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THE COMBINATORICS OF q-CHARLIER POLYNOMIALS

arXiv:math/9307208v1 [math.CA] 9 Jul 1993THE COMBINATORICS OF q-CHARLIER POLYNOMIALSAnne de M´edicis*Dennis Stanton†Dennis WhiteAbstract. We describe various aspects of the Al-Salam-Carlitz q-Charlier polyno-mials.

These include combinatorial descriptions of the moments, the orthogonalityrelation, and the linearization coefficients.1. Introduction.The Charlier polynomials Can(x) are well-known analytically [4], and have beenstudied combinatorially by various authors [8], [12], [16], [17], [20].

The momentsfor the measure of these orthogonal polynomials are(1.1)µn =nXk=1S(n, k)ak,where S(n, k) are the Stirling numbers of the second kind. The purpose of this paperis to study combinatorially an appropriate q-analogue of Can(x), whose moments area q-Stirling version of (1.1).

While studying these polynomials, we use statistics onset partitions which are q-Stirling distributed.Our main result (Theorem 3) is the combinatorial proof of the linearizationcoefficients for these polynomials. In the q = 1 case, the linearization coefficientsare given as a polynomial in a, whose coefficients are quotients of factorials (see(4.4)).

This has a simple combinatorial explanation. However, in the q-case thecoefficients are not the analogous quotients of q-factorials.

They are alternatingsums of quotients of q-factorials, and thus a combinatorial explanation is muchmore difficult. From the combinatorial interpretations of the polynomials and theirmoments, in terms of weighted partial permutations and set partitions, we deduce acombinatorial interpretation for the linearization coefficients of a product of three q-Charlier polynomials.

We then apply a weight-preserving sign-reversing involutiondefined in five steps. Theorem 3 is obtained by enumerating the remaining fixedpoints.

Some of the steps of the involution are quite straight-forward, but someothers are more complicated.They use more sophisticated techniques such asencoding of permutations or set partitions into 0–1 tableaux (cf [6], [18]), whichare fillings of Ferrers diagrams with 0’s and 1’s such that there is exactly one 1in each column. They also use interpolating statistics on set partitions, as were*This work was supported by NSERC funds.†This work was supported by NSF grant DMS-9001195.Tt bAMS T X

introduced by White in [22]. Indeed, the characterization of the final set of fixedpoints uses a bijection ΨS of White between interpolating statistics, making theirenumeration all the more complicated.It turns out that our q-Charlier polynomials are not what have classically beencalled q-Charlier; in fact they are rescaled versions of the Al-Salam-Carlitz polyno-mials [4, p.196].

Some comparisons to the classical q-Charlier are given in §7. Zeng[24] has also studied both families of polynomials from the associated continuedfractions.The basic combinatorial interpretation of the polynomials is given in Theorem1.

Several facts about the polynomials can be proven combinatorially. The combi-natorics of set partitions, restricted growth functions and 0–1 tableaux is discussedin §3, and the statistic for the moments is given in Theorem 2.

In §4, we stateour main theorem, Theorem 3, giving the linearization coefficient for a product ofthree q-Charlier polynomials, and we set up the general combinatorial context forits demonstration. The five steps of the weight-preserving sign-reversing involu-tion proving Theorem 3 are given in §5, and the combinatorial evaluation of theremaining fixed points is the subject of §6.We use the standard notation for q-binomial coefficients and shifted factorialsfound in [11].

We will also need[n]q = 1 −qn1 −q ,and[n]!q = [n]q[n −1]q · · ·[1]q.2. The q-Charlier polynomials.We define the q-Charlier polynomials by the three term recurrence relation(2.1)Cn+1(x, a; q) = (x −aqn −[n]q)Cn(x, a; q) −a[n]qqn−1Cn−1(x, a; q),where C−1(x, a; q) = 0 and C0(x, a; q) = 1.It is not hard to show that these polynomials are rescaled versions of the Al-Salam Carlitz polynomials [4, p.196](2.2)Cn(x, a; q) = anUn(xa −1a(1 −q),−1a(1 −q)).Since the generating function of the Un(x, b) is known [4], we see that(2.3)∞Xn=0Cn(x, a; q) tn(q)n=(at)∞(−t1−q)∞(t(x −11−q))∞.This gives the explicit formula(2.4)Cn(x, a; q) =nXk=0nkq(−a)n−kq(n−k2 )k−1Yi=0(x −[i]q).Clearly, we want a q-version of [16], which gives the Charlier polynomials asa generating function of weighted partial permutations, i.e.pairs (B, σ), where2

B ⊆{1, 2, · · · , n} = [n], and σ is a permutation on [n] −B. Thus we need onlyinterpret the individual terms in (2.4) for a combinatorial interpretation.

The insideproduct can be expanded in terms of the q-Stirling numbers of the first kind. Welet cyc(σ) be the number of cycles of a permutation σ and inv(σ) be the number ofinversions of σ written as a product of disjoint cycles (increasing minima, minimafirst in a cycle).k−1Yi=0(x −[i]q) =Xσ∈Sk(−1)k−cyc(σ)qinv(σ)xcyc(σ).For the sum over k in (2.4), we sum over all (n −k) subsets B ⊆[n].

Letinv(B) =Xb∈B(b −1),so that the generating function for these subsets isnkqq(n−k2 ).We have established the following theorem.Theorem 1. The q-Charlier polynomials are given byCn(x, a; q) =XB⊆[n]Xσ∈Sn−Bqinv(σ)+inv(B)(−1)n−cyc(σ)a|B|xcyc(σ),=XB⊆[n]Xσ∈Sn−Bωq(B, σ)xcyc(σ).A combinatorial proof of the three-term recurrence relation (2.1) can be givenusing Theorem 1.

An involution is necessary. For more details, we refer the readerto [5].3.

The moments.An explicit measure for the q-Charlier polynomials is known, [4, p.196]. It isnot hard to find the nth moment of this measure explicitly.

The result is a perfectq-analogue of (1.1)(3.1)µn =nXk=1Sq(n, k)ak,where Sq(n, k) is the q-Stirling number of the second kind, given by the recurrence(3.2)Sq(n, k) = Sq(n −1, k −1) + [k]qSq(n −1, k),where Sq(0, k) = δ0,k. In fact, one sees that [13](3.3)Sq(n, k) =1(1 −q)n−kn−kXj=0nk + j k + jjq(−1)j.3

Clearly (3.1) suggests that there is some statistic on set partitions, whose gener-ating function is µn. This statistic, rs, arises from the Viennot theory of Motzkinpaths associated with the three-term recurrence (2.1) [20].

We do not give thedetails of the construction here.However, let us review some combinatorial facts about q-Stirling numbers. Setpartitions of [n] = {1, 2, .

. .

, n} can be encoded as restricted growth functions (orRG-functions) as follow: if the blocks of π are ordered by increasing minima, theRG-function w = w1w2 . .

. wn is the word such that wi is the block where i islocated.

For example, if π = 147|28|3|569, w = 123144124. Note that set partitionson any set A can be encoded as RG-functions as long as A is a totally ordered set.In [21], Wachs and White investigated four natural statistics on set partitions,called ls, lb, rs and rb.

They are defined as follow:ls(π) = ls(w) =nXi=1|{j : j < wi, j appears to the left of position i}|,lb(π) = lb(w) =nXi=1|{j : j > wi, j appears to the left of position i}|,rs(π) = rs(w) =nXi=1|{j : j < wi, j appears to the right of position i}|,rb(π) = ls(w) =nXi=1|{j : j > wi, j appears to the right of position i}|.Thus in the example, ls(π) = 13, lb(π) = 7, rs(π) = 7 and rb(π) = 11. Theyshowed, using combinatorial methods, that each had the same distribution (up toa constant) on the set RG(n, k) of all restricted growth functions of length n andmaximum k, and that their generating function was indeed Sq(n, k) for rs and lb(respectively q(k2)Sq(n, k) for ls and rb).We also use another encoding of set partitions in terms of 0–1 tableaux.

