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.
284 questions
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assigning men and women to teams, each team has odd # of women and men, and any intersection has even # women OR men
I am completely lost. Or I have a ton of ideas but I get confused as hell and then I cannot finish a thought. So here is what I got:
Exercise
Given $n$ women and $n$ men, we form $m$ teams fulfilling ...
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Counting problem similar to Eventown
The following problem is from my "algebraic methods in combinatorics" course.
Let $\mathcal{A}$ be a family of subsets of $[n]$ such that $|A|$ and $|A \cap B|$ are even for every $A, B \in \...
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Grothendieck polynomials and divided difference operators
Let $$\pi_{i}(f) = \frac{(1+\beta x_{i+1})f - (1 + \beta x_i)s_if}{x_i - x_{i+1}},$$ where $s_if(..., x_i, x_{i+1}, ...) = f(..., x_{i+1}, x_i, ...)$, and $\beta$ is a real number. The isobaric ...
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Combinatorial understanding of a specific power of monomial ideals
Definition. Let $I$ be a squarefree monomial ideal (that is, an ideal in $K[x_1,\dots,x_n]$ generated by squarefree monomials). The $k$-th squarefree power $I^{[k]}$ of $I$ is the ideal generated by ...
user969954
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Taft Hopf Algebra is self dual
I'm trying to prove that Taft Hopf Algebra is self-dual, i.e., there is an isomorphism of Hopf algebras from $H_{n,q}$ to its dual. Here's the definition:
Let $n \geq 1$ and suppose that $k$ contains ...
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Counting sub-squares in convex polyominoes
Let $ G $ be an $ m \times n $ grid in the plane, and consider an embedded diagram $P$ that is a Ferrers diagram or, more generally, a convex polyomino. For a fixed positive integer $ h $, I am ...
user969954
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Finding coefficients $c_k$ using series solutions for polynomial expression
I am working on a problem where I need to express a polynomial in terms of binomial coefficients. Specifically, to find $c_k$ such that\begin{equation}
n^{K-1} = \sum_{k=1}^K c_k \binom{N-n+k-2}{k-...
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Proof of Theorem on Punctured Combinatorial Nullstellensatz
I am reading the proof of this corollary from "Punctured Combinatorial Nullstellensatz" by Ball and Serra, and there is a crucial step in the proof that I am struggling to understand.
Let $F$...
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Multiplication of Pairs of Semi-Standard Young Tableaux that Agrees with Robinson-Schested-Knuth (RSK) Matrix Multiplication?
The RSK correspondence tells us that nonegative integer valued matricies are in bijective correspondance with pairs of Semi-Standard Young Tableaux (P,Q) of the same shape. Taking two pair of SSYT, ...
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Punctured Combinatorial Nullstellensatz Proof
I am reading the proof of a theorem in "Punctured Combinatorial Nullstellensatz" by Ball and Serra, and there are a few things that I am struggling to understand. In this proof, $T(n,t)$ ...
- 285
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Is this a counterexample or no?
Here is the Matroid intersection property as stated by Gary Gordon in “Metroids a geometric introduction”
Let $M_1$ and $M_2$ be Metroids on the same ground set $E.$ We attempt to define a new matroid ...
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counting $k$-dimensional subspaces of $\mathbb{F}_q^n$ not containing any of $V_1,\dots,V_{\ell}$
Let $q$ be a prime power and let $\mathbb{F}_q$ denote the finite field of order $q$.
Let $V_1,\dots,V_{\ell}$ be $1$-dimensional linear subspaces of $\mathbb{F}_q^n$ such that the sum $V_1\oplus\dots\...
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Rank2 cluster algebras and subpaths of the maximal Dyck path $D_6$
Here is some context: Following Section 2, Example 10, page 6 here https://arxiv.org/pdf/1106.0952. I tried to find the cluster variable $x_6$ by using the formula from $\textbf{Theorem 9}$ for the ...
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The number of non-real quadratic algebraic integers with the given norm in the first quadrant and the upper z-axis.
We know that the norm of an algebraic integer is an integer, so I want to go a step further and know how many quadratic algebraic integers there are with a specific norm. Of course, I want to ...
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maximal length of saturated chains with a given terminal point in the lattice of partitions of an integer ordered by dominance
Let $Pr(n)$ be the set of partitions of the positive integer $n$. This is a lattice with respect to the dominance order: if $\lambda=(\lambda_1\geq\lambda_2\geq\cdots)$, $\mu=(\mu_1\geq\mu_2\geq\cdots)...
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