# Questions tagged [generating-functions]

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### What alternatives are there to the binomial poset theory of generating function families?

A natural question in combinatorics is, why are certain families of generating functions combinatorially useful, like $\Sigma_n a_n x^n$ and $\Sigma_na_n\frac{x^n}{n!}$, why are other families are not,...
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### Conjecture on bernoulli numbers and binomial coefficients

Crossposted from https://math.stackexchange.com/questions/4116414/conjecture-on-bernoulli-numbers-and-binomial-coefficients In playing around with some formulas, I have come up with the following ...
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### Fibonacci-Motzkin paths and J-type continued fractions

Recall that a Motzkin path is a piece-wise linear planar path connecting points in the integer lattice quadrant $\Bbb{Z}_{\geq 0} \times \Bbb{Z}_{\geq 0}$ beginning at the origin $(0,0)$ and ending at ...
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### Counting unions of unlabelled connected graphs

My question can be stated as follows: let $X$ be a hereditary family of unlabelled graphs closed under disjoint unions. Suppose we know, for each $n$, the number $c_n$ of connected graphs in X on $n$ ...
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### Is there a bijective proof of an identity enumerating independent sets in cycles?

Let $C_m$ be the cycle with $m$ vertices, defined so that $C_1$ has a self-loop on its unique vertex. Let $p_m$ be the generating function enumerating the number of ways to choose $k$ vertices in $C_m$...
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### Lie theoretic meaning to $e^{\text{cycle}} = \text{permutation}$?

It is well known that exponentiating the EGF(exponential generating function) for cycles gives the EGF for permutations: link here. Usually summarized under the catchy slogan ...
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### Induction step in Bóna and Ehrenborg's proof that the generating function of the alternating runs has -1 as a root of a certain multiplicity

This is a crosspost of a question I asked on Mathematics SE four months ago. Periodically bumping it and placing a bounty on it to attract more attention were to no avail. There are some comments ...
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### Counting planar trees with the same underlying tree

$T$ be a tree, and $P_T$ denote the number of planar rooted trees whose underlying tree is $T$. Here, a planar rooted tree is a rooted tree such that the children of every vertex are totally ordered. ...
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### General upper bound of extinction probability

We consider here a Galton–Watson process with an offspring distribution $X$, where $\mathbb{E}X = \mu$ and $\operatorname{Var} X = \sigma^{2} < \infty$ and $q = \mathbb{P}(\text{extinction})$, i.e.,...
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### Meinardus theorem at use: problems with conditions

I am working on an enumerative problem related to knot theory, and I have found the following generating function $$F(z)=\prod_{n\geq 1} \frac{1}{(1-z^{2n+1})^2}.$$ I am interested on getting ...