Bounded compositions and products of chains, or bi(s)nomials everywhere from de Moivre's formula to spin multiplicities and constant-sum codes

Source abstract

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 $\prod_{i=1}^{n}(1+x+\cdots+x^{s_{i}})$. 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 $\mathrm{SU}(2)$ 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 $s \geq 1$ 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 $k \sim λn^{s/(s+1)}$, with factor $e^{-αλ^{s+1}}$ when the proportion of bounds equal to the fixed minimum $s \geq 1$ 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.