UNI DEERMINANS IN GENERALIZED PASCAL RIANGLES MARJORf E BtCKNELL and V. E. HOGGA, JR. San Jose State University, San Jose, California here are many ways in which one can select a square array from Pascal's triangle which will have a determinant of value one. What is surprising is that two classes of generalized Pascal triangles which arose in [ l] also have this property: the multinomial coefficient triangles and the convolution triangles formed from sequences which.are found as the sums of elements appearing on rising diagonals within the binomial and multinomial coefficient triangles. he generalized Pascal triangles also share sequences of k x k determinants whose values are successive binomial coefficients in the k column of PascaPs triangle.. UNI DEERMINANS WIHIN PASCAL'S RIANGLE When Pascal's triangle is imbedded in matrices throughout this paper, we will number the rows and columns in the usual matrix notation, with the leftmost column the first column. If we refer to Pascal's triangle itself 9 then the leftmost column is the zero column, and the top row is the zero row. First we write Pascal's triangle in rectangular form as the n x n matrix P = (p.,). (.) 7 J i i X n where the element p.. in the i row and j column can be obtained by p ij (i + j - \ = p i-i,j + p i. M = { i - ) and the generating function for the elements appearing in the j column is /( - x)\ j =,,, n. A = (a..), ij Pascal's triangle written in left-justified form can be imbedded in the n X n matrix
UNI DEERMINANS IN GENERALIZED PASCAL RIANGLES [Apr. i (.) A = where L " n X n and the column generators are, - (! :! ) x j ~V(l x) J he sums of the elements found on the rising diagonals of A, found by beginning at the leftmost column and moving right one and up one throughout the array, are,,,,, 8,, F, - + F, the Fibonacci numbers. "n+ n+ n heorem.. Any k X k submatrix of A which contains the column of ones and has for its first row the i row of A has a determinant of value one. his i s easily proved, for if the preceding row is subtracted from each row successively for i = k, k -, " ",, and then for i = k, k -,,,, and finally for i = k, the matrix obtained equivalent to the given submatrix has ones on its main diagonal and zeroes below. For example, for k = and i =, which is an interesting process in itself, since each row becomes the same as the first row except moved successively one space right. Let the n X n matrix A be the transpose of A, so that Pascal's triangle appears on and above the main diagonal with its rows in vertical position. hen AA = P, the r e c - tangular Pascal's matrix of (.). hat i s, for n =, AA ] As proof, the generating functions for the columns of A are
97] UNI DEERMINANS IN GENERALIZED PASCAL RIANGLES and for A, ( + x)"* \ j =,, -, n, of A A are so that the generating functions for the columns ^ (>* ^)'" = (r^j j =,, which we recognize as the generating functions for the columns of P. Notice that det A = det A =, so that det AA = det P = for any n hat is, heorem.. he k X k submatrix, formed from Pascal*s triangle written in rectangular form to contain the upper left-hand corner element of P, has a determinant value of one, In P, if the (n - ) column is subtracted from the n column, the (n - ) n from the (n - ), -, the from the, and the first from the n, a new array is formed: n x n In the new array, the determinant is still one but it equals the determinant of the (n - l ) x (n - ) array formed by deleting the first row and first column, which a r r a y can be found within the original array P by moving one column right and using the original row of ones as the top row. If we again subtract the (k - ) column from the k column successively for k = n - l, n -, e,,, the new a r r a y has the determinant value one and can be found as the (n - ) X (n - ) a r r a y using the original row of ones as its top row and two columns right in P. By the law of formation of Pascal s triangle, we can subtract thusly, k times beginning with an (n + k) X (n + k) matrix P to show that the determinant of any n X n array within P containing a row of ones has determinant equivalent to det P and thus has value one. By the symmetry of Pascal's triangle and P, any n X.n square array taken within P to include a column of ones as its edge also has a unit determinant. We have proved heorem.. he determinant of any n X n array taken with its first row along the row of ones, or with its first column along the column of ones in Pascal's triangle written in rectangular form, is one. Returning to the matrices A and A, AA gave us Pascal's triangle in rectangular m form. Consider A A
