Memoir on the Theory of the Partitions of Numbers Volume 1

Memoir on the Theory of the Partitions of Numbers Volume 1

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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1899 edition. Excerpt: ...bu, + b, + &, +1_m which = 1 i I--s z- m = 0 if Z + sm; whilst (s) occurs to the power----b, _!_m--bt + b, _, -f b, _u which = 1 if s =. m = 0 if.9. I and /ii =-1 if s==Z; the product is, therefore, (/ + 1) (/ + 2)... (to)) 0 (to + 1) (to + 2)... (/ + m) _ (/_+a) (1) (2)... (/) (/ + 1)(/ + 2)... (to) 0, =1" (2) Art. 97. When I = m = n = oo the generating function is 1 (f_, r) (i-., 2)2 (i _ xy (i _ xy...' which may be written ft 1 (l/l-il.l-Vl-;'. 1-.1-(...)... -h -j / -S -5 / from which is deduced a graphical representation in two dimensions involving units and zeros. The graph is regular, and the successive rows involve the numbers 1; 1, 0; 1, 0, 0; 1, 0, 0, 0;... respectively. In the general case there is a similar representation, proper restrictions being placed upon the numbers of figures in the rows. Section 7. Art. 98. It might have been conjectured that the lattice in solido would have afforded results of equal interest, but this on investigation does not appear to be the case. The simplest of such lattices is that in which the points are the summits of a cube and the branches the edges of the cube. a, a, a, a4 a;, ac a; a8 is a partition of a number into eight parts, satisfying the conditional relations indicated by the symbols as shown. The descending order is in the positive direction parallel to each axis. The Q function a2 a a, a5 (1) (2) (3) (4) (5) (G) (7) (8) a7 =-a6 a, 0 + a;7 4-a'" -3 a, -3 a8 (1) (2) (3) (4) (5) (6) (7) (8) and by addition the resulting generating function is 1 + 2a-+ 2, r2 + 3xl + 3 + 5a8 + 4a;7 + 8a? + 4g + 5.-J1" + 3a11 + 3.r13 + 2x" + 2g" + (1) (2) (3) (4) (5) (6) (7) (8) Art. 99. By analogy with the lattice in Plano one might have conjectured...show more

Product details

  • Paperback | 36 pages
  • 189 x 246 x 2mm | 82g
  • Rarebooksclub.com
  • Miami Fl, United States
  • English
  • black & white illustrations
  • 123654109X
  • 9781236541093