Part-Products of 1-Free Integer Compositions

Abstract

If $\vec{\lambda}$ is a composition of the positive integer $n$, define ${\bf B}(\vec{\lambda})$ to be the product of the parts of $\vec{\lambda}$. We present a modified version of Hitczenko's stopped sequence construction that leads to a proof of the asymptotic lognormality of ${\bf B}$ for random 1-free compositions (compositions containing no parts of size 1).

Part-products of S-restricted integer compositions

“…However, due to the length and messiness of the calculations, we record the following result without proof. More details can be found in [30].…”

“…In this section we combine generating function identities from [29,30] with singularity analysis [10,11] to estimate the moments of log B. Recall that r is the magnitude of the second smallest root of f , that 0 < p < 1, and that p < r.…”

“…In her thesis [30], Shapcott considered the random variable B( λ) = i λ i , the product of the parts of a composition λ. (See Hahn [13] for analogous results in a different context.)…”

“…In [29], Shapcott studied the asymptotic distribution of B for uniform random 1-free compositions of n, i.e. the case S = {k ∈ Z : k ≥ 2}.…”