# Questions tagged [algebraic-combinatorics]

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### Motivation behind Panyushev's "constant-averages-along-orbits" conjecture

In his article "On orbits of antichains of positive roots" (European Journal of Combinatorics 30 (2009) 586–594, Dmitri Panyushev discusses an interesting self-map on the set of antichains of a finite ...
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### Ubiquitous Zimin words

Let $w$ be a word in letters $x_1,...,x_n$. A value of $w$ is any word of the form $w(u_1,...,u_n)$ where $u_1,...,u_n$ are words. For example, $abaaba$ is a value of $x^2$. A word $u$ is called ...
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### Arithmetic product of symmetric functions: why is it integral?

For every commutative ring $A$, let $\mathbf{Symm}_A$ be the ring of symmetric functions over $A$. Let $\mathbf{Symm}$ without a subscript denote $\mathbf{Symm}_{\mathbb{Z}}$. We can define a ...
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### Is this sum of cycles invertible in $\mathbb QS_n$?

I am interested the following element of the group algebra $\mathbb{Q}S_n$: \begin{align} \phi_n=2e+(1\ 2)+(1\ 2\ 3)+\dotsb+(1\ldots n) \end{align} where $e$ is the identity permutation. My question ...
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### Smith Normal Form of a Cayley Graph of the Symmetric Group

Let $A_n$ be the adjacency matrix of the Cayley graph $\text{Cay}(S_n,C_n)$ where $C_n \subseteq S_n$ is the conjugacy class of $n$-cycles of the symmetric group $S_n$. Since the generating set of ...
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### Software for Combinatorial Algebra sought

I am looking for software which helps me do straightforward tasks in combinatorial algebra. Let me give an example of what I mean by a straightforward task: I have two graded (generally ...
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### What is the smallest cardinality of a self-linked set in a finite cyclic group?

A subset $A$ of a group $G$ is defined to be self-linked if $A\cap gA\ne\emptyset$ for all $g\in G$. This happens if and only if $AA^{-1}=G$. For a finite group $G$ denote by $sl(G)$ the smallest ...
I've noticed that for every natural number $n\in\mathbb{N}$, there is a finite Heyting algebra with cardinality $C(n)$, where $C(n)$ is the $n$th Catalan number, $$1,1,2,5,14,42,132,\ldots$$ I'm ...