A 0–1tableau is a pair ϕ = (λ, f) where λ = (λ1 ≥λ2 ≥. .

. ≥λk) is a partition ofan integer m = |λ| and f = (fij)1≤j≤λi is a “filling” of the corresponding Ferrersdiagram of shape λ with 0’s and 1’s such that there is exactly one 1 in each column.0–1 tableaux were introduced by Leroux in [18] to establish a q-log concavity resultconjectured by Butler [3] for Stirling numbers of the second kind.There is a natural correspondence between set partitions π of [n] with k blocksand 0–1 tableaux with n −k columns of length less than or equal to k. Simplywrite the RG-function w = w1w2 .

. .

wn associated to π as a k × n matrix, with a1 in position (i, j) if wj = i, and 0 elsewhere. The resulting matrix is row-reducedechelon, of rank k, with exactly one 1 in each column.

A 0–1 tableau (in the thirdquadrant) is then obtained by removing all the pivot columns and the 0’s that lieon the left of a 1 on a pivot column. Figure 1 illustrates these manipulations forπ = 1247|39(12)|568(11)|(10).We define two statistics on 0–1 tableaux ϕ: first, the inversion number, inv(ϕ),which is equal to the number of 0’s below a 1 in ϕ; and the non-inversion number,nin(ϕ), which is equal to the number of 0’s above a 1 in ϕ.

For example, for ϕin Figure 1, inv(ϕ) = 7 and nin(ϕ) = 8. Note that an easy involution on thecolumns of 0–1 tableaux sends the inversion number to the non-inversion numberand vice-versa.

We call this map the symmetry involution.4

110100100000001000001001000011010010000000000100⇕00001011101000001010100Figure 1: Correspondence between partitions and 0–1 tableauxIt is not hard to see that the inversion number (respectively non-inversion num-ber) on 0–1 tableaux corresponds to the statistic lb (resp. ls−k2) on set partitions.Similarly, permutations σ of [n] in k cycles can be encoded as 0–1 tableaux withn−k columns of distinct lengths less than or equal to n−1.

The correspondence isdefined by recurrence on n. Suppose σ is written as a standard product of cycles.If n = 1, then σ = (1) corresponds to the empty 0–1 tableau ϕ = ∅. Otherwise,let σ ∈Sn+1 and let ϕ denote the 0–1 tableau associated to the permutation σ inwhich (n + 1) has been erased.

There are two cases. If (n + 1) is the minimum ofa cycle in σ, then σ corresponds to ϕ.

If (n + 1) is not the minimum of a cycle,then it appears in σ at a certain position i, 2 ≤i ≤n + 1. The permutation σthen corresponds to the 0–1 tableau ϕ plus a column of length n with a 1 in the(i −1)-th position (from top to bottom).

For example, σ = (1, 3, 4, 7, 2)(5, 6)(8)corresponds to the following 0–1 tableau.00011001010000010Figure 2: Correspondence between permutations and 0–1 tableauxIt is not hard to see that under this transformation, the inversion number on0–1 tableaux corresponds to the inversion number on permutations, as defined in§2. Thus, their generating functions are the q-Stirling numbers of the first kindcq(n, k).In [6], de M´edicis and Leroux investigated q and p, q-Stirling numbers from thepoint of view of the unified 0–1 tableau approach.

In particular, they proved com-binatorially or algebraically a number of identities involving q-Stirling numbers.For the combinatorial interpretation of the moments of the q-Charlier polyno-mials in terms of set partitions π, we need two statistics. The number of blocks#blocks(π) is one, and the other statistic is rs(π).5

Theorem 2. The nth moment for the q-Charlier polynomials is given byµn =Xπ∈P (n)a#blocks(π)qrs(π).As we mentioned, many other q-Stirling distributed statistics have been found[21].

It is surprising that the Viennot theory naturally gives a so-called “hard”statistic (rs), not an easy one (e.g. lb, [21]).

Other variations on the rs-statisticcan be given from the Motzkin paths, although the lb-statistic is not among them.It can be derived from the Motzkin paths associated with the “odd” polynomialsfor (2.1).4. The orthogonality relation and the linearization of products.Let L be the linear functional on polynomials that corresponds to integratingwith respect to the measure for the Charlier polynomials.

The orthogonality rela-tion is(4.1)L(Can(x)Cam(x)) = ann! δm,n.The q-version of (4.1) is(4.2)Lq(Cn(x, a; q)Cm(x, a; q)) = anq(n2)[n]!qδm,n.Since the polynomials Cn(x, a; q) and Lq have combinatorial definitions from The-orems 1 and 2, it is possible to restate (4.2) as a combinatorial problem.

We willgive an involution which then proves (4.2) in this framework.A more general question is to find L(Can1(x)Can2(x) · · ·Cank(x)) for any k.Asolution is equivalent to finding the coefficients ank in the expansionCan1(x)Can2(x) · · ·Cank−1(x) =XnkankCank( x).This had been done bijectively for some classes of Sheffer orthogonal polynomials in[5], [7], [9], [10]. Moreover, in the q-case of Hermite polynomials, some remarkableconsequences have been found [15].For the Charlier polynomials, it is easy to see that(4.3)∞Xn1,··· ,nk=0L(Can1(x)Can2(x) · · ·Cank(x))tn11n1!

· · · tnkknk! = ea(e2(t1,··· ,tk)+···+ek(t1,··· ,tk)),where ei is the elementary symmetric function of degree i, [19].In this caseL(Can1Can2 · · · Cank) is a polynomial in a with positive integer coefficients; a combi-natorial interpretation of this coefficient has been given ([12] and [23]).

For k = 3,(4.3) is equivalent to(4.4)L(Can1(x)Can2(x)Can3(x)) =⌊(n1+n2−n3)/2⌋Xl=0an3+ln1!n2!n3!l! (n3 −n2 + l)!

(n3 −n1 + l)! (n1 + n2 −n3 −2l)!.One can hope that Lq(Cn1(x)Cn2(x)Cn3(x)) is simply a weighted version, withan appropriate statistic, of the q = 1 case.

However this is false. For example,Lq(C2(x)C2(x)C1(x)) = q(q2 + 2q + 1)a2 + q(q3 + q2 −q −1)a3.Nonetheless, we have an exact formula for Lq(Cn1(x, a, q)Cn2(x, a, q)Cn3(x, a, q)),which is equivalent to one of Al-Salam-Verma [1].6

Theorem 3. Let n3 ≥n1 ≥n2 ≥0.

ThenLq(Cn1(x)Cn2(x)Cn3(x)) =n1+n2−n3Xl=0lXj=0an3+lqK(q −1)l−j [n1 −j]!q[n1 −l]!qn2l −jq[n3]!qn1jqn2 −l + jn3 −n1 + jq[j]!q[n1 −j]!q[n3 −n2 + l]!qn1 + n2 −n3 −ljq,(4.5)whereK =l −j2+n12+ j(−n3 −j + 1) +j2+n2 −l + j2+ (n3 −n1 + j)(n3 −n2 + l) + j(n3 −n2 + l).The generating function of Lq(Cn1(x)Cn2(x)Cn3(x)) can be evaluated from The-orem 3, yieldingXn1,n2,n3Lq(Cn1(x)Cn2(x)Cn3(x)) tn11[n1]!qtn22[n2]!qtn33[n3]!q=(−t3; q)∞(−at1t2(1 −q); q)∞2φ1at1(1 −q), at2(1 −q)−at1t2(1 −q); q, −t3. (4.6)Letting q →1 in (4.6) gives back (4.3) for k = 3.

This generating function can alsobe evaluated directly using the measure ([4, p.196]), the generating function (2.3)for the polynomials and a 3φ2 transformation.More generally, for k ≥4, the generating function of Lq(Cn1(x) . .

