## 5.279. no_valley

Origin
Constraint

$\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi ’}\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\right)$

Argument
 $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$
Restrictions
 $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>0$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi \pi \pi }\right)$
Purpose

A variable ${V}_{k}$ $\left(1 of the sequence of variables $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }={V}_{1},\beta ―,{V}_{m}$ is a valley if and only if there exists an $i$ $\left(1 such that ${V}_{i-1}>{V}_{i}$ and ${V}_{i}={V}_{i+1}=\beta ―={V}_{k}$ and ${V}_{k}<{V}_{k+1}$. The total number of valleys of the sequence of variables $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ is equal to 0.

Example
$\left(β©1,1,4,8,8,2βͺ\right)$

The $\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi ’}$ constraint holds since the sequence $114882$ does not contain any valley.

Typical
 $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>3$ $\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\right)>1$
Symmetries
• Items of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ can be reversed.

• One and the same constant can be added to the $\mathrm{\pi \pi \pi }$ attribute of all items of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$.

Arg. properties

Contractible wrt. $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$.

Counting
 Length ($n$) 2 3 4 5 6 7 8 Solutions 9 50 295 1792 11088 69498 439791

Number of solutions for $\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi ’}$: domains $0..n$

generalisation: $\mathrm{\pi \pi \pi \pi \pi \pi ’}$Β (introduce a $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ counting the number of valleys).
FigureΒ 5.279.2 depicts the automaton associated with the $\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi ’}$ constraint. To each pair of consecutive variables $\left({\mathrm{\pi  \pi °\pi }}_{i},{\mathrm{\pi  \pi °\pi }}_{i+1}\right)$ of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ corresponds a signature variable ${S}_{i}$. The following signature constraint links ${\mathrm{\pi  \pi °\pi }}_{i}$, ${\mathrm{\pi  \pi °\pi }}_{i+1}$ and ${S}_{i}$: $\left({\mathrm{\pi  \pi °\pi }}_{i}<{\mathrm{\pi  \pi °\pi }}_{i+1}\beta {S}_{i}=0\right)\beta §\left({\mathrm{\pi  \pi °\pi }}_{i}={\mathrm{\pi  \pi °\pi }}_{i+1}\beta {S}_{i}=1\right)\beta §\left({\mathrm{\pi  \pi °\pi }}_{i}>{\mathrm{\pi  \pi °\pi }}_{i+1}\beta {S}_{i}=2\right)$.