Generalized Binomial Coefficients and the Subset-Subspace Problem

ADVANCES IN APPLIED MATHEMATICS 21, 228᎐240Ž. 1998 ARTICLE NO. AM980598

Generalized Binomial Coefficients and the Subset᎐Subspace Problem

John Konvalina

Department of Mathematics, Uni¨ersity of Nebraska at Omaha, Omaha, Nebraska View metadata, citation and similar papers at core.ac.uk68182-0243 brought to you by CORE Received April 8, 1998; accepted May 5, 1998 provided by Elsevier - Publisher Connector

Generalized binomial coefficients of the first and second kind are defined in terms of object selection with and without repetition from weighted boxes. The combinatorial definition unifies the binomial coefficients, the Gaussian coefficients, and the Stirling numbers and their recurrence relations under a common interpretation. Combinatorial proofs for some Gaussian coefficient identities are derived and shown to reduce to the ordinary binomial coefficients when q s 1. This approach provides a different perspective on the subset᎐subspace analogy problem. Generating function relations for the generalized binomial coefficients are derived by formal methods. ᮊ 1998 Academic Press

1. INTRODUCTION

The binomial coefficients, the Gaussian coefficients, and the Stirling numbers are three fundamental classes of numbers arising frequently in enumerative combinatoricsŽ see, for example,wx 14. . The binomial coefficients have a well-known interpretation in terms of subset selection with or without repetition. The Gaussian coefficients have a classical interpretation related to counting subspaces of a finite vector space, as well as an interpretation in terms of row-reduced echelon form matrices. The Stirling numbers of the first kind count cycles in the cycle decomposition of permutations, while the Stirling numbers of the second kind count set partitions. Despite the various combinatorial interpretations of these classical numbers, the question still remains: given the algebraic similarities of these classes is there a unifying combinatorial generalization that captures the intrinsic properties of these numbers? The algebraic unification was started by Rotawx 12, 13 , who gave the general definition of the characteristic polynomial of a geometric lattice. The binomial coefficients, the Gauss-

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0196-8858r98 $25.00 Copyright ᮊ 1998 by Academic Press All rights of reproduction in any form reserved. GENERALIZED BINOMIAL COEFFICIENTS 229 ian coefficients, and the Stirling numbers appear as the coefficients in the characteristic polynomials of the following geometric lattices: subsets, subspaces, and set partitions, respectively. In this algebraic interpretation the generalized coefficients vary with the lattice and are called the Whitney numbers of the first and second kind Žnumbers of the first kind are associated with the characteristic polynomial, and numbers of the second kind with the rank polynomial, seewx 1 and w 14 x. . In this paper we will define two kinds of generalized binomial coeffi- cientsŽ. the combinatorial analogues of the Whitney numbers and unify combinatorially the basic properties of the binomial coefficients, the Gaussian coefficients, and the Stirling numbers. Specifically, the generalized binomial coefficients of the first kind will be defined in terms of selections from weighted boxes with the following constraints: no repeated boxes are selected and only one object is chosen from each selected box Ž.i.e., we are counting choice functions on weighted boxes . The generalized binomial coefficients of the second kind are similarly defined except box repetition is allowed. By varying the weights of the boxes we obtain as special cases the classical combinatorial numbers. The generalized coeffi- n cients of the first kind include the ordinary binomial coefficientsŽ.Žk box weights are constant. , the Gaussian coefficients of the first kind, denoted

k n qž/2 ž/k q

Ž.box weights are exponential , and the Stirling numbers of the first kind, n denotedwxk Ž. box weights are linear . These numbers can be interpreted combinatorially in terms of general selection without repetition. The generalized coefficients of the second kind can be interpreted in terms of selection with repetition allowed. These coefficients include the binomial coefficients of the second kindŽ. combinations with repetition , de- RŽ. Žn q k y 1 . noted here as Cn,k, ork , the Gaussian coefficients of the n second kind, denotedŽ.k q , and the Stirling numbers of the second kind, n denotedÄ4k . Besides providing a combinatorial foundation for these classical numbers, we also hope to shed some significant light on the subset᎐subspace problemŽ seewxwx 6 , 9 , and wx 3. . The traditional approach to the subset᎐subspace problem has been to draw the following analogy: the binomial n coefficientŽ.k counts k-subsets of an n-set, while the analogous Gaussian n coefficientŽ.k q counts the number of k-dimensional subspaces of an n-dimensional finite vector space over the field of q elements. The implication from this analogy is that the Gaussian coefficients and related identities tend to the analogous identities for the ordinary binomial coefficients as q approaches 1. The proofs are often algebraic or mimic 230 JOHN KONVALINA subset proofs. But what is the combinatorial reason for the striking parallels between the Gaussian coefficients and the binomial coefficients? We will show that interpreting the Gaussian coefficients as generalized binomial coefficients of the second kind Ž.combinations with repetition reveals the combinatorial connections between not only the binomial coefficients and the Gaussian coefficients, but the Stirling numbers as well. n Thus, the ordinary Gaussian coefficientŽ.k q tends to be an algebraic n generalization of the binomial coefficient of the first kindŽ.k , and a combinatorial generalization of the binomial coefficient of the Ž.n q k y 1 second kindk . Before defining the generalized binomial coefficients we summarize some of the fundamental properties of the combinatorial numbers under discussion for future referenceŽ see, for example,wx 4 , wx 7 , w 11 x , w 14 x , and wx10. :

