# Questions tagged [co.combinatorics]

Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.

6,793
questions

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21 views

### The number of hamiltonian circuits on a convex polytope embedded in $\mathbb{R}^N$

Recently I wondered whether there might be a natural topological complexity measure for convex polytopes embedded in $\mathbb{R}^N$. After some reflection it occurred to me that the number of distinct ...

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130 views

### A conjectural trigonometric identity

Recently, I formulated the following conjecture which seems novel.
Conjecture. For any positive odd integer $n$, we have the identity
$$\sum_{j,k=0}^{n-1}\frac1{\cos 2\pi j/n+\cos 2\pi k/n}=\frac{n^2}...

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46 views

### Linear coefficient of chromatic polynomial

I am interested in the combinatorics of the linear coefficient of the chromatic polynomials. I have the following questions.
What are some class of graphs for which it is possible to calculate this ...

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66 views

### Decomposing tensor powers of the fundamental representation of exceptional Lie algebras

For the $A$-series, tensor powers of the fundamental representation of $\frak{sl}_n$ decompose into irreducibles according to a certain Young diagram/ partition formula. This inspires, for example, ...

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232 views

### Сlosed formula for $(g\partial)^n$

The objective is to obtain a closed formula for:
$$
\boxed{A(n)=\big(g(z)\,\partial_z\big)^n,\qquad n=1,2,\dots}
$$
where $g(z)$ is smooth in $z$ and $\partial_z$ is a derivative with respect to $z$.
...

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41 views

### Unified framework for posets with order polynomial product formulas

One of the most celebrated results in algebraic combinatorics is the Hook Length Formula of Frame-Robinson-Thrall which counts the number of standard Young tableaux of given partition shape. Such SYTs ...

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57 views

### Domination relationship between generalized Dyck Paths

In short, we are seeking an injection between generalized Dyck paths that end at a certain height into the set of paths of the same length that end at a lower height such that the image path stays ...

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74 views

### Is it possible to stab every permutation of any four element subset of $D_n$ with less than $n/2$ elements?

Say for a permutation group $G$ over $n$ that a set $S\subset \{1,\ldots,n\}$ is G-stabbed by $X\subset \{1,\ldots,n\}$ if for every $g\in G$ we have $gS\cap X\ne \emptyset$.
Is there for every $|S|...

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112 views

### A combinatorial / geometric interpretation of compositional inversion via matrix inversion

There are several ways of finding the power or Taylor series for the compositional inverse of a function $f(x)$ with $f(0)=0\;$ given its series expansion, e.g., by using the classic Lagrange ...

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156 views

### Chromatic number and graph polynomial

If $\prod_{i=1}^t x_i^{e_i}$ is a monomial, define
$$rad\biggl(\prod_{i=1}^t x_i^{e_i}\biggr)$$
to be the number of distinct (nonzero) values of $e_i$.
Now let $G$ be a simple graph with vertices ...

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34 views

### A weakened form of list coloring

The list coloring of a simple loopless graph is the assigning of a color from a certain list of colors to every vertex. The list coloring chromatic number of a graph is the minimal cardinality of the ...

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182 views

### Polynomial defined recursively by a resultant

Cross posting from MSE.
Definition:
For any natural number $n\ge 3$, define the polynomial $P_{n}\left(x_1,x_2,...,x_{n-1},x_{n} \right)$, with indeterminates $x_{i}$, where $i\in\{1,2,...,n-1,n\}$, ...

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10 views

### If I have 3 different tee pads for a 9 hole golf course how may different combinations are there and how do you calculate it in equation form [migrated]

If I have 3 different tee pads for a 9 hole golf course how may different combinations are there and how do you calculate it in equation form.

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54 views

### Large subsets of the Hamming cube with small intersections with all spheres of given radius

What is the maximal cardinality of a subset $A$ of $\{-1,1\}^n$ such that any Hamming sphere with radius $r$ contains at most $k$ elements of $A$?
Are explicit constructions with large cardinality ...

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40 views

### Number of maximal independent sets in a simple graph

Consider a simple regular graph on $n$ vertices and size $E$. How many distinct maximal independents can we find at the least in the graph?
I think we can always find at least two maximal ...