.Cnk(x)) canbe expressed as a difference of two basic hypergeometric series. This has been doneby Ismail and Stanton [14] for the Al-Salam Carlitz polynomials, so an equivalentformula can be deduced for the q-Charlier polynomials using (2.2).Let us set up the combinatorial context in which Theorem 3 will be proven.

Wefirst introduce notations and conventions that will be used throughout the proof.DefineLq(n1, n2, n3) = {((Bi, σi); π) = ((B1, σ1), (B2, σ2), (B3, σ3); π)|(Bi, σi) is a partial permutation on the set {i} × [ni],and π is a partition on the cycles of σ1, σ2 and σ3 }.We will say that an element of the set {i}×[ni] is of color i. When giving exam-ples of elements of Lq(n1, n2, n3), to simplify notation, pairs (1, i), (2, i) and (3, i)will always be denoted i, i and i respectively.

Thus a typical element of Lq(8, 7, 10)would be described in the following way: B1 = {2, 3}, B2 = ∅, B3 = {5, 9, 10}, andπ = (1, 5, 7)(8)(1)|(4)|(6)(3, 5)(3, 7)|(1, 4, 2)(6, 7)|(2, 8, 4, 6) (the underlying permu-tations σ1 = (1, 5, 7)(4)(6)(8), σ2 = (1, 4, 2)(3, 5)(6, 7) and σ3 = (1)(2, 8, 4, 6)(3, 7)can be recovered from π).Note that the lexicographic order on pairs (i, j) induces a total order on thecycles of σ1, σ2 and σ3, according to their minima. Therefore we can talk about RG-functions.

We will always use the letter w to denote the RG-function associated toπ. In the above example, w = 1231434153.

The first cyc(σ1) letters of w correspond7

to the positions of cycles of color 1 in π, the next cyc(σ2) to the positions of cyclesof color 2, and the last cyc(σ3) letters to the positions of cycles of color 3. We willdenote by wa, wb and wc respectively these portions of w. In the above example,we have wa = 1231, wb = 434, wc = 153, and w = wawbwc, the concatenation ofwords wa, wb and wc.Finally, we will use the notation Supp(w) (or Supp(σ) or Supp(πi)) to denotethe underlying set of letters of a word w (or a permutation σ or a block πi of apartition π respectively).From Theorems 1 and 2, we deduce that(4.7)Lq(Cn1(x)Cn2(x)Cn3(x)) =X((Bi,σi);π)∈Lq(n1,n2,n3)ωq((Bi, σi); π),where(4.8)ωq((Bi, σi); π) = ωq(B1, σ1)ωq(B2, σ2)ωq(B3, σ3)qrs(π)a#blocks(π),and ωq(B, σ) was defined in Theorem 1, as a signed monomial in the variables aand q.

This gives a combinatorial interpretation of the right-hand side of (4.5).For q = 1, the negative coefficients of a are counterbalanced by the positivecoefficients of a, and (4.7) is a polynomial with positive coefficients. Indeed, inthat case, it is not hard to find a weight-preserving sign-reversing involution onLq(n1, n2, n3) (cf [5]) whose fixed points ((Bi, σi); π) are characterized byi)Bi = ∅and σi = Identity, for i = 1, 2, 3;ii) the word wa (respectively wb and wc) contains all distinct letters, and Supp(wa) ⊆Supp(wbwc) (respectively Supp(wb) ⊆Supp(wawc) and Supp(wc) ⊆Supp(wawb)).Identity (4.4) easily follows from ω1-counting these fixed points.However, the general q-case is much harder, and some negative weights remain.The sign of ωq((Bi, σi); π) comes from the cardinalities of the sets Bi and the signsof the permutations σi.

In our proof, we successively apply five weight-preservingsign-reversing involutions Φi to Lq(n1, n2, n3), each one acting on the fixed pointsof the preceding one. Φ1 forces σ3 = Id, Φ2 forces B3 = ∅, Φ3 forces σ1 = Id, Φ4forces B1 = ∅, and Φ5 forces σ2 = Id, leaving B2 arbitrary.

Hence the negativepart of (4.7) is due only to B2.The final set of fixed points, FixΦ5, does not contain the fixed point set (above)for q = 1. Instead there is a bijection from a subset of FixΦ5 to this set, but itdoes not preserve the powers of q.The five weight-preserving sign-reversing involutions Φi and their respective fixedpoints sets FixΦi are given in the next section and the complete characterizationof FixΦ5 is given by the conditions Fix.1 through Fix.4, stated at the beginning of§6.

In §6, we show that the ωq-weight of FixΦ5 is equal to the right-hand side of(4.5), thus establishing Theorem 3.5. The weight-preserving sign-reversing involutions Φi.Let us recall that a weight-preserving sign-reversing involution (or WPSR-invo-lution) Φ with weight function ω is an involution such that for any e ̸∈FixΦ,ω(Φ(e)) = −ω(e).Involution Φ1.

This WPSR-involution will kill any ((Bi, σi); π) such that σ3 isnot the identity.8

Remember that the cycles of σ3 are ordered by increasing minima. Find thegreatest cycle ci0 such that either this cycle is of length ≥2 or it lies in the sameblock πi of π as some other 1-cycle greater than it.

If ci0 satisfies the latter condition,the 1-cycle greater than ci0 in the leftmost block πh of partition π is glued to theend of ci0. Then, if h = h0 < h1 < .

. .

< hm = i denote the indices of the blocksbetween πh and πi containing 1-cycles greater than ci0, these 1-cycles are movedfrom block πhl to block πhl−1.For example, for ((Bi, σi); π) ∈Lq(9, 0, 10) such that B1 = B2 = ∅, B3 ={10} and π = (1)(1, 2)(6)|(2, 8)(5)|(3)(4)|(5)(3, 9)(8)|(6, 9)(7)(7)|(4), we have σ3 =(1, 2)(3, 9)(4)(5)(6)(7)(8), ci0 = (3, 9), and Φ1((Bi, σi); π) is given by the same Bi’s,σ3 becomes (1, 2)(3, 9, 6)(4)(5)(7)(8), and π = (1)(1, 2)(5)|(2, 8)(8)|(3)(4)|(5)(3, 9, 6)|(6, 9)(7)(7)|(4).Note that the number of inversions gained in σ3 is counterbalanced by the lossin the statistic rs(π). Conversely, if ci0 is of length ≥2 and does not lie in thesame block as any other greater cycles, its image is defined in the obvious way sothat Φ1 is an involution.

For more details, see [5].Fixed points for Φ1. The cycle ci0 is not defined if and only if σ3 contains only1-cycles which all lie in different blocks of π. Therefore,FixΦ1 = {((Bi, σi); π) ∈Lq(n1, n2, n3) | σ3 is the identityand wc contains all distinct letters}.Involution Φ2.

This WPSR-involution is designed to discard all ((Bi, σi); π) ∈FixΦ1 such that B3 is not empty.Let ((Bi, σi); π) ∈FixΦ1 and let k = #blocks(π).Denote by j0, 0 ≤j0 ≤(n3 −1), the integer such that j0 + 1 = min(B3). IfB3 = ∅, we let j0 = ∞.

Likewise, denote by j1, 1 ≤j1 ≤n3, the maximum integersuch that the 1-cycle (j1) forms a singleton block in π. Remember that σ3 = Idand wc contains all distinct letters.

By maximality, (j1) lies in the k-th block of π.Denote by j′1 its contribution to the statistic rs, that is the number of (different)letters after the only occurrence of k in wc (and in w). If there are no such singletonblocks in π, let j1 = j′1 = ∞.There are two cases: j0 ≤j′1, or j0 > j′1.

