## 5.254. orchard

Origin
Constraint

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi ½\pi \pi Ύ\pi },\mathrm{\pi \pi \pi ΄\pi ΄\pi }\right)$

Arguments
 $\mathrm{\pi ½\pi \pi Ύ\pi }$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi \pi \pi ΄\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi ‘}-\mathrm{\pi \pi \pi },\mathrm{\pi ‘}-\mathrm{\pi \pi \pi \pi },\mathrm{\pi ’}-\mathrm{\pi \pi \pi \pi }\right)$
Restrictions
 $\mathrm{\pi ½\pi \pi Ύ\pi }\beta ₯0$ $\mathrm{\pi \pi \pi ΄\pi ΄\pi }.\mathrm{\pi \pi \pi \pi \pi ‘}\beta ₯1$ $\mathrm{\pi \pi \pi ΄\pi ΄\pi }.\mathrm{\pi \pi \pi \pi \pi ‘}\beta €|\mathrm{\pi \pi \pi ΄\pi ΄\pi }|$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi \pi \pi ΄\pi ΄\pi },\left[\mathrm{\pi \pi \pi \pi \pi ‘},\mathrm{\pi ‘},\mathrm{\pi ’}\right]\right)$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi \pi \pi ΄\pi ΄\pi },\mathrm{\pi \pi \pi \pi \pi ‘}\right)$ $\mathrm{\pi \pi \pi ΄\pi ΄\pi }.\mathrm{\pi ‘}\beta ₯0$ $\mathrm{\pi \pi \pi ΄\pi ΄\pi }.\mathrm{\pi ’}\beta ₯0$
Purpose

Orchard problemΒ [Jackson1821]:

βYour aid I want, Nine trees to plant, In rows just half a score, And let there be, In each row, threeβSolve this: I ask no more!β

Example
$\left(\begin{array}{c}10,β©\begin{array}{ccc}\mathrm{\pi \pi \pi \pi \pi ‘}-1\hfill & \mathrm{\pi ‘}-0\hfill & \mathrm{\pi ’}-0,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-2\hfill & \mathrm{\pi ‘}-4\hfill & \mathrm{\pi ’}-0,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-3\hfill & \mathrm{\pi ‘}-8\hfill & \mathrm{\pi ’}-0,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-4\hfill & \mathrm{\pi ‘}-2\hfill & \mathrm{\pi ’}-4,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-5\hfill & \mathrm{\pi ‘}-4\hfill & \mathrm{\pi ’}-4,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-6\hfill & \mathrm{\pi ‘}-6\hfill & \mathrm{\pi ’}-4,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-7\hfill & \mathrm{\pi ‘}-0\hfill & \mathrm{\pi ’}-8,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-8\hfill & \mathrm{\pi ‘}-4\hfill & \mathrm{\pi ’}-8,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-9\hfill & \mathrm{\pi ‘}-8\hfill & \mathrm{\pi ’}-8\hfill \end{array}βͺ\hfill \end{array}\right)$

The 10 alignments of 3 trees correspond to the following triples of trees: $\left(1,2,3\right)$, $\left(1,4,8\right)$, $\left(1,5,9\right)$, $\left(2,4,7\right)$, $\left(2,5,8\right)$, $\left(2,6,9\right)$, $\left(3,5,7\right)$, $\left(3,6,8\right)$, $\left(4,5,6\right)$, $\left(7,8,9\right)$. FigureΒ 5.254.1 shows the 9 trees and the 10 alignments corresponding to the example.

Symmetries
• Items of $\mathrm{\pi \pi \pi ΄\pi ΄\pi }$ are permutable.

• Attributes of $\mathrm{\pi \pi \pi ΄\pi ΄\pi }$ are permutable w.r.t. permutation $\left(\mathrm{\pi \pi \pi \pi \pi ‘}\right)$ $\left(\mathrm{\pi ‘},\mathrm{\pi ’}\right)$ (permutation applied to all items).

• One and the same constant can be added to the $\mathrm{\pi ‘}$ attribute of all items of $\mathrm{\pi \pi \pi ΄\pi ΄\pi }$.

• One and the same constant can be added to the $\mathrm{\pi ’}$ attribute of all items of $\mathrm{\pi \pi \pi ΄\pi ΄\pi }$.

Keywords
Arc input(s)

$\mathrm{\pi \pi \pi ΄\pi ΄\pi }$

Arc generator
$\mathrm{\pi Ά\pi Ώ\pi Ό\pi \pi \pi Έ}$$\left(<\right)\beta ¦\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi }\mathtt{1},\mathrm{\pi \pi \pi \pi \pi }\mathtt{2},\mathrm{\pi \pi \pi \pi \pi }\mathtt{3}\right)$

Arc arity
Arc constraint(s)
$\beta \left(\begin{array}{c}\mathrm{\pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi ‘}*\mathrm{\pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi ’}-\mathrm{\pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi ‘}*\mathrm{\pi \pi \pi \pi \pi }\mathtt{3}.\mathrm{\pi ’},\hfill \\ \mathrm{\pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi ’}*\mathrm{\pi \pi \pi \pi \pi }\mathtt{3}.\mathrm{\pi ‘}-\mathrm{\pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi ’}*\mathrm{\pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi ‘},\hfill \\ \mathrm{\pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi ‘}*\mathrm{\pi \pi \pi \pi \pi }\mathtt{3}.\mathrm{\pi ’}-\mathrm{\pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi ’}*\mathrm{\pi \pi \pi \pi \pi }\mathtt{3}.\mathrm{\pi ‘}\hfill \end{array}\right)=0$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$=\mathrm{\pi ½\pi \pi Ύ\pi }$

Graph model

The arc generator $\mathrm{\pi Ά\pi Ώ\pi Ό\pi \pi \pi Έ}\left(<\right)$ with an arity of three is used in order to generate all the arcs of the directed hypergraph. Each arc is an ordered triple of trees. We use the restriction $<$ in order to generate one single arc for each set of three trees. This is required, since otherwise we would count more than once a given alignment of three trees. The formula used within the arc constraint expresses the fact that the three points of respective coordinates $\left({\mathrm{\pi \pi \pi \pi \pi }}_{1}.\mathrm{\pi ‘},{\mathrm{\pi \pi \pi \pi \pi }}_{1}.\mathrm{\pi ’}\right)$, $\left({\mathrm{\pi \pi \pi \pi \pi }}_{2}.\mathrm{\pi ‘},{\mathrm{\pi \pi \pi \pi \pi }}_{2}.\mathrm{\pi ’}\right)$ and $\left({\mathrm{\pi \pi \pi \pi \pi }}_{3}.\mathrm{\pi ‘},{\mathrm{\pi \pi \pi \pi \pi }}_{3}.\mathrm{\pi ’}\right)$ are aligned. It corresponds to the development of the expression:

$\left|\begin{array}{ccc}{\mathrm{\pi \pi \pi \pi \pi }}_{1}.\mathrm{\pi ‘}& {\mathrm{\pi \pi \pi \pi \pi }}_{2}.\mathrm{\pi ’}& 1\\ {\mathrm{\pi \pi \pi \pi \pi }}_{2}.\mathrm{\pi ‘}& {\mathrm{\pi \pi \pi \pi \pi }}_{2}.\mathrm{\pi ’}& 1\\ {\mathrm{\pi \pi \pi \pi \pi }}_{3}.\mathrm{\pi ‘}& {\mathrm{\pi \pi \pi \pi \pi }}_{3}.\mathrm{\pi ’}& 1\end{array}\right|=0$