I was recently playing around with look-and-say sequences and variations upon them. One of the sequences I came across started with a seed sequence $S$, and quickly collapsed into an $n$-cycle of sequences for some $n$ for every $S$ I have checked.
Intuitively, while in look-and-say you say "02241" as "1 zero, then 2 twos, then 1 four, then 1 one," in this cousin you say "02241" as "1 zero, 1 one, 2 twos, and 1 four." That is, you completely disregard the order of the numbers in the previous sequence.
Here is the formal version, given some sequence $S$ of cardinalities with supremum $\kappa$:
$$\mathfrak{K}(S)=(|\{\alpha:S_\alpha=\lambda\}|,\lambda)_{\lambda\leq\kappa}$$
Restricting ourselves to finite sequences of finite natural numbers, there are still many interesting properties of this function. For example, starting with the sequence (5) and iterating the $\mathfrak{K}$ function on the sequence reveals an eventual 3-cycle of:
$$(1, 0, 4, 1, 4, 2, 1, 3, 1, 4, 2, 5, 1, 6)$$ $$(1, 0, 5, 1, 2, 2, 1, 3, 3, 4, 1, 5, 1, 6)$$ $$(1, 0, 5, 1, 2, 2, 2, 3, 1, 4, 2, 5, 1, 6)$$
Here are my questions:
- Is there a sequence $S$ such that iterating $\mathfrak{K}$ on $S$ never produces an $n$-cycle for some $n$?
- Are there some $m$ and $n$ such that for any $X\ge m$ iterating $\mathfrak{K}$ on the sequence $S=(X)$ eventually produces an $n$-cycle? (I believe that $n=2$ if so)
- If 2. is yes, then what is (a lower bound for) $m$?
- Does this already have a name? What is it?