# Kaprekar's mapping fixed points

Jens Kruse Andersen in his comment in OEIS's A099009 noticed 3 families of numbers among Kaprekar's fixed mapping points (otherwise known as kernels of the Kaprekar's routine):

"Let $d(n)$ denote $n$ repetitions of the digit $d$. The sequence includes the following for all $n\ge0$: $5(n)499(n)4(n)5, 63(n)176(n)4, 8643(n)1976(n)532$."

The above comment, made by Jens Kruse Andersen, is missing one more family of terms (which starts with one or more digits "$9$" and ends with the digit "$1$"): 97508421, 9753086421, 9975084201, 975330866421, 997530864201, 999750842001, ... .

This family could be generalized (using the same method as in Andersen's comment) and it is actually covered by Syed Iddi Hasan in A214559: $9(x_1+1)//8(x_2)//7(x_3+1)//6(x_2)//5(x_3+1)//4(x_2)//3(x_4)//2(x_2)//1(x_3)//0//9(x_2)//8(x_3+1)//7(x_2)//6(x_4)//5(x_2)//4(x_3+1)//3(x_2)//2(x_3+1)//1(x_2)//0(x_1)//1$ where the sign // denotes concatenation of digits in the definition, $d(x)$ denotes $x$ repetitions of $d$, $x\ge0$.

NB - in his OEIS wiki page Syed Iddi Hasan wrote: "I narrowed it down to four parameters. I ordered the digits from largest to smallest and smallest to largest, and by comparing them I was able to find the interdependent pairs of numbers. However, these four parameters seem to be independent of each other."

Also A214557 and A214558 (both by Syed Iddi Hasan) are two variants relevant to Andersen's 8643(n)1976(n)532 - those two should be somehow combined, in my opinion, for the purpose of identifying unique families of Kaprekar mapping fixed points.

Could someone finalize classification of distinct families for Kaprekar's fixed mapping points and prove that each of Kaprekar's fixed mapping points belong ONLY to the one of the above mentioned families ?

• Hasan considered sequences defined by 4 parameters, with digits in decreasing order. It is possible that the number of sequences of fixed point for the Kaprekar mapping, with digits not in order (that is, with many subsequences of the same digit) is inifinite (with many more parameters). I can't offer any proof, just a reasonable doubt. – Mauro Fiorentini Feb 14 '16 at 18:01
• Like I mentioned in the beginning of my question, Jens Kruse Andersen in his comment in OEIS's A099009 noticed 3 families of numbers among Kaprekar's fixed mapping point, which are not in ascending or descending order of digits: 5(n)499(n)4(n)5, 63(n)176(n)4, 8643(n)1976(n)532. Together with the fourth family, added by Hasan, they cover all Kaprekar's fixed mapping points, listed in A099009 "b-list" oeis.org/A099009/b099009.txt – Alex Feb 22 '16 at 12:19