LATTICE POINT SOLUTION OF THE GENERALIZED PROBLEM OF TERQUEi. AND AN EXTENSION OF FIBONACCI NUMBERS.

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1 i LATTICE POINT SOLUTION OF THE GENERALIZED PROBLEM OF TERQUEi. AND AN EXTENSION OF FIBONACCI NUMBERS. C. A. CHURCH, Jr. and H. W. GOULD, W. Virginia University, Morgantown, W. V a. In this paper we give a siple lattice point solution to a generalized perutation proble of Terque and develop soe eleentary results for the extended Fibonacci nubers associated with the perutation probles The classical perutation proble of Terque [12] has been stated by Riordan [10, p 17, ex 15] in the following anner,. Consider cobinations of n nubered things in natural (rising) order, with f (n,r) the nuber of r- cobinations with odd eleents in odd position and even eleents in even positions, or, what is equivalent, with f (n,r) the nuber of cobinations with an equal nuber of odd and even eleents for r even and with the nuber of odd eleents one greater than the nuber of even for r odd. It is easy to show that f(n,r) = f (n - 1, r - 1) + f(n - 2, r), with f(n, 0) = 1, and explicitly (1) f(n,r) = Moreover, n (2) f (n) = ^T f (n, r) = f (n - 1) + f (n - 2) r=o so that f(n) is an ordinary Fibonacci nuber with f(0) = 1 and f(l) = 2 A detailed discussion of Terque f s proble is given by Netto [8, pp ] and Thoralf Skole [8, pp ] has appended notes on an extension of the proble in which the even and odd question is replaced by the ore general question of what happens when one uses a odulus to deterine the position of an eleent in the perutation. "^Research supported by National Science Foundation Grant GP

2 60 LATTICE POINT SOLUTION OF THE GENERALIZED [ F e b. More precisely, for a odulus > 2, Skole T s generalization a y b e stated as follows. F r o aong the first n natural nubers let f(n,r;) d e - note the nuber of cobinations in natural order of r of these nubers such that the j eleent in the cobination is congruent to j odulo. That is, o (3) f(n,r;) = N < a 1 a e 2 a r : 1 < a* < a 2 < * " < a r < n, a. = j (od )> 1< B, t < az 2 < < a r < n a. = j (od ) Consider the a r r a y in Fig. 1, where the last entry is r + k, with k = since r + k < n iplies that the l a r g e s t integral value of k cannot exceed (n - r ) / 0 This a r r a y contains those, and only those, eleents fro aong 1, 2,, n which ay appear in a cobination,, That i s, the j colun consists of all those eleents < n in the s a e congruence c l a s s (od ) which ay appear in the j position. (0,0) - 1 #-$*» X r + l i-1 + k i k i. 3 + k -i. Fig 0 1 -r + k # B = (r,s) F r o the lattice appended to the a r r a y In Fig 1, we can systeatically write out the desired cobinations, and evaluate f(n, r;) To get the desired result, let "a path fro A to B ft ean a path along the vertical and horizontal segents of the lattice, always oving downward or fro left to right (we take the positive x-axis to the right, the positive y-axis

3 1967] PROBLEM OF TERQUEM AND AN EXTENSION 61 OF FIBONACCI NUMBERS downward, thus agreeing with the inforal way of writing down the perutations)* Each such path will generate a cobination of the desired type, and conversely, as follows? Starting at A each horizontal step picks up an entry and vertical steps line up entries 0 Now, It is well known how any lattice paths there are fro a = (0,0) to B = (r,s). MacMahon [7, VoL I, p 167] shows that this nuber is precisely ( r ) hi our case s = [(n - r)/] c Thus we have at once that (4) f (n, r;) r + \[- ( - l)r" HI r as found by Skole 0 Terque! s (1) follows when = 2 To illustrate, we consider soe exaples 0 Exaple 1. Let n = 12, r = 3, = 4 0 Then the corresponding array i s 1 1 E c «3 _ J r fi D 2 i ^ -» * and the ten cobinations are and the particular cobination 5, 10, 11 corresponds to the path indicated by arrows. Inforally, one writes out the cobinations by paths fro the left colun to the right colun, oving horizontally and/or diagonally,, The clue to a systeatic count is found by superiposing the rectangular grid* Exaple 2 Let n = 12, r = 4, = 3 0 Then the corresponding array is

