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The symmetric group $S_n$ is the group of permutations of the set of integers $\{1,\dots,n\}$. This has $n!$ elements and is generated by the $n-1$ involutions exchanging consecutive integers. The symmetric groups form the simplest family of Coxeter groups.

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The number of permutations with a special condition

Suppose we are considering $S_n$. For any permutation, let $h$ be the number of derangement and $N$ be the number of cycles with length no less than 2. I'm interested in the number of permutations ...
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The number of permutations with specified number of cycles and fixed points

I'm interested in the number of permutations for a specified number of fixed points and cycles. Suppose we are in $S_n$. For any permutation in $S_n$, let $h$ be the number of changed points (the ...
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Is this sum of cycles invertible in QSn?

I am interested the following element of the group algebra $\mathbb{Q}S_n$: \begin{align} \phi_n=2e+(12)+(123)+\dots+(1\dots n) \end{align} where $e$ is the identity permutation. My question is ...
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Cycle generating function of permutations with only odd cycles

Let $\mathrm{ODD}(n)$ be the set of permutations in $\mathfrak{S}_n$ whose cycle lengths are all odd. It is known that $$ \#\mathrm{ODD}(n) = \begin{cases} ((n-1)!!)^2 &\textrm{ if $n$ is even}; \\...
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Ref. request: Enumerating elements of Bruhat cells

Given a field $F$ and a natural number $n$, let $B$ be the group of lower triangular, invertible $n \times n$ matrices over $F$. Then $$GL_n(F) = \biguplus_{\pi \in S_n} B \pi B,$$ where we embed the ...
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Combinatorics of $p$-Kazhdan--lusztig polynomials

When can we (and can we not!) understand the dimensions of simple modules, $D(\lambda)$, of symmetric groups in a combinatorial fashion? Let's assume that I'm going to try to do this using the theory ...
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Can $S_n$ be partitioned into subsets containing an involution and satisfying $∀σ≠τ, ∃j$ s.t. $σ(j)≠τ(j),σ^{−1}(j)=τ^{−1}(j)$?

Background Let $\sigma, \tau \in S_n$. We will say that $\sigma$ and $\tau$ are locally orthogonal and write $\sigma \perp \tau$ if there exists $j \in \{1, 2, \ldots, n\}$ such that $\sigma(j) \neq \...
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Cycle Structure of a Permutation Based on the Binary Representation

This is a question I posted on math.stackexchange.com before but never got an answer. I am cross-posting it here. Define a permutation $\sigma$ on the set $X=\{1,2,...,n\}$, $n$ is a natural number ...
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Reference request: Coxeter length and irreducible characters

Let $S_n$ be the symmetric group on $\{1,2,\ldots, n\}$ and $\ell$ the Coxeter length on $S_n$. There is a well-known formula to compute this length, namely for a $\pi \in S_n$ we have $$\ell(\pi) = |\...
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Character theoretic proof of the Littlewood–Richardson rule?

The Littlewood–Richardson coefficients are the multiplicities $$ c(\lambda,\mu,\nu)= \dim_{\mathbb{C}}\operatorname{Hom}_{S_n}(S(\nu),S(\lambda/\mu)) $$ and the Littlewood–Richardson rule says that ...
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Combinatorial model for twisted involutions in $S_n$

Let $(W,S)$ be a Coxeter group and $*:S \to S$ be an automorphism of the Dynkin diagram of $W$ so that $*^2$ is the identity. This induces a bijection $*:W \to W$ mapping $w = s_1 \dots s_n$ to $w^* = ...
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Large subgroups of $S_n$ without large symmetric or alternating subgroups

I'm interested in determining the existence of a permutation group $G\subseteq S_n$ of the following form. $G$ is large. Meaning that $G$ have at least $n!/2^{o(n)}$ elements. Equivalently, their ...
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working with symmetric groups presented via nonstandard generators

This is follow-up to my earlier question. Suppose that we have elements $\sigma_1,\ldots,\sigma_k\in S_n$, and that we established that these elements actually generate $S_n$. Since that previous ...
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A dimension formula for generalised symmetric powers of the natural module

I need a reference for the following well-known statement - does anyone know one? Let $\mu$ a partition of $n$ into at most $d$ parts. We let $${\rm Sym}^\mu(\Bbbk^d)={\rm Sym}^{\mu_1}(\Bbbk^d) \...
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Permutation factorizations according to number of generated orbits

Let $\pi$ be a permutation in $S_n$ with cycle type $\lambda$. How many factorizations into two factors $\pi=\sigma_1\sigma_2$ are there, such that the subgroup $\langle \sigma_1,\sigma_2\rangle$ ...
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Homotopy type of the semi-simplicial set of symmetric groups

Consider the collection of symmetric groups $\{\Sigma_n\}_{n\geq1}$ as a semi-simplicial set (i.e. a simplicial set without degeneracies) as follows. Consider $i\in\{1,\dots,n+1\}$ and $\pi\in\Sigma_{...
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What are all the transitive extensions of cyclic groups?

