# Sequence defined on multisets

 Importance: Medium ✭✭
 Author(s): Erickson, Martin
 Subject: Combinatorics
 Keywords: multiset sequence
 Posted by: Martin Erickson on: June 29th, 2010

\begin{conjecture} Define a $2 \times n$ array of positive integers where the first row consists of some distinct positive integers arranged in increasing order, and the second row consists of any positive integers in any order. Create a new array where the first row consists of all the integers that occur in the first array, arranged in increasing order, and the second row consists of their multiplicities. Repeat the process. For example, starting with the array $[1; 1]$, the sequence is: $[1; 1]$ -> $[1; 2]$ -> $[1, 2; 1, 1]$ -> $[1, 2; 3, 1]$ -> $[1, 2, 3; 2, 1, 1]$ -> $[1, 2, 3; 3, 2, 1]$ -> $[1, 2, 3; 2, 2, 2]$ -> $[1, 2, 3; 1, 4, 1]$ -> $[1, 2, 3, 4; 3, 1, 1, 1]$ -> $[1, 2, 3, 4; 4, 1, 2, 1]$ -> $[1, 2, 3, 4; 3, 2, 1, 2]$ -> $[1, 2, 3, 4; 2, 3, 2, 1]$, and we now have a fixed point (loop of one array).

The process always results in a loop of 1, 2, or 3 arrays. \end{conjecture}

% You may use many features of TeX, such as % arbitrary math (between $...$ and $$...$$) % \begin{theorem}...\end{theorem} environment, also works for question, problem, conjecture, ... % % Our special features: % Links to wikipedia: \Def {mathematics} or \Def[coloring]{Graph_coloring} % General web links: \href [The On-Line Encyclopedia of Integer Sequences]{http://www.research.att.com/~njas/sequences/}

## Bibliography

* Erickson, Martin J., "Introduction to Combinatorics," Wiley, 1996.

* indicates original appearance(s) of problem.

### Solution

This problem has recently been solved by the author.