# Tagged Questions

The tag has no usage guidance.

661 views

### What is the probability that two random permutations have the same order?

I am interested in the orders of random permutations. Since the law of the logarithm of the order of a permutation converges to a normal law (for instance Erdös-Turan Statistical group theory III), ...
97 views

### irregular LDPC code construction algorithm

I want to construct a sparse random binary matrix ${{\bf{H}}_{m \times n}}$ that has the following properties 1- Faction of columns of weight $i$ is ${v_i}$ . 2- Fraction of rows of weight $i$...
622 views

### fixed points of permutation groups

As is well-known (see, for example, a nice exposition by our own Qiaochu: https://qchu.wordpress.com/2012/11/07/fixed-points-of-random-permutations/) that the distribution of the number of fixed ...
207 views

### variance of the number of fixed points for a permutation group

It is reasonably well-known that the variance of the number of fixed points for $S_n$ equals $1.$ Now, what about other transitive permutation groups on $\{1, \dotsc, n\}?$ Presumably much is known. I ...
44 views

### Smooth bivariate functions identifiable under permutations

Consider a "smooth" $f : [0,1]^2 \to [0,1]$ and let $\pi : [n] \to [n]$ be a permutation of $[n] =\{1,\dots,n\}$. Let us define  A^{f,\pi} := \Big( f\Big(\frac{\pi(i)}{n},\frac{\pi(j)}{n}\Big) \...
126 views

### Horizontal vs Vertical sides Exit from a Rectangle for simple symmetric Random Walk on $\textbf{Z}^{2}$

Consider simple symmetric random walk, $X_{n} = (X_{n}^{(1)}, X_{n}^{(2)})$ with $X_0= (0,0)$, on the 2 dimensional integer lattice, $\textbf{Z}^{2}$. Let $T_{M}, T_{N}$ be the smallest $n$ such ...
2k views

### Combinatorial Morse functions and random permutations

This question has its origin in combinatorial topology. In the 90s R. Forman proposed a discrete counterpart of Morse theory. In his case, a Morse function on a triangulated space is a function ...
126 views

### What is the probability of a given induced ordering of a random permutation?

I ran into the following problem in a calculation involving permutations. Let $[n] = \{1,...,n\}$, and assume that $[n]$ is partitioned into equivalency classes. That is, $[n]$ is the disjoint union ...
199 views

### Partial sums of partitions

Suppose I have two random partitions of $N.$ ("random" really means "the cycle type of a random permutation", but if there is an answer with any definition, I am interested). The question is: what is ...
723 views

### Number of Permutations with k-inversions and with a single clamped value

This question is cross-posted from math.stackexchange because it might be too technical. Let $S_n$ be the symmetric group. Recall that the number of inversions of a permutation $\sigma\in S_n$ is the ...
164 views

### Factorization of permutations.

Let $n,k$ be positive integers such that $3n=2k$ and $N = \lfloor \alpha n\rfloor$ for some constant $0<\alpha<1$. Let $S_{3n}$ denote the permutation group of order $3n$. Consider the following ...
Suppose $S_{n,N}$ be the set of $n$ elements with $N$ many cycles where $N$ is proportional to $n$. $U_{n,N}$ is an element picked randomly from this. It is known that the length of any cycle cannot ...