UNI DEERMINANS IN GENERALIZED PASCAL RIANGLES [Apr. 8 8 8 9 9 J n X n where the first column is,,, 8,,,, with column generators ( - x) j - l ( - x) j =,,, n. Notice that the sums of elements appearing on the rising diagonals are, 8,,, the Fibonacci numbers with even subscripts. In general, A, ma has first column, m, m, k, with column generators [ - (m - l ) x ] j " X ( - mx) =,,-,n. m Notice that any k X k submatrix of A A containing the first row (which is a row of ones) has a unit determinant, since that submatrix is the product of submatrices with unit determinants from A and A ~ A, where A A = P. In general, if we consider determinants whose rows are composed of successive m e m- bers of an arithmetic progression of the r order, we can predict the determinant value as well as give a second proof of the foregoing unit determinant properties of Pascal s triangle, he following theorem is due to Howard Eves [ ]. Let us define an arithmetic progression of the r order, denoted by (AP), to be a sequence of numbers whose r row of differences is a row of constants, but whose (r - ) row is not. For example, the third row of Pascal's triangle in rectangular form is an (AP), since A row of repeated constants will be called an (AP). he constant in the r row of differences of an (AP) will be called the constant of the progression. heorem.. (Eves heorem). Consider a determinant of order n whose i row (i =,,, n) is composed of any n successive terms of an ( A P ). - with constant a.. hen the value of the determinant is the product a-^ * a n.
97] UNI DEERMINANS IN GENERALIZED PASCAL RIANGLES he proof is an armchair one, since, by a sequence of operations of the type where we subtract from a column of the determinant the preceding column, one can reduce the matrix of the determinant to one having a l9 a,, a n alongthe main diagonal and zeros e v e r y - where above the main diagonal. Eves f heorem applies to all the determinants formed from Pascal's triangle discussed in this section, with aj = a = B = a =, whence the value of each determinant is one. As a corollary to Eves heorem, notice that the n order determinants containing a row of ones formed from Pascal's rectangular array still have value one if the k row is shifted m spaces left, k =,,, n, m ^, where m is arbitrary and can be a different value for each row! Corollary... If the k row of an n x n determinant contains n consecutive nonst zero members of the (k - ) row of Pascal's triangle written in rectangular form for each k =,,, n, the determinant has value one, Since all arrays discussed in this section contain a row (or column) of ones, we also have infinitely many determinants that can be immediately written with arbitrary value c: merely add (c - ) to each element of the matrix of any one of the unit determinant arrays! he proof is simple: since every element in the first row equals c, factor out c. hen subtract (c - ) times the first row from each other row, returning to the original array which had a determinant value of one. hus, the new determinant has value c. his is a special case of a wide variety of problems concerning the lambda number of a matrix [], [], Eves' heorem also applies to the a r r a y s of convolution triangles and multinomial coefficients which follow, but the development using other methods of proof is more informative.. OHER DEERMINAN VALUES FROM.PASCAL'S RIANGLE Return again to matrix P of (.). Suppose that we remove the top row and left column, and then evaluate the k Xk determinants containing the upper left corner. hen =, =,, and the,k>< k determinant has value (k + ). Proof is by mathematical induction. Assume that the (k - ) X (k - ) determinant has value k. In the k X k determinant, subtract the preceding column from each column s u c- cessively for j = k, k -, k -,,. hen subtract the preceding row from each row successively for i = k, k -, k -,,, leaving = l + k.