Binomial coefficients:

nny1ny1 nn sq,s.1Ž. ž/ž/ž/ž/ž/kkky1knyk Combinations with repetition:

R R R C Ž.n, k s C Žn y 1, k .q C Žn, k y 1. . Ž. 2 Gaussian coefficients:

nnky1ny1nn sq q , s , ž/ž/ž/ž/ž/kkkqqqqqy1knyk Ž.3 nny1nykny1 sqq . ž/kkqq ž / ž ky1 / q Stirling numbersŽ. first kind :

nny1ny1 sŽ.ny1q.4Ž. kkky1 Stirling numbersŽ. second kind : nny1ny1 sk q .5Ž. ½5kkk ½ 5 ½y1 5 What is the combinatorial significance of the coefficients in these recurrence relations? The recurrence relations are strikingly similar, but the combinatorial interpretations of these numbers are diverse. What are the generalized binomial coefficients that unify these numbers and relations? GENERALIZED BINOMIAL COEFFICIENTS 231

2. THE GENERALIZED BINOMIAL COEFFICIENTS

Suppose we are given n distinct labeled boxes with box i Ž.1 F i F n containing wi distinct objects, where we have 1 F w12F w F иии F wn. The number wi is called the weight of box i. We assume that collectively the objects are distinct. Also, let w denote the sequence of given weights: wsŽ.w12,w,...,wn. The generalized binomial coefficient of the first kind with weight w, n denoted Ck Ž.w , is defined as the number of ways to select k objects from k of the n boxes with the following constraints: select distinct boxesŽ order not important.Ž and choose one object from each selected box i.e., we are counting choice functions with domain size k.. Suppose we choose k boxes i22- i - иии - ik. Then by the fundamental counting principle there are ww иии wk-selections. Thus, summing over all ways of choosing k ii12 ik distinct boxes we have

C n w ww иии w .6 kiŽ.s Ý 12iik Ž. 1Fi12-i-иии -ikFn

The generalized binomial coefficient of the second kind with weight w, n denoted Sk Ž.w , is defined as the number of ways to select k objects from k boxesŽ. not necessarily distinct with the following constraint: choose one object from each selected boxŽ i.e., sampling with replacement and box n repetition allowed. . The formula for Sk Ž.w is similar to Ž. 6 but this time we sum over all ways of choosing k numbers fromÄ4 1, 2, . . . , n with repetition allowed, that is, over 1 F i12F i F иии F ikF n. Thus, Snw ww иии w .7 kiŽ.s Ý 12iik Ž. 1Fi12FiFиии FikFn

First we give combinatorial proofs for the general recurrence relations n n for CkkŽ.w and S Ž.w . They are generalizations of the recurrence relations for the ordinary binomial coefficients of the first and second kindŽ see Eqs. Ž.1 and Ž.. 2 .

THEOREM 1. Let w s Ž.w12, w ,...,wn with 1 F w12F w F иии F wn. Then

n ny1 ny1 I. CkkŽ.w s C Ž.w q wC nky1 Ž.w ,8Ž. n ny1 n II. SkkŽ.w s S Ž.w q wS nky1 Ž.w.9Ž.

