COMBINATORIAL
158 71
3
5
aa:2s+1
of oAc in the product (1+ axXl+ aa3 )(l+«2r ) (1+ ) and thence the coefficient of aO in this product is xr and we have the expansion 1 — X2 . 1 — X4 . 1 - x6. ...1 — x2d 3 (l + ax)(l + <xx )(l+ax5)... ad inf. ia3+ ... “■^l-x2® + l-x2. l-x4" + 1-x2. 1-x4. l-xe Again, if we restrict the part magnitude to i, the largest angle of nodes contains at most 2£ — 1 nodes, and based upon a square of 62 nodes we have partitions enumerated by the coefficient of a x in the product (l+ax)(l+aa?)(l+aa^)/v (l + ax2i-l);l moreover the same number enumerates the partition of {n - d‘ ) into. 9 or fewer parts, of which the largest part is equal to or less than i — 9, and is thus given by the coefficient of x^-*2) in the expansion of 1 _a;i-0+l. l-xi~e+2 . 1 -xi-e+3. ... 1 -x4 1 -x. 1 — x2. 1 — x3. ... 1 — x0 1_x2i-2e+21_x2i-2e+l' ' 1_, -x* ; or of x" in 1 - X2 . 1 - X4.1 — X6 . ... 1 - X20 hence the expansion (l + ax)(l+ax3)(l+ax5)... (l+ax2i-l) 0 i 2i_20 + 2 2i_2e+4 ....1-X2j2^ = 1„ = ^1-X 2 . 1-X n % M• 4 o=i 1 -x .1 —x . 1 — x6. ...1 — x20 There is no difficulty in extending the graphical method to . three dimensions, and we have then a theory of a p to three*0 sPeC4a^ kind of partition of multipartite numbers. Of Dimen= such kind is the partition sioas. (a1a2a3 ..., ..., c1c2c3 ..., ...) of the multipartite number + ^ + , a^+b^+C2+ j ®s + ^3 + c3 + if aa20^3... 5 — ^3^^ *** ? *** ..., for then the graphs of the parts a1«2!X3 ••• > &1&2&3 ••• > ••• ar® superposable, and we have what we may term a regular graph in three dimensions. Thus the partition (643,632,411) of the multipartite (16,8,6) leads to the graph O — x ® © ® ®
ANALYSIS
every line of route which proceeds from the origin in the positive directions of the axes. This brings in view the modern notion of a partition, which has enormously enlarged the scope of the theory. We consider any number of points in piano or in solido connected (or not) by lines in pairs in any desired manner and fix upon any condition, such as is implied by the symbols, ^ , >, = < , ^, ^, as affecting any pair of points so connected. Thus in ordinary unipartite partition we have to solve in integers such a system as ^an cti + 0,2 4* ctg + ... + an = 7iy the points being in a straight line. In the simplest example of the three-dimensional graph we have to solve the system — a2 tti 4- a2 4- a3 4- a4 — Ml 1^ Cto ^ a. and a system for the general lattice constructed upon the same principle. The system has been discussed by MacMahon, Phil. Trans. B. S. vol. clxxxvii. A, 1896, pp. 619-673, with the conclusion that if the numbers of nodes along the axes of x, y, z be limited not to exceed the numbers m, n, l respectively, then writing for brevity l-x* = (s), the generating function is given by the product of the factors (Z + l) (1) {1 + 2) (2)
{1 + 2) (2) (Z + 3) (3)
{l + n) (Z + Ti + l) {n) * (« + l)
{l + m) {m) (l + m+1) (m + 1) (l + m + n-1) {m + n- 1)
y one factor appearing at each point of the lattice. In general, partition problems present themselves which depend upon the solution of a number of simultaneous relations in integers of the form X1a1 + Agtq + Xsa3 + ... ^ 0, the coefficients X being given positive or negative integers, and in some cases the generating function has been determined in a form which exhibits the fundamental solutions of the problems from which all other solutions are derivable by addition. (See MacMahon, Phil. Trans. R.S. vol. cxcii. (1899), pp. 351-401 ; and Trans. Camb. Phil. Soc. vol. xviii. (1899), pp. 12-34.) The number of distributions of n objects {piP^Pz---) into parcels (m) is the coefficient of bm {PiPTPz ...)*”in the develop- Metbod of ment of the fraction. Symmetric 1 Functions. y (1 — bax2 .2 1 — 5/3x . 12— byx ...2 2 x (1 - 5ct 3x 3.1 - 5a/3x . 1 - 5/3 x ... ) and every such graph is readable in six ways, the axis of z being x (1 — ba x . 1 — btpfix?. 1 - bafiyo?...) perpendicular to the plane of the paper. Ex. Gr. and if we write the expansion of that portion which involves proPlane parallel to xy, direction Ox reads (643,632,411) ducts of the letters a, /3, 7, ... of degree r in the form Oy „ (333211,332111,311100) xy, l+hrlbxr + hr.2b2x2r +..., Oy „ (333,331,321,211,110,110) yz, Oz „ (333,322,321,310,200,200) we may write the development yz, r=co Oz „ (333322,322100,321000) zx, 2 2r +...), II (1 + hrlbxr + hj/x Ox ,, (664,431,321) zx, r=l the partitions having reference to the multipartite numbers 16,8,6, and picking out the coefficient of bm xn we find 976422,13, 11, 6, which are brought into relation through the Tih h h T ..., medium of the graph. The graph in question is more conveniently UHlH? represented by a numbered diagram, viz. 3 3 3 3 2 2 'Zr — m, 'Lrt = n. where 3 2 2 1 The quantities h are symmetric functions of the quantities a,/3,7,... 3 2 1 which in simple cases can be calculated without difficulty, and and then we may evidently regard it as a unipartite partition on then the distribution function can be formed. the points of a lattice, jsX' gT'—Required the enumeration of the partitions of all multipartite numbers {piPTPs •••) 4nto exactly two parts. We find htf—h^ - hfi-L + h^ Aj2 = ^6 — ^5^1 hfl^ A 42=Ag — lifi + /ig/i2 — hfi% + /i2, and paying attention to the fact that in the expression of term A", is absent when r is uneven, the law is clear. the descending order of magnitude of part being maintained along generating function is
the The
