The tensor package provides routines which calculate quantities that are specific to the Theory of Relativity.  Many of these quantities possess certain symmetries in the indices of their components.  For this reason, the tensor package provides and uses some specific indexing functions to implement these symmetries.  In some cases, the package also makes use of Maple symmetric and antisymmetric indexing functions.

The following indexing functions are implemented in the tensor package:

$\mathrm{with}\left(\mathrm{tensor}\right)\:$

$\mathrm{c1}≔\mathrm{array}\left(\mathrm{cf1}\,1..4\,1..4\,1..4\right)$

$\mathrm{c1}≔array\left(\mathrm{cf1}\,1..4\,1..4\,1..4\,\left[\left(1\,1\,1\right)={\mathrm{`?`}}_{1,1,1}\,\left(1\,1\,2\right)={\mathrm{`?`}}_{1,1,2}\,\left(1\,1\,3\right)={\mathrm{`?`}}_{1,1,3}\,\left(1\,1\,4\right)={\mathrm{`?`}}_{1,1,4}\,\left(1\,2\,1\right)={\mathrm{`?`}}_{1,2,1}\,\left(1\,2\,2\right)={\mathrm{`?`}}_{1,2,2}\,\left(1\,2\,3\right)={\mathrm{`?`}}_{1,2,3}\,\left(1\,2\,4\right)={\mathrm{`?`}}_{1,2,4}\,\left(1\,3\,1\right)={\mathrm{`?`}}_{1,3,1}\,\left(1\,3\,2\right)={\mathrm{`?`}}_{1,3,2}\,\left(1\,3\,3\right)={\mathrm{`?`}}_{1,3,3}\,\left(1\,3\,4\right)={\mathrm{`?`}}_{1,3,4}\,\left(1\,4\,1\right)={\mathrm{`?`}}_{1,4,1}\,\left(1\,4\,2\right)={\mathrm{`?`}}_{1,4,2}\,\left(1\,4\,3\right)={\mathrm{`?`}}_{1,4,3}\,\left(1\,4\,4\right)={\mathrm{`?`}}_{1,4,4}\,\left(2\,1\,1\right)={\mathrm{`?`}}_{1,2,1}\,\left(2\,1\,2\right)={\mathrm{`?`}}_{1,2,2}\,\left(2\,1\,3\right)={\mathrm{`?`}}_{1,2,3}\,\left(2\,1\,4\right)={\mathrm{`?`}}_{1,2,4}\,\left(2\,2\,1\right)={\mathrm{`?`}}_{2,2,1}\,\left(2\,2\,2\right)={\mathrm{`?`}}_{2,2,2}\,\left(2\,2\,3\right)={\mathrm{`?`}}_{2,2,3}\,\left(2\,2\,4\right)={\mathrm{`?`}}_{2,2,4}\,\left(2\,3\,1\right)={\mathrm{`?