CHAP, ill] PARTITIONS. 147 



distinct positive summands in which the sum of the absolutely least residues 

 modulo 3 of the summands equals a given positive or negative number h, 

 there occur as many partitions into an even number of summands as into 

 an odd number, except only when s is the pentagonal number (3/i 2 h)/2, 

 for which there exists an additional partition into an even or odd number of 

 parts according as h is even or odd. Also a purely arithmetical proof is 

 given. If we employ this theorem for each of the permissible values of 

 h and add the results, we get Legendre's 23 result: 



These theorems are extended (pp. 16-17) to w-gonal numbers. 



T. P. Kirkman 151 took all partitions of x into k parts = 0, as 5, 

 1 1 3, 2 2 1, 1 4, 2 3 for x = 5, k = 3, formed their permutation 

 symbols, 3a 2 6 + 2abc, counted their permutations 3-3 + 2-6 = 21 = (2), 

 and stated that the result is always Cl^r 1 )- There is a question on the 

 partition of a polygon of r sides into k parts, treated later (ibid., 8, 1894, 

 109-129); cf. Cayley. 152 



P. Bachmann 153 gave an exposition of the work by Euler. 



P. A. MacMahon 154 considered compositions, i. e., partitions in which 

 the arrangement of the parts is essential. The number of compositions of 

 n into p parts > is the binomial coefficient (ll). The total number 

 of compositions of n is 2 n ~ 1 . If the parts are ^ s, the number is the 

 coefficient of x n in (x + x 2 + + z s ) p . A multipartite number p\pz' 

 specifies pi + p 2 + numbers (or things), pi of one sort, p 2 of a second 

 sort, etc. The number of its compositions into r parts is the number of 

 distributions of the pi + p z + numbers into r parcels and is the coeffi- 

 cient of ai pl a 2 pit - - in the expansion of (h^ + /i 2 + -) r , where h s is the sum 

 of the homogeneous products of degree s of a\, a 2j . The graph of a com- 

 position (2, 1, 4) of 7 is given by placing nodes at points P, Q on the line 

 A B divided into 7 equal segments, so that in moving from A to B by steps 

 proceeding from node to node, 2, 1 and 4 segments of the line are passed 

 over in succession. The graph of a composition of a bipartite number pq 

 is derived by placing nodes at suitable points on q + 1 similar graphs of p 

 placed parallel and equidistant and with corresponding points joined by a 

 second set of parallels. Let A and B be opposite vertices of the resulting 

 total parallelogram [see figure, MacMahon 168 ]. Pass from A to B by 

 successive steps, each consisting in moving a certain number of segments 

 parallel to AK and then moving a certain number of segments parallel 

 to KB. The successive steps are marked by nodes, which define the graph 

 of a composition. An essential node is where the course changes from the 



161 Mem. and Proc. Manchester Lit. Phil. Soc., (4), 7, 1893, 211-3. Math. Quest. Educ. 



Times, 60, 1894, 98-102. 

 152 Proc. London Math. Soc., 22, 1891, 237-262; Coll. Math. Papers, XIII, 93-113. 



163 Zahlentheorie, 2, 1894, Ch. 2, 13-45. 



164 Phil. Trans. Roy. Soc. London for 1893, 184, A, 1894, 835-901. 



