# Tag Info

Accepted

### Why 'excedances' of permutations?

Mea culpa. Comtet used the term excédence. When writing EC1 I needed an English term for this concept. For some reason I didn't like the word exceedance. I thought it looked better without the double ...
• 43.8k
Accepted

### Multiplying all the elements in a group

Yes, your $G!$ (as a set) is always either $[G,G]$ (if the order of $G$ is odd, or its Sylow $2$-subgroup is non-cyclic) or $z[G,G]$ if $G$ has cyclic Sylow $2$-subgroup, where $z$ is the involution ...
• 1,837
Accepted

• 12.9k

### Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power

Not an answer, but maybe a start: It is fairly clear why trivial cases like $n=18,$ power$=2$ don't work, after all of the sum-pairs $\neq$ a power of $2$ that are $\leq2n$ are stripped away: ...
• 1,801
Accepted

### Linear permutations commuting with $x\rightarrow x^{-1}$

The answer is no: a linear transformation of $F$ which commutes with $\phi$ is an automorphism of $F$. This is a seemingly inelegant but simple argument. Any map $\psi: F \rightarrow F$ can be ...
• 4,661

### Is the Number of Carries in Integer-Addition Associative?

For any base $b$, if we add $a$ and $c$ with $k$ carries, then $S_b(a+c)=S_b(a)+S_b(c)-(b-1)k$, where $S_b$ denotes the sum of digits. Since the resulting sum is independent of the order of addition, ...
• 18.9k
Accepted

### Is $(\mathbb{R},+)$ isomorphic to a subgroup of $S_\omega$?

See Theorem 4.3 of this paper by De Bruijn. Any abelian group of order $2^\kappa$ can be embedded in $Sym(\kappa)$ when $\kappa$ is infinite. (There is also an addendum to the paper which corrects ...
• 2,925

• 88.5k
Accepted

### Constructing permutations avoiding a pattern

As far as "combining pattern foo and bar makes the set empty for large $n$", there is an answer and it is fairly trivial. There are no permutations of length longer than $(k-1)(\ell-1)+1$ ...
• 2,251

### Permutations with all cycles odd length and permutations with all cycles even length

Here is Eytan's proof in more detail. First, there is a canonical way to write the cycle decomposition of a permutation. You order the cycles in descending order based on the largest member they ...
• 107k

### How many rearrangements must fail to alter the value of a sum before you conclude that none do?

This is really a comment on Joel's answer, but apparently too long. Let P be the forcing which adds a permutation of $\mathbb{N}$ by finite pieces (so $P$ is forcing-equivalent to Cohen forcing). ...
• 2,430
This already fails for the second-smallest sporadic group $M_{12}$. A simple subgroup $G$ of $S_N$ cannot contain an $n$-cycle with $n$ even (unless $|G|=2$...), because then $G \cap A_N$ would be an ...
### Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power
This is a to long for a comment: Let $G(n,N)$ Micah's graph with vertices the numbers $1,..,n$ and edges $\{i,j\}$ if $i+j$ is a power of $N$. Your condition is satisfied if and only if $G$ contains ...