All Questions
11 questions
17
votes
1
answer
1k
views
Can this probability be obtained by a combinatorial/symmetry argument?
Suppose that $a_1,\dots,a_n,b_1,\dots,b_n$ are iid random variables each with a symmetric non-atomic distribution.
Let $p$ denote the probability that there is some real $t$ such that $t a_i \ge b_i$ ...
- 128k
4
votes
1
answer
225
views
Integer-valued polynomials from Pólya counting
Let finite group $G$ act on a finite set $X$ and hence on colorings $Y^X$, where $Y=\{1,2,\ldots,k\}$ is a set of colors. The Burnside-Pólya-Redfield-etc. counting theorem says that the number of ...
- 24.2k
7
votes
2
answers
713
views
Total sum of squares of characters of the symmetric group $\mathfrak{S}_n$
In my earlier MO post, I proposed the double sum $\sum_{\mu\vdash n}\sum_{\lambda\vdash n}\chi_{\mu}^{\lambda}$ regarding characters of the symmetric group $\mathfrak{S}_n$. Soon after, I started ...
- 43.2k
6
votes
0
answers
479
views
Darkness in the lamplighter group
Consider paths through the lamplighter group $\mathbb{Z}_n\wr\mathbb{Z}$ with steps consisting of moving left, moving right, and toggling the lamp at the current position. How many paths of length $m$ ...
- 2,203
5
votes
1
answer
764
views
Conjugacy classes in $GL_{n}(Z / pZ)$
Let $p$ be a prime number and $G=GL_n ( \mathbb{Z} / p \mathbb{Z}
)$. Consider the set $U$ of upper-triangular matrices of $G$
having entries of $1$ on the diagonal. The cardinality of $U$ is $p^{\...
- 463
4
votes
0
answers
165
views
Counting "deflected" permutations: Part II
This is the second sequel to my earlier question on MO. Although the the current problem appears very similar, the answer is certainly different as experiments indicate.
As usual, let $\mathfrak{S}_n$...
- 43.2k
4
votes
1
answer
158
views
Counting "deflected" permutations: Part I
Let $\mathfrak{S}_n$ denote the group of permutations on $\{1,2,\dots,n\}$. Now, introduce the sets
$$\mathcal{A}_n^{(k)}:=\{\pi\in\mathfrak{S}_n: -1\leq \pi(j)-j\leq k,\,\forall j\}.$$
I would like ...
- 43.2k
13
votes
1
answer
651
views
The Möbius number of the nonabelian finite simple groups
Let $L$ be a finite lattice with minimum $\hat{0}$ and maximum $\hat{1}$. The Möbius function $\mu$ for $L$ is defined recursively by: for $\forall a,b \in L$ with $a<b$, $\mu(b,b) = 1$ and $\mu(...
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
21
votes
1
answer
2k
views
Are there enough additive permutations?
I am hoping to learn enough about additive permutations to help with a number theory problem. These permutations also have connections to difference sets, orthomorphisms, transversals, and other ...
- 3,209
5
votes
2
answers
567
views
Orbits of independent sets of the hypercube
How does one enumerate the distinct orbit classes of independent sets of the hypercube modulo symmetries of the hypercubes?
The counting of the number of independent sets in an $n$-dimensional ...
- 171