Proof. Ž.I Given a k-selection from distinct boxes, either the last box Ž.box n is selected or not selected. If not selected, we have a k-selection from Ž.n y 1 boxes; otherwise, there are wn objects to choose from box n 232 JOHN KONVALINA and Ž.k y 1 other selections from the remaining Ž.n y 1 boxes. Thus, Ž. 8 follows. Ž.II Given a k-selection with box repetition allowed, either the last box selected was box n or not. If not we have a k-selection with box repetition allowed from Ž.n y 1 boxes; otherwise, there are wn objects to choose from box n and Ž.k y 1 other selections with box repetition allowed on all n boxes. Thus,Ž. 9 follows. Next we derive the recurrence relationsŽ.Ž. 1 ᎐ 5 as corollaries of this n n theorem. We will also use the notation CwkiŽ.and Sw kiŽ.to denote the generalized binomial coefficients with weight wi for 1 F i F n. We can n n interpret CwkiŽ.and Sw kiŽ.as strict selection and weak selection, respectively. Observe that if wi s 1 for all i, then the generalized coefficientsŽ. 6 andŽ. 7 reduce to the ordinary binomial coefficients of the first and second kind:

nnk1 CnnŽ.1,SŽ.1qy. kksž/kks ž /

n Note.InSk Ž.1 if we replace n by n y k q 1 we obtainŽ see Gaussian coefficients below.

n S nykq1 Ž.1. Ž. ž/k s k )

If wi s i we obtain the Stirling numbers of the first and second kind. n The Stirling number of the first kindwxk has a well-known representation as the sum of all products of n y k different integers taken from Ä41, 2, . . . , n y 1Ž seewx 4 and wx 8. . Thus, we have the relations

nnny1nq1 sCin kkŽ.,Ci Ž.s .10Ž. kny q1yk

n Similarly, the Stirling number of the second kindÄ4k can be expressed as the sum of all products of n y k integers taken fromÄ4 1, 2, . . . , k with repetition allowed. Thus,

n SiknŽ.,Si Ž. n q k .11Ž. ½5k snykks½5n

In terms of the generalized binomial coefficient, the Stirling number of n the first kindwxk can be interpreted as the number of ways to select n y k distinct objects from n y k distinct boxesŽ. one object selected per box chosen from a total of n y 1 boxes with box i having weight i. Similarly, GENERALIZED BINOMIAL COEFFICIENTS 233

n the Stirling number of the second kindÄ4k is the number of ways to select n y k objects from n y k boxes with box repetition allowedŽ one object selected per box. chosen from a total of k boxes with box i having weight i.

Applying the theorem with wi s i andŽ.Ž. 10 , 11 we obtain the recurrence relationsŽ. 4 and Ž. 5 :

COROLLARY 1. If wi s i for i G 1, then

ny1 ny2 ny2 I. CnyknŽ.i s C ykn Ž.i q Žn y 1 .C yky1 Ž.i , nny1 ny1 sqŽ.ny1; kky1 k

k ky1 k II. SnyknŽ.i s S ykn Ž.i q kS yky1 Ž.i , nny1 ny1 sqk. ½5kk ½y1 5 ½k 5

Note. The combinatorial interpretation of the coefficient Ž.Ž.n y 1in4 is now clear. It is the weight of the last box in the Ž.n y k strict selection used for the Stirling numbers of the first kind. Similarly, the coefficient k inŽ. 5 is the weight of the last box in the Žn y k . weak selection used for the Stirling numbers of the second kind.

3. COMBINATORIAL PROOFS FOR GAUSSIAN COEFFICIENTS

iy1 If q is a positive integer and wi s q for 1 F i F n, then the generalized binomial coefficients become the Gaussian coefficients of the first and second kind. We will study the Gaussian coefficients of the second kind in detail since these turn out to be the ordinary Gaussian coefficients. n If q is a prime power, then the Gaussian coefficientŽ.k q counts the number of k-dimensional subspaces of an n-dimensional finite vector space over the finite field with q elements. If q is an integer Ž.q ) 1 , then the Gaussian coefficient can also be interpreted as the number of k = n row-reduced echelon matrices with no zero rows. However, neither one of these two interpretations seems to adequately explain why the Gaussian coefficients tend to the ordinary binomial coefficients as q approaches 1. In both cases q s 1 results in an ill-defined interpretation. We will try to gain some insight into this problem from a purely combinatorial perspective. 234 JOHN KONVALINA

n The Gaussian coefficientŽ.k q has a known representation as the sum of all products of k integers taken fromÄ 1, q, q 2,...,qnyk4 with repetition allowed. Thus, it can be expressed and interpreted in terms of the iy1 generalized binomial of the second kind with wi s q Žusing vector notation we let w s q.:

n i i иии i s Ý q12q q q k ž/k q 0Fi12FiFиииFikFnyk

Ži1.Ži1.иии Ži 1. s Ý q1yq2yqq ky 1Fi12Fi FиииFikFnykq1

nykq1 sSk Ž.q.12Ž.

