# Questions tagged [enumerative-combinatorics]

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### Catalan and path pairs in polynomials

Define $\mathbf{K}_n$ to be the set of all $(2n+1)$-tuple sequences $\mathbf{a}=(a_0,a_1,\dots,a_{2n})\in\{-1,1\}^{n+1}$ satisfying: (a) there are $n$ occurrences of $-1$ and $n+1$ of $+1$; (b) all ...
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### Super Catalan (super ballot) numbers

We refer to this article by Ira Gessel. In section 6, page 10, equation (28), the Super Catalan numbers are defined as $$S(m,n)=\frac{(2m)!(2n)!}{m!n!(m+n)!}.$$ On page 12, equation (31), there goes ...
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### Transitive action on domino tilings

Fix a $n \times m$ rectangle and consider the set $S_{n,m}$ of all its dominos tilings. Here are examples with $n=m=8$. The set $S_{n,m}$ is empty if and only if $nm$ is odd, and for small $nm$, its ...
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### An atlas for the enumeration of planar maps

The theory of planar map enumeration was started by Tutte and iniciated the theory of map enumeration when trying to solve the 4-colour theorem by enumerative arguments. Nowadays a wide diversity of ...
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### Asymptotics for enumerating graphs

Let $K>k$ be positive integers. Now assume that $G$ is a $K$-connected graph with $n$ vertices and $m$ edges. I would like to ask: QUESTION. Is there an asymptotic for the number of $k$-connected ...
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### Convergency radius of the generating series for A93637

Sequence A93637 of the OEIS (https://oeis.org/A093637) starting as $1,1,2,4,9,20,49,117,297,746,1947,\ldots$ is defined by the coefficients $a_0,a_1,\ldots$ of the unique formal power series defined ...
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### Upper bound of the number of oscillatory sequences

Let $$A_n=\{(x_1,x_2,x_3,\cdots,x_n):x_i \in [q] \text{ for } i \in [n], x_1 < x_2, x_2 > x_3, x_3 < x_4, \cdots , (-1)^{n}x_{n-1} < (-1)^{n} x_n\}.$$ What is the cardinality of $A_n$? I ...
The present quest emanates from this study by R. Stanley, including his recent MO question. Define the product (polynomials after full expansion) $$I_n(x)=\prod_{i=1}^n(1+x^{F_{i+1}})$$ based on the ...