# 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.

123 questions
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### Unbounded, Repeated Figures in Non-periodic Tilings

I was just wondering something about non-periodic tilings (of the plane, though I imagine the dimension is irrelevant for finite dimensions; would be interesting if it wasn't!). I assume we know what ...
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376 views

### Powers of a simple matrix and Catalan numbers

Consider $m \times m$ anti-bidiagonal matrix $M$ defined as: $$M_{ij} = \begin{cases} -1, & i+j=m\\ \,\,\ 1, & i+j=m+1\\ \,\,\, 0, & \text{otherwise} \end{cases}$$ Let $S_n$ stand ...
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### Interpretation of the numerator of the Hilbert series?

Let $R$ be a finitely generated graded ring over a field $k$. Let $R_\ell$ be the degree-$\ell$ homogeneous component of $R$. By the Noether normalization theorem, $R$ is finite over a graded ...
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### How many ways to place $m$ labelled balls into $n$ labelled boxes such that at least one box has more than $k$ balls

There are $n^m$ ways to place $m$ labelled balls into $n$ labelled boxes with no constraints. When it comes to the problem such that at least one box has more than $k$ balls. My thought was Step 1....
1answer
84 views

### Is there any proof of this identity?

$$\prod\limits_{i=1}^{\infty}\frac{1}{1-yx^{i}}=\sum\limits_{k=0}^{\infty}\frac{y^k x^k}{(1-x)(1-x^2)...(1-x^k)}$$ I know its combinatorial proof through "inspection",but is it true when $x,y$ are ...
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### Representation theory of the symmetric group

I'm trying to understand the representations of $k\mathfrak{S}_3$, where $k$ is a field of characteristic $2$. Could someone explain the permutation module $M^{(1^3)}$ has a filtration layers ...
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33 views

### Is there any way to simplify $\frac{1}{2^{n - 1}}\sum_{i=0}^{n - 1} {\binom{n - 1}{i}}{a[i]}$?

Here $a[i]$ denotes the i'th element (0 indexed) of a tuple. The goal is to avoid the huge coefficients since I know the final result won't be very big (in proportion to the elements of the tuple). I ...
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### Invariants in the ring of coinvariants

This seems to be part of folklore, but I can't find a proof anyqhere, only references to "a very nice proof of a special case by Arkady Berenstein in a seminar I attended in 1989". Okay, here it goes....