n The Gaussian coefficients of the first kind Ck Ž.q can be expressed in terms Ž.n Ž ofkqm note the substitution j s imy m q 1 for 1 F m F k when going from the strict to the weak inequality. :

n Ži1y1 .qŽi2y1 .qиииqŽiky1. CkŽ.q s Ý q 1Fi12-i-иии-ikFn

Žj1.Žj1.иии Žj 1. 1 2 иии Žk 1. s Ý q1yq2yqq ky q q q q y 1Fj12Fj FиииFjkFnykq1 k n sqž/2 . ž/k q

Ž. Ž.n nykq1 Ž. Ž. Note.Ifqs1 in 12 we obtain k s Sk 1 ; see Eq. ) . Thus, the combinatorial interpretation of the Gaussian coefficient as a generalized binomial coefficient of the second kind reduces to the ordinary binomial coefficient when q is replaced by 1. Now we use our combinatorial interpretation to derive some classical Gaussian coefficient identities. There are typically two recurrence relations for the Gaussian coefficients of the second kind. The first is an immediate consequence of Theorem 1 and the relation fromŽ. 12 :

n nykq1 s Sk Ž.q ž/k q

Žfor other proofs seew 2, p. 35 xwxwx , 5 , 6 , and w 10, p. 295 x. . GENERALIZED BINOMIAL COEFFICIENTS 235

iy1 COROLLARY 2. If wi s q for i G 1 Ž.or w s q , then

nykq1 nyk nyk nykq1 SkkŽ.q s S Ž.q q q S ky1Ž.q ,

nny1nykny1 sqq . ž/ž/kkqq ž/ ky1 q

The symmetry property of the binomial coefficients Ž.nn Ž .has a kns yk simple combinatorial proof: a selection of k objects chosen from an n-set determines a selection of n y k objects not chosen. However, this proof does not seem to generalize to the Gaussian coefficients for all q G 1. The symmetry property of the Gaussian coefficientsŽ. 3 can be proved algebraically as a generalization of the symmetry property of the binomial coefficients or by noting that the lattice of subspaces of a finite vector space is self-dualŽ. when q is a prime power . Surprisingly, we will now show that this symmetry property can be obtained as a combinatorial generalization of the symmetry property of the combinations with repeti- R tion numbersŽ. binomial coefficients of the second kind : CnŽ.ykq1, k R sCkŽ.q1, n y k . Observe that algebraically this symmetric identity reduces to the symmetry property of the ordinary binomial coefficientsŽ. 1 .

THEOREM 2. ŽCombinatorial proof of symmetry property of Gaussian coefficients..

nn s for all q G 1. ž/knqq žyk /

Proof. We must showŽŽ. by 12 above .

nykq1 kq1 SknŽ.q s S yk Ž.q for any integer q G 1, that is,

Ý qŽi1y1 .qŽi2y1 .qиииqŽiky1. 1Fi12Fi FиииFikFnykq1

Žj1.Žj1.иии Žj 1. s Ý q1yq2yqq nyky . иии 1Fj12Fj F FjnykFkq1

If we replace k on the left-hand side by Ž.n y k we obtain the dual coefficient on the right-hand side. Also, when q s 1 we obtain the symmetry property for the combinations with repetition numbers. Now we detail the combinatorial proof. A selection of k boxes with repetition from the setÄ4 1, 2, . . . , n y k q 1 , say 1 F i12F i F иии F ikF n y k q 1, can be uniquely associated with an n y k selection with repetitionŽ the dual or 236 JOHN KONVALINA conjugate selection. from the setÄ4 1, 2, . . . , k q 1 , say 1 F j12F j F иии F jnyk Fkq1, by a sequence of k 0s and n y k 1s as follows. The number of 0s sandwiched between two 1s represents the number of times a respective box is chosen from the setÄ4 1, 2, . . . , n y k q 1 . In other words the 1s act as separators. For example, if n s 7 and k s 4, then the sequenceŽ. read from left to right 0110010 represents the selection Ä41, 3, 3, 4 . Thus, the number of 0s to the left of the first 1 is the number of times box 1 is selected, while the number of 0s between the first and second 1 is the number of times box 2 is selected, etc. The dual selection is defined by interchanging the roles of 0 and 1 and then reading the sequence from right to left. Thus, in the dual selection the 0s act as separators. In our example the dual selection with n y k s 3is2,4,4.Ä4 We will now show that a selection and its dual have the same q-weight. Consider our binary representation of a general k-selection with repetition fromÄ4 1, 2, . . . , n y k q 1 , where we have grouped the consecutive blocks of 0s and 1s,

