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The sum of the signs of conjugacy classes in the symmetric group S_n [duplicate]

Let $r$ be the number of conjugacy classes of the symmetric group $S_n$ whose sign is $1$, i.e. \begin{equation} r := \#\{c \in \text{Conj} (S_n): \text{sgn} (c) = 1 \}. \end{equation} Let $s$ be the ...
alpha2357alpha's user avatar
2 votes
0 answers
352 views

On characters of the symmetric group: Part 1

Given an integer partition $\lambda$, denote $\ell(\lambda)=$ length, $\vert\lambda\vert=$ size and $\lambda=$ conjugate of $\lambda$. Allow to write $\lambda\vdash n$ either as $(\lambda_1,\dots,\...
T. Amdeberhan's user avatar
4 votes
0 answers
313 views

What is $\dim D^{\lambda}$ for the symmetric group?

What are the dimensions of the simple modules $D^{\lambda}=S^{\lambda}/S^{\lambda}\cap (S^{\lambda})^{\perp}$ for the modular representation theory of $S_n$, i.e. $\operatorname{char}(k)=p>0$? I ...
Jackson Walters's user avatar
3 votes
1 answer
221 views

Asymptotics for number of $p$-regular partitions of $n$

The number of simple modules $D^{\lambda}=S^{\lambda}/S^{\lambda}\cap (S^{\lambda})^{\bot}$ of the symmetric group over a field $k$ such that $\text{char}(k)=p > 0$ is the number of $p$-regular ...
Jackson Walters's user avatar
8 votes
3 answers
2k views

Bijective proof for a partition identity

I came across the following cute fact about partitions: \begin{align} & |\{\lambda \vdash n \text{ with an even number of even parts}\}| \\[8pt] & {} - |\{ \lambda \vdash n \text{ with an odd ...
Nate's user avatar
  • 2,242
1 vote
0 answers
82 views

How to obtain explicit formula for this sum over Young diagram?

Consider the next essence $$ B_N (r, q) =\sum_{\tau \vdash r} d^2 (\tau ) \prod_{i = 1}^{r} \frac{\Gamma [N + \tau_i - i +1]}{\Gamma [N + \tau_i - i +1+q]} $$ where $d(\tau)$ is dimension of ...
Sergii Voloshyn's user avatar
2 votes
1 answer
137 views

How to re-expand the sum of Schur function?

Consider next sum \begin{eqnarray} \label{PF_spindef} Z = \sum_{r=0}^{N N_f} h^{2r} \ Q(r) . \end{eqnarray} and \begin{equation} Q(r) \ = \ \sum_{\sigma \vdash r} s_{\sigma}(1^{N_f}) \ s_{\sigma}...
Sergii Voloshyn's user avatar
2 votes
0 answers
168 views

New identity for sum over Young diagram of symmetric group?

Consider the next identity $$ \sum_{\tau \vdash r} d^2 (\tau ) \! \prod_{i = -(r-1)}^{r-1} \! \! \left( N+i \right)^{t_i^r} =\sum_{\tau \vdash r} d^2 (\tau ) \prod_{i = 1}^{r} \Gamma [N + \tau_i - ...
Sergii Voloshyn's user avatar
5 votes
0 answers
76 views

Finite trees with forests realizing all partitions

Removing interiors of some edges in a tree with $n$ vertices leaves a spanning-forest with $k$ connected components (given by subtrees) having respectively $\lambda_1,\ldots,\lambda_k$ vertices. We ...
Roland Bacher's user avatar
2 votes
0 answers
89 views

Upper bound on decomposition numbers for the symmetric group in a block of weight $w$

The $p$-blocks of the symmetric group $S_n$ are labelled by pairs $(\gamma, w)$ where $\gamma$ is a $p$-core partition, $w \in \mathbb{N}_0$ is the weight of the block, and $|\gamma| + p w = n$. ...
Mark Wildon's user avatar
  • 11.2k
10 votes
1 answer
274 views

When are immanants irreducible?