`}}_{2,3,1}\,\left(2\,3\,2\right)={\mathrm{`?`}}_{2,3,2}\,\left(2\,3\,3\right)={\mathrm{`?`}}_{2,3,3}\,\left(2\,3\,4\right)={\mathrm{`?`}}_{2,3,4}\,\left(2\,4\,1\right)={\mathrm{`?`}}_{2,4,1}\,\left(2\,4\,2\right)={\mathrm{`?`}}_{2,4,2}\,\left(2\,4\,3\right)={\mathrm{`?`}}_{2,4,3}\,\left(2\,4\,4\right)={\mathrm{`?`}}_{2,4,4}\,\left(3\,1\,1\right)={\mathrm{`?`}}_{1,3,1}\,\left(3\,1\,2\right)={\mathrm{`?`}}_{1,3,2}\,\left(3\,1\,3\right)={\mathrm{`?`}}_{1,3,3}\,\left(3\,1\,4\right)={\mathrm{`?`}}_{1,3,4}\,\left(3\,2\,1\right)={\mathrm{`?`}}_{2,3,1}\,\left(3\,2\,2\right)={\mathrm{`?`}}_{2,3,2}\,\left(3\,2\,3\right)={\mathrm{`?`}}_{2,3,3}\,\left(3\,2\,4\right)={\mathrm{`?`}}_{2,3,4}\,\left(3\,3\,1\right)={\mathrm{`?`}}_{3,3,1}\,\left(3\,3\,2\right)={\mathrm{`?`}}_{3,3,2}\,\left(3\,3\,3\right)={\mathrm{`?`}}_{3,3,3}\,\left(3\,3\,4\right)={\mathrm{`?`}}_{3,3,4}\,\left(3\,4\,1\right)={\mathrm{`?`}}_{3,4,1}\,\left(3\,4\,2\right)={\mathrm{`?`}}_{3,4,2}\,\left(3\,4\,3\right)={\mathrm{`?`}}_{3,4,3}\,\left(3\,4\,4\right)={\mathrm{`?`}}_{3,4,4}\,\left(4\,1\,1\right)={\mathrm{`?`}}_{1,4,1}\,\left(4\,1\,2\right)={\mathrm{`?`}}_{1,4,2}\,\left(4\,1\,3\right)={\mathrm{`?`}}_{1,4,3}\,\left(4\,1\,4\right)={\mathrm{`?`}}_{1,4,4}\,\left(4\,2\,1\right)={\mathrm{`?`}}_{2,4,1}\,\left(4\,2\,2\right)={\mathrm{`?`}}_{2,4,2}\,\left(4\,2\,3\right)={\mathrm{`?`}}_{2,4,3}\,\left(4\,2\,4\right)={\mathrm{`?`}}_{2,4,4}\,\left(4\,3\,1\right)={\mathrm{`?`}}_{3,4,1}\,\left(4\,3\,2\right)={\mathrm{`?`}}_{3,4,2}\,\left(4\,3\,3\right)={\mathrm{`?`}}_{3,4,3}\,\left(4\,3\,4\right)={\mathrm{`?`}}_{3,4,4}\,\left(4\,4\,1\right)={\mathrm{`?`}}_{4,4,1}\,\left(4\,4\,2\right)={\mathrm{`?`}}_{4,4,2}\,\left(4\,4\,3\right)={\mathrm{`?`}}_{4,4,3}\,\left(4\,4\,4\right)={\mathrm{`?`}}_{4,4,4}\right]\right)$