If j0 ≤j′1, Φ2((Bi, σi); π) is obtainedby inserting the 1-cycle (j0 + 1) in σ3 and by inserting the letter (k + 1) in wc atthe (j0 + 1)-th position from the end of wc, leaving everything else fixed.For example, for ((Bi, σi); π) defined by B1 = ∅= B2, B3 = {2, 6, 8} and π =(1, 6)(5)(7)|(2)(1, 3, 2)(4)|(3, 5, 4)|(4)(9)|(1)|(3)|(5), wc = 562714, j0 = 1, j1 = 5 andj′1 = 2. Then the new wc in Φ2((Bi, σi); π) is wc = 5627184, and Φ2((Bi, σi); π) isdefined by B1 = ∅= B2, B3 = {6, 8} and π = (1, 6)(5)(5)|(2)(1, 3, 2)(3)|(3, 5, 4)|(4)(9)|(1)|(2)|(4)|(7).Note that Φ2((Bi, σi); π) has its j′1 equal to the j0 associated to ((Bi, σi); π).Conversely, if j′1 < j0, the image of ((Bi, σi); π) is defined in the obvious way sothat Φ2 is an involution.

Φ2 is also weight-preserving and sign-reversing. For moredetails, see [5].Fixed points for Φ2.

Fixed points correspond to the case j0 = j′1 = ∞. Thismeans that B3 = ∅and there are no singleton blocks in π of color 3.

Therefore,FixΦ2 = {((Bi, σi); π) ∈FixΦ1 | B3 = ∅and Supp(wc) ⊆Supp(wawb)}.Note that Supp(wc) ⊆Supp(wawb) is equivalent to the condition that the wawbis an RG-function whose maximum equals #blocks(π).9

To do Φ3 and later Φ5, we need to describe the contribution to the statistic rsof the elements of color 1 and 2 in partition π. Let w be a word on the alphabet[k].

Let wij denote the subword of w obtained by discarding letters not equal to ior j, 1 ≤i < j ≤k. For instance, if w = 123144124, w12 = 12112.

Then we canwriters(w) =X1≤i

Then v isan RG-function andrs(w) =X1≤i

He provides a bijection on RG(n, k) such that the mixed statistic is sent to theeasy statistic ls (up to a constant). More precisely,Lemma 4.

Let S = {s1 < s2 < . .

. < sm} ⊆[k].There is a bijection ΨS :RG(n, k) →RG(n, k) such that for any w ∈RG(n, k),(5.1)X1≤i

Define Ψi : RG(n, k) →RG(n, k), 1 ≤i ≤k −1 as follow:i) if w ∈RG(n, k) has a letter i to the right of the first occurrence of (i + 1), thenthe rightmost letter i is switched to (i+1) and any (i+1) to its right is changedto i. For example, Ψ1(111212332122) = 111212332211.ii) if w does not have a letter i to the right of the first occurrence of (i+1), then all(i + 1)’s to its right are switched to i’s.

For example, Ψ1(1112232) = 1112131.For convenience, we will set Ψk : RG(n, k) →RG(n, k) to be the identity. Now,given S = {s1 < s2 < .

. .

< sm} ⊆[k], ΨS is defined as follow:ΨS = (Ψk ◦Ψk−1 ◦. .

. ◦Ψs1) ◦(Ψk ◦Ψk−1 ◦.

. .

◦Ψs2) ◦. .

. ◦(Ψk ◦.

. .

◦Ψsm).Note that ΨS preserves the positions of the first occurrences. For more details, thereader is referred to [22].□Involution Φ3.

This next involution is designed to kill any element ((Bi, σi); π)such that σ1 is not the identity. Note that since the interpolating statistics on waare q-Stirling distributed, it reduces to proving the orthogonality relationnXk=m(−1)n−kcq(n, k)Sq(k, m) = δn,m.10

But this formula was deduced in Proposition 3.1 of [6] from a weight-preservingsign-reversing involution on appropriate pairs of 0–1 tableaux. The general idea isto map σ1 and wa bijectively into a pair of 0–1 tableaux, using ΨS defined in theprevious lemma and the correspondences described in §1.

Then we can apply theWPSR-involution, essentially shifting the rightmost shortest column from one 0–1tableau to the other. Φ3((Bi, σi); π) is then obtained by replacing σ1 and wa bythe new decoded pair of 0–1 tableaux.

Involution Φ5 will use similar ideas.We need only specify the bijective coding of (σ1, wa) into a pair of 0–1 tableaux.Let ((Bi, σi); π) ∈FixΦ2 and let n = n1−|B1|, k = cyc(σ1) and m = max(Supp(wa)).i) For σ1, simply use the correspondence described in §3 to get a 0–1 tableau ϕ1with (n −k) columns of distinct length ≤(n −1). Note that inv(σ1) = inv(ϕ1).ii) For wa, we first want to reduce the interpolating statistic rs(w)|wa to the easystatistic ls(wa).

This is done by applying ΨS defined in the previous lemma towa, for S = [m] \ Supp(wbwc). We then use the correspondence described in §3to get a 0–1 tableau ϕ2 with (k −m) columns of length ≤m.

There is one lasttechnicality: the statistic ls is sent to the non-inversion statistic on 0–1 tableaux(up to the constantm2), therefore we will apply to ϕ2 the symmetry involutionexchanging non-inversions and inversions, so that for its image ˜ϕ2, we havers(w)|wa = inv( ˜ϕ2) +m2−Xi∈S(m −i).Note that m is not modified by the WPSR-involution applied to pairs of 0–1tableaux, thus insuring that the overall involution Φ3 is well-defined (the new w isstill an RG-function) and weight-preserving. It is also sign-reversing.

Details areleft to the reader.Fixed points for Φ3. At the 0–1 tableau level, the only fixed pair of 0–1 tableauxis (∅, ∅), because in that case, it is impossible to move columns.

But this canhappen if and only if (n −k) = (k −m) = 0, and therefore n = k = m = n1 −|B1|,σ1 is the identity on [n1] −B1, and wa = 12 . .

. (n1 −|B1|).

ThereforeFixΦ3 = {((Bi, σi); π) ∈FixΦ2 | σ1 is the identity and wa = 12 . .

. (n1 −|B1|)}.Involution Φ4.

This involution is the simplest. Its task is to eliminate elements((Bi, σi); π) such that B1 ̸= ∅.Let ((Bi, σi); π) ∈FixΦ3 and let i0 be the smallest integer, 1 ≤i0 ≤n1, suchthat either i0 ∈B1, or the 1-cycle (i0) forms a singleton block in π.

Then if i0 ∈B1,insert it as a 1-cycle in σ1 and as a singleton block in π, and vice-versa.For example, if B1 = {2}, B2 = B3 = ∅, and π = (1)(1, 3)|(3)|(1, 2)(2), theni0 = 2 and the image of ((Bi, σi); π) under Φ4 is B1 = ∅, B2 = B3 = ∅, andπ = (1)(1, 3)|(2)|(3)|(1, 2)(2). Details are left to the reader.Fixed points for Φ4.FixΦ4 = {((Bi, σi); π) ∈FixΦ3 | B1 = ∅and Supp(wa) = [n1] ⊆Supp(wbwc)}.Involution Φ5.

This final WPSR-involution will annihilate the remaining ((Bi, σi); π)such that σ2 is not the identity. It is the only one using the hypothesis n3 ≥n1 ≥n2.The principle of the involution is similar to Φ3: we will reduce the problem to find-ing an involution for the easy statistic ls.11

Let ((Bi, σi); π) ∈FixΦ4, and let #blocks(π) = n3 +s. First, encode σ2 as a 0–1tableau ϕ with (n2 −|B2| −cyc(σ2)) columns of distinct lengths ≤(n2 −|B2| −1),using the correspondence described in §3.

Note that inv(σ2) = inv(ϕ) and that theshortest column of ϕ is of length at most cyc(σ2).For wb, we reduce the interpolating statistic rs(w)|wawb to the easy statistic ls byapplying ΨS defined in Lemma 4 to wawb, with S = [n3 +s]−Supp(wc). Note thatsince wa = 12 .

. .n1 and ΨS preserves first occurrences, ΨS(wawb) = wa ˜wb for someword ˜wb = ˜b1˜b2 .

