How many permutations of $a_1, a_2, a_3, ..., a_{2020}$ of $1,2,\ldots,2020$ s.t. $|a_1-1|=|a_2-2|=\ldots=|a_{2020}-{2020}|$?

Question

Let $a_1,a_2, ...,a_{2020}$ be a permutation of $1,2,3,...,2020$. How many permutations of $a_1, a_2, a_3, ..., a_{2020}$ are there such that $$ |a_1-1|=|a_2-2|=\ldots=|a_{2020}-{2020}|? $$

I thought 2020; if a_1=1, then a_2020=2020,.. a_1=2, then a_2020=2019, a_2018=2017,.... a_1=3, then a_2020=2018,.... but it seemed wrong

Answer 1

Note that if $a_k = 1$, then $\lvert 1 - k \rvert = k - 1 = \lvert a_1 - 1 \rvert = a_1 - 1$, so $a_1 = k$. Then $\lvert a_2 - 2 \rvert$ must be $k - 1$, so $a_2$ must be $k+1$, and so on.

If we let $l = k - 1$, the permutation must consist of disjoint involutions swapping $m$ and $m + l$ for each $m$ equivalent to $1, \dotsc, l$ modulo $2l$:

$$ (1,1+l)(2,2+l)\dotsb(l,2l)\,(2l+1,3l+1)(2l+2,3l+2)\dots(3l,4l)\,(4l+1,5l+1)\dotsb $$

For this to work, $n$ must split into chunks of size $2l$. For $n = 2020$, this works with $l \in \{1, 2, 5, 10, 101, 202, 505, 1010\}$. Along with the identity permutation, this gives 9 solutions:

• $(1)$
• $(1,2)(3,4)\dotsb(2019,2020)$
• $(1,3)(2,4)\,(5,7)(6,8)\dotsb(2017,2019)(2018,2020)$
• $(1,6)(2,7)(3,8)(4,9)(5,10)\,(11,16)\dotsb(2011,2016)(2012,2017)(2013,2018)(2014,2019)(2015,2020)$
• $(1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)\,(21,31)\dotsb(2010,2020)$
• $(1,102)(2,103)\dotsb(101,202)\,(203,304)\dotsb(1919,2020)$
• $(1,203)(2,204)\dotsb(202,404)\,(405,607)\dotsb(1818,2020)$
• $(1,506)(2,507)\dotsb(505,1010)\,(1011,1516)\dotsb(1515,2020)$
• $(1,1011)(2,1012)\dotsb(1010,2020)$