Part-products of S-restricted integer compositions

Abstract

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.

Asymptotic normality of integer compositions inside a rectangle

“…The latter, with various types of restrictions, have attracted much attention in recent years (cf. [2], [4], [8], [10], [13], [15], [16]). For example, Malandro [13] determines asymptotic formulas for L-restricted integer compositions -L being an arbitrary finite set -and Shapcott [16] and Schmutz and Shapcott [15] find a lognormal distribution for part products of restricted integer compositions.…”

“…[2], [4], [8], [10], [13], [15], [16]). For example, Malandro [13] determines asymptotic formulas for L-restricted integer compositions -L being an arbitrary finite set -and Shapcott [16] and Schmutz and Shapcott [15] find a lognormal distribution for part products of restricted integer compositions. Hitczenko and Stengle [11] derive the expected number of distinct part sizes of unrestricted random compositions.…”