UNI DEERMINANS IN GENERALIZED PASCAL RIANGLES [Apr. Returning to matrix P, take x determinants along the second and third rows:, =,, giving the values found in the second column of Pascal s left-justified triangle, for ( (':') C-) by simple algebra. Of course, I x l determinants along the second row of P yield the successive values found in the first column of Pascal s triangle. aking x determinants yields =, =,, the successive entries in the third column of Pascal's triangle. In fact, taking successive nd rd st k x k determinants along the,,, and (k + ) rows yields the successive entries of the k column of Pascal's triangle. o formalize our statement, heorem.: he determinant of the k x k matrix R(k,j) taken with its first column the j column of P, the rectangular form of Pascal's triangle imbedded in a matrix, and its first row the second row of P, is the binomial coefficient (! ~r) o illustrate,
97] UNI DEERMINANS IN GENERALIZED PASCAL RIANGLES 7 detr(,) 7 = det R(,) + det R(,) -OHO-(- First, the preceding column was subtracted from each column successively, j = k, k -,, j and then the preceding row was subtracted from each row successively for i = k, k -,,. hen the determinant was made the sum of two determinants, one bordering R(,) and the other equal to R(,) by adding the j column to the (j + l ) s, j =,,, k -. By following the above procedure, we can make + 7 detr(k s j) = detr(k -, j) + det R(k, j - ). We have already proved that d e tr(k,l) = = ( k k ) - det R(k,) = k + = k * X J for all k, deth(l.j) = j = ( j J j, detr( 9 j) = M J ) for all j. If then dete(k -, j) = ( J - ' ) and detr(k, j - ) = ( ; i + ^ ~ )
8 UNI DEERMINANS IN GENERALIZED PASCAL RIANGLES [Apr. «H *.» - (, ; v ) * (, + i - * ) - (, + r i ) for all k and all j by mathematical induction. Since P is its own transpose, heorem. is also true if the words "column' "row" are everywhere exchanged. Consider PascaPs triangle in the configuration of A, which is just PascaPs rectangular array P with the i row moved (i - ) spaces right, i =,,,. Form k x k matrices R (k, j) such that the first row of R (k, j) is the second row of A beginning Hi with the j column of A. hen AR f (k, j - ) = R(k,j) as can be shown by considering their column generating functions, and since deta =, detr! (k,j - ) = detr(k,j), leading us to the following theorems. heorem.. Let A be the n x n matrix containing PascaPs triangle on and above its main diagonal so that the rows of PascaPs triangle are placed vertically. Any k x k sub- matrix of A selected with its first row along the second row of A and its first column Hi the j column of A, has determinant value / k + j - \ I k ) Since A is the transpose of A, wording heorem. in terms of the usual Pascal triangle provides the following. heorem.. If PascaPs triangle is written in left-justified form, any k x k and matrix selected within the array with its first column the first column of Pascal s triangle and its Hi first row the i row has determinant value given by the binomial coefficient (*-i-'). MULINOMIAL COEFFICIEN ARRAYS he trinomial coefficients arising in the expansions of ( + x + x ), written in leftjustified form, are (.) 7 9 where the entry t.. in the i row and j column is obtained by the relationship
97] UNI DEERMINANS IN GENERALIZED PASCAL RIANGLES 9 i] -, -.- -,- he sums of elements on rising diagonals formed by beginning in the left-most column and counting up one and right one are the ribonacci numbers,,,, 7,, n+ n+ + n+ + n" ^ s i n P a s c a * f s triangles the left-most column is the zero column and th the top row the zero row.) If the summands for the Fibonacci numbers from the rising diagonals of P a s c a l' s t r i - angle (.) in left-justified form are used in reverse order as the rows to form an n x n matrix Ft = ( y, f ij = ( i - j j which has the rows of Pascal' s triangle written in vertical position on and below the main di- agonalj and the matrix A is written with Pascal's triangle on and