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Since the Laplace operator $$ \Delta=\frac{\partial^2}{\partial x_1^2}+\cdots+\frac{\partial^2}{\partial x_n^2} $$ preserves symmetric polynomials in $n$ variables, its action on the Schur polynomial …

Davis-Slepian-Polya formula for the number of simple graphs on $n$ nodes $$ \frac{1}{n!} \sum_{j_1+2j_2+\cdots+n j_n=n}\frac{n!}{\prod\limits_{k=1}^n k^{j_k} j_k!} 2^{\displaystyle \frac{1}{2}\left( \ …

Let $\mathcal{Y}$ be a vector $\mathbb{Q}$-space of all Young diagrams. Denote by $\delta_\lambda$ the Young diagram of the partition $\lambda$ and $c(\square)$ be the content of the square $\squar …

Consider the generating function $$ G_n(x_1,x_2,\ldots,x_n, t_1,t_2,\ldots,t_n) =\sum_{\lambda}s_{\lambda}(x_1,x_2,\ldots, x_n) t_1^{\lambda_1}t_2^{\lambda_2} \cdots t_n^{\lambda_n}, $$ where the sum …

Suppose we have a complete graph invariant $I$ i.e. for any two graphs $G$ and $H$ we have $ I(G)=I(H) \iff G \cong H $. Suppose now that the invariant $I$ is reconstructible from the desk of $G.$ …

Let $\Lambda_n$ be the algebra of all symmetric polynomials in $n$ variables, which we also consider as an infinite-dimensional vector $\mathbb{Q}$-space, whose basis is the Schur polynomials. The pro …

Let $f_i=f_i(x_1,x_2,\ldots, x_n),i=0,1,2, \ldots $ be a family of symmetric polynomials. For the partition $\lambda=(\lambda_1,\lambda_2, \ldots, \lambda_n)$ consider the determinant $$ D_\lambda(f …