$\mathrm{c1}\left[1,2,1\right]≔\frac{m}{r^2}$

${\mathrm{c1}}_{1,2,1}≔\frac{m}{r^2}$

$\mathrm{c1}\left[2,1,1\right]$

$\frac{m}{r^2}$

$\mathrm{c2}≔\mathrm{array}\left(\mathrm{cf2}\,1..4\,1..4\,1..4\right)$

$\mathrm{c2}≔array\left(\mathrm{cf2}\,1..4\,1..4\,1..4\,\left[\left(1\,1\,1\right)={\mathrm{`?`}}_{1,1,1}\,\left(1\,1\,2\right)={\mathrm{`?`}}_{1,1,2}\,\left(1\,1\,3\right)={\mathrm{`?`}}_{1,1,3}\,\left(1\,1\,4\right)={\mathrm{`?`}}_{1,1,4}\,\left(1\,2\,1\right)={\mathrm{`?`}}_{1,1,2}\,\left(1\,2\,2\right)={\mathrm{`?`}}_{1,2,2}\,\left(1\,2\,3\right)={\mathrm{`?`}}_{1,2,3}\,\left(1\,2\,4\right)={\mathrm{`?`}}_{1,2,4}\,\left(1\,3\,1\right)={\mathrm{`?`}}_{1,1,3}\,\left(1\,3\,2\right)={\mathrm{`?`}}_{1,2,3}\,\left(1\,3\,3\right)={\mathrm{`?`}}_{1,3,3}\,\left(1\,3\,4\right)={\mathrm{`?`}}_{1,3,4}\,\left(1\,4\,1\right)={\mathrm{`?`}}_{1,1,4}\,\left(1\,4\,2\right)={\mathrm{`?`}}_{1,2,4}\,\left(1\,4\,3\right)={\mathrm{`?`}}_{1,3,4}\,\left(1\,4\,4\right)={\mathrm{`?`}}_{1,4,4}\,\left(2\,1\,1\right)={\mathrm{`?`}}_{2,1,1}\,\left(2\,1\,2\right)={\mathrm{`?`}}_{2,1,2}\,\left(2\,1\,3\right)={\mathrm{`?`}}_{2,1,3}\,\left(2\,1\,4\right)={\mathrm{`?`}}_{2,1,4}\,\left(2\,2\,1\right)={\mathrm{`?`}}_{2,1,2}\,\left(2\,2\,2\right)={\mathrm{`?`}}_{2,2,2}\,\left(2\,2\,3\right)={\mathrm{`?`}}_{2,2,3}\,\left(2\,2\,4\right)={\mathrm{`?`}}_{2,2,4}\,\left(2\,3\,1\right)={\mathrm{`?`}}_{2,1,3}\,\left(2\,3\,2\right)={\mathrm{`?`}}_{2,2,3}\,\left(2\,3\,3\right)={\mathrm{`?`}}_{2,3,3}\,\left(2\,3\,4\right)={\mathrm{`?`}}_{2,3,4}\,\left(2\,4\,1\right)={\mathrm{`?`}}_{2,1,4}\,\left(2\,4\,2\right)={\mathrm{`?`}}_{2,2,4}\,\left(2\,4\,3\right)={\mathrm{`?`}}_{2,3,4}\,\left(2\,4\,4\right)={\mathrm{`?`}}_{2,4,4}\,\left(3\,1\,1\right)={\mathrm{`?`}}_{3,1,1}\,\left(3\,1\,2\right)={\mathrm{`?`}}_{3,1,2}\,\left(3\,1\,3\right)={\mathrm{`?`}}_{3,1,3}\,\left(3\,1\,4\right)={\mathrm{`?`}}_{3,1,4}\,\left(3\,2\,1\right)={\mathrm{`?`}}_{3,1,2}\,\left(3\,2\,2\right)={\mathrm{`?`}}_{3,2,2}\,\left(3\,2\,3\right)={\mathrm{`?`}}_{3,2,3}\,\left(3\,2\,4\right)={\mathrm{`?`}}_{3,2,4}\,\left(3\,3\,1\right)={\mathrm{`?`}}_{3,1,3}\,\left(3\,3\,2\right)={\mathrm{`?`}}_{3,2,3}\,\left(3\,3\,3\right)={\mathrm{`?`}}_{3,3,3}\,\left(3\,3\,4\right)={\mathrm{`?`}}_{3,3,4}\,\left(3\,4\,1\right)={\mathrm{`?`}}_{3,1,4}\,\left(3\,4\,2\right)={\mathrm{`?`}}_{3,2,4}\,\left(3\,4\,3\right)={\mathrm{`?`}}_{3,3,4}\,\left(3\,4\,4\right)={\mathrm{`?`}}_{3,4,4}\,\left(4\,1\,1\right)={\mathrm{`?`}}_{4,1,1}\,\left(4\,1\,2\right)={\mathrm{`?`}}_{4,1,2}\,\left(4\,1\,3\right)={\mathrm{`?`}}_{4,1,3}\,\left(4\,1\,4\right)={\mathrm{`?`}}_{4,1,4}\,\left(4\,2\,1\right)={\mathrm{`?`}}_{4,1,2}\,\left(4\,2\,2\right)={\mathrm{`?`}}_{4,2,2}\,\left(4\,2\,3\right)={\mathrm{`?`}}_{4,2,3}\,\left(4\,2\,4\right)={\mathrm{`?`}}_{4,2,4}\,\left(4\,3\,1\right)={\mathrm{`?`}}_{4,1,3}\,\left(4\,3\,2\right)={\mathrm{`?`}}_{4,2,3}\,\left(4\,3\,3\right)={\mathrm{`?`}}_{4,3,3}\,\left(4\,3\,4\right)={\mathrm{`?`}}_{4,3,4}\,\left(4\,4\,1\right)={\mathrm{`?`}}_{4,1,4}\,\left(4\,4\,2\right)={\mathrm{`?`}}_{4,2,4}\,\left(4\,4\,3\right)={\mathrm{`?`}}_{4,3,4}\,\left(4\,4\,4\right)={\mathrm{`?`}}_{4,4,4}\right]\right)$

