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This paper is devoted to a systematic study of combinatorial identities which assert the equality of different sets of compositions, or ordered partitions, of integers. The proofs are based on properties of zig-zag graphs the graphical representations of compositions introduced by Percy A. MacMahon in his classic book Combinatory Analysis. In particular it(More)
A collection {S 1 ,S 2 ,...} of nonempty sets is called a complementing system of subsets for a set X of nonnegative integers if every element of X can be uniquely expressed as a sum of elements of the sets S 1 ,S 2 ,.... We present a complete characterization of all complementing systems of subsets for the set of the first n nonnegative integers as well as(More)
The labeled factorizations of a positive integer n are obtained as a completion of the set of ordered factorizations of n. This follows a new technique for generating ordered factorizations found by extending a method for unordered factorizations that relies on partitioning the multiset of prime factors of n. Our results include explicit enumeration(More)