All Questions
Tagged with symmetric-groups partitions
24 questions
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_{...
- 26.5k
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 ...
- 2,242
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 | \...
- 7,952
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}^...
- 211
7
votes
1
answer
582
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}...
- 2,137
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 ...
- 43.2k
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 ...
- 43.2k
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 ...
- 17.9k
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 ...
- 1,013
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^{\...
- 499
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 ...
- 4,952
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 ...
- 571
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 ...
- 571
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,
- 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 $\...
- 4,952
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}...
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 ...
- 1,261
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,\...
- 43.2k
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 - ...
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 $|\gamma| + p w = n$. ...
- 11.2k
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
$$\sum_{all \ k's}^\infty f(k_1,k_2,k_3,...,k_s)=\sum_{n=s}^\infty \sum_{\lambda\vdash n}\frac{s!\prod_l \lambda_l}{z_\lambda} f(\...
- 151
2
votes
0
answers
161
views
Partitions limit shape and LDP
Hello!
I am trying to understand the paper of Dembo-Vershik-Zeitouni, Large deviations for integer partitions. I am only interested in Theorem 2, which deals with the case of the uniform distribution....
1
vote
1
answer
82
views
The sum of the signs of conjugacy classes in the symmetric group S_n [duplicate]
Let $r$ be the number of conjugacy classes of the symmetric group $S_n$ whose sign is $1$, i.e.
\begin{equation}
r := \#\{c \in \text{Conj} (S_n): \text{sgn} (c) = 1 \}.
\end{equation}
Let $s$ be the ...
1
vote
0
answers
82
views
How to obtain explicit formula for this sum over Young diagram?
Consider the next essence
$$
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 ...