$\mathrm{c2}\left[4,3,4\right]≔\frac{\cos \left(\mathrm{\theta }\right)}{\sin \left(\mathrm{\theta }\right)}$

${\mathrm{c2}}_{4,3,4}≔\frac{\cos \left(\mathrm{\theta }\right)}{\sin \left(\mathrm{\theta }\right)}$

$\mathrm{c2}\left[4,4,3\right]$

$\frac{\cos \left(\mathrm{\theta }\right)}{\sin \left(\mathrm{\theta }\right)}$

$R≔\mathrm{array}\left(\mathrm{cov\_riemann}\,1..4\,1..4\,1..4\,1..4\right)$

$R≔array\left(\mathrm{cov\_riemann}\,1..4\,1..4\,1..4\,1..4\,\left[\left(1\,1\,1\,1\right)=0\,\left(1\,1\,1\,2\right)=0\,\left(1\,1\,1\,3\right)=0\,\left(1\,1\,1\,4\right)=0\,\left(1\,1\,2\,1\right)=0\,\left(1\,1\,2\,2\right)=0\,\left(1\,1\,2\,3\right)=0\,\left(1\,1\,2\,4\right)=0\,\left(1\,1\,3\,1\right)=0\,\left(1\,1\,3\,2\right)=0\,\left(1\,1\,3\,3\right)=0\,\left(1\,1\,3\,4\right)=0\,\left(1\,1\,4\,1\right)=0\,\left(1\,1\,4\,2\right)=0\,\left(1\,1\,4\,3\right)=0\,\left(1\,1\,4\,4\right)=0\,\left(1\,2\,1\,1\right)=0\,\left(1\,2\,1\,2\right)={\mathrm{`?`}}_{1,2,1,2}\,\left(1\,2\,1\,3\right)={\mathrm{`?`}}_{1,2,1,3}\,\left(1\,2\,1\,4\right)={\mathrm{`?`}}_{1,2,1,4}\,\left(1\,2\,2\,1\right)=-{\mathrm{`?`}}_{1,2,1,2}\,\left(1\,2\,2\,2\right)=0\,\left(1\,2\,2\,3\right)={\mathrm{`?`}}_{1,2,2,3}\,\left(1\,2\,2\,4\right)={\mathrm{`?`}}_{1,2,2,4}\,\left(1\,2\,3\,1\right)=-{\mathrm{`?`}}_{1,2,1,3}\,\left(1\,2\,3\,2\right)=-{\mathrm{`?`}}_{1,2,2,3}\,\left(1\,2\,3\,3\right)=0\,\left(1\,2\,3\,4\right)={\mathrm{`?`}}_{1,2,3,4}\,\left(1\,2\,4\,1\right)=-{\mathrm{`?`}}_{1,2,1,4}\,\left(1\,2\,4\,2\right)=-{\mathrm{`?`}}_{1,2,2,4}\,\left(1\,2\,4\,3\right)=-{\mathrm{`?`}}_{1,2,3,4}\,\left(1\,2\,4\,4\right)=0\,\left(1\,3\,1\,1\right)=0\,\left(1\,3\,1\,2\right)={\mathrm{`?`}}_{1,2,1,3}\,\left(1\,3\,1\,3\right)={\mathrm{`?`}}_{1,3,1,3}\,\left(1\,3\,1\,4\right)={\mathrm{`?`}}_{1,3,1,4}\,\left(1\,3\,2\,1\right)=-{\mathrm{`?`}}_{1,2,1,3}\,\left(1\,3\,2\,2\right)=0\,\left(1\,3\,2\,3\right)={\mathrm{`?`}}_{1,3,2,3}\,\left(1\,3\,2\,4\right)={\mathrm{`?