Augustine O. Munagi

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The number of ways of partitioning a set of m elements into k nonempty subsets (called classes or blocks) is given by s2(m,k), the Stirling number of the second kind. Without loss of generality we assume that the m elements have been labeled 1,2, . . . ,m and consider k-partitions of the set [m] = {1,2, . . . ,m}. Substantial information on set partitions(More)
Let S = {S1,S2, . . .} represent a collection of nonempty sets of nonnegative integers in which each member contains the integer 0. Then S is called a complementing system of subsets for X ⊆ {0,1, . . .} if every x ∈ X can be uniquely represented as x = s1 + s2 + ··· with si ∈ Si. We will also write X = S1 ⊕ S2 ⊕ ··· and, when necessary, refer to X as the(More)
We consider words over the alphabet [k] = {1, 2, . . . , k}, k ≥ 2. For a fixed nonnegative integer p, a p-succession in a word w1w2 · · ·wn consists of two consecutive letters of the form (wi, wi + p), i = 1, 2, . . . , n − 1. We analyze words with respect to a given number of contained p-successions. First we find the mean and variance of the number of(More)
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)
We study a special partial fraction technique which is designed for rational functions with poles on the unit circle, known as q-fractions. Even though the theory of q-partial fractions has already been applied to the Rademacher Conjecture, no systematic computational development appeared. In this paper we present two algorithms for the computation of(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)