We say that $N$ is writable as a sum of permutation pair $\{a,b\}$ if $a+b=N$, $a\neq b$ and $a$ and $b$ are permutations of each other (e.g. $321 = 156 + 165 = 147 + 174 = ... $).
Looking at 3-digit numbers, 386 of them are writable as a sum of permutations.
- What is the density of numbers which are writable as a sum of permutations? Does it go to $1$ as $N$ increases?
Another interesting question is the number of different permutations that a number can be written as.
Looking at 3 digit numbers:
- $321$ is writable as sum of 5 different permutation pairs.
- $666$ is writable as sum of 7 different permutation pairs.
- $888$ is writable as sum of 10 different permutation pairs.
And for 4 digits:
- $5555$ is writable as sum of 28 different permutation pairs.
- $7777$ is writable as sum of 58 different permutation pairs.
- $9999$ is writable as sum of 96 different permutation pairs.
2.What is the best upper bound we can give on the number of different number permutation pairs that sums up to $N$? ($\frac{N}{2}$ is trivial, is there a better bound?)
If carry is not allowed, better bound would be $5* 9^{\lceil log_{10}^n\rceil - 1}$ + some delta. But, what happens when carry is there in addition?