00 иии 011иии 100иии 011иии 1 иии 00 иии 011иии 100иии 0 , ^`_^`_^`_^`_ ^`_^`_^`_ abab1122 abry1ry1 ar and where a1 G 0, arijG 0 and b G 1, a G 1 for 1 F i F r y 1, and 2 F j r ry1 Fry1. We have Ýis1 aiis k for the selection and Ý s1bis n y k for the dual. Observe that a1 is the number of times box 1 is chosen, while a2 is the number of times box b1 q 1 is chosen, and in general ai is the Ž.иии Ž number of times box b12q b q qbiy1q 1 is chosen note that the иии b1qb2qqbiy1 weight of this box is q with the convention b0s 0.. In the dual bry1 is the number of times box arrq 1 is chosen, while b y2 is the number of times box arrq a y1q 1 is chosen, and in general biis Ž.иии the number of times box arrq a y1q qaiq11q 1 is chosen. Let S s riy1ry1 r Ýis1aijÝs1bjand S2s Ýis1bijÝ siq1aj. But S11s Ý Fj-iFrijabs Ý1F i- jF rijba sS2, so the q-weights of a selection and its dual are the S S same; that is, q 1 s q 2 . Hence, summing over all k-selections we have Ý qŽi1y1 .qŽi2y1 .qиииqŽiky1. 1Fi12Fi FиииFikFnykq1

S s Ý q1 1Fi12FiFиииFikFnykq1

S s Ý q2 иии 1Fj12FjFFjnykFkq1 Žj1.Žj1.иии Žj 1. s Ý q1yq2yqq nyky . иии 1Fj12Fj F FjnykFkq1 GENERALIZED BINOMIAL COEFFICIENTS 237

From Corollary 2 and Theorem 2 we obtain the second recurrence relation for the Gaussian coefficients:

iy1 COROLLARY 3. If wi s q for i G 1, then

nykq1 k nyk nykq1 SkkkŽ.q s q S Ž.q q S y1 Ž.q ,

nnky1ny1 sq q . ž/ž/ž/kkkqqqy1

Next, we give combinatorial proofs for a couple of Gaussian coefficient identities that are q-analogs of binomial coefficient identitiesŽ seew 2, p. 37x. . For example, we can prove the known result

nnq1 jjmqjnqmq1y1jnq1 ÝÝqs or qSmmŽ.qsSq1 Ž.q ž/mqž/m1q js0 q js1 as follows. The number of Ž.m q 1 -selections with box repetition from Ä41, 2, . . . , n q 1 having j as the last box selectedŽ. 1 F j F n q 1 is the weight of box j times the number of m-selections with box repetition from Ä41, 2, . . . , j . Thus, the identity follows. The q-Vandermonde identity is

h Žnyk.Ž hyk. nm nqm Ý q s . ž/žkhqqkh / ž / q ks0 y To prove this we must show

h Žnyk.Ž hyk. nq1yk Žmq1.yŽhyk. nqmq1yh ÝqSkhŽ.qSykhŽ.qsSŽ.q. ks0 The combinatorial proof runs as follows. Consider the h-selection with box repetition 1 F i12F i F иии F ihF n q m q 1 y h.If i1Gnq1, then we have an h-selection with box repetition from the set Än q 1, n q hn mq1yh 2,...,nqmq1yh4. Factoring in the weights this equals qSh Ž.q nq1 Žthis is the term with k s 0 and the convention S01Ž.q s 1.If . i FnF i2, then we have a 1-selection from the setÄ4 1, 2, . . . , n and an Ž.h y 1- selection from the set Ä4n, n q 1,...,nqmq1yh Žthis is the term with . Žny1.Ž hy1. nŽ. Žmq1.yŽhy1.Ž. ks1 . Factoring in the weights this equals qS1qShy1 q. Otherwise, we must have i12F i F n y 1 F i3. In this case we have a 2-selection from the setÄ4 1, 2, . . . , n y 1 and an Ž.h y 2 -selection from the Žn2.Ž h 2. set Ä4n y 1, n,...,nqmq1yh and the term is q yy и ny1Ž. Žmq1.yŽhy2.Ž. S2 qShy2 q. Thus, continuing for 0 - k - h and i12F i иии Ä FF ikkF n q 1 y k F i q1we have a k-selection from 1, 2, . . . , n q 1yk4and an Ž.h y k -selection from Än q 1 y k, n q 2 y k,...,nqm 238 JOHN KONVALINA

Ž ny k .Žhy k. nq 1y k q 1 y h4. The weighted result equals qSkи Ž. Žmq1.yŽhyk.Ž. qShyk q. The case k s h follows similarly, and the combinatorial proof is complete.

