Symmetrically constrained compositions

Symmetrically Constrained Compositions

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 ...

Compositions and samples of geometric random variables with constrained multiplicities

We investigate the probability that a random composition (ordered partition) of the positive integer n has no parts occurring exactly j times, where j belongs to a specified finite ‘forbidden set’ A of multiplicities. This probability is also studied in the related case of samples Γ = (Γ1,Γ2, . . . ,Γn) of independent, identically distributed random variables with a geometric distribution. Résu...

Compositions with Parts Constrained by the Leading Summand

Many symmetrically indivisible structures

A structureM in a first-order language L is indivisible if for every coloring of M in two colors, there is a monochromatic M′ ⊆ M such that M′ ∼=M. Additionally, we say thatM is symmetrically indivisible ifM′ can be chosen to be symmetrically embedded inM (that is, every automorphism of M′ can be extended to an automorphism ofM). In the following paper we give a general method for constructing ...

Compositions versus Cyclic Compositions

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 ...,...