. .˜bk.

Note also that we must have {n1+1, . .

. , n3+s} ⊆Supp( ˜wb)(because wawb has maximum (n3 + s)).For example, if ((Bi, σi); π) ∈FixΦ4 is defined by B1 = B2 = B3 = ∅, andπ = (1)(2)|(2)(3)(5)|(3)(2)|(4)(4)|(5)(5)(1)|(1)(4)(3), we have wa = 12345, wb =61265, wc = 53642, and σ2 = (1)(2)(3)(4)(5).

Then σ2 corresponds to the empty0–1 tableau ϕ = ∅, and we successively compute S = [6] −Supp(53642) = {1},Ψ{1}(wawb) = 1234566154, and ˜wb = 66154.Let i0 denote the length of the shortest column in ϕ, 1 ≤i0 ≤cyc(σ2). If ϕ = ∅,let i0 = ∞.

Likewise, let h0 denote the smallest integer, 1 ≤h0 ≤cyc(σ2), suchthat ˜bh0 < h0. If no such ˜bi exists, set h0 = ∞.There are two cases: i0 ≥h0 or i0 < h0.

If i0 ≥h0, then delete ˜bh0 from theword ˜wb and add a column of length (h0 −1) to ϕ, with a 1 in position ˜bh0, frombottom to top, thus obtaining a new pair ( ˜w′b, ϕ′). Since the letter removed from˜wb is at most equal to (cyc(σ2) −1) < (n2 −|B2|) < (n1 + 1), wa ˜w′b is still an RG-function of maximum (n3 + s), and the new i0 associated to ϕ′ is equal to (h0 −1).Φ5((Bi, σi); π) is then obtained by applying Ψ−1Sto wa ˜w′b and by decoding the 0–1tableau ϕ′.In the above example, i0 = ∞and h0 = 3.

Hence ϕ′ =01(corresponding tothe new permutation σ2 = (1)(2, 3)(4)(5)) and ˜w′b = 6654. From Ψ−1{1}(wa ˜wb) =123456165, we get Φ5((Bi, σi); π) equals B1 = B2 = B3 = ∅, and π = (1)(2, 3)|(2)(5)|(3)(2)|(4)(4)|(5)(5)((1)(4)(3).If i0 < h0, the image of ((Bi, σi); π) is defined in the obvious way so that Φ5 isan involution.

The proof that Φ5 is weight-preserving and sign-reversing is quitestraight-forward, and the details will be left to the reader.It remains to showthat Φ5 is well-defined. Remember that if ((Bi, σi); π) ∈FixΦ4, we must haveSupp(wa) ⊆Supp(wbwc).

We have to show that Φ5 preserves this property. Whatcomplicates matters is the application of ΨS and Ψ−1Sto the RG-functions wawb.In Lemma 5, we explicitly find the set of images wa ˜wb (which we will denote by˜W(S)) of all possible wawb under ΨS.We will then show that the deletion orinsertion of a letter whose value is strictly less than its position in ˜wb yields newRG-functions wa ˜w′b which remain in the set ˜W(S).Fix n3 ≥n1 ≥n2 ≥0, 0 ≤t ≤n2, and 0 ≤s ≤n1 + n2 −n3.

Let S ⊆[n3 + s]such that |S| ≤s, and fix wa = 12 . .

.n1. We denote byW(S) = {wb | wawb ∈RG(n1 + n2 −t, n3 + s), and[n1] ⊆([n3 + s] −S) ∪Supp(wb)},˜W(S) = { ˜wb | ΨS(wawb) = wa ˜wb for wb ∈W(S)},12

andwaW(S) = {wawb | wb ∈W(S)},wa ˜W(S) = {wa ˜wb | ˜wb ∈˜W(S)}.In particular, when |S| = s, if wc is a word containing the letters in ([n3+s]−S) inany order, with no repetition, and (B2, σ2) is a partial permutation of {2}×[n2] withcyc(σ2) = n2 −t, W(S) contains all possible words wb such that w = 12 . .

.n1wbwcis the RG-function associated to some ((Bi, σi); π) ∈FixΦ4 having these fixed(B2, σ2) and wc.Lemma 5. (characterization of˜W(S)) Let S ⊆[n3 + s] such that |S| ≤s.

Theset ˜W(S) depends only upon the cardinality j = |S ∩[n1]|. More precisely, we have(i) ˜W(S) = ˜W(S ∩[n1]),(ii) If j = 0, ˜W(∅) = W(∅), and ˜wb ∈˜W(∅) has the following form:(5.2)˜wb =∗.

. .∗| {z }entries ≤n1(n1 + 1) ∗.

. .∗| {z }≤(n1+1)(n1 + 2) .

. .

(n3 + s −1)∗. .

.∗| {z }≤(n3+s−1)(n3 + s) ∗. .

.∗| {z }≤(n3+s). (iii) If j = 1, then ˜W({i}) = ˜W({1}) is obtained from ˜W(∅) by keeping only thewords ˜wb of the form (5.2) such that one of the stars ∗is set to its maximumand the maximum value of all the stars to its right is lowered by 1.

So any ˜wbhas the form˜wb =∗. .

.∗| {z }entries ≤n1(n1 + 1) ∗. .

.∗| {z }≤(n1+1)(n1 + 2) . .

. (n1 + h) ∗.

. .∗| {z }≤(n1+h)(n1 + h)∗.

. .∗| {z }≤(n1+h−1)(n1 + h + 1) .

. .

(n3 + s −1)∗. .

. ∗| {z }≤(n3+s−2)(n3 + s)∗.

. .∗| {z }≤(n3+s−1).

(5.3)(iv) If j ≥2, then ˜W(S) = ˜W({1, 2, . .

. , j}) is obtained from ˜W({1, 2, .

. .

, j −1}) bythe same construction as the one described in (iii).Proof.(i). First we show that ˜W(S) = ˜W(S∩[n1]).

From the definition of W(S), it is clearthat W(S) = W(S ∩[n1]). Moreover, if S = {s1 < .

. .

< sj < sj+1 < . .

. < sn},where sj ≤n1 and sj+1 > n1, since ΨS\[n1] is a bijection on RG(n1+n2 −t, n3 +s),preserving first occurrences and leaving all letters ≤n1 fixed, we must haveΨS\[n1](waW(S)) = waW(S).Therefore,wa ˜W(S) = ΨS(waW(S)) = ΨS∩[n1] ◦ΨS\[n1](waW(S))= ΨS∩[n1](waW(S ∩[n1])) = wa ˜W(S ∩[n1]).(ii).

If j = 0, Ψ∅is the identity map and˜W(∅) = W(∅) = {wb | 12 . .

.n1wb ∈RG(n1 + n2 −t, n3 + s)},13

in which typical elements (tails of RG-functions) are given by (5.2).(iii). If j = 1, suppose S = {i}, 1 ≤i ≤n1.

ThenW(S) = {wb | wawb ∈RG(n1 + n2 −t, n3 + s), and i ∈Supp(wb)}.Let wb ∈W(S) and suppose the rightmost occurrence of i lies in position p of wb,between the first occurrence of (n1 + h) and the first occurrence of (n1 + h + 1).Thus wawb has the formwawb =12 . .

.n1∗. .

.∗| {z }entries ≤n1(n1 + 1) ∗. .

.∗| {z }≤(n1+1)(n1 + 2) . .

. (n1 + h) ∗.

. .

∗| {z }≤(n1+h)i|{z}position(n1+p)∗. .

. ∗| {z }≤(n1+h),entries ̸=i(n1 + h + 1) .

. .

(n3 + s −1)∗. .

.∗| {z }≤(n3+s−1),̸=i(n3 + s) ∗. .

.∗| {z }≤(n3+s),̸=i. (5.4)Apply Ψ{i} = Ψn3+s ◦Ψn3+s−1 ◦.

. .