For a partition $\lambda$ let $\chi_\lambda$ be the corresponding irreducible representation of the symmetric group $S_n$. Let $\mathrm{Imm}_\lambda(x) = \sum\limits_{\pi \in S_n} \chi_\lambda(\pi) x_{...
Mare's user avatar
  • 26.5k
4 votes
1 answer
324 views

What is the $p$-regular partition corresponding to the sign representation of $S_{n}$ over a field of characteristic $p$?

I'm now interested in the modular representation of symmetric groups. It is well-known that for a fixed prime $p$, there is a bijection between the irreducible representations of $S_{n}$ over a field ...
gualterio's user avatar
  • 1,013
7 votes
1 answer
583 views

Hurwitz numbers and $t$-cores

For integers $k \geq 0$ and $d \geq 1$ let $H(k,d)$ be the Hurwitz number which, for the purposes of this posting, will be defined by: \begin{equation} H(k,d) \, := \ d! \, \sum_{\lambda \, \vdash d}...
Jeanne Scott's user avatar
  • 2,137
7 votes
4 answers
1k views

Maximum conjugacy class size in $S_n$ with fixed number of cycles

Context: It is well known that given a permutation in $S_n$ with $a_i$ $i$-cycles (when written as a product of disjoint cycles), the size of the conjugacy class is given by $$ \frac{n!}{\prod_{j=1}^...
metallicmural99's user avatar
2 votes
0 answers
90 views

How to prove this identity on summations and partitions?

Let $f$ be a symmetric function of $s$ variables. The identity is $$\sum_{all \ k's}^\infty f(k_1,k_2,k_3,...,k_s)=\sum_{n=s}^\infty \sum_{\lambda\vdash n}\frac{s!\prod_l \lambda_l}{z_\lambda} f(\...
Anthonny's user avatar
  • 151
4 votes
1 answer
528 views

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^{\...
Māris Ozols's user avatar
6 votes
0 answers
196 views

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 ...
T. Amdeberhan's user avatar
6 votes
1 answer
407 views

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 ...
T. Amdeberhan's user avatar
3 votes
2 answers
924 views

sum over all integer partitions, of the product of the factorials of the terms

I'm looking for something making tractable the sum, over all partitions into k terms of an integer n, of the product of the factorials of all the terms. Thanks,
Eric T's user avatar
  • 31
3 votes
0 answers
153 views

Adding a row to a Young Tableau via Novelli-Pak-Stoyanovskii

Let $T_{\lambda}$ be the set of standard young tableaux (SYT) of shape $\lambda_1\geq \lambda_2\cdots\geq \lambda_n$. Now consider pushing a row $\mu$ with $\mu\geq \lambda_1$ onto $Y$ to give shape $\...
Alex R.'s user avatar
  • 4,952
4 votes
1 answer
149 views

Counting a Modified Class of Standard Young Tableau

Let $\lambda=(\lambda_1,\cdots,\lambda_n)$ be a partition, with $|\lambda|:=N$. Attach an extra box to $\lambda$ to the right end of the $r$'th row. In coordinate form, the last box on row $r$ has ...
Alex R.'s user avatar
  • 4,952
2 votes
1 answer
4k views

Cyclic Subgroups of the Symmetric Group

If we write a partition $n=k_1+...+k_r$, then we can create a $(k_1,...,k_r)$-cycle in $S_n$ with order equal to the least common multiple of the $k_i$'s. It is clear that every cyclic subgroup will ...
Jon Cohen's user avatar
  • 1,261
8 votes
2 answers
742 views

A product identity for partitions

For a partition $\lambda=(\lambda_1\ge \lambda_2\ge \dots)$, let $m_\lambda=\prod_i (\lambda_i-\lambda_{i+1})!$ be the product of factorials of consecutive differences and let $v_\lambda=\prod_{i | \...
Dmitry Vaintrob's user avatar
2 votes
0 answers
161 views

Partitions limit shape and LDP

Hello! I am trying to understand the paper of Dembo-Vershik-Zeitouni, Large deviations for integer partitions. I am only interested in Theorem 2, which deals with the case of the uniform distribution....