Unified derivation of the limit shape for multiplicative ensembles of random integer partitions with equiweighted parts

• Published in: Random structures & algorithms Vol. 47; no. 2; pp. 227 - 266
• Main Author: Bogachev, Leonid V.
• Format: Journal Article
• Language: English
• Published: Hoboken Blackwell Publishing Ltd 01.09.2015; Wiley Subscription Services, Inc
• Subjects: Algorithms; Combinatorial analysis; Counting; cumulants; Derivation; generating functions; integer partitions; Integers; limit shape; local limit theorem; Mathematical analysis; Partitions; Probability; Randomization; Young diagrams
• ISSN: 1042-9832; 1098-2418
• DOI: 10.1002/rsa.20540

We derive the limit shape of Young diagrams, associated with growing integer partitions, with respect to multiplicative probability measures underpinned by the generating functions of the form ℱ(z)=∏ℓ=1∞ℱ0(zℓ) (which entails equal weighting among possible parts ℓ∈ℕ). Under mild technical assumptions on the function H0(u)=ln(ℱ0(u)), we show that the limit shape ω*(x) exists and is given by the equation y=γ−1H0(e−γx), where γ2=∫01u−1H0(u) du. The wide class of partition measures covered by this result includes (but is not limited to) representatives of the three meta‐types of decomposable combinatorial structures — assemblies, multisets, and selections. Our method is based on the usual randomization and conditioning; to this end, a suitable local limit theorem is proved. The proofs are greatly facilitated by working with the cumulants of sums of the part counts rather than with their moments.Copyright © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 227–266, 2015