"Let $G$ be a transitive group of permutations on a given set of letters. Let a new fixed letter be adjoined to every permutation of $G$. Then a transitive group $H$ of permutations on the combined ...
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Edge-transitive Cayley graphs of $S_n$

I came across the following question which I haven't seen before: Question. Fix $k\ge 3$. For infinitely many $n$, does there exists a generating set $\langle R_n \rangle = S_n$, $|R_n|=k$, such ...
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Tabloid Construction of Permutation Representation of Hyperoctahedral Group

For a partition $\lambda \vdash n$, the permutation representation $M^{\lambda}$ of the symmetric group can be constructed in two ways. First, it may be written as the induced representation $M^{\...
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Is the Normal centralizer problem in P?

Notation $\le$ is used for the subgroup relation; $P$ means polynomial time in input size; $\Omega = \{1,2,3,\cdots,n\}$ is a input domain; $\mathrm{Sym}(\Omega)$ means the symmetric group on $\...
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Low-dimensional irreducible 2-modular representations of the symmetric group

I apologize if this question is a little too basic for MathOverflow, but it's somewhat outside of my background and I'm frustrated that the answer doesn't seem to be explicit in the literature even ...
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Symmetric functions of eigenvalues

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a smooth function, which is symmetric under the action of the symmetric group (acting on $\mathbb{R}^n$ by permuting the variables). Let $M_{n\times n}$...
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Number of permutations in a set in an algorithm

Previously I asked a question about the space usage in an algorithm of mine: Upper bound on the number of permutations in a set during an algorithm. This question does not depend on the previous one, ...
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Murnaghan-Nakayama rule when all cycles have same size

Let $\lambda \vdash nk$. Let $n^k$ denote the partiton with $k$ parts of size $n$. We can compute $\chi^\lambda(n^k)$ by using the Murnaghan-Nakayama rule, as a signed sum over border-strip tableaux, (...
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Dimension of irreducible representation associated to a Young tableau

This might be a classic question, but since I am new to representation theory of the symmetric group, I am asking it here. Suppose that $\lambda_1 \geq \lambda_2 \geq \dots \lambda_k$ and $\rho$ be ...
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Upper bound on the number of permutations in a set during an algorithm

Fix $n\geq 2$ and let $S_n$ be the symmetric group on $n$ letters with identity $e$. We consider elements of $S_n$ to be bijections $[n]\to [n]$ as well as sequences (one line notation). For $1\leq i&...
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234 views

Generating symmetric groups with small cycles

This was asked but never answered at MSE. Let $S_n$ denote the symmetric group and let $H$ be a subgroup which contains an $n$-cycle. If $n$ is prime, and if $H$ also contains a 2-cycle, then ...
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Natural maps between Schur functors: understanding the image

Let $V$ be a finite dimensional representation of symmetric group $\mathbb{S}_n.$ Consider a natural map $$\pi \colon \Lambda^2 V \otimes \Lambda^2 V \longrightarrow \Lambda^4 V.$$ Let $[\Lambda^2 V]...
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alternating and symmetric powers of the standard representation of the symmetric group

Let $n \geq 7$ and $V = \mathbb{C}^n$ be the standard representation for $S_{n+1}$, the symmetric group of cardinal $(n+1)!$ Let $k$ be an integer such that $2 \leq k \leq n$. Is it true or false ...
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Question on a reduction in Kirillov's paper on positivity of divided difference operators

As the title says, my question is on a specific argument in Kirillov - Skew divided difference operators and Schubert polynomials (journal, MSN) on positivity of divided difference operators. I recall ...
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Characterization of permutations which have at most one successor in the covering relation of the weak Bruhat order

Let $W$ be the symmetric group on $n+1$ letters. Let $\ell$ be the length function on $W$. As the title says, can we characterize all $v\in W$ such that there exists a $w\in W$ such that for all ...
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Generating sets of the symmetric group that yield isomorphic Cayley graphs

Let $S$ and $S'$ be subsets of size $k$ of $\mathfrak{S}_n$. Are there any necessary or sufficient conditions to determine whether or not $S$ and $S'$ yield isomorphic Cayley graphs? Assuming we ...
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A question about set of inversion

Let $w \in S_n$ and $inv(w) = \{(i,j): i,j \in \{1,\ldots,n\}, i<j, w(i)>w(j)\}$ the inversion set of $w$. Let ${\bf i}=(i_1,\ldots,i_m)$ be a sequence such that $s_{i_1}\cdots s_{i_m}$ is a ...
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Permanent of the symmetric group

