Intereting Posts

Proof Using the Monotone Convergence Theorem for the sequence $a_{n+1} = \sqrt{4 + a_n}$
Does the improper integral $\int_0^\infty e^{-x^2}dx$ converge?
Proving that $\pi(2x) < 2 \pi(x) $
Path space of $S^n$
How to parameterize an orange peel
Linear transformation preserving orthogonality
Superelliptical Trig Functions
Eigenvalues of product of two hermitian matrices
Order of matrix and monic irreducible polynomial over finite field
Prove that $\displaystyle\int_{x=-1}^{1}P_L(x)P_{L-1}\acute (x)\,\mathrm{d}x=\int_{x=-1}^{1}P_L\acute(x)P_{L+1} (x)\,\mathrm{d}x=0$
A samurai cuts a piece of bamboo
Continuity and Joint Continuity
Noetherian ring whose ideals have arbitrarily large number of generators
substituting a variable in a formula (in logic)
Additive form of a spectral decomposition?

let $\mathbf{A}$ be the set of all $n\times n$ matrices on $\mathbb{N}_{n^2}=\{1,2,…,n^2\}$ with distinct entries.

let $T$ be the set of all permutations of $\mathbf{A}$ which swap the entry 1 with one of its adjacent entries. adjacent means one is above/below/right/left of the other (and both are neighbors).

- Elementary formula for permutations?
- Missing step in Galois theory proofs
- Number of zeroes at end of factorial
- Can someone explain how the Schreier-Sims Algorithms works on a permutation group with a simple example?
- Proving that $A_n$ is the only proper nontrivial normal subgroup of $S_n$, $n\geq 5$
- Does every automorphism of a permutation group preserve cycle structure?

Let $S$ be the permutation subgroup generated by $T$.

Show that $S$ acts ** intransitively** on $\mathbf{A}$.

I’m a.s. it’s correct for even $n$.

- Arrange $n$ people so that some people are never together.
- Identical balls arrangement in a circle
- Combinatorics: How to find the number of sets of numbers in increasing order?
- How many ways to put 20 things to different 4 boxes?
- An Example for a Graph with the Quaternion Group as Automorphism Group
- $S_n$ acting transitively on $\{1, 2, \dots, n\}$
- How to see that the polynomial $4x^2 - 3x^7$ is a permutation of the elements of $\mathbb{Z}/{11}\mathbb{Z}$
- Problem with random permutation and conditional probability

This appears to be a generalization of the famous 15-puzzle. The proof that it is not transitive is the same. To each matrix, associate the number given by

\begin{equation}

\text{(block distance of 1 from upper left corner)} +

\text{(parity of underlying permutation of $\mathbb{N}_{n^2}$)}.

\end{equation}

The parity of this number is an invariant under the allowed moves, that is, it does not change when you make a swap. So if you start with the numbers neatly ordered row by row, you will never get the matrix with 2 and 3 exchanged.

- Calculating volume of convex polytopes generated by inequalities
- How to calculate $\int_{0}^{1}(\arcsin{x})(\sin{\frac{\pi}{2}x})dx$?
- Generalized Euler sum $\sum_{n=1}^\infty \frac{H_n}{n^q}$
- Continuity of the function $x\mapsto d(x,A)$ on a metric space
- Group of order 4k+2. Prove that the following permutation is odd.
- Show that $x^4 + 8$ is irreducible over Z
- Solve the roots of a cubic polynomial?
- What is so special about negative numbers $m$, $\mathbb{Z}$?
- Is an irreducible element still irreducible under localization?
- Do all CDFs eventually become Horizontal?
- Prove that $\overline{A \cup B} = \overline A\cup \overline B$.
- Riemann-integrable (improperly) but not Lebesgue-integrable
- Category theory text that defines composition backwards?
- Algebraic proof of $\sum_{i=0}^k{{n \choose i}{m \choose {k-i}}}= {{m+n}\choose k}$
- How to solve $n < 2^{n/8}$ for $n$?