`}}_{1,3,2,4}\,\left(1\,3\,3\,1\right)=-{\mathrm{`?`}}_{1,3,1,3}\,\left(1\,3\,3\,2\right)=-{\mathrm{`?`}}_{1,3,2,3}\,\left(1\,3\,3\,3\right)=0\,\left(1\,3\,3\,4\right)={\mathrm{`?`}}_{1,3,3,4}\,\left(1\,3\,4\,1\right)=-{\mathrm{`?`}}_{1,3,1,4}\,\left(1\,3\,4\,2\right)=-{\mathrm{`?`}}_{1,3,2,4}\,\left(1\,3\,4\,3\right)=-{\mathrm{`?`}}_{1,3,3,4}\,\left(1\,3\,4\,4\right)=0\,\left(1\,4\,1\,1\right)=0\,\left(1\,4\,1\,2\right)={\mathrm{`?`}}_{1,2,1,4}\,\left(1\,4\,1\,3\right)={\mathrm{`?`}}_{1,3,1,4}\,\left(1\,4\,1\,4\right)={\mathrm{`?`}}_{1,4,1,4}\,\left(1\,4\,2\,1\right)=-{\mathrm{`?`}}_{1,2,1,4}\,\left(1\,4\,2\,2\right)=0\,\left(1\,4\,2\,3\right)={\mathrm{`?`}}_{1,4,2,3}\,\left(1\,4\,2\,4\right)={\mathrm{`?`}}_{1,4,2,4}\,\left(1\,4\,3\,1\right)=-{\mathrm{`?`}}_{1,3,1,4}\,\left(1\,4\,3\,2\right)=-{\mathrm{`?`}}_{1,4,2,3}\,\left(1\,4\,3\,3\right)=0\,\left(1\,4\,3\,4\right)={\mathrm{`?`}}_{1,4,3,4}\,\left(1\,4\,4\,1\right)=-{\mathrm{`?`}}_{1,4,1,4}\,\left(1\,4\,4\,2\right)=-{\mathrm{`?`}}_{1,4,2,4}\,\left(1\,4\,4\,3\right)=-{\mathrm{`?`}}_{1,4,3,4}\,\left(1\,4\,4\,4\right)=0\,\left(2\,1\,1\,1\right)=0\,\left(2\,1\,1\,2\right)=-{\mathrm{`?`}}_{1,2,1,2}\,\left(2\,1\,1\,3\right)=-{\mathrm{`?`}}_{1,2,1,3}\,\left(2\,1\,1\,4\right)=-{\mathrm{`?`}}_{1,2,1,4}\,\left(2\,1\,2\,1\right)={\mathrm{`?`}}_{1,2,1,2}\,\left(2\,1\,2\,2\right)=0\,\left(2\,1\,2\,3\right)=-{\mathrm{`?`}}_{1,2,2,3}\,\left(2\,1\,2\,4\right)=-{\mathrm{`?`}}_{1,2,2,4}\,\left(2\,1\,3\,1\right)={\mathrm{`?`}}_{1,2,1,3}\,\left(2\,1\,3\,2\right)={\mathrm{`?`}}_{1,2,2,3}\,\left(2\,1\,3\,3\right)=0\,\left(2\,1\,3\,4\right)=-{\mathrm{`?`}}_{1,2,3,4}\,\left(2\,1\,4\,1\right)={\mathrm{`?`}}_{1,2,1,4}\,\left(2\,1\,4\,2\right)={\mathrm{`?`}}_{1,2,2,4}\,\left(2\,1\,4\,3\right)={\mathrm{`?