We explore random decomposable combinatorial structures $\sigma$ of size $n\geq 1$ having $k_j(\sigma)$ components of size $j$, $1\leq j\leq n$, sampled according to the frequency $\nu_n$ from their class ${\mathbb S}_n$. The relation $1k_1(\sigma)+\cdots+nk_n(\sigma)=n$ holding for all $\sigma\in {\mathbb S}_n$ induces stochastic dependence of the entries of the structure vector $\bar k(\sigma):=\big(k_1(\sigma),\dots, k_n(\sigma)\big)$. The analysis of its covariance matrix is a worthy task. So, by finding the extreme eigenvalue in the case of uniform permutations, J. Klimavičius and E. Manstavičius (2018) established the optimal variance estimate
$$\begin{equation*}\frac1{n!}\sum_{\sigma \in \mathbb{S}_n}\bigg(\sum_{j\leq n} y_j\Big(k_j(\sigma)- \frac{1}{j}\Big)\bigg)^2 \leq \frac{3}{2} \sum_{j\leq n} \frac{y_j^2}{j}, \quad y_j\in{\mathbb R},\; n\geq 2. \end{equation*}$$
The success has been extended by Ž. Baronėnas, E. Manstavičius and P. Šapokaitė (2021) for the permutations when $\nu_n$ is the Ewens probability, and for the uniformly drawn polynomials over a finite field and other weighted multisets by A. Karbonskis and E. Manstavičius (2020). Apart from such linear statistics, we examine additive functions defined via vectors $\big(h_1(k_1(\sigma)), \dots, h_n(k_n(\sigma))\big)$, where $h_j: {\mathbb R} \to{\mathbb R}$ are some mappings. The transformations drastically change covariances; nevertheless, one can hope to obtain asymptotically sharp variance estimates in particular cases when $n\to\infty$. Our recent result (see, Lithuanian Math. J. , 2024) witnesses that. This paper contains a vast historical account.
In localising the eigenvalues, we propose constructions based upon special functions of the approximate eigenvectors. The methodology takes its origin in the papers by J. Kubilius (1985) and J. Lee (1989) exploring additive number theoretical functions. The continuing frequent applications of the Turán-Kubilius inequality also support the belief in the perspective of our research.
This is a joint work with E. Manstavičius.