Bounded compositions and products of chains, or bi(s)nomials everywhere from de Moivre's formula to spin multiplicities and constant-sum codes
Abstract
How many ways are there to select k objects from n types when type i is available in only si copies? For equal bounds, this is de Moivre's problem of obtaining a prescribed total with n dice, now a textbook application of the principle of inclusion-exclusion. For arbitrary bounds, the answer is encoded by the rank polynomial ∏i=1n(1+x+⋯+xsi). Its coefficients enumerate bounded compositions and submultisets, give, after normalization, the probability mass function of a sum of independent discrete uniform random variables, and are the weight multiplicities of a tensor product of SU(2) representations. Most of these interpretations are classical but dispersed; here we place them in a common notation and document their provenance. A complement involution proves symmetry and boundary identities. For arbitrary bounds, lower-half first differences give both symmetric-chain starts and mixed-spin multiplicities, and the chain-length and irreducible-dimension profiles agree. Convolution gives log-concavity and the central mode. For equal bounds with fixed s ≥ 1 and even ns, local expansions give central asymptotics with explicit 1/n and 1/n2 corrections for the coefficient and its first difference. A Poisson limit gives the first boundary crossover of these differences at scale k ∼ λns/(s+1), with factor e−αλs+1 when the proportion of bounds equal to the fixed minimum s ≥ 1 tends to α>0. Central first differences include the s-Catalan numbers, themselves Littlewood-Richardson coefficients; central layers yield maximum-size constant-sum codebooks for hash-based signatures. Under the polynomial binomial convention, the full inclusion-exclusion transform vanishes, and the omitted terms sum to the negative of the number of bounded compositions.
Related
- Linear Independence of Polynomial Compositions with Bounded Input Degree
- Algebras with bounded multiplicities of the central cocharacters
- Beyond Constant Truncations: The Bounded Sobolev Approximation Gap
- Bounded cohomology and optimal separating constant for representations
- Bounded Harmonic Functions on Products with a Parabolic Factor
BibTeX
@misc{belbachir2026bounded,
title = {{Bounded compositions and products of chains, or bi(s)nomials everywhere from de Moivre's formula to spin multiplicities and constant-sum codes}},
author = {Hacène Belbachir and José Ricardo G. Mendonça},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31865},
eprint = {2610.31865},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31865},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bounded compositions and products of chains, or bi(s)nomials everywhere from de Moivre's formula to spin multiplicities and constant-sum codes
AU - Hacène Belbachir
AU - José Ricardo G. Mendonça
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31865
UR - https://arxiv.org/abs/2610.31865
AB - How many ways are there to select k objects from n types when type i is available in only s_i copies? For equal bounds, this is de Moivre's problem of obtaining a prescribed total with n dice, now a textbook application of the principle of inclusion-exclusion. For arbitrary bounds, the answer is encoded by the rank polynomial ∏_i=1^n(1+x+⋯+x^s_i). Its coefficients enumerate bounded compositions and submultisets, give, after normalization, the probability mass function of a sum of independent discrete uniform random variables, and are the weight multiplicities of a tensor product of SU(2) representations. Most of these interpretations are classical but dispersed; here we place them in a common notation and document their provenance. A complement involution proves symmetry and boundary identities. For arbitrary bounds, lower-half first differences give both symmetric-chain starts and mixed-spin multiplicities, and the chain-length and irreducible-dimension profiles agree. Convolution gives log-concavity and the central mode. For equal bounds with fixed s ≥ 1 and even ns, local expansions give central asymptotics with explicit 1/n and 1/n^2 corrections for the coefficient and its first difference. A Poisson limit gives the first boundary crossover of these differences at scale k ∼ λn^s/(s+1), with factor e^−αλ^s+1 when the proportion of bounds equal to the fixed minimum s ≥ 1 tends to α>0. Central first differences include the s-Catalan numbers, themselves Littlewood-Richardson coefficients; central layers yield maximum-size constant-sum codebooks for hash-based signatures. Under the polynomial binomial convention, the full inclusion-exclusion transform vanishes, and the omitted terms sum to the negative of the number of bounded compositions.
ER -