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Similar questions can also be dealt with using generating functions and Lagrange inversion. Let $T(z)$ (the "tree function") be the formal power series satisfying $T(z)=z\,e^{T(z)}$. If $F$ is a fo …

Since $a_0\neq 0$ there is (by the (formal) Lagrange theorem) a unique formal power series $x=x(z)$ solving $x={z \over A^3(x)}$, and $A(x(z))=B(z)$ because $A(x)=B(x\,A^3(x))$. Expanding $A(x)$ using …

I add two hopefully useful remarks. I consider the general situation. In the sequel $k\geq 1$ and $\lambda=(\lambda_1,\ldots,\lambda_{k+1})$ are fixed, $s:=\sum_{i=1}^{k+1}\lambda_i$ and $n\geq s$. …

I reformulate slightly, please check. You are considering a sequence $X_1,X_2,\ldots$ of (discrete) i.i.d random variables and want an upper bound for the probability $\mathbb{P}(R>n)$ in terms of $ …

Here is a generating-function proof of your conjectured identity (and an answer to question 2). The main ingredient is a formula for the appearing symmetric sums. Let $T(z)$ (the ``tree function'') …

Here is an approach via Lagrange inversion. Let $N$ denote the time of the first repeat, and let $T(z)$ (the ``tree function'') be the formal power series satisfying $T(z)=z\,e^{T(z)}$. If $F$ is a …

Here is a proof using generating functions. Let $c\geq 2$ and $k\geq 2$ be fixed. Let $X=(X_1(n),\ldots,X_k(n))$ be the $k$-tuple of occupancy numbers at time $n$, i.e. $X_i(n)$ = number of bins of ty …

This question was completely settled by J.A. Fill here: https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.8.641

The following inequality holds: $$\mathbb{P}(C_k(m)\geq 1)\geq \mathbb{P}(\mathrm{Bin}(m, \lVert A\rVert_k)\geq k)$$ where here and in the sequel $\mathrm{Bin}(n,p)$ denotes a binomially distributed r …

Here is another way of studying these processes. In the sequel $k=3$ but it is clear how to generalize. The idea is to look at the matrices formed by three consecutive triples used in the insertion p …

Let $n\geq 3$, let $Z$ be the matrix of the cyclic shift (the companion matrix of $X^n-1$), and for $\mathbf{d}\in \mathbb{C}^n$ let $\operatorname{diag}(\mathbf{d})$ be the diagonal matrix with $\ma …

Let $F(n,\ell)$ be the matrix with coefficients $$F_{i,j}(n,\ell)=[t^{\ell j-i}] \left(\frac{1-t^\ell}{1-t}\right)^n,\,\;\;\;1\leq i,j \leq n$$ Above Pat Devlin pointed out that it suffices to show t …

There a well known generating function methods (the ''symbolic method'' and ''Poissonization'') which can be used to deal with this kind of question. However, I am unable to point to a reference for …

(You are considering the uniform distribution on a discrete simplex. I'm not aware of specific results in the literature, and a brief internet search didn't reveal anything.) A simple way is to use …

As in Timothy Budd's answer let $w=w(q)$ denote the (formal) solution of $q=\frac{w}{f(w)}$. Let $p$ be another variable and consider the sum \begin{align*} S(p,q):=\sum_{n,m>0} \sum_{j>0} j[z^{n+j}]\ …