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        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.

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        1answer
        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 ...
        2
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        2answers
        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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        0answers
        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 ...
        3
        votes
        1answer
        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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        1answer
        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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        1answer
        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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        0answers
        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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        votes
        1answer
        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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        0answers
        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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        0answers
        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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        0answers
        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.
        1
        vote
        1answer
        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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        1answer
        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 ...

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