Large Sidon Subsets and Pair-Sum Multiplicities of Distinct Multinomial Coefficients

For a positive integer $n$, let ${\mathcal{M}}_n$ be the set of distinct multinomial coefficient values $n!\mathrm{/}(p_1!\cdots p_t!)$, where $(p_1,\dots ,p_t)$ ranges over the integer partitions of $n$. We study two complementary aspects of the additive structure of ${\mathcal{M}}_n$: the maximum cardinality $s(n)$ of a Sidon subset and the multiplicities of unordered pair sums in the full set. A product embedding, strongly Sidon subsets, and estimates for prime partitions give $$\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{\log (s(n)\mathrm{/}n)\log \log n}{\sqrt{\log n}}\ge \frac{\pi }{\sqrt{3}}\mathrm{.}$$ Long arithmetic progressions and separated copies yield a complementary lower bound $2\mathrm{/}(3\sqrt{3})$ for the normalized Sidon defect. Writing $r_n(t)$ for the number of unordered representations $t=x+y$ with $x,y\in {\mathcal{M}}_n$, we study the collision excess $C(n)$, the number $D(n)$ of multiply represented sums, the maximum multiplicity $\mu (n)$, and the cumulative profile $T(n,k)$. We prove $$\mathrm{lim\hspace{0.17em}inf}\frac{C(n)}{n^2\log n}\ge \frac{1}{8},\mathrm{lim\hspace{0.17em}inf}\frac{D(n)}{n^{3\mathrm{/}2}\sqrt{\log n}}\ge \frac{4}{3\sqrt{3}},\mathrm{lim\hspace{0.17em}inf}\frac{\mu (n)}{\sqrt{n\log n}}\ge \frac{1}{2}\mathrm{.}$$ We also obtain a scaled lower envelope for the full profile, an additive-energy bound, stabilization with respect to the number of variables, and an explicit family of trinomial collisions. Exact Sidon values through $n=16$, the lower bound $s(17)\ge 89$, and pair-sum statistics through $n=20$ are reported with reproducible computational material.