## 5.150. highest_peak

Origin
Constraint

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }\left(\mathrm{\pi ·\pi ΄\pi Έ\pi Ά\pi ·\pi },\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\right)$

Arguments
 $\mathrm{\pi ·\pi ΄\pi Έ\pi Ά\pi ·\pi }$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$
Restrictions
 $\mathrm{\pi ·\pi ΄\pi Έ\pi Ά\pi ·\pi }\beta ₯0$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\beta ₯0$ $\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 $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }={V}_{1},...,{V}_{m}$ is a peak if and only if there exists an $i$ $\left(1 such that ${V}_{i-1}<{V}_{i}$ and ${V}_{i}={V}_{i+1}=...={V}_{k}$ and ${V}_{k}>{V}_{k+1}$. $\mathrm{\pi ·\pi ΄\pi Έ\pi Ά\pi ·\pi }$ is the maximum value of the peak variables. If no such variable exists $\mathrm{\pi ·\pi ΄\pi Έ\pi Ά\pi ·\pi }$ is equal to 0.

Example
$\left(\begin{array}{c}8,β©\begin{array}{c}\mathrm{\pi \pi \pi }-1,\hfill \\ \mathrm{\pi \pi \pi }-1,\hfill \\ \mathrm{\pi \pi \pi }-4,\hfill \\ \mathrm{\pi \pi \pi }-8,\hfill \\ \mathrm{\pi \pi \pi }-6,\hfill \\ \mathrm{\pi \pi \pi }-2,\hfill \\ \mathrm{\pi \pi \pi }-7,\hfill \\ \mathrm{\pi \pi \pi }-1\hfill \end{array}βͺ\hfill \end{array}\right)$

The $\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$ constraint holds since 8 is the maximum peak of the sequence $11486271$.

Typical
 $\mathrm{\pi ·\pi ΄\pi Έ\pi Ά\pi ·\pi }>0$ $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>2$ $\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\right)>1$
Symmetry

Items of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ can be reversed.

FigureΒ 5.150.2 depicts the automaton associated with the $\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$ 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 ${\mathrm{\pi }}_{i}$. The following signature constraint links ${\mathrm{\pi  \pi °\pi }}_{i}$, ${\mathrm{\pi  \pi °\pi }}_{i+1}$ and ${\mathrm{\pi }}_{i}$:
${\mathrm{\pi  \pi °\pi }}_{i}>{\mathrm{\pi  \pi °\pi }}_{i+1}\beta {\mathrm{\pi }}_{i}=0\beta §{\mathrm{\pi  \pi °\pi }}_{i}={\mathrm{\pi  \pi °\pi }}_{i+1}\beta {\mathrm{\pi }}_{i}=1\beta §{\mathrm{\pi  \pi °\pi }}_{i}<{\mathrm{\pi  \pi °\pi }}_{i+1}\beta {\mathrm{\pi }}_{i}=2$.