)n � � � (ys; ps)n.Theorem 8 Provided that U and V are not both empty and their comple-ments are not both empty, the sequence fMn+1(t; U; V )gn�0 is generated byXn�0 Mn+1(t; U; V )zn+1(x;y;q;p)cn+2(x;y;q;p)n+1 = Xi; j�0 xiyj K(i ; j)1 (z(1 � t))K(i(U) ; j)0 (z(1� t);U; V )� twhere (x;y;q;p)cn = Ql=2U (xl; ql)nQm=2V (ym; pm)n and(x;y;q;p)n = Ql2U(xl; ql)nQm2V (ym; pm)n.Taking the limit as x; y ! 1 and setting qi = 1, 1 � i � r, in Theorem 7gives a result equivalent to one obtained by Stanley [18].References[1] G. E. Andrews, The Theory of Partitions, Addison-Wesley 1976.[2] L. Carlitz, q-Bernoulli numbers and Eulerian numbers, Trans. Amer.Math. Soc. 76 (1954) 332-350.

the electronic journal of combinatorics 1 (1994), #R11 16[3] L. Carlitz, A combinatorial property of q-Eulerian numbers, Amer.Math. Monthly 82 (1975) 51-54.[4] L. Carlitz, R. Scoville, and T. Vaughan, Enumeration of pairs of permu-tations, Discrete Math. 14 (1976) 215-239.[5] M. P. Delest and J. M. Fedou, Enumeration of skew Ferrers' diagrams,Discrete Math. 112 (1993) 65-79.[6] J. M. Fedou and D. P. Rawlings, Statistics on pairs of permutations,Discrete Math. (to appear).[7] J. M. Fedou and D. P. Rawlings, Adjacencies in words, Adv. in Appl.Math. (to appear).[8] D. Foata, Distributions Eul�eriennes et Mahoniennes sur le groupe despermutations, Higher Combinatorics, Proceedings, ed. M. Aigner, ReidelPubl. Co., Dodrecht, Holland (1977) 27-49.[9] A. M. Garsia, On the maj and inv q-analogues of Eulerian polynomials,J. Lin. and Multilin. Alg. 8 (1980) 21-34.[10] A. M. Garsia and I. Gessel, Permutation statistics and partitions, Adv.in Math. 31 (1979) 288-305.[11] G. Gasper and M. Rahman, Basic Hypergeometric Series, CambridgeUniv. Press 1990.[12] I. M. Gessel, Generating Functions and Enumeration of Sequences, Doc-toral thesis, Massachusetts Institute of Technology, Cambridge, Mas-sachusetts 1977.[13] I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, JohnWiley & Sons 1983.[14] M. E. H. Ismail, The zeros of basic Bessel functions, : : : , J. Math. Annal.and Appl. 86 (1982) 1-19.[15] F. H. Jackson, The basic gamma function and elliptic functions, Proc.Royal Soc. A 76 (1905) 127-144.

the electronic journal of combinatorics 1 (1994), #R11 17[16] M. P. MacMahon, Combinatory Analysis, Cambridge Univ. Press 1915(reprinted by Chelsea 1960).[17] D. P. Rawlings, Generalized Worpitzky identities with applications topermutation enumeration Europ. J. Comb. Theory 2 (1981) 67-78.[18] R. P. Stanley, Binomial posets, M�obius inversion, and permutation enu-meration, J. Comb. Theory (A) 20 (1976) 336-356.[19] R. P. Stanley, Enumerative Combinatorics Vol. I, Wadsworth & Brooks1986.[20] X. G. Viennot, Heaps of pieces I: Basic de�nitions and combinatoriallemmas, in Combinatoire �Enum�erative, eds. G. Labelle and P. Leroux,Lect. Notes in Math. 1234, Springer-Verlag, Berlin 1986.[21] D. Zeilberger, Enumeration of words by their number of mistakes, Dis-crete Math. 34 (1981) 89-92.