EVALUATION OF A FAMILY OF BINOMIAL DETERMINANTS
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1 EVALUATION OF A FAMILY OF BINOMIAL DETERMINANTS CHARLES HELOU AND JAMES A SELLERS Abstract Motivated by a recent work about finite sequences where the n-th term is bounded by n, we evaluate some classes of determinants such as the n n determinant n xn x k + h n k k n, for n 3, 0 h n 3 and more generally the n n determinant D n xi + j i i n, for n, where n, k, h, i, j are integers, x k k n is a sequence of indeterminates over C and A B is the usual binomial coefficient We thus prove that D n and n n n 3 Keywords: Determinants, binomial coefficients, row reduction AMS Classification Numbers: Primary C0; Secondary 05A0, B65, 5B36 Introduction In a recent work [9] about finite sequences whose n-th term does not exceed n, there appeared the determinant n n k + h n k k n, 0 h n 3 with an integer n 3 One of the authors of [9], L Haddad, conjectured after some computations that n ± The authors of the present paper first proved that n n n 3, essentially by a process of row reduction Then, following a suggestion by G E Andrews that this result should be true in a more general context, namely upon replacing n by x n and k by x k, where x k k n is an arbitrary sequence of indeterminates over C, the proof was extended to this general case We then
2 CHARLES HELOU AND JAMES A SELLERS realized that the problem can be reduced to the evaluation of a simpler, more general family of determinants, namely D n xi + j i i n, for all integers n In what follows, we will establish that D n and deduce that n n n 3 Results of a similar nature, involving determinants of matrices whose entries involve binomial coefficients, can be found in [,, 3, 4, 6, 0, ] In contrast to these papers, we note that our determinant evaluations are strikingly simple and easy to state In fact, our result is a special case of the results contained in [0], but our proof is more elementary, using only row reduction and induction We are thankful to L Haddad for his conjecture and to G E Andrews for his insightful suggestion The method of proof First recall eg [7] or [8] that for an indeterminate x over C and an integer n 0, the binomial coefficient x n is defined by x x x x x n +, n n! with the convention that x 0 It satisfies the fundamental recurrence relation x x x + +, for all n 0 n n + n + Let x k k n be a sequence of indeterminates over C, with n 3, and consider the n n determinant n n x,, x n xn x k + h n k k n 0 h n 3 First, setting i k and j h + allows one to rewrite n as n xn x i+ + j n i i n Second, the substitution i n i transforms n into n nx,, x n xn x n i + j i i n,
3 EVALUATION OF A FAMILY OF BINOMIAL DETERMINANTS 3 which has the same rows as n but in reverse order This order reversal consists in respectively swapping each row of n with all the rows above it The total number of those row swaps is n n 3 n 3 + n Therefore, n n n 3 n 3 The problem is thus reduced to the determination of n Third, setting x x n gives n x xn i + j i i n 4 Fourth, setting y i x x n i yields n yi + j i i n 5 Finally, setting m n leads to the equality n yi + j i i m 6 j m The problem is thus reduced to the evaluation of the family of determinants D n D n x,, x n xi + j i i n, 7 for n, where x, x,, x n are arbitrary indeterminates Our primary result is now the following: Theorem For any positive integer n, we have D n The proof proceeds by row reduction and is presented in the next section Corollary For any integer n 3, we have n Proof This follows from 6 and Theorem Corollary 3 For any integer n 3, we have n n n 3 Proof This follows from 3 and Corollary
4 4 CHARLES HELOU AND JAMES A SELLERS Remark 4 An alternative method for deriving the last result is to set y k x n x n k, and i k, j h + Then n yn i + j n i i n Now, reversing the order of the rows, which consists in replacing i by n i, transforms n into n yi + j i i n Moreover, the permutation ρ that reverses the n rows of n has n orbits, namely {, n }, {, n 3}, It follows that see, eg, [] the sign of ρ is ɛ ρ n n n Hence n n n n, in view of our main theorem, which yields n 3 The proof of the main Theorem We start with two results about binomial coefficients that will be used in the proof of Theorem Lemma 3 For any integers 0 a b and n, and any indeterminate x over C, we have x + b x + a b x + h n n n Proof By the fundamental recurrence relation for binomial coefficients, x + x x, n n n for n Hence, by iterating this recurrence numerous times, we have x + b x + a b x + h + x + h b x + h n n n n n ha km ha Lemma 3 For any integers 0 m n, we have n k n + m m + ha