◦Ψi to wawb. The last occurrence of i inwawb (in position (n1 + p)) lies to the right of the first occurrence of (i + 1) (case(i) in the definition of Ψm), so it is changed to (i + 1) by Ψi, and any (i + 1) to itsright is changed to i.

Thus the last occurrence of (i + 1) in Ψi(wawb) now appearsin position (n1 + p), again to the right of the first occurrence of (i + 2). So all(i+2)’s to its right are changed to (i+1)′s by Ψi+1, the (i+1) in position (n1 +p)is switched to (i + 2), and every other letter remain fixed.The same argument applies until we reach Ψn1+h.

At this point in Ψn1+h−1 ◦. .

.◦Ψi(wawb), there is a (n1 +h) in position (n1 +p) and no occurrence of (n1 +h)to its right. This means that there are no letters (n1 + h) to the right of the firstoccurrence of (n1 + h + 1) (case (ii) in the definition of Ψm).

Hence Ψn1+h changesevery occurrence of (n1 + h + 1), except for the first one, to (n1 + h)′s, and fixeseverything else. Once again in the RG-function obtained, there are no occurrencesof (n1 + h + 1) to the right of the first occurrence of (n1 + h + 2).

It is clear thatby applying successively Ψn1+h+1, . .

. , Ψn3+s respectively, we will get Ψ{i}(wawb)exactly of the form (5.3).

This shows that the set defined in (iii) is equal to ˜W({i}).Note that the definition of the set ˜W({i}) is independent of the actual value of i,so ˜W({i}) = ˜W({1}). (iv).The proof is an easy induction based on the proof of (iii).Note that ifS = {s1 < s2 < .

. .

< sj}, sj ≤n1, the positions of the last occurrences ofs1, s2, . .

. , sj respectively in wawb correspond exactly to the positions of the starssuccessively fixed to their maximum in Ψs(wawb).□We can show now that Φ5 is well-defined.Let ˜wb ∈˜W({1, 2, .

. .

, j}). The letters of ˜wb can be divided into two categories:the fixed letters (first occurrences of (n1 + 1) up to (n3 + s), and j stars that werefixed to their maximum in the construction described in the preceding lemma), andthe free letters (corresponding to stars in the description of ˜wb in Lemma 5).

So inorder to be in ˜W({1, 2, . .

. , j}), a word ˜wb must have (n3 + s −n1 + j) fixed letters(appearing in some fixed relative order), and possibly some free letters, dependingon its length.On one hand, note that the fixed letters of ˜wb are always greater or equal totheir positions in ˜wb.

Indeed, we have already seen that the first occurrences were14

necessarily greater than their position p ((n3 + s) ≥(n1 + 1) > n2 ≥p). As for thej stars fixed to their maximum, the way to minimize their value in the constructionof Lemma 5 is to fix them successively by increasing order of their positions.

Then,if they all lie before the first occurrences of (n1 + 1) up to (n3 + s), the j-th starfixed will have minimum value (n1 −j + 1), and the rightmost position where itcan be located is, for example, the one in the following word:˜wb = ∗∗. .

. ∗| {z }entries≤n1n1 (n1 −1) .

. .

(n1 −j + 1)(n1 + 1)(n1 + 2) . .

. (n3 + s).But from the relations n3 ≥n2, t ≥0, and j ≤s, we deduce that its position p,p = | ˜wb| −(n3 + s −n1) = n1 + (n2 −n3) −t −s ≤n1 −j + 1.Therefore in that case, all fixed stars are greater or equal to their positions.

Moregenerally, if a fixed star is rather located to the right of a first occurrence, its valueis increased by one, so the letter remains greater or equal to its position.On the other hand, note that the allowed maxima for the free letters are alsogreater or equal to their positions in ˜wb. The same type of argument (with sameinequalities) applies.

Details are left to the reader.Now, the “involutive step” of Φ5 was to add or to delete a letter from ˜wb, andthis letter had the property of being strictly smaller than its position in ˜wb.If the involutive step deleted a letter from ˜wb (wa ˜wb ∈RG(n1 + n2 −t, n3 + s)),then it had to be one of its free letters because the fixed ones are greater or equal totheir positions. Therefore the new ˜w′b obtained is in the set ˜W({1, 2, .

. .

, j}) (withwa ˜w′b ∈RG(n1 + n2 −(t + 1), n3 + s)). Likewise, if the involutive step added aletter to ˜wb, the new letter is in the right range to be considered a free letter, andthe fixed letters (and their relative order) are not modified, so the new ˜w′b is in theset ˜W({1, 2, .

. .

, j}) as well (with wa ˜w′b ∈RG(n1 + n2 −(t −1), n3 + s)).Fixed points for Φ5. The fixed points of Φ5 correspond to the case i0 = h0 = ∞.Clearly,FixΦ5 = {((Bi, σi); π) ∈FixΦ4 | σ2 = Id and for S = [#blocks(π)] −Supp(wc),the word ˜wb in ΨS(wawb) = wa ˜wb has its i-th letter ≥i, ∀i}.6.

Combinatorial evaluation of Lq(Cn1(x)Cn2(x)Cn3(x)).An expression of Lq(Cn1(x)Cn2(x)Cn3(x)) can now be computed by ωq-countingof the remaining fixed points FixΦ5. More precisely, ((Bi, σi); π) ∈FixΦ5 if andonly ifFix.1 B1 = B3 = ∅,Fix.2 σi = Id for i = 1, 2, 3,Fix.3 wa (respectively wc) has all distinct letters and Supp(wa) ⊆Supp(wbwc)(respec-tively Supp(wc) ⊆Supp(wawb)),Fix.4 for S = [#blocks(π)]−Supp(wc), the word ˜wb = ˜b1˜b2 .

. .˜bn2−|B2| in ΨS(wawb) =wa ˜wb has all ˜bi’s ≥i, where ΨS was defined in Lemma 4.Clearly, for such elements, the weight (as was defined in (4.8)) reduces to(6.1)ωq((Bi, σi) : π) = (−1)|B2|qinv(B2)+rs(π)a|B2|+#blocks(π).15

By ωq-counting this fixed point set, we will show thatLq(Cn1(x)Cn2(x)Cn3(x)) =n1+n2−n3Xl=0lXs=0sXj=0an3+l(−1)l−sqL[n3]!qn2l −sq(6.2)n1jqn3 −n1 + ss −jqn2 −l + sn3 −n1 + sq[j]!q[n1 −j]!q[n3 −n2 + l]!qn1 + n2 −n3 −ljq,whereL =n12+l −s2+ j(−n3 −s + 1) +j2−s −j2−(s −j)(n3 −n1 + j)+n2 −l + s2+ (n3 −n1 + s)(n3 −n2 + l) + j(n3 −n2 + l).Evaluating the s-sum by the q-binomial theorem (which has a simple bijectiveproof) gives the right-hand side of (4.5) and thus Theorem 3.The main difficulty here is to transpose the condition Fix.4 into the ωq-counting.Using Lemmas 4 and 5, we will see that this corresponds to the q-counting of somespecial sets of RG-functions according to the statistic ls, which is the object ofLemma 6.Let us first group the elements of FixΦ5 by powers of a. The power of a rangesfrom a minimum of n3 (expressing the fact that wc has n3 distinct letters) to amaximum of (n1 + n2) (being the maximum value of {max(Supp(wawb)) + |B2|}).Now,Lq(Cn1Cn2Cn3) =n1+n2−n3Xl=0an3+llXs=0(−1)l−sq(l−s2 )n2l −sqXπqrs(π),(6.3)where the last sum ranges over all partitions π corresponding to ((Bi, σi); π) ∈FixΦ5 such that #blocks(π) = (n3 + s) and B2 is any fixed subset of {2} × [n2] ofcardinality (l −s).

The s-sum is the generating function for the subsets B2, as wasestablished in §2. Butrs(π) = rs(w) = rs(wc) + rs(w)|wawb= rs(wc) + ls(wa ˜wb) −Xu∈([n3+s]−Supp(wc))(n3 + s −u).

