A PROPERTY OF MULTINOMIAL COEFFICIENTS

Transcription

1 A PROPERTY OF MULTINOMIAL COEFFICIENTS V.E. HOGGATT, JR. San Joe State College, San Joe, California and G. L ALEXANDERSON Univerity of Santa Clara, Santa Clara, California ABSTRACT The ultinoial coefficient "urrounding" a given ultinoial coefficient in a generalized Pacal pyraid are partitioned into ubet uch that the product of the coefficient In each ubet i a contant N and uch that the product of all the coefficient "urrounding" a given -noial coefficient i N. The reult i then generalized to other nuerical triangle or pyraid,,. INTRODUCTION In the paper by Hanell and Hoggatt [] the following i proved: Theore., The product of the ix binoial coefficient urrounding each binoial coefficient [ k > (n ^ 2; < k < n), in PacaP triangle i a perfect integer quare, N 2» Further^ each triad fored by taking alternate binoial coefficient ha product N. Further reult In the plane are obtained by Gould In [4], In thi paper, we generalize thi theore to generalized Pacal pyraid in -pace* 2 S SELECTING THE MULTINOMIAL COEFFICIENTS Let u expand (x t + x 2 + x x ), ( ^ 2; n =, l, 2, - -» ) : frl * > = 2-f W k k J W " - * - kjfh e +k =n (k^o) Here 35

2 352 A PROPERTY OF MULTINOMIAL COEFFICIENTS [Oct. / k i + k k \ ^ (ki + k k ) \ k l 9 k 29..., k / ktskaska!... k! The r e c u r r e n c e relation i / ^ + k k \ _ J ^ / kj + k k - \ y k l f k 2,..., k y Z l ^ \ k i " c ij» k 2 " 2j> *'» k " e j / j=l where 5.. = if i ^ j and 6.. = if i = j. Given a ultinoial coefficient (inide the pyraid, i. e., k ^, =, 2,,) / l q + k k \ \ki, k 2J -, k y there are ultinoial coefficient / k + k 2 + k k - l \ (. = ^ \k t - dij, k 2 - $ 2 j»» k - 5 j / 2, '.., ) which "contribute to A" by ean of r e c u r r e n c e relation (R); that i, lie d i - rectly above A in the pyraid. Thee ae ultinoial coefficient contribute to ( - ) ultinoial coefficient k-t + k ( k _- + k \ k i - «i j + C i k. k 2 ^ 2 j k - - k 9 ^ j + S k / J ( j ' k = ' 2 ' ' ' ' ' ; 3 ^ k ) which are all on the ae level a A. T h e r e are alo ultinoial coefficient which are contributed to by A, naely thoe of the for / ki + k k + \ \ ki + Cij, k j,, k + 5j / ' (j =, 2,..., )

3 97] A PROPERTY OF MULTINOMIAL COEFFICIENTS 353 Thu there are above A, ( - ) on the ae level a A and below A. Thee ( + ) ultinoial coefficient we ay are adjacent to A, and geoetrically urround A. Theore. The product of the ( + ) ultinoial coefficient adjacent to A i a perfect integer 3. THE PRINCIPAL RESULT power. Proof. In the following, =, 2,,. On the level above A, the nuber k - appear once; on the level with A the nuber k - appear - tie; k - doe not appear in the level below A. (In the level with A, k appear ( - l)( - 2) tie; and on the level below A, k appear - tie. On the level above A, k + doe not appear; on the level with A, k + appear ( - ) tie; and on the level below A, k + appear once. Thu? in the denoinator of the product, (k - ) appear tie, (k )! appear ( - ) tie, and (k +) appear tie. The product, therefore, nv of all ( + ) n ultinoial coefficient adjacent to A i: / \ I T Zh 2>, & v^^'-r^vi + i)'-] n -, 4 n TT (k. - D K k. D ^ ^ k. + ) Z^J i + I I -i HI = N N. B. N i an integer, ince (p/q) reduced to lowet ter with q f I not an integer when raied to the power. But the product i an integer, ince each ultinoial coefficient factor i an integer. We next prove the following rather urpriing reult.

4 354 A PROPERTY OF MULTINOMIAL COEFFICIENTS [Oct Theore: The ( + ) ultinoial coefficient adjacent to A with product N, can be decopoed into et of ( + ) ultinoial coefficient uch that the product over each et i N. Furtherore, the contruction yield et of ( + ) ultinoial coefficient uch that peruting the ubcript cyclically on any one et - tie produce all the other et. Thu the et are congruent by rotation. Proof. We now decribe a contruction for the et. product within each et ut be -, IS k -- HI k ')! ( I H! N = = Recall that the coefficient For convenience we introduce the following notation for the ultinoial ( ki + kg + + k \ k, k 2>... r j = (,,,... ) ) o that by introducing - or + a entrie in the -tuple, we can raie or lower one of the k. and thu repreent adjacent coefficient. For exaple: and / k t + k k - \ \ k i -, k 2, k 3,, k y ~ (-,,,, ) / kj + k k \ ( k, k 2, k 3 + l, k 4,..., k n i i! k - l j = ^ ' ' ' - - ' ' - ) Thu a ubet of ultinoial coefficient of the type deired could be repreented a an ( + ) x atrix, where each row i a vector a decribed above, each row repreenting one of the adjacent ultinoial coefficient. Each ubet 5 in order to have the proper nuerator in the product, ut have

5 97 A PROPERTY OF MULTINOMIAL COEFFICIENTS 355 one coefficient fro above, one fro below, and - fro the ae level a the given coefficient. We hall adopt the convention that the firt row r e p- t reent the coefficient above and the ( + ) row the coefficient below* Let the ( + ) x atrix have entrie a... two eparate cae. For odd 5 let It i neceary to conider a.. = - 3J a l.. = + a.. = otherwie 3, 2 5, 2 f, +, We illutrate with -, f Spacing between - and + 4 I on the iddle - row. 2 Thi correpond to ix ultinoial coefficient whoe 5 lower arguent are given in row of thi atrix. The other four et are obtained by rotating cyclically the colun vector of atrix C. We note that (k - )! appear once 5 k I appear - tie, and (k + ) appear once in each of the five e t, =,- 2,,. F o r even, let a J a k + l 5 + l - k = + a + l - k 9 k = + 3 =, k =, k =, 2,, 2,, (/2) - 2,, (/2) + a + l ( / 2 ) + l = X a.. = otherwie 3 h = i. 2,, + 2,

6 356 A PROPERTY OF MULTINOMIAL COEFFICIENTS Oct. 97 We illutrate for = 6: C f = 4 2 J Spacing between - 5 } + on the iddle 3 I - row and In both atrice C and C f a cyclic perutation of the colun vector doe not produce a duplication before tep. Thu each et of + eleent are ditinct and the et exhaut the collection of ( + ) ultinoial coefficient adjacent to A. It hould be noted that the above contruction doe not yield the only poible partitioning. There exit other partitioning into ubet with the deired property in both the even and odd cae. For exaple, for = 3, the above contruction yield but - - / I - V o alo have the deired property. For = 4, the following i a partitioning different fro that yielded by the above proce: [Continued on page 42. ]