All Questions
8 questions with no upvoted or accepted answers
6
votes
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answers
196
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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 ...
- 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
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
0
answers
153
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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
0
answers
352
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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
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New identity for sum over Young diagram of symmetric group?
Consider the next identity
$$
\sum_{\tau \vdash r} d^2 (\tau ) \! \prod_{i = -(r-1)}^{r-1} \! \! \left( N+i \right)^{t_i^r} =\sum_{\tau \vdash r} d^2 (\tau ) \prod_{i = 1}^{r} \Gamma [N + \tau_i - ...
2
votes
0
answers
161
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Partitions limit shape and LDP
Hello!
I am trying to understand the paper of Dembo-Vershik-Zeitouni, Large deviations for integer partitions. I am only interested in Theorem 2, which deals with the case of the uniform distribution....
1
vote
0
answers
82
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How to obtain explicit formula for this sum over Young diagram?
Consider the next essence
$$
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 ...