above the main diagonal, then the matrix product FjA is the trinomial coefficient array (.) but written vertically rather than in the horizontal arrangement just given. o illustrate, when n = 7, FiA J " " = " 7 9 " 9 Generating functions give an easy proof. he generating function for the j column of F t is [ x ( l + x ) ] J _, j =,, ( + x ) j - l,, B 8 e. he generating function for the j column of A is so the generating function for the j th column of F^P is [ + x(l + x)]* or ( + x + x r, so that F A A has the rows of the trinomial triangle as its columns. Since det Fj det A =, det FjA =, and n x n determinants containing the upper left corner of the trinomial triangle have value one. Further, any k x k sub- matrix of FjA containing a row of ones has a unit determinant, for any k x k submatrix of FjA selected with a row of ones is always the product of a submatrix of F t and a submatrix
UNI DEERMINANS IN GENERALIZED PASCAL RIANGLES [Apr. of A containing a row of ones, both of which here have unit determinants. Lastly, any k x k submatrix of F t A having its first row along the second row of FjA and its first th column the j column of F A is the product of a submatrix of F t with a unit determinant and a k x k submatrix of A satisfying heorem., and thus has determinant value H" ) We can form matrices F analogous to F A using the rising diagonals of the triangle of multinomial coefficients arising from the expansions of ( + x + x + + x m ), m >, n ^, as the rows of F, so that the rows of the multinomial triangle appear as the columns of F written on and below the main diagonal. (In each case, the row sums of F m & m are = r, = r = = r m, r n = V l + r Q _ + + r Q _ m, n > m +.) he matrix product F A will contain the coefficients arising from the expansions of ( + X + X +... + X m + ) written in vertical position on and above the main diagonal. As proof, the generating functions of columns of F m are [x(l + x + + x * )] ", j =,,, n, while the g e n e r - ating function for the j t n column of A is (+x), j =,,, n, so that the genth rating function for the j column of F A is [ + ( x ( l + x +... H - X ) ) ] " = [ + x + x +... + x m + ] ", j = l,,..., n. If a submatrix of F m A is the product of a submatrix of F m with unit determinant and a submatrix of F-^Q^A- - taken in the corresponding position, it will thus have the same determinant as the submatrix of F m _ A. Notice that this is true for submatrices taken across the first or second columns of F m A. Since FjA has its submatrices with the same determinant values as the submatrices of A taken in corresponding position, and these values are given in heorem., we have proved the following theorems by mathematical induction if we look at the transpose of F m A. heorem.. If the multinomial coefficients arising from the expansions of ( + x + m n + x ), m ^ l, n ^ O, are written in left-justified form the determinant of the k x k matrix formed with its first column taken anywhere along the leftmost column of ones of the array is one. heorem,. If the multinomial coefficients arising from the expansions of ( + x + - + x m ) n, m ^, n ^, are written in left-justified form the determinant of the k x k matrix formed with its first column the first column of the array (the column of successive whole numbers) and its first row the i row of t the multinomial coefficient a r r a y, has d e- terminant value given by the binomial coefficient ( k+ M rr>
