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Since you mentioned Cayley's theorem for spanning trees, I believe one important example is its higher dimensional analogue due to Kalai. Indeed the number of simplicial spanning trees of the k-skelet …

I'm not entirely sure of what you are asking, but note that Erdos and Lehner proved here that $$p(n,m)\sim \frac{n^{m-1}}{m!(m-1)!}$$ holds for $m=o(n^{1/3})$. In generality for any finite set $A$, wi …

To answer the question on the origin of $\alpha_q$. The original proof goes back to Manin's article "What is the maximum number of points on a curve over $\mathbb F_2$", J. Fac. Sci. Univ. Tokyo, 198 …

Markus Brede proves the following formula in the paper "On the convergence of the sequence defining Euler’s number". Let $$\left(1+\frac{1}{z}\right)^z=\sum_{n\geq 0} \frac{a_n}{z^n}$$ then we have $$ …

Andrew is right, the following limit seems to be what you are looking for $$\lim_{q\uparrow 1}\left(\log(1-q)-\log q \sum_{n\geq 0}\frac{q^{n+1}}{1-q^{n+1}}\right)=\gamma$$ See , for example theorem 1 …

This is a result of Pillai. Indeed we have $\text{rphi}(n)=\frac{\log n}{\log 2}$ when $n$ is a power of 2, and we have $\text{rphi}(n)=\lceil\frac{\log n}{\log 3}\rceil$ when $n$ is twice a power of …

Yes, the conjectured limit is true. Let $d=\gcd (m,n)$ and $m=m_1d, n=n_1d$. Suppose $a_k$ denotes the number of solutions to $k=m_1r+n_1s$ with $r,s\geq 0$, so that $$a_0+a_1x+a_2x^2+\cdots =\frac{1} …

I don't know if this is optimal, but here is a pair of plane partitions of $n=53$ with no symmetries that have equal projections on each individual axis: $$A=\begin{matrix} 5 4 4 4 4 \\ 5 4 1 1 \ \ \ …

These were enumerated by Harary and Prins in Frank Harary and Geert Prins, "The number of homeomorphically irreducible trees and other species" Acta Math., 101 (1959), 141-162 and for the asymptotics …

See for example Nathanson's paper: "Asymptotic density and the asymptotics of partition functions". …