We prove that the sum of greatest common divisors of parts in all compositions of n equals the sum of lengths of all cyclic compositions of n. The proof highlights structural similarities between the set of compositions of n and the set of cyclic compositions of n. 1. Definitions and Statement A composition of the positive integer n is an ordered partition of n; that is, a vector ( ) r p p ...,...

In this paper we study the class of m-row matrix compositions (m-compositions, for short), i.e., m-row matrices with nonnegative integer entries in which every column has at least one non-zero element. We provide several enumerative results, various combinatorial identities, and some combinatorial interpretations. Most of these properties are an extension to matrix compositions of the combinato...

background & aim: drought stress, as a main abiotic stress, has a significant influence on growth and metabolic activities of plant species. in this study, the influence of reduced irrigation on dry herbage, essential oil yield, and chemical components of two thyme species were investigated. experimental: this study was done in an experimental greenhouse, i.a.u., shahrekord branch, iran at...

Concave compositions are compositions (i.e. ordered partitions) of a number in which the parts decrease up to the middle summand(s) and increase thereafter. Perhaps the most surprising result is for even length, concave compositions where the generating function turns out to be the quotient of two instances of the pentagonal number theorem with variations of sign. The false theta function disco...

We consider the problem of reconstructing compositions of an integer from their subcompositions, which was raised by Raykova (albeit disguised as a question about layered permutations). We show that every composition w of n ≥ 3k+1 can be reconstructed from its set of k-deletions, i.e., the set of all compositions of n− k contained in w. As there are compositions of 3k with the same set of k-del...