97] UNI DEERMINANS IN GENERALIZED PASCAL RIANGLES. HE FIBONACCI CONVOLUION ARRAY AND RELAED CONVOLUION ARRAYS by If {a.} and {b.} are two sequences, the convolution sequence {c} is given, _-! J -I i=l = Ci = atbi, c = a bi + a ^, c = a b x + a b + a ^, c = V^ a b n JL-i k n- k+ k=l If g(x) is the generating function for { a. }, then [g(x)] is the generating function for the k convolution of {a.} with itself. he Fibonacci sequence, when convoluted with itself j - times, forms the sequence in the j column of the matrix C = ( c. ) below [ l ], where the original sequence is in the leftmost column C = 8 8 9 9 7 98 9 n x n For formation of the Fibonacci convolution array, Let the n x n c. c... + c.. + c.. -. -, i - j i, j - l matrix Fj be formed as in Section with the rows of Pascal s triangle in vertical position on and below the main diagonal, and let the n x n matrix P be Pascal's rectangular array (.). hat FjP is the Fibonacci convolution array in rectangular form is easily proved. he generating functions for the columns of P A are [x(l + x)] J ~, while those for P are /( - x) J, j =,,, n, so that the generating functions for the columns of FjP are l / [ l - x(l + x)] J or /( - x - x )\ the generators of C, the Fibonacci convolution array. Since det F t = det P =, det F*P =, and any n x n matrix formed using the upper left corner of the Fibonacci convolution array has a unit determinant* Further, since submatrices of C taken along either the first or second row of C are the product of submatrices of F t with unit determinants and similarly placed submatrices of P whose determinants are given in heorems. and., we have the following theorem. heorem.. Let the Fibonacci convolution triangle be written in rectangular form and imbedded in an n >< n matrix C. hen the determinant of any k x k submatrix of C placed
UNI DEERMINANS OF GENERALIZED PASCAL RIANGLES [Apr. to contain the row of ones is one. Also, the determinant of any k x k submatrix of C s e - lected with its first row along the second row of C and its first column the j column of C has determinant value given by the binomial coefficient I?~ J he generalization to convolution triangles for sequences which are found as sums of rising diagonals of multinomial coefficient triangles written in left-justified difficult Form the matrix F diagonals of the left-justified form is not as in Section to have its rows the elements found on the rising multinomial coefficient triangle induced by expansions of ( + x +... + x m ), m ^, n ^. hen F P is the convolution triangle for the sequence of sums of the rising diagonals just described, for F has column generators [x(l + x + + x m )]*'~ and P has column generators /( - x)j, making the column generators of F P be /( - x - x -. - x m + ), j = l,,, n, which for j = is known to be the generating function for the sequence of sums found along the rising diagonals of the given multinomial coefficient triangle []. As before, since a submatrix of F P is the product of a submatrix of F with unit H m m determinant and a similarly placed submatrix of P with determinant given by heorems. and. when we take submatrices along the first or second column of F P, we can write the following. heorem.. Let the convolution triangle for the sequences of sums found along the rising diagonals of the left-justified multinomial coefficient array induced by expansions of ( + x +... + x m ), m ^, n ^, be written in rectangular form and imbedded in an n x n matrix C*. hen the determinant of any k x k submatrix of C* placed to contain the row of ones is one. Also, the determinant of any k x k submatrix of C* selected with its first row along the second row of C* and its first column the j column of C* has determinant value given by the binomial coefficient i [ " he convolution triangle imbedded in the matrix C* = ( a. ) of heorem. has c. = c. -. + c. _. + + c.. - c.... If we form an array J B = (b..) with rule of ij l - l, j -,] i - m, j,- i j ' formation b.. = b... + b. n. +... + fc>. + h. + b.. -, B becomes the powers F of l - l, ] -,],,,-' convolution array. 8 8 9 8 9 7 8 7 9......... rising diagonal sums are,,,,,, the Fibonacci numbers with odd subscripts. his convolution a r r a y has both unit determinant properties d i s- cussed in Section because it fulfills Eves heorem. In fact, all the convolution arrays of this section fulfill Eves' heorem, so that any k x k matrix placed to contain a row of ones in any one of these convolution arrays has determinant value one.
