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9

The $T_n$'s are equal to the product of $C$ and the Fubini numbers: number of ordered partitions of $n$, also known as ordered Bell numbers. The generating function is $(2-e^x)^{-1}$ and the large-$n$ asymptotics is $$T_n\sim C \frac{n!}{2(\ln 2)^{n+1}}.$$


3

The answer is no. This is equivalent to stating that for any linear hypergraph $H=(\kappa,E)$ with $\kappa$ infinite (note that for $\kappa$ finite, $G=I(H)$ would be finite), we have $\chi(I(H))\leq\kappa$. This follows immediately once we show $|V(I(H))|=|E|\leq\kappa$. For $\alpha<\kappa$ let $E_\alpha=\{e\in E:\alpha\in e\}$. If we remove $\alpha$ ...


2

Here is a way to formulate it as a convex optimization problem, which can then be solved in polynomial time. Your variables are $a_{i,j}$, one for each position in the matrix. The problem is then: $$0\le a_{i,j} \le 1$$ $$\forall_j \sum_i a_{i,j} = 1$$ $$\forall_i \sum_j a_{i,j} = 1$$ $$\textrm{Minimize } M\bullet a$$ The first constraint defines $a$ as ...


1

The specialization $P_\lambda(x;-1)$ is what is referred to as Schur's P functions. They can be described combinatorially using shifted tableaux, and are Schur-positive. See also slides here by S. Cho.


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