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Tagged with symmetric-groups partitions
7 questions
6
votes
1
answer
407
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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 ...
- 43.2k
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
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
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
2
votes
0
answers
168
views
New identity for sum over Young diagram of symmetric group?
Consider the next identity
$$
\sum_{\tau \vdash r} d^2 (\tau ) \! \prod_{i = -(r-1)}^{r-1} \! \! \left( N+i \right)^{t_i^r} =\sum_{\tau \vdash r} d^2 (\tau ) \prod_{i = 1}^{r} \Gamma [N + \tau_i - ...
1
vote
0
answers
82
views
How to obtain explicit formula for this sum over Young diagram?
Consider the next essence
$$
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 ...