Definition:Pea Pattern/Descending
Definition
The descending pea pattern is defined as follows:
The first term $p_0$ is an arbitrary integer.
$p_n$ is formed from $p_{n - 1}$ as follows.
First, the distinct digits in $p_{n - 1}$ are counted, and the number of each is noted.
The count of each distinct digit is concatenated with an instance of the digit itself.
Then those concatenations are themselves concatenated into $p_n$ according to a predetermined order.
For the descending pea pattern, the concatenation into $p_n$ is in descending order of the distinct digits of $p_{n - 1}$.
Examples
Descending Pea Pattern on $111$
| \(\ds \) | \(\) | \(\ds 111\) | ||||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 31\) | three $1$s | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 1311\) | one $3$, one $1$ | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 1331\) | one $3$, three $1$s | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 2321\) | two $3$s, two $1$s | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 132211\) | one $3$, two $2$s, one $1$ | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 132231\) | one $3$, two $2$s, three $1$s | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 232221\) | two $3$s, two $2$s, two $1$s | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 134211\) | one $3$, four $2$s, one $1$ | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 14131231\) | one $4$, one $3$, one $2$, three $1$s | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 14331241\) | one $4$, two $3$, one $2$, four $1$s | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 24231231\) | two $4$s, two $3$, one $2$, three $1$s | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 14233221\) | one $4$, two $3$, three $2$, two $1$s | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 14233221\) | one $4$, two $3$, three $2$, two $1$s |
and a fixed point has been reached.
Descending Pea Pattern on $231$
| \(\ds \) | \(\) | \(\ds 231\) | ||||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 131211\) | one $3$, one $2$, one $1$ | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 131241\) | one $3$, one $2$, four $1$s | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 14131231\) | one $4$, one $3$, one $2$, three $1$s | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 14231241\) | one $4$, two $3$s, one $2$, four $1$s | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 24132231\) | two $4$s, one $3$, two $2$s, three $1$s | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 14233221\) | one $4$, two $3$s, three $2$s, two $1$s | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds 14233221\) | one $4$, two $3$s, three $2$s, two $1$s |
and a fixed point has been reached.
Linguistic Note
The origin of the term pea pattern is unclear.
The terms ascending pea pattern, descending pea pattern and sequential pea pattern appear not to be standard, as many sources focus on one of the variants and ignore the others.