Let $A$ be the algebra corresponding to a representation-finite block of a Schur algebra. See for example 6.1. of https://arxiv.org/pdf/1607.05965.pdf for quiver and relations and some relevance of ...
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The number of permutations of a given cycle type that fix a string with a given histogram

Let $\lambda$ and $\mu$ be partitions of some integer $n \geq 1$. Let $d$ be the number of parts in $\mu$ and let $\bar{\mu} \in \{1,\dotsc,d\}^n$ denote the string $1^{\mu_1} 2^{\mu_2} \dotsb d^{\...
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Induced representation of a Young subgroup

This might be a classic question, but since I am new to representation theory of the symmetric group, I am asking it here. Suppose that $n=k+l+r$ where $k\geq l\geq r\geq 0$. Let $G$ be the symmetric ...
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Classification of transitive subgroups of finite symmetric groups generated by double transpositions

I want to classify (up to isomorphism) all transitive subgroups of symmetric group $S_n$ which are generated by double transpositions (product of two transpositions). Is there a characterization for ...
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Character values at a cyclic permutation of a symmetric group

Let $S_n$ be the symmetric group of degree $n$ and $\sigma\in S_n$ be a cyclic permutation of order $p$, where $p$ is a prime and $p>n/2$. Consider ordinary irreducible characters of $S_n$. Are ...
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Bijective proof of an identity involving number of standard Young tableaux and semistandard tableaux

Question. Can you find a bijective proof of the identity $$ \operatorname{dim}(S^{\lambda} \mathbb{C}^m)\ \operatorname{dim}(S^{\lambda'} \mathbb{C}^n) \ f^{n^m} = \dim \Lambda^p (\mathbb{C}^m \...
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1answer
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Reference request: Reduced reflection length in Coxeter groups

I recently read this paper, where the authors define on page 26 what they call the reduced reflection length. For that we take a Coxeter group $G$ with Coxeter generators $S$ and transpositions $T$. ...
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Provoking involutions further

Let $\mathfrak{S}_n$ denote the permutation group, and $I_0(n)=\sum_{j\geq0}\binom{n}{2j}\frac{(2j)!}{2^jj!}$ stand for involutions see A000085 for more interpretations. There is also these numbers $...
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Irreducible representations occuring in $\mathrm{Ind}_G^{S_{|G|}}1$ for $G$ finite group

Let $G$ be a finite group with $|G|=n$, let $S_G=S_n$ be the group of $n!$ permutations of the set $G$. Then $G$ is a subgroup of $S_G$ via left-translation (i.e. $g\in G$ corresponds to the ...
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A constraint satisfaction problem on matrix sum involving symmetric group

Given $n\in\Bbb N$ what is the smallest with $m>n$ we need such that there is a non-negative $\epsilon<1$ and $\Phi_i,\Psi_j\in\Bbb C^{m\times m}$ at every $i,j\in[n]$ ($[n]=\{1,\dots,n\}$) such ...
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What is this symmetric simplex category, concretely?

Let $\Delta_+$ denote the category of finite ordinal numbers with monotonic maps (the subscript indicates that $0$ is included, so this is the augmented simplex category). This has a monoidal ...
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trace and involution permutations: Part II

This is a follow up on my earlier MO question. Let $\operatorname{Inv}(\mathfrak{S}_n):=\{\pi\in\mathfrak{S}_n: \pi^2=1\}$ be the set of involutions in the symmetric group $\mathfrak{S}_n$. Denote $...
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trace and involution permutations: Part I

Let $\operatorname{Inv}(\mathfrak{S}_n):=\{\pi\in\mathfrak{S}_n: \pi^2=1\}$ be the set of involutions in the symmetric group $\mathfrak{S}_n$. Denote $I_n:=\#\operatorname{Inv}(\mathfrak{S}_n)$. Let $\...
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hooks and contents: Part II

This is a 2nd installment to my earlier MO question for which Mark Wildon furnished a clean answer. $\mathcal{O}(\pi)$ and $\mathcal{E}(\pi)$ stand for the number of odd and even cycles of a ...
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hooks and contents: Part I

For a cell $\square$ in the Young diagram of a partition $\lambda$, let $h_{\square}$ and $c_{\square}$ denote the hook length and content of $\square$, respectively. R Stanley proved the following ...
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Solutions of $x^d=1$ in the symmetric group

L Moser and M Wyman, On solutions of $x^d = 1$ in symmetric groups, Canad. J. Math., 7 (1955), pages 159-168, explored asymptotic behavior of the cardinality of such permutations: $$f_d(n):=\#\{\pi\in\...
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What can be said of a $6$-core Young diagram whose $2$-and $3$-cores are empty

Let $\lambda$ be a partition. Suppose that $\lambda$ is both $2$- and $3$-decomposable, in the sense that $\lambda$ admits a total decomposition by both $2$-rim hooks (aka dominos) and $3$-rim hooks. ...