(6.4)Note that for any fixed set Supp(wc), there are no constraints on the positions ofthe letters in wc, so rs(wc) is simply the number of inversions of the word wc, whosedistribution is mahonian (i.e. the generating function equals [n3]!q).

From Lemma5, we also know that the possible choices for ˜wb only depend on the cardinality j ofthe set ([n3 + s] −Supp(wc)) ∩[n1], not on the actual set Supp(wc) itself. Hence,if we letFix ˜W(j) = { ˜wb | ˜wb = ˜b1 .

. .˜bn2−l+s ∈˜W({1, 2, .

. .

, j}) and ˜bi ≥i, ∀i},16

where ˜W({1, 2, . .

. , j}) was characterized in Lemma 5, we get that the last sum onthe right-hand side of (6.3) equalsXπqrs(π) = [n3]!qsXj=0X˜wb∈F ix ˜W(j)qls(wa ˜wb)qj(−n3−s+1)+(j2)n1jqq−(s−j2 )−(s−j)(n3−n1+j)n3 −n1 + ss −jq.

(6.5)Finally, we show thatLemma 6. If wa = 12 .

. .n1, ˜wb = ˜b1 .

. .˜bn2−l+s and max (Supp(wa ˜wb)) = n3 +s,then(6.6)X˜wb∈F ix ˜W(j)qls(wa ˜wb) = qAn2 −l + sn3 −n1 + sq[j]!q[n1 −j]!q[n3 −n2 + l]!qn1 + n2 −n3 −ljq,whereA =n12+n2 −l + s2+ (n3 −n1 + s + j)(n3 −n2 + l).Proof.

Note that the statistic ls of any RG-function is just the sum of the valuesof the letters minus one, sols(wa ˜wb) =n12+n2−l+sXi=1(˜bi −1).To visualize more easily where the various factors of (6.6) come from, let usencode ˜wb as a 0–1 tableau ϕ in the following manner: start with a (n3 +s)×(n2 −l + s) rectangular Ferrers diagram. Fill it with a 1 in position j (from bottom totop) of column i if ˜bi = j, and with 0’s elsewhere.

For example, if (n3 + s) = 8,n1 = 6 = (n2 −l + s) and ˜wb = 175787, ϕ is the 0–1 tableau on the left of Figure 3.ϕ =000010010101000000001000000000000000000000100000−→˜ϕtyp =00001∗01∗∗0∗∗0∗∗0∗∗0∗∗0∗0∗∗∗0∗∗0∗Figure 3: Encoding of ˜wb as a 0–1 tableauObviously, we have P(˜bi −1) = inv(ϕ). Note also that the 0’s in the shadedstaircase shape of ϕ in Figure 3 always count as inversions, expressing the factthat ˜bi ≥i.

They account for the factor q(n2−l+s2) in (6.6). We can therefore dropthem from ϕ without loss of generality, and compute the inversion number of thereduced 0–1 tableau ˜ϕ.

We now use Lemma 5 to characterize the possible fillingsof ˜ϕ according to j.17

Case 1: j = 0. From Lemma 5 (ii), ˜wb ∈˜W(∅) simply means that it is the tailof an RG-function.

Thus the only restrictions on ˜wb are that the first occurrencesof (n1 + 1), (n1 + 2), . .

. , (n3 + s) appear in the right order.If we set x = (n3 + s), y = n1, and z = (n2 −l + s), in the context of 0–1tableaux, we want to q-count all 0–1 tableaux ˜ϕ with z columns of lengths x, (x −1), .

. .

, (x −z + 1) respectively, such that when we look at the top (x −y) rows of˜ϕ from left to right, the leftmost 1 in any row must always occur before the ones inthe rows above it. Grouping the tableaux according to these leftmost occurrencesof 1’s, we get “typical” 0–1 tableaux ˜ϕtyp, corresponding exactly to the typicalwords ˜wb described in (5.2) of Lemma 5.

For instance, the typical 0–1 tableau ˜ϕtypcontaining our previous example is illustrated in Figure 3 (stars ∗correspond topossible positions of 1’s). Carrying out the q-counting, observe that1.1 Each column containing a number m of stars contributes a factor [m]q to theq-counting of inversions.

No matter which (x −y) columns are chosen to befirst occurrences of upper 1’s, the number of stars in the remaining columns isy, (y −1), . .

. and (x −z + 1) respectively, contributing to an overall factor of[y]!q[x −z]!q=[n1]!q[n3 −n2 + l]!q.1.2 The 0’s below the leftmost occurrences of upper 1’s (shaded in Figure 3) form apartition µ with (x−y) parts of length at least (x−z) and at most y, determinedby the positions of the first occurrences.

Summing over all possible choices, itcontributes a factorq(x−y)(x−z)zx −yq= q(n3−n1+s)(n3−n2+l)n2 −l + sn3 −n1 + sq.Case 2: j ≥1. Recall that Lemma 5 (iii) and (iv) provides a method to constructall the elements of ˜W({1, 2, .

. .

, j}) uniquely from ˜W(∅). In the 0–1 tableau con-text, if we extract only the cells filled with stars in ˜ϕtyp (hence obtaining a tableau˜ψtyp with (n1 + n2 −n3 −l) columns of lengths n1, (n1 −1), .

. .

, (n3 −n2 + l + 1)respectively), the manipulation described in Lemma 5 (iii) corresponds to replacingthe top star of a column by a 1, and all the top stars to its right and the stars belowit by a 0. Repeating this procedure j times and reinserting the columns of ˜ψtyp in˜ϕtyp yields to “typical” 0–1 tableaux that correspond to the elements of Fix ˜W(j).For example, Figure 4 shows the above manipulations on the third and the firstcolumns respectively of the stars extracted from ˜ϕtyp of Figure 3.∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗−→∗∗10∗∗0∗∗∗0∗∗∗0∗∗∗−→10100∗000∗0∗0∗00∗0=⇒00001001010010∗00∗00∗0000∗000∗000Figure 4: Manipulations of Lemma 5 in the context of 0–1 tableaux18

Proceeding to q-counting, the part (1.2) of case 1 is left unchanged and the part(1.1) is replaced by the contribution of the different choices of ˜ψtyp. But observethat2.1 All the 0–1 tableaux ˜ψtyp such that a star has been changed to a 1 in columnsc1, c2, .

. .

and cj contribute to [j]!q times the q-counting of the 0–1 tableaux ˜ψtypsuch that this procedure was done in increasing order of the ci’s. Therefore, wecan restrict to this latter case.

This explains the factor [j]!q in (6.6).2.2 It is not hard to see that in that case, we are q-counting all 0–1 tableaux ˜ψcontaining (n1 +n2 −n3 −l) columns of lengths n1, (n1 −1), . .

. , (n3 −n2 +l +1)respectively, such that when we look at the top j rows from right to left, therightmost 1 in any row has to occur before the ones in the rows above it.

Thereis a simple weight-preserving bijection between this class of 0–1 tableaux and theone that was q-counted in case 1, for x = n1, y = (n1−j) and z = (n1+n2−n3−l)(this class is defined by interchanging “left” and “right”). Given ˜ψ in the firstclass of 0–1 tableaux, just leave all the 1’s below the j-th row fixed and “reversethe order” of the 1’s in the top j rows, within the columns where they appear.Figure 5 gives an example of this for j = 2.011000000001000100000010000100000←→001001010000000100000010000100000Figure 5: Bijection between two classes of 0–1 tableauxThis is clearly an involution that preserves the number of 0’s below 1’s.