4 62 LATTTCE POINT SOLUTION OF THE GENERALIZED PROBLEM [Feb* FH 1 n 1 1 2» - 5 * o 4 *7 t *». Q, J L^^io ^and the fifteen cobinations are and the cobination 4, 5, 9, 10 corresponds to the path indicated by arrows, It is felt that our proof shows atruis of atheatics: one ay often find a sipler proof by ebedding a given proble (Terque f s) in a ore general setting., The lattice point enueration we used is well known, but ay not be apparent in the original proble because of its specialized for 0 The extended Fibonacci nubers, in analogy to (2), are now defined by (5) f(n) = f (n) = n r=o j*n + ( - l)r" and it is not difficult to verify that they satisfy the recurrence relation (6) f (n) - f (n - 1) + f (n - ) For exaple, with = 3 we have the sequence 1, 2, 3, 4, 6, 9,13,19, 28, By well-known theores in the theory of linear difference equations, if the distinct roots, of the equation (7) t - t 111 " 1 - i - o are ti, t 2, t * then there exist constants C r such that

5 1967] OF TERQUEM AND AN EXTENSION OF FIBONACCI NUMBERS 63 (8) f (n) = \ ^ C t n ZJ r r This generalizes the failiar forulas HI -o a - b, n,, n F = -~, L = a + b, n a - b n * for the Fibonacci-Lucas nubers. The constants C the syste of linear equations in C : a y b e deterined fro (9) 7JC r t 3 r = j + 1,- for j = 0, 1, 2,, - 1. r=i F o r exaple 9 when = 3, an approxiate solution of the equation (7) is given by I tj = , t o , t 3 = -0 o o , where i 2 = ' - 1. Relations (5) through (9) a r e given by Skole [ 8, ]* When = 3 the exact solution of (7) is given by ti = A + B + - (11) t 2 = J - A + i + A^B ^ p 1 A + B A - B. / - o w h e r e

6 64 LATTICE POINT SOLUTIONS OF THE GENERALIZED PROBLEM [ F e b. As a partial check on the values of the roots, we note the following t h e o r e fro the theory of equations. Let (12) n (t - t.) = t - t - z Then (13) - ll l ] = \b> X " ) ^ ' n > 1 5 j=i where A / i.\ a / a + bk \ A k ( a ' b ) = r r b k ( k ) This ay be copared with the well-known [2, 3, 4, 5] expansion (14) x a = ^ T A k ( a, b ) z k, with z = x - 1 b, x which was actually found by Lagrange in his great e o i r of 1770 (Vol, 24 of P r o c. of the Berlin Acadey of Sciences) and which leads at once to the general addition t h e o r e discussed in [2, 3, 4, 5 ] a s first noted by H. A. Rothe See relation (20), this paper,, l F o r the equation t - t - z = 0, we define the power sus of the roots t. by (15) S(n) - l )

7 1967] OF TERQUEM AND AN EXTENSION OF FIBONACCI NUMBERS 65 Since t. " + z = t., we find that 3 3 S(n - 1) + zs(n - ) = {t*" 1 + z t ^ J = t ^ / t ^ 1 + z \ - J ] t ^ t * 1, j=i j=i ^ / j=i so that S(n) itself also satisfies a Fibonacci-type r e c u r r e n c e (16) S(n) = S(n - 1) + zs(n - ). Using the values z = 1, = 3, the previous roots (10) yield the approxiate values (by log tables): S(l) - 1, S(2) = 0 e 9998, S(3) - 3,9995, and S(4) = 5 v e r y nearly c This gives a partial check on (10) o In any event 5 we ay consider the sequence defined by (13), (15), (16) a s a kind of extended Fibonacci sequence,, hi particular, (17) S(n) Z^ n - ( - l)k ( n - ( k l)k n > 1, satisfies (16) just as (5) satisfies (6). T h e r e a r e siilarities and contrasts if we copare (17) and (5) 0 We also call attention to another such result given recently by J A 9 Raab [Vj, who found that the sequence defined by (18) x = [r+ij \ - ^ / n - r k \ n- k(r+i) b k satisfies (19) x = ax J + bx n n - l n - r - i

