Combinatorial bijections are given from the set of n-color compositions of ν, for which a part of size n can take on n colors, to the set of compositions of 2ν − 1 having only parts of size 1 or 2, the set of compositions of 2ν having only odd parts, and the set of compositions of 2ν + 1 having no parts of size 1. A generalized bijection based on similar ideas is then given between the set of compositions of ν into parts congruent to a(mod b) and the set of compositions of ν + b − a into parts congruent to b(mod a) with each part greater than b − a.

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).

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If S is a cofinite set of positive integers, an "S-restricted composition of n" is a sequence of elements of S, denoted λ = (λ1, λ2, . . . ), whose sum is n. For uniform random S-restricted compositions, the random variable B( λ) = i λi is asymptotically lognormal. (A precise statement of the theorem includes an error term to bound the rate of convergence.) The proof is based upon a combinatorial technique for decomposing a composition into a sequence of smaller compositions.2010 Mathematics Subject Classification. 05A16, 60C05, 60F05.

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