There-fore, we can simply use case 1 to compute the q-contribution of the ˜ψ’s. We obtainqj(n3−n2+l)[n1 −j]!q[n3 −n2 + l]!qn1 + n2 −n3 −ljq.□Finally, putting together Lemma 6, identities (6.5) and (6.3) yields identity (6.2),thus completing the proof of Theorem 3.Note that if we take n2 = 0 and apply Φ1 and Φ2 to Lq(n1, 0, n3) (assumingn3 ≥n1), the set FixΦ2 is easily seen to be empty unless n1 = n3, in which case itcan be proven to be ωq-counted by the right-hand side of (4.2) (n = n1, m = n3),thus proving orthogonality and Theorem 2.

These results can also be obtained byapplying directly Φ3 and Φ4 to the set Lq(n1, 0, n3), assuming this time n1 ≥n3.We also have a weight-preserving sign-reversing involution proving orthogonalitywhen the q-statistic for the moments is taken to be lb instead of rs, but we do notknow how to generalize it to the linearization problem.Corollary 7. Let n1 ≥n2 ≥.

. .

≥nk. The coefficient of the lowest power of a,an1 in Lq(Cn1Cn2 .

. .

Cnk) is a polynomial in q with positive coefficients.19

Proof. The proof of Theorem 3 can be generalized to a product of k q-Charlierpolynomials, any additional color being treated as was color 2, the middle color.

Itis easy to see then that the fixed points contributing to the lowest power of a musthave all Bi = ∅, and therefore have all positive weights.□Corollary 8. Let n3 ≥n1 ≥n2.

The coefficient of an1+n2−i in Lq(Cn1Cn2Cn3) isequal to (q −1)n1+n2−n3−2i times the coefficient of an3+i, for 0 ≤i ≤⌊(n1 + n2 −n3)/2⌋.Our proof of Corollary 8 is analytical, but we would like to have a combinatorialexplanation of this “symmetry” property.Note that FixΦ5 is not an optimal set of fixed points, in the sense that there arestill some terms that cancel each other when we proceed to ωq-counting of FixΦ5.For example, for n1 = n2 = n3 = 2, the two elements of FixΦ5 such that B2 = {2},w = 12121 and B2 = ∅, w = 123123 have weight −a3q3 and a3q3 respectively.However, we do not believe that an attempt to reduce FixΦ5 would be worthwhile.Corollary 9. Let n1 ≥n2 ≥.

. .

≥nk. If q = 1 + r, Lq(Cn1Cn2 .

. .

Cnk) is apolynomial in r with positive coefficients.7. The classical q-Charlier polynomials.We contrast the results of the previous sections with those for the classical q-Charlier polynomials [11, p.187](7.1)cn(x; a; q) =2φ1(q−n, x; 0; q, −qn+1/a).The monic form of these polynomials, ccn(x; a; q) satisfiesccn+1(x; a; q) = (x −bn)ccn(x; a; q) −λnccn−1(x; a; q),whereλn = −aq1−2n(1 −q−n)(1 + aq−n),bn = aq−1−2n + q−n + aq−2n −aq−n.A calculation (see [11, p.187]) shows that the moments for these polynomials areµn =nYi=1(1 + aq−i).We need to rescale x and a so that bn and λn are q-analogues of a + n and anrespectively.

If we put x = 1 + z(1 −q), and multiply a by (1 −q), and call theresulting monic polynomials ˆCn(z; a; q), the explicit formula from (7.1) is(7.2)ˆCn(z; a; q) = q−n2nXk=0nkq(−a)n−kq(k+12 )k−1Yi=0(qiz −[i]q)The three term recurrence relation coefficients are(7.3) bn = q−n[n]q(1+a(1−q)q−n)+aq−1−2n,λn = aq1−3n[n]q(1+a(1−q)q−n).20

A calculation using the measure in [11, p.187] gives(7.4)µn =nXj=1q−(j2)−nS1/q(n, j)aj.Again we find q-Stirling numbers for the moments. Zeng [24] has also derived (7.2)and (7.3) from the continued fraction for the moment generating function.We see that the individual terms in (7.3) do not have constant sign.

This meansthat the Viennot theory must involve a sign-reversing involution for its combinato-rial versions of (7.3) and (7.4). Nonetheless we can give combinatorial interpreta-tions of (7.2) and (7.4), but have no perfect analog of Theorem 3.Acknowledgement.

Theorems 1 and 2 were originally found in joint work withMourad Ismail.References1. W. Al Salam and D. Verma, private communication (1988).2.

R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that generalizeJacobi polynomials, Memoirs Amer. Math.

Soc. 319 (1985).3.

L. Butler, The q-log concavity of q-binomial coefficients, J. of Comb. Theory A 54 (1990),53-62.4.

T.S. Chihara, An Introduction to orthogonal polynomials, Gordon and Breach, New York,1978.5.

A. de M´edicis, Aspects combinatoires des nombres de Stirling, des polynˆomes orthogonauxde Sheffer et de leurs q-analogues, ISBN 2-89276-114-X, vol. 13, Publications du LACIM,UQAM, Montr´eal, 1993.6.

A. de M´edicis and P. Leroux, A unified combinatorial approach for q-(and p, q-)Stirling num-bers, J. of Stat. Planning and Inference 34 (1993), 89–105.7.

M. de Sainte-Catherine and G. Viennot, Combinatorial interpretation of integrals of productsof Hermite, Laguerre and Tchebycheffpolynomials, Polynˆomes Orthogonaux et Applications,Lecture Notes in Math., vol. 1171, Springer-Verlag, 1985, pp.

120–128.8. D. Foata, Combinatoire des identit´es sur les polynˆomes orthogonaux, Internat.

Congress Math. (1983), Warshaw, Poland.9.

D. Foata and D. Zeilberger, Laguerre polynomials, weighted derangements and positivity,SIAM J. Discrete Math. 1 (1988), 425–433.10., Linearization coefficients for the Jacobi polynomials, Actes 16e S´eminaire Lotharingien(1987), I.R.M.A., Strasbourg, 73–86.11.

G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of mathematics andits applications, vol. 35, Cambridge University Press, New York, 1990.12.

I. Gessel, Generalized rook polynomials and orthogonal polynomials, q-Series and Partitions(D. Stanton, ed. ), IMA Volumes in Math.

and its Appl., vol. 18, Springer-Verlag, New York,1989, pp.

159–176.13. H.W.

Gould, The q-Stirling Numbers of First and Second Kinds, Duke Math. J.

28 (1961),281–289.14. M. Ismail and D. Stanton, On the Askey-Wilson and Rogers polynomials, Can.

J. Math.

XL,no.5 (1988), 1025–1045.15. M. Ismail, D. Stanton and X.G.

Viennot, The combinatorics of q-Hermite polynomials and theAskey-Wilson integral, Europ. J. Comb.

8 (1987), 379–392.16. J. Labelle and Y.N.

Yeh, The combinatorics of Laguerre, Charlier and Hermite polynomials,Studies in Applied Math. 80 (1989), 25–36.17., Combinatorial proofs of some limit formulas involving orthogonal polynomials, Dis-crete Math.

79 (1989), 77–93.18. P. Leroux, Reduced matrices and q-log concavity properties of q-Stirling numbers, J. of Comb.Theory A 54 (1990), 64–84.21

19. I.G.

Macdonald, Symmetric functions and Hall polynomials, Clarendon Press, Oxford, 1979.20. X.G.

Viennot, Une Th´eorie Combinatoire des Polynˆomes Orthogonaux, Lecture Notes, Pub-lications du LACIM, UQAM, Montr´eal, 1983.21. M. Wachs and D. White, p, q-Stirling Numbers and Set Partition Statistic, J. Comb.

TheorySer. A 56 (1991), 27–46.22.

D. White, Interpolating Set Partition Statistics, preprint (1992).23. J. Zeng, Weighted derangements and the linearization coefficients of orthogonal Sheffer poly-nomials, Proc.

London Math. Soc.

65 (1992), 1–22.24., The q-Stirling numbers, continued fractions and the q-Charlier and q-Laguerre poly-nomials, preprint (1993).School of Mathematics, University of Minnesota, Minneapolis, MN 55455.22

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