All Questions
Tagged with symmetric-groups enumerative-combinatorics
7 questions
15
votes
1
answer
749
views
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 ...
- 1,413
10
votes
1
answer
358
views
Induction step in Bóna and Ehrenborg's proof that the generating function of the alternating runs has -1 as a root of a certain multiplicity
This is a crosspost of a question I asked on Mathematics SE four months ago. Periodically bumping it and placing a bounty on it to attract more attention were to no avail. There are some comments ...
user82537
7
votes
0
answers
183
views
Explaining $\left(a-1\right)^n \cdot n! \mid a^{n-1} \prod_{i=1}^n \left(a^i-1\right)$ by a free $S_n$-action
Here is an olympiad-level problem on elementary number theory:
Let $a$ be an integer and $n$ a positive integer. Prove that
\begin{align}
\left(a-1\right)^n \cdot n! \mid a^{n-1} \prod_{i=1}^n \left(...
- 33.8k
4
votes
1
answer
296
views
Number of paths in the Bruhat order in the symmetric group
Let $\mathbb{S}_m$ the symmetric group on $m$ letters. Let $v\in\mathbb{S}_m$, and consider paths in the Bruhat order like this: $1\lessdot v_1\lessdot\cdots\lessdot v$, where $\lessdot$ means the ...
- 1,066
2
votes
1
answer
199
views
Sequence of monotone tuples and permutation condition for rotation
I was doing some counting in $S_n$ symmetric group I encountered the following problem, which also someway related to central factorial number.
So given a $n$ cycle say $(1,2,\ldots,n)$, what are the ...
- 685
2
votes
1
answer
238
views
"flavored" equivalence classes of permutations
We say two permutations $\pi_1$ and $\pi_2$ in the symmetric group $\mathfrak{S}_n$ are $k$-equivalent, denoted
$\pi_1 \sim_k \pi_2$, if one can be
determined from the other after a finite number of ...
- 43.2k
1
vote
0
answers
177
views
Combinatorial bijection on monotone sequences
Let $(n),\mu$ be the partition of $n$ define $H_g^{m}((n);\mu)$ count's the number of tuples $(\tau_1,\ldots,\tau_r)$ of transposition in symmetric group $S_n$ with the following conditions
$$ (1,2,\...
- 685