# Odd & even permutations and unit fractions

One more motivated by recent questions of Zhi-Wei Sun.

Let $$S_n$$ be the group of permutations of $$\{1,2,\ldots, n\}$$.

Is it true that, for every $$n \ge 8$$, there is at least one even permutation $$\pi \in S_n$$ and at least one odd permutation $$\tau \in S_n$$ with $$\sum_{k=1}^n \frac{1}{k \, \pi(k)} = \sum_{k=1}^n \frac{1}{k \, \tau(k)} = 1?$$

One case for each $$n$$ is not hard; I made it a math.stackexchange question that was successfully answered in 12 minutes. Hopefully the other case is more interesting.

Clarification: As per the MSE question referenced above and Zhi-Wei's comment, the $$n$$-cycle $$(1,2, \dots, n) \in S_n$$ satisfies the sum condition. An $$n$$-cycle is an odd permutation for even $$n$$ and an even permutation for odd $$n$$.

Here are the remaining parts of the conjecture.

a. For $$n$$ even and $$n \ge 8$$, there is an even $$\pi \in S_n$$ satisfying $$\sum_{k=1}^n \frac{1}{k \, \pi(k)} = 1$$.

b. For $$n$$ odd and $$n \ge 9$$, there is an odd $$\tau \in S_n$$ satisfying $$\sum_{k=1}^n \frac{1}{k \, \tau(k)} = 1$$.

Here are the numbers of even and odd permutations satisfying the sum condition for small $$n$$.

$$\begin{array}{c|rr} n\backslash \text{sgn} & +1 & -1 \\ \hline 1 & 1 & 0 \\ 2 & 0 & 1 \\ 3 & 2 & 0 \\ 4 & 0 & 2 \\ 5 & 4 & 0 \\ 6 & 0 & 2 \\ 7 & 4 & 0\\ 8 & 6 & 4\\ 9 & 12 & 24\\ 10 & 90 & 88 \end{array}$$

One of the first interesting'' permutations is the even permutation (in cycle notation) $$(1,2,5,8,7,6)(3,4) \in S_8$$ which gives \begin{align*} \frac{1}{1\cdot2} + \frac{1}{2\cdot5}+ \frac{1}{3\cdot4}+ \frac{1}{4\cdot3}+ \frac{1}{5\cdot8}+ \frac{1}{6\cdot1}+ \frac{1}{7\cdot6}+ \frac{1}{8\cdot7}\\ = \frac{1}{2} + \frac{1}{10}+ \frac{1}{12}+ \frac{1}{12}+ \frac{1}{40}+ \frac{1}{6}+ \frac{1}{42}+ \frac{1}{56}=1. \end{align*} Not coincidentally, $$n=8$$ is the smallest value for which there are non-$$n$$-cycle permutations that satisfy the sum condition.

Clearly, \begin{align}&\frac1{1\cdot2}+\frac1{2\cdot3}+\ldots+\frac1{(n-1)n}+\frac1{n\cdot1} \\=& 1-\frac12+\frac12-\frac13+\ldots+\frac1{n-1}-\frac1n+\frac1n=1. \end{align} If $$n=2k+1$$, then we also have \begin{align}&\frac1{1\cdot2}+\frac1{2\cdot3}+\ldots+\frac1{k(2k+1)}+\frac1{(2k+1)(k+1)} \\&+\frac1{(k+1)(k+2)}+\cdots+\frac1{(2k-1)2k}+\frac1{2k\cdot1} \\=&1.\end{align} See also my related conjectures available from http://oeis.org/A322069 and http://oeis.org/A322070.
• @ Zhi-Wei Your first permutation, $\pi = (2,3,\ldots, n, 1)$, is the not hard'' $n$-cycle solution from the math.stackexchange problem. When $n$ is even, $\pi$ is odd and vice versa. Your second permutation is a different $n$-cycle for $n$ odd, giving another even permutation. The remaining parts of the question ask for an even permutation of $S_{2k}$ (with $k \ge 4$) and an odd permutation of $S_{2k+1}$ (also with $k \ge 4)$ that satisfy the sum condition. – Brian Hopkins Nov 26 '18 at 4:04