Showing papers in "Duke Mathematical Journal in 1941"

The distribution of the number of summands in the partitions of a positive integer

Abstract: It is easily seen that the number of partitions of n having k or less summands is equal to the number of partitions of n in which no summand exceeds k . Thus the preceding results can be applied to this case also . In §3 we consider P(n), the number of partitions of n into unequal parts. (By a theorem of Euler, P(n) is also equal to the number of partitions of n into odd summands with repetitions allowed .) We obtain results similar to the above for pk(n), but we shall not give all details of the proof . In §4 we derive an asymptotic formula for pk(n),

The Arithmetical Theory of Birkhoff Lattices

Abstract: In the development of lattice theory considerable work has been devoted to the study of the arithmetical properties of modular and distributive lattices. Indeed most of the decomposition theorems of abstract algebra have been extended to these more general domains. Nevertheless, there are lattices with very simple arithmetical properties which come under neither of these classifications. For example, the lattices with unique irreducible decompositions, which were studied by the author in a previous paper [3]1 satisfy the Birkhoff condition2 which is even less restrictive than the modular axiom. Furthermore, there are important algebraic systems which give rise to non-modular, Birkhoff lattices. Thus, since every exchange lattice (Mac Lane [4]) is a Birkhoff lattice, the systems which satisfy Mac Lane’s exchange axiom form lattices of the type in question. In this paper we shall study the arithmetical structure of general Birkhoff lattices and in particular determine necessary and sufficient conditions that certain important arithmetical properties hold.