# Questions tagged [algebraic-combinatorics]

For problems involving algebraic methods in combinatorics (especially group theory and representation theory) as well as combinatorial methods in abstract algebra.

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### How many partitions of $n$ different objects into equinumerous parts are there?

How many partitions of $n$ distinct objects are there, given that all parts are equinumerous? (Let us not consider the empty partition and the identity partition.) The cases when $n$ is the unit, or ...
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### On the geometric realization of a finite abstract simplicial complex which is connected, orientable $3$-manifold without boundary

Let $\Delta$ be an abstract simplicial complex on finitely many vertices and $|\Delta|$ be it's geometric realization. (https://en.m.wikipedia.org/wiki/Abstract_simplicial_complex) If $|\Delta|$ is ...
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### Decomposing into irreducible $S_n$ modules, aka Specht modules.

Let $S_{\pi}$ where $\pi$ is an integer partition of $n$, denote the Specht module corresponding to $\pi$. I am trying to decompose the set of all homogeneous polynomials in $x_1,x_2,...,x_n$ ...
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### Minimum distance of a code given its basis

I know that to determine the minimum distance of a code given its basis we can follow this procedure: Let $\{ c_1, \ldots, c_k \}$ be a basis for a code of length $n$ and dimension $k$. Then the ...
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### On a particular kind of simplicial complex with maximum facet size of $3$.

Let $\Delta$ be an abstract simplicial complex on $n$ vertices such that $\max \{|F| : F$ is a face of $\Delta \}=3$. Let $f_2$ be the number of faces of size (cardinality) $3$ and $f_1$ be the ...
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### Is there generalization of the natural numbers?

Natural numbers are defined inductively https://softwarefoundations.cis.upenn.edu/lf-current/Basics.html#lab30 as s(s(...s(0)...)). Such definition is nothing special, especially when one can ...
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### Combinatorial bijection of primitive factorization

Let $\nu$ is a partition on $n$. Given a $n$ cycle $(1,2,\ldots,n)\in S_n$. Let define $H_g^{m}((n);\mu)$ count the number of tuples $(\tau_1,\ldots,\tau_r)$ in symmetric group $S_n$.Let $\beta$ ...
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### On finding a finite set of generators for a certain semigroup

Let $A$ be a finite subset of $\mathbb Z^2$. Let $\mathbb ZA$ be the subgroup of $\mathbb Z^2$ generated by $A$. Let $\mathbb R_{+}, \mathbb Q_{+}$ denote the set of non-negative real and rational ...
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### A collection of sets that cover all edges in Kn?

The problem is the following: Let $\mathcal{F}$ be a family of distinct proper subsets of {1,2,...,n}. Suppose that for every $1\leq i\neq j\leq n$ there is a unique member of $\mathcal{F}$ that ...
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Show that the number of monic irreducible polynomials of degree $n$ over a finite field of size $q$ is the same as the number of primitive necklaces of size $n$ with $q$ colors. I have the formula $$... 1answer 188 views ### Vishnoi's Proof of Combinatorial Nullstellensatz The question was already asked here: Question regarding a proof of the Combinatorial Nullstellensatz, but I am having trouble understanding the document in the comment, and so I was wondering if ... 2answers 119 views ### Applying Burnside's lemma to show there are k+n-1 \choose k-1 ways to store n stars in k bins? Usually it is proven using multisets I think, but I wondered how Burnside's lemma could be applied. Everytime I tried to wrap it around my head the indices didn't seem to fit. So I TeXed it and ... 3answers 271 views ### How many n-pointed stars are there? Say we have n distinct points spaced evenly in a circle. Define a star as a connected graph with these points as vertices and with n edges, no two having the same endpoints. We think of two stars ... 2answers 157 views ### Proof that the Lascoux-Schützenberger involutions satisfies the braid-relations I am interested in the Lascoux-Schützenbereger involutions \theta_i, defined on semistandard Young tableaux. See this paper for definitions, page 4. These involutions satisfy the braid relations: (... 0answers 64 views ### Hook-length formula [closed] Let \lambda=(\lambda_1,\ldots,\lambda_n) be a partition of d. Then hook length formula gives us$$dim(\lambda)=\chi_{1^d}^{\lambda}=\frac{d!}{\prod h_{\lambda}(i,j)} where $\chi_{a}^{b}$ denote ...
What is the motivation behind using linear algebra in these three problems ? A pair $(m,n)$ is called nice if there is a directed graph with (self edge are allowed, but multiple edge are not allowed) ...