97] U N I DEERMINAN IN GENERALIZED PASCAL RIANGLES. CONVOLUION ARRAYS FOR DIAGONAL SEQUENCES FROM MULINOMIAL COEFFICIEN ARRAYS Let the n x n matrix D be D = n x n where the row sums of D are the rising diagonal sums from Pascal s triangle (.) found by beginning at the leftmost column and going up and to the right one, namely,,,,,,,, 9,,. he column generators of D are [ x ( l + x ) ], j =,,. hen, the matrix product DP, where P is the rectangular Pascal matrix (.), is the convolution t r i - angle for the sequence,,,,,,, 9,,, u n +i = u + u, for the column n n ^ generators of DP are l / [ l - x ( l + x ) ] J = /( - x - x ) J, j =,,, n, where the generating function for j = is given as the generating function for the sequence discussed in []. he results of heorem. and. can be extended to cover the matrix DP as before. Now, consider the sequence of elements in Pascal's triangle that lie on the diagonals found by beginning at the leftmost column of (.) and going up p and to the right throughout the a r r a y (.). (he sum of the elements on these diagonals are the generalized Fibonacci numbers u(n; p, l ) of Harris and Styles [].) Place the elements from successive diagonals on rows of a matrix D^p, ) in reverse order such that the column of ones in A lies on the diagonal of D (p,l). hen the columns of D^pjl) are the rows of Pascal's triangle in left-justified form but with the entries separated by (p - ) zeroes. Notice that det D^p,!) =. he column generators of D^p,!) are [ x ( l + x P ) ] j, j =,,, so that the column generators of D^p.DP are l / [ l - x(l + x' p )]^ = /( - x - x P + ) \ j =,,, n, the generating functions for the convolution array for the numbers u(n; p, l ) (see []). he same arguments hold for Di(p,l)P as for DP = ^, )?, so that D ^ p. D P has the determinant properties which extend from heorems. and.. Now, the same techniques can be applied to the elements on the rising diagonals in all the generalized Pascal triangles induced by expansions of ( + x + + x m ), m ^,. Form the n x n matrix D (p, ) so that elements on the diagonals formed by beginning in the leftmost column and going up p and right one throughout the left-justified multinomial coefficient array lie in reverse order on its rows. D (p, ) will have a one for each element on its main diagonal and each column will contain the corresponding row of the multinomial array but with (p - ) zeroes between entries. Generating functions for the columns of n ^
UNI DEERMINAN IN GENERALIZED PASCAL RIANGLES Apr. 97 D (p,l) are [x(l + x p + x p +.. - + x ( m ~ ) p ) ] j s j =,,., n, while for P they are, of course, ( - x). he generating functions for the columns of D (p,l)p are then or /[ - x { l + X P + x P + -., + x ( m - ) P ) ] /[ - x - x? + - x? +.tm-wp+lj, j. i, a,.... n. which, for j =, is known to be the generating function for the diagonal sums here considered []. Since again the k x k submatrices of the product D (p, )P taken along either the first or second row of D (p, )P have the same determinants as the correspondingly placed k x k submatrices of P, we look again at heorems. and. to write our final theorem. heorem.. Write the convolution triangle in rectangular form imbedded in an n x n matrix C* for the sequence of sums found on the rising diagonals formed by beginning in the leftmost column and moving up p and right one throughout any left-justified multinomial coefficient array., he k x k submatrix formed to include the first row of ones has determinant one. he k x k submatrix formed with its first row the second row of C* and its first th m column the j column of C * has determinant given by the binomial coefficient ( l *r') REFERENCES. Verner E. Hoggatt, J r., and Marjorie Bicknell, "Convolution riangles," Fibonacci Quarterly, Vol., No., Dec. 97, pp. 99-8.. Howard Eves in a letter to V. E. Hoggatt, J r., January, 97.. Marjorie Bicknell and V. E. Hoggatt, Jr., "Fibonacci Matrices and Lambda Functions," Fibonacci Quarterly, Vol., No., April, 9, pp. 7-.. Marjorie Bicknell, "he Lambda Number of a Matrix: he Sum of Its n Cofactors," American Mathematical Monthly, Vol. 7, No., March, 9, pp. -.. V. E. Hoggatt, J r., and Marjorie Bicknell, "Diagonal Sums of Generalized Pascal r i - angles," Fibonacci Quarterly, Vol. 7, No., November, 99, pp. -8.. V. C. Harris and Carolyn C. Styles, "A Generalization of Fibonacci Numbers," Fibonacci Quarterly, Vol., No., December, 9, pp. 77-89.