Noting that the usual ‘propositionally’ based way of composing retrenchments can yield many ‘junk’ cases, alternative approaches to composition are introduced (via notions of tidy, neat, and fastidious retrenchments) that behave better in this regard. These alternatives do however make other issues such as associativity harder. The technical details are presented for vertical composition of ret...

Given integers a1, a2, . . . , an, with a1 + a2 + · · ·+ an ≥ 1, a symmetrically constrained composition λ1 + λ2 + · · ·+ λn =M of M into n nonnegative parts is one that satisfies each of the the n! constraints {∑n i=1 aiλπ(i) ≥ 0 : π ∈ Sn } . We show how to compute the generating function of these compositions, combining methods from partition theory, permutation statistics, and lattice-point ...

We study formulas for Fibonacci numbers as sums over compositions. The Fibonacci number Fn+1 is the number of compositions of n with parts 1 and 2. Compositions with parts 1 and 2 form a free monoid under concatenation, and our formulas arise from free submonoids of this free monoid.

Feature modeling is a popular domain analysis method for describing the commonality and variability among the domain products. The current formalisms of feature modelling do not have enough support for automated domain product configuration and validation. We have developed a theory of feature modeling: a feature model is analogous to a definition of a language; a particular feature composition...

In this paper we introduce a new graph composition, called 2-amalgam, and we prove that the 2-amalgam of perfect graphs is perfect. This composition generalizes many of the operations known to preserve perfection, such as the clique identification, substitution, join and amalgam operations. We give polynominal-time algorithms to determined whether a general graph is decomposable with respect to...

In this paper we consider the absolute variation statistics of a composition σ = σ1 · · ·σm of n which is a measure of the sum of absolute differences between each consecutive pair of parts in a composition. This and some related statistics which we discuss, can also be interpreted within the context of bargraph polygons and bargraph walks of area n. In this context, absolute variation correspo...

A (1, k)-composition of a positive integer n consists of an ordered sequence of the integers 1 and k whose sum is n. A palindromic (1, k)-composition is one for which the sequence is the same from left to right as from right to left. We give recursive equations and generating functions for the total number of such compositions and palindromes, and for the number of 1’s, k’s, “+”signs and summan...

A composition is a sequence of positive integers, called parts, having a fixed sum. By an m-congruence succession, we will mean a pair of adjacent parts x and y within a composition such that x ≡ y (mod m). Here, we consider the problem of counting the compositions of size n according to the number of m-congruence successions, extending recent results concerning successions on subsets and permu...

This paper deals with Carlitz compositions of natural numbers (adjacent parts have to be different). The following parameters are analysed: number of parts, number of equal adjacent parts in ordinary compositions, largest part, Carlitz compositions with zeros allowed (correcting an erroneous formula from Carlitz). It is also briefly demonstrated that so-called 1-compositions of a natural number...