8 6^ LATTICE POINT SOLUTIONS OF THE GENERALIZED PROBLEM [Feb Forula (13) is substantially that given by Arthur Cayley [!]<, The classical Lagrange inversion forula for series is inherent in all these forulas,, One should also copare the Fibonacci-type relations here with the expansions given in [5] For = 3, (17) gives the sequence 1,1, 4, 5 5 6,10,15, 21, 31, We als.o call attention to the two well-known special cases and ( n i k ) r,, n+i n+i -2k z k = x - y x - y V^ / n - k \ n n-2k-i k = x 11 n +, y n =r \ k / n - k z ' + y x + y where x = 1 + Vz + 1, y = 1 -v/z + 1 F and L occur when z = 4 9 J v n n Relations (17) and (5) differ because the initial conditions differ., z = 1, (17) satisfies precisely the sae recurrence as (5) c If the initial values were the sae then we would have found a forula for the perutation proble not unlike (17) There are any papers (too nuerous to ention) in which coplicated binoial sus are found by lattice point enuerations,, The convolutions in 2, 3,4, 5J ay ostly be found by such counting ethods,, We also note the recent papers of Greenwood jf] and Stocks [ll] wherein the Fibonacci nubers occur 0 The convolution addition theore [2, 3,4, 5] of H A e Rothe (1793) For (20) ^ A k ( a, b ) A n _ k ( c s b ) - A n (a + c,b), valid for all real or coplex a, b, c (being a polynoial identity in these), has been derived several ties by lattice point ethods e We ention only a novel

9 1967] OF TERQUEM AND AN EXTENSION OF FIBONACCI NUMBERS 67 derivation by Lyness [13j 0 Relation (20) has been rediscovered dozens of ties since 1793, and its application in probability., graph theory, analysis, and the enueration of flexagons, etc, shows that the theore is very useful In fact, it is a natural source of binoial identities,, We should like to raise the question here as to whether any analogous relation involving the generalized Terque coefficients (4) existso It sees appropriate to study the generating function defined by fa + (b - l)n1\ (21) T(x;a,b) = J^ ( n=o for as general a and b as possible., If b is a natural nuber and a is an integer >0, the series terinates with that ter where n = a, as is evident fro the fact that a + (b - l)n < bn for n > a and the fact that 1 = 0 for k < n when n > 0, provided k > 0 o We also note that for arbitrary coplex a and x < 1 T(x;a,l) = 2 (») *" - {1 + x)" n=o * ' so that in this case we do have an addition theore: T(x;a 9 l)t(x;c 5 l) = T(x ; a + c,l). This, of course, corresponds to the case b = 0 in forula (20); the relation iplies the failiar Vanderonde convolution or addition theore. There does not see to be any especially siple closed su for the series n / f a + ( b - l ) k l \ / f c + (b - l')(n - k)] \ (22) C n (a,c,b) - ^ which occurs in

10 68 LATTICE POINT SOLUTION OF THE GENERALIZED PROBLEM OF TERQUEM AND AN EXTENSION OF FIBONACCI NUMBERS Feb T(x;a,b)T(x,c,b) = V x n C n ( a s c? b ), n=o for a r b i t r a r y b 6 REFERENCES 1. A Cayley, "On the Sus of Certain Series Arising fro the Equation x = u + f x, " Quart, J, Math., 2(1857), Ho Wo Gould, "Soe Generalizations of Vanderond 1 s Convolution," A e r. Math, Monthly, 63(1956), e H 0 Wo Gould, "Final Analysis of Vanderonde T s Convolution," A e r. Math. Monthly, 64(1957), C Ho W Gould, "Generalization of a T h e o r e of Jensen Concerning Convolutions, " D u ^ e M a t h o J o, 27(1960), c 5. Ho W 0 Gould, "A New Convolution F o r u l a and Soe New Orthogonal Relations for Inversion of Series, " Duke Matho J 0, 29(1962), o 6. R 0 Ec Greenwood, "Lattice P a t h s and Fibonacci N u b e r s, " Fibonacci Quarterly, VoL 2, 1964, pp Po A. MacMahon, Cobinatory Analysis, Cabridge, , 2 V o l s., Chelsea Reprint, N e Y o 8. E. Netto, Lehrbuch der Cobinatorik, Leipzig, 1927, 2nd E d., Chelsea Reprint, N. Y., J. A. Raab, "A Generalization of the Connection Between the Fibonacci Sequence and P a s c a l ' s Triangle, Fibonacci Quarterly, Vol 1, 1963, No 3, pp o J. Riordan, An Introduction to Cobinatorial Analysis, N 0 Y», D 0 Ro Stocks, J r c, "Concerning Lattice Paths and Fibonacci N u b e r s, " Fibonacci Quarterly, VoL 3, 1965, pp c o Oo Terque, "Sur un sybole cobinatoire d f Euler et son Utilite dans PAnalyse, " J 0 Math 0 P u r e s Appl., 4(1839), Ro Co Lyness, "Al Capone and the Death R a y, " Matho Gazette, 25(1941), o * * * * *

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