`}}_{1,2,3,4}\,\left(2\,1\,4\,4\right)=0\,\left(2\,2\,1\,1\right)=0\,\left(2\,2\,1\,2\right)=0\,\left(2\,2\,1\,3\right)=0\,\left(2\,2\,1\,4\right)=0\,\left(2\,2\,2\,1\right)=0\,\left(2\,2\,2\,2\right)=0\,\left(2\,2\,2\,3\right)=0\,\left(2\,2\,2\,4\right)=0\,\left(2\,2\,3\,1\right)=0\,\left(2\,2\,3\,2\right)=0\,\left(2\,2\,3\,3\right)=0\,\left(2\,2\,3\,4\right)=0\,\left(2\,2\,4\,1\right)=0\,\left(2\,2\,4\,2\right)=0\,\left(2\,2\,4\,3\right)=0\,\left(2\,2\,4\,4\right)=0\,\left(2\,3\,1\,1\right)=0\,\left(2\,3\,1\,2\right)={\mathrm{`?`}}_{1,2,2,3}\,\left(2\,3\,1\,3\right)={\mathrm{`?`}}_{1,3,2,3}\,\left(2\,3\,1\,4\right)={\mathrm{`?`}}_{1,4,2,3}\,\left(2\,3\,2\,1\right)=-{\mathrm{`?`}}_{1,2,2,3}\,\left(2\,3\,2\,2\right)=0\,\left(2\,3\,2\,3\right)={\mathrm{`?`}}_{2,3,2,3}\,\left(2\,3\,2\,4\right)={\mathrm{`?`}}_{2,3,2,4}\,\left(2\,3\,3\,1\right)=-{\mathrm{`?`}}_{1,3,2,3}\,\left(2\,3\,3\,2\right)=-{\mathrm{`?`}}_{2,3,2,3}\,\left(2\,3\,3\,3\right)=0\,\left(2\,3\,3\,4\right)={\mathrm{`?`}}_{2,3,3,4}\,\left(2\,3\,4\,1\right)=-{\mathrm{`?`}}_{1,4,2,3}\,\left(2\,3\,4\,2\right)=-{\mathrm{`?`}}_{2,3,2,4}\,\left(2\,3\,4\,3\right)=-{\mathrm{`?`}}_{2,3,3,4}\,\left(2\,3\,4\,4\right)=0\,\left(2\,4\,1\,1\right)=0\,\left(2\,4\,1\,2\right)={\mathrm{`?`}}_{1,2,2,4}\,\left(2\,4\,1\,3\right)={\mathrm{`?`}}_{1,3,2,4}\,\left(2\,4\,1\,4\right)={\mathrm{`?`}}_{1,4,2,4}\,\left(2\,4\,2\,1\right)=-{\mathrm{`?`}}_{1,2,2,4}\,\left(2\,4\,2\,2\right)=0\,\left(2\,4\,2\,3\right)={\mathrm{`?`}}_{2,3,2,4}\,\left(2\,4\,2\,4\right)={\mathrm{`?`}}_{2,4,2,4}\,\left(2\,4\,3\,1\right)=-{\mathrm{`?`}}_{1,3,2,4}\,\left(2\,4\,3\,2\right)=-{\mathrm{`?`}}_{2,3,2,4}\,\left(2\,4\,3\,3\right)=0\,\left(2\,4\,3\,4\right)={\mathrm{`?`}}_{2,4,3,4}\,\left(2\,4\,4\,1\right)=-{\mathrm{`?`}}_{1,4,2,4}\,\left(2\,4\,4\,2\right)=-{\mathrm{`?`}}_{2,4,2,4}\,\left(2\,4\,4\,3\right)=-{\mathrm{`?`}}_{2,4,3,4}\,\left(2\,4\,4\,4\right)=0\,\left(3\,1\,1\,1\right)=0\,\left(3\,1\,1\,2\right)=-{\mathrm{`?`}}_{1,2,1,3}\,\left(3\,1\,1\,3\right)=-{\mathrm{`?`}}_{1,3,1,3}\,\left(3\,1\,1\,4\right)=-{\mathrm{`?`}}_{1,3,1,4}\,\left(3\,1\,2\,1\right)={\mathrm{`?