COMETS h2T? + IQi-pt? + (/i4 + Aa)*4 + + 7i3/(2)a;5 + (/iG + 2/i47i2) x6 + (7^67^ + 7i-5^2 + + (7i.8 + 27i6^2 + /i4)a;8 + ... 2 2 Taking ^ + 7^=+ {(2) + (l )} 2 = 2(4) + 3(31) + 4(2 ) + 5(212) + 7(14), 2 the term 5(21 ) indicates that objects such as a, a, b, c can be partitioned in five ways into two parts. These are aa, b, c ; ba, a, c; ca, a, b ; a, ab, c; a, ba, c. The function Ay; has been, studied. (See MacMahon, Proc. Lond. Math. Soc. vol. xix.) Putting x equal to unity, the function may be written (A2 + A.4 + A6+ ...)(1+7i1 + A2 + A3 + A4+ ...), a convenient formula. The method of Differential Operators, of wide application to problems of combinatorial analysis, has for its leading idea the designing of a function and of Method of a differential operator, so that when the opera-
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and the law by which the operation is performed upon the product shows that the solutions of the given problem are enumerated by the number A, and that the process of operation actually represents each solution. Ex. Gr.—Take X^S, X2 = 2, X1 = l, ih = 2, P2 = 2»i,3=1Ji,4=1> 0201^3^2*1 = 8, and the process yields the eight diagrams :—
Operators^ tor Performed upon the function a number is reached which enumerates the solutions of the given problem. Generally speaking, the problems considered are such as are connected with lattices, or that it is possible to connect with lattices. To take the simplest possible example, consider the problem of finding the number of permutations of n different letters. The viz., every solution of the problem. Observe that transposition of / d
n
function is here xn, and the operator (^“J —<>x> yielding the diagrams furnishes a proof of the simplest of the laws of symmetry in the theory of Symmetric Functions. Snxn = n ! the number which enumerates the permutations. In For the next example we have a similar problem, but no restriction is placed upon the magnitude of the numbers which fact— may appear in the compartments. The function is now dxxn = dx. x. x. x. x. x...., Aaj 71 a0 • - ■ hmi hXm being the homogeneous product sum of the and differentiating we obtain a sum of n terms by striking out an quantities a, of order X. The operator is as before a; from the product in all possible ways. Fixing upon any one of DjqD^ ... T)pm, these terms, say x.tp.x.x , we again operate with bx by striking out an x in all possible ways, and one of the terms so reached is and the solutions are enumerated by x.fy.x.'b.x Fixing upon this term, and again operating and continuing the process, we finally arrive at one solution of the •" ^Pn ••• ^ problem, which (taking say ?i = 4) may be said to be in correPutting as before Xi = 3 , X2 = 2, X3 = l, p1 = 2, p2 = 2, jp3=1 ,p4 = l, spondence with the operator diagram— the reader will have no difficulty in constructing the diagrams of the eighteen solutions. The next and last example of a multitude that might be given shows the extraordinary power of the method by solving the famous problem of the “Latin Square,” which for hundreds of years had proved beyond the powers of mathematicians. The problem consists in placing n letters a, b, c, ...» in the compartments of a square lattice of »2 compartments, no compartment being empty, so that no letter occurs twice either in the same the number in each row of compartments denoting an operation row or in the same column. The function is here of bx. Hence the permutation problem is equivalent to that of -1 n-2 a o anv placing n units in the compartments of a square lattice of order (Saf 0-;2 - n-i ) n in such manner that each row and each column contains a single unit. Observe that the method not only enumerates, but and the operator D n the enumeration being given by also gives a process by which each solution is actually formed. The 2same problem is that of placing n Rooks upon a chess-board 1 211-2 a 2 a y of ?i compartments, so that no Rook can be captured by any ct n-l M other Rook. Regarding these elementary remarks as introductory, we See Trans. Camb. Phil. Soc. vol. xvi. pt. iv. pp. 262-290. proceed to give some typical examples of the method. Take a Authorities.—MacMahon. “Combinatory Analysis: A lattice of m columns and n rows, and consider the problem of Review of the Present State of Knowledge,” Proceedings London placing units in the compartments in such wise that the sth Mathematical Society, vol. xxviii. London, 1897. Here will be column shall contain X, units (s=:l, 2, 3,...m), and the <th row found a bibliography of the Theory of Partitions.—Whitworth. units (7 = 1, 2, 3,... n). Choice and Chance.—Lucas. TMorie des Nombres. Paris, 1891. Writing —Cayley. Collected Mathematical Papers, Cambridge, 1898, vol. l + a1x + azxz+ ... +... = (1 +axa;)(l +a2:r)(l +a3a;)... ii. p. 419 ; vol. iii. pp. 36, 37 ; vol. iv. pp. 166-70 ; vol. v. pp. 62-65, 617 ; vol. vii. p. 575 ; vol. ix. pp. 480-83 ; vol. x. pp. 16, and DP=^|(5ai + «15a<>-|-a25a3+...)p, the multiplication being sym- 38, 611 ; vol. xi. pp. 61, 62, 357-64, 589-91 ; vol. xii. pp. 217-19, 273-74 ; vol. xiii. pp. 47, 93-113, 269.—Sylvester. American bolic, so that 0^ is an operator of order p, the function is Journal of Mathematics, vol. v. pp. 119, 251.-'-MacMahon. Proceedings London Mathematical Society, vol. xix. p. 228 et seq. ; - «Am, Phil. Trans. Royal Society of London, vol. clxxxiv. pp. 835-901 ; and the operator Dp1Dp2Dp3... Dpn. The number vol. clxxxv. pp. 111-60 ; vol. clxxxvii. pp. 619-73 ; vol. cxcii. 262-90. D^D^ ... Di5nax1'xA2aA3 ••• aAm enumerates the solutions. For the pp. 351-401 ; Trans. Camb. Phil. Soc. vol. xvi. pp. (p. A. M.) mode of operation of upon a product reference must be made Comets.—All comets move in orbits that are conic to the section on “Differential Operators” in the article Algesections—ellipses, parabolas, or hyperbolas. The periodic braic Forms. Writing comets move in elliptic orbits, and are usually seen at more Pi P2 Pn axA1axA ...«> =... + A2a12 a i..an +...,’ than one return. Non-periodic comets move in parabolas 2 (or in hyperbolas), and are seen at only one return. The or, in partition notation, comets of each calendar year are provisionally designated Al A2 (l )(l )...(lS = ... + A(plP2...^)... + , as 1895 A, 1895 B, 1895 C, &c., the letters A, B, C, &c. D2>1Dp2...D^(Di)(lA2)...(iAm) = A, indicating the order of discovery. The name of the dis-
