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Sequence defined on multisets

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


Bibliography

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


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Solution

On July 18th, 2011 Anonymous says:

This problem has recently been solved by the author.

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