410 



SCIENCE 



[N. S. Vol. XLV. No. 1165 



functions. A distinction is drawn between 

 the parcel of objects, in which the order of 

 arrangement in the parcel is immaterial, and 

 the group of objects, in which the order of 

 arrangement in the group is material. For 

 instance if we sort 3 a's 2 ^'s, 1 y, 1 8 into 

 seven boxes of which four boxes are exactly 

 alike, and three boxes alike but difPerent from 

 the first four, we find that we haye a problem 

 of distribution of objects of type (321-) into 

 parcels of type (43), which can be done in 11 

 ways. This number 11 may be found by a dis- 

 tribution function, derived from the theory of 

 symmetric functions. This function gives, for 

 instance, for the various types of 4 objects dis- 

 tributed into parcels of type (2) these results: 

 for type (4), 2 ways, for type (31) 3 ways, 

 for type (22) 4 ways, for type (211) 5 ways, 

 for type (1111) 7 ways. If, however, the dis- 

 tribution is into groups rather than into par- 

 cels, we have for type (4) 2 ways, for type 

 (31) 6 ways, for type (22) 10 ways, for type 

 (211) 18 ways, and for type (1111) 36 ways. 

 The determination by the function consists 

 in finding the coefficients in formulse that 

 arise from the theory of symmetric functions. 

 These coefiicients may be found directly for 

 the individual terms by using the operators 

 referred to. 



Section two considers the theory of separa- 

 tions, a separation being a distribution of the 

 numbers constituting a partition of some 

 integer into parcels, or groups. Extensive 

 generalizations are possible from the formulae 

 and the operators produced. The application 

 to sets of objects of given types and their dis- 

 tributions resolves more complicated problems 

 than those given before. For instance, with 

 a set of four threefold objects, afl„a^, afl,a^, 

 hJ)J}„ cfi„c^ can be formed 38 cases of distri- 

 bution into the types (211), (22), (211), 

 namely the objects aflfi^c^, a.a.J}.fi,, a^afi^c^, 

 and the different permutations of these ar- 

 rangements. 



Section three deals with permutations, par- 

 ticularly with points useful in the general 

 theory of combinations and distributions. A 

 certain master theorem is deduced which has 

 great resolving power. In particular it solves 



the problem of ascertaining the number of per- 

 mutations in which every letter occupies a 

 new place, and in expressing sums of powers 

 of binomial coefficients. The notion of lattice 

 permutation is introduced, by which is meant 

 that if any permutation be made of a a's, 

 h p's, c y's, etc., to be a lattice permuta- 

 tion it must be such that reading it from left 

 to right, at no point of it will the number of 

 a's so far written be less than the number of 

 ^'s, nor number of j3's less than the number 

 of y's, etc. For instance, for 2 a's and 2 j8's 

 the lattice permutations are aa^fi, and a/3a8. 

 The permutation afS/ia is not a lattice permu- 

 tation because when we arrive at the third 

 letter, we shall have 2 j3's and only 1 a- These 

 are called lattice permutations because they 

 serve to handle arrangements of integers at 

 the nodes of a rectangular lattice in a plane, 

 or in space. 



Section four considers compositions of in- 

 tegers, by which is meant the permutations 

 of the partitions of the integer. In connection 

 with these some new symmetric functions are 

 introduced. An application to Newcomb's 

 problem is made. It is this : given p cards 

 marlced 1, q marked 2, r marked 3, etc., which 

 are shuffled and dealt in such wise, that as 

 long as a card is not of lower number than the 

 preceding it is placed upon the preceding, but 

 if lower it must start a new pile; what is the 

 probability that there are at most m piles 

 when all have been dealt? 



Section five introduces the notion of perfect 

 partition, that is partitions such that each 

 contains only one partition of each lower num- 

 ber. For instance, for 7, a perfect partition is 

 (4111), since we have only one partition of 

 1, 2, 3, 4, 5, 6. These are then applied to dis- 

 tributions upon a chess board. A connection 

 is thus arrived at between magic squares and 

 the general theory. The enumeration of Latin 

 squares is effected by generating functions, 

 thus solving a long-standing problem. For 

 instance, the number of reduced Latin squares 

 of order 1, is 1, of order 2, is 1, of order 3, 

 is 1, of order 4, is 4, and of order 5, is 52. 



Section six enumerates the partitions of 

 multipartite numbers. A multipartite num- 



