# Numbers which are writable as a sum of permutation pairs

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.

1. 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?)

• Surely every even number is a sum of a permutation pair; $2n=n+n$ --- the identity is a permutation! If you mean to exclude the identity permutation, you should explicitly say so in the body of the question. – Gerry Myerson Sep 1 '14 at 13:04
• @GerryMyerson - You are correct, I meant different a,b, I'll edit the question, thanks. – R B Sep 1 '14 at 13:09
• Do you allow leading zeroes? E.g., $101=100+001$? – Gerry Myerson Sep 2 '14 at 3:12
• @GerryMyerson - yes, leading zeros are fine. – R B Sep 2 '14 at 5:48
• Then aren't there more than 10 pairs for 888? $840+048,741+147,642+246,543+345,804+084,714+174,624+264,534+354,480+408,471+417,462+426,453+435$ – Gerry Myerson Sep 2 '14 at 6:45

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?