5 EVALUATION OF A FAMILY OF BINOMIAL DETERMINANTS 5 Proof For a fixed integer m 0, the proof can be completed by induction on n m Such an argument can be found in [5, p 38] This result also follows from Lemma 3 by taking n m +, a 0, x m, b n m + We now proceed to prove Theorem by row reduction We start with D n d i n, where d xi + j for i, j n, 3 i ie D n x + x +3 x +j x3 + x3 +3 x3 +j x4 + x4 +3 x4 +j x + x3 + x4 + 3 xi + i xn+ n 3 xi + i xn+ n 3 xi +3 i xn+3 n 3 xi +j i xn+j n x +n x3 +n x4 +n 3 xi +n i xn+n n 3 Denoting the i-th row of D n by R i, the first row reduction step consists in replacing R i by R i d i R for i n This gives D n d i n, where { d d d i x i +j i xi + i j xi h i, if i n, j n, d j, if i, j n, 33 the last expression, for i, is obtained by using Lemma 3, with the usual convention that an empty sum is equal to 0 Thus
6 6 CHARLES HELOU AND JAMES A SELLERS 0 j n 0 x3 + x3 h j x3 h n h 0 x4 + x4 D n h j x4 h n h 0 xi + xi i h j xi i h n i h 0 xn+ xn n h j xn n h n n h Moreover, we obviously have Proposition 33 For k n, let D n k x3 x4 xi i xn n 34 D n D n 35 d k i n be the n n determinant obtained from D n by applying k row reduction steps, each of which consists in replacing the i-th row R k i of the determinant, obtained after k such row reduction steps, by the row D k n R k i R k i while the first k rows are unchanged, ie Then d k { j k h d k R k i j h xi k d k ik R k k, for k + i n, R k i for i k i k, if k + i n, j n,, if i k, j n, with the convention that an empty sum here, when j k is equal to 0 36 Proof The proof is by induction on k The first row reduction step was applied to D n right before this Proposition, and it gave D n d i n, which satisfies the stated equalities for d as shown in 33 above So the property holds for k Assume that it holds for k, where k n, ie assume that D n k d k i n satisfies d k { j k h j h k x i i k, if k i n, j n, d k, if i k, j n 37
7 EVALUATION OF A FAMILY OF BINOMIAL DETERMINANTS 7 Now, as for k + i n, we have R k i d k d k R k i d k ik d k ik d k kj, for j n, and by the induction assumption, there hold j k+ d k j h k d k kj Therefore we get d k d k ik j k+ h j k+ h j k+ h k k+ h h k h j h j h k j h k k Moreover, by Lemma 3, xi + h for i > k and h Hence j k+ d k j h h k h r Furthermore, by Lemma 3, j k+ j h k Thus hr+ k xi + h xi + h xk + h k k xi + h xi + h j k+ h xi + h r j k xi + r j r sk j k+ h, R k k, ie xi +, j h k xi + r s k xi + r, xi + r j h k j r j k d k j r xi + r, k r for k + i n and j n Also, for i k, since R k i R k i, we have d k k d k, for i k, j n xi + j k+ hr+ j h This shows that the property holds for k, and completes the induction k
8 8 CHARLES HELOU AND JAMES A SELLERS Corollary 34 For k n, the determinant D n k d k i n obtained from D n by applying k row reduction steps as described in Proposition 33 is given by d k { j i, if i k, j n, 38, if k + i n, j n, j k h j h k xi i k where if 0 m < n are integers then m n 0, and an empty sum is equal to 0 Proof Only the expression of d k for i k and j n needs to be proved The rest is contained in Proposition 33 This expression holds for k since by 33 d j d j j, for j n Assume that the expression holds for k, where k < n, ie d k { j i, if i k, j n,, if k i n, j n j k+ h j h k xi i k Then, by Proposition 33 and Lemma 3, we have d k d k j, for i k, j n, i and Hence d k kj d k kj j sk j k+ h s k d k and the expression holds for k j h xk + h k 0 h j, for j n k j k+ j, for i k, j n, i j h k Remark 35 We have D n D n k, since the determinant is invariant under the row reduction steps consisting of adding to a row a multiple of another row
9 In particular, EVALUATION OF A FAMILY OF BINOMIAL DETERMINANTS 9 D n n j i i n is the determinant of an upper triangular n n matrix whose diagonal entries are d n ii i i for i n Therefore D n D n n This concludes the proof of Theorem Remark 36 As noted in the Introduction, our result is a special case of the results contained in [0] Indeed, in [0], Proposition, taking p j x x j +x j, which is a polynomial of degree j in x, with leading coefficient a j, j! for j n, we get xj + X i n j i n j! X j X i, j j i<j n then specializing to X i i for i n, we get xj + i n j i n j! i<j n j i 4 Acknowledgment We are thankful to the referee for a careful, thorough reading of the paper, and for many helpful and interesting suggestions References [] T Amdeberhan and D Zeilberger, Determinants through the looking glass, Special issue in honor of Dominique Foata s 65th birthday Philadelphia, PA, 000, Adv in Appl Math 7 00, no -3, 5-30 [] G E Andrews, E Pfaff s method I The Mills-Robbins-Rumsey determinant, Selected papers in honor of Adriano Garsia Taormina, 994, Discrete Math , no 3, [3] G E Andrews and W H Burge, Determinant identities, Pacific J Math , no, -4 [4] G E Andrews and D W Stanton, Determinants in plane partition enumeration, European J Combin 9 998, no 3, 73-8 [5] R Brualdi, Introductory Combinatorics, Fifth Edition, Pearson Prentice Hall, 00 [6] W Chu and L V di Claudio, Binomial determinant evaluations, Ann Comb 9 005, no 4, [7] L Comtet, Advanced Combinatorics, D Reidel Publishing Co, Dordrecht, 974 [8] R Graham, D Knuth, O Patashnik, Concrete Mathematics, second edition, Addison- Wesley, 994
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