All Questions
7 questions
10
votes
3
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1k
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Number of permutations with longest increasing subsequences of length at most $n$
Is there a known expression for, or a nontrivial upper bound on, the number of permutations in $S_k$ with longest increasing subsequence of length at most $n$?
Let $l(\sigma)$ denote the length of the ...
- 201
14
votes
2
answers
514
views
Classification of shod Dyck paths
A sequence $[c_0,c_1,...,c_{n-1}]$ with $n \geq 2$ is called a Dyck path in case $c_{n-1}=1$, $c_i \geq 2$ for $i \neq n-1$ and $c_i-1 \leq c_{i+1}$ for each $i$.
For example the Dyck paths for $n=4$ ...
- 26.5k
3
votes
0
answers
190
views
Orthogonal basis for decomposition of induced representation of derangements
Background
Let $V_n$ be the $\mathbb{C}$-module spanned by the set of derangements (permutations with no fixed points) inside the group ring of $S_n$. We make $V_n$ into a $\mathbb{C}S_n$-module ...
- 185
3
votes
2
answers
216
views
trace and involution permutations: Part II
This is a follow up on my earlier MO question.
Let $\operatorname{Inv}(\mathfrak{S}_n):=\{\pi\in\mathfrak{S}_n: \pi^2=1\}$ be the set of involutions in the symmetric group $\mathfrak{S}_n$. Denote $...
- 43.2k
13
votes
1
answer
637
views
trace and involution permutations: Part I
Let $\operatorname{Inv}(\mathfrak{S}_n):=\{\pi\in\mathfrak{S}_n: \pi^2=1\}$ be the set of involutions in the symmetric group $\mathfrak{S}_n$. Denote $I_n:=\#\operatorname{Inv}(\mathfrak{S}_n)$. Let $\...
- 43.2k
9
votes
0
answers
398
views
When do almost all these invariants of tensors vanish?
Let $A,B,C,D$ be $n$-dimensional vector spaces over a field $k$.
There is a natural homomorphism from the $mn^m$th tensor power $A^{\otimes (m n^m)} $ of $A$ to $k$ given by the determinant map $A^{\...
- 148k
9
votes
0
answers
275
views
pattern-avoiding permutations vs multi-core partitions
Let $\mathfrak{S}_n$ be the permutation group on $[n]$. Given the pattern $\sigma=k(k-1)\cdots321$, let $I_n(\sigma)$ be the number of involutions in $\mathfrak{S}_n$ that avoid the pattern $\sigma$. ...
- 43.2k