4. GENERATING FUNCTIONS

The formal properties of the generalized binomial coefficients can be studied in terms of generating functions. The coefficients of the first kind n Ck Ž.wcan be formally represented as elementary symmetric functions. Thus, if w s Ž.w12, w ,...,wnand n k k fxŽ.s Ž1ywx12 .Ž1ywx .иии Ž1 y wxnk .syÝ Ž1 .␴x ks0 then ␴0 s 1 and, for k G 1, ␴ ww иии w kis Ý 12iik 1Fi12-i-иии -ikFn

n denotes the kth elementary symmetric function. But, byŽ. 6 , CkkŽ.w s ␴ ,so n fxŽ.is a generating function for Ck Ž.w , n k nk fxŽ.s Ž1ywx12 .Ž1ywx .иии Ž1 y wxnk .syÝ Ž1 .C Ž.wx.13 Ž . ks0 What is the corresponding generating function for the generalized binomial coefficients of the second kind? We quickly show it is the reciprocal of fxŽ.: 11n ϱϱ ii k sswxjks ax, fx 1 wx 1 wx иии 1 wx ŁÝ Ý Ž. Žy 12 .Žy . Žy n . js1is0ks0 where a Ý w i1 w i2 иии w i n with i 0, i 0,..., i 0. kis 12qiqиииqinsk 12 n 12G G nG n But akkis just S Ž.w :

i1i2in ak sÝ w12wиии wn i12qiqиииqinsk ww иииw Sn w . s Ý ii12 iks kŽ. 1Fi12FiFиииFikFn Thus, we have 11ϱ nk ssSkŽ.wx.14 Ž . fx 1 wx 1 wx иии 1 wx Ý Ž. Žy 12 .Žy . Žy n . ks0 GENERALIZED BINOMIAL COEFFICIENTS 239

iy1 Setting wi s 1, i,orq inŽ. 13 and Ž. 14 we obtain generating function relations for the binomial coefficients, Stirling numbers, and Gaussian coefficients, respectively:

Binomial coefficients Ž.wis 1: n n k n Ž.1 x Ž.1 xk; y syÝž/k ks0 ϱ 1 n k 1 q y xk. nsÝž/k Ž.1yx ks0

Stirling numbers Ž.wi s i : n k n 1 Ž.Ž.Ž.1 x 1 2 x иии 1 nx Ž.1 q xk ; y y y syÝ n1k ks0 qy 1 ϱ nkk s qx. 1x12xиии 1 nx Ý½5n Ž.Ž.Ž.y y y ks0

iy1 Gaussian coefficients Žwi s q .: n k n 1 k n k Ž.Ž.1yx 1yqx иии Ž.1 y qxy syÝŽ.1qxž/2 ; ž/k q ks0 1 ϱ nqky1 k n 1 sÝ x. 1x1qx иии 1 qxy ž/k q Ž.Ž.yy Ž.y ks0 The generalized binomial coefficients satisfy orthogonality relations obtained by multiplying the generating functions fxŽ.and 1rfx Ž..Soby Ž.13 and Ž. 14 we have 1 ϱ m fxŽ. s1s cxm . fx Ý Ž. ms0 m Ž.knŽ.n Ž. Thus, if m ) 0, Ýks0 y1 Ckmw S ykw s 0. If m s n, then n knn ÝŽy1 .Ckn Ž.wSyk Ž.ws0.Ž. 15 ks0

Replacing k by n y k we get a dual toŽ. 15 : n nyk nn ÝŽ.y1Cnykk Ž.Ž.wSws0.Ž. 16 ks0 240 JOHN KONVALINA

For example, if wi s 1, then fromŽ. 16 we obtain an orthogonality relation for the binomial coefficients of the first and second kind:

n n k nnk1 Ž.1y qy 0. Ýy ž/ž/nkks ks0 y

iy1 If wi s q , we obtain the q-orthogonality relation: n n k nyk y nnqky1 ÝŽ.y1 qž/2 s0. ž/ž/nkkqq ks0 y

Finally, wi s i results in an orthogonality relation for the Stirling numbers: n n k n 1 Ž.1 y q n q k 0. Ýy k1½5ns ks0 q

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