`}}_{1,2,1,3}\,\left(3\,1\,2\,2\right)=0\,\left(3\,1\,2\,3\right)=-{\mathrm{`?`}}_{1,3,2,3}\,\left(3\,1\,2\,4\right)=-{\mathrm{`?`}}_{1,3,2,4}\,\left(3\,1\,3\,1\right)={\mathrm{`?`}}_{1,3,1,3}\,\left(3\,1\,3\,2\right)={\mathrm{`?`}}_{1,3,2,3}\,\left(3\,1\,3\,3\right)=0\,\left(3\,1\,3\,4\right)=-{\mathrm{`?`}}_{1,3,3,4}\,\left(3\,1\,4\,1\right)={\mathrm{`?`}}_{1,3,1,4}\,\left(3\,1\,4\,2\right)={\mathrm{`?`}}_{1,3,2,4}\,\left(3\,1\,4\,3\right)={\mathrm{`?`}}_{1,3,3,4}\,\left(3\,1\,4\,4\right)=0\,\left(3\,2\,1\,1\right)=0\,\left(3\,2\,1\,2\right)=-{\mathrm{`?`}}_{1,2,2,3}\,\left(3\,2\,1\,3\right)=-{\mathrm{`?`}}_{1,3,2,3}\,\left(3\,2\,1\,4\right)=-{\mathrm{`?`}}_{1,4,2,3}\,\left(3\,2\,2\,1\right)={\mathrm{`?`}}_{1,2,2,3}\,\left(3\,2\,2\,2\right)=0\,\left(3\,2\,2\,3\right)=-{\mathrm{`?`}}_{2,3,2,3}\,\left(3\,2\,2\,4\right)=-{\mathrm{`?`}}_{2,3,2,4}\,\left(3\,2\,3\,1\right)={\mathrm{`?`}}_{1,3,2,3}\,\left(3\,2\,3\,2\right)={\mathrm{`?`}}_{2,3,2,3}\,\left(3\,2\,3\,3\right)=0\,\left(3\,2\,3\,4\right)=-{\mathrm{`?`}}_{2,3,3,4}\,\left(3\,2\,4\,1\right)={\mathrm{`?`}}_{1,4,2,3}\,\left(3\,2\,4\,2\right)={\mathrm{`?`}}_{2,3,2,4}\,\left(3\,2\,4\,3\right)={\mathrm{`?`}}_{2,3,3,4}\,\left(3\,2\,4\,4\right)=0\,\left(3\,3\,1\,1\right)=0\,\left(3\,3\,1\,2\right)=0\,\left(3\,3\,1\,3\right)=0\,\left(3\,3\,1\,4\right)=0\,\left(3\,3\,2\,1\right)=0\,\left(3\,3\,2\,2\right)=0\,\left(3\,3\,2\,3\right)=0\,\left(3\,3\,2\,4\right)=0\,\left(3\,3\,3\,1\right)=0\,\left(3\,3\,3\,2\right)=0\,\left(3\,3\,3\,3\right)=0\,\left(3\,3\,3\,4\right)=0\,\left(3\,3\,4\,1\right)=0\,\left(3\,3\,4\,2\right)=0\,\left(3\,3\,4\,3\right)=0\,\left(3\,3\,4\,4\right)=0\,\left(3\,4\,1\,1\right)=0\,\left(3\,4\,1\,2\right)={\mathrm{`?`}}_{1,2,3,4}\,\left(3\,4\,1\,3\right)={\mathrm{`?`}}_{1,3,3,4}\,\left(3\,4\,1\,4\right)={\mathrm{`?`}}_{1,4,3,4}\,\left(3\,4\,2\,1\right)=-{\mathrm{`?`}}_{1,2,3,4}\,\left(3\,4\,2\,2\right)=0\,\left(3\,4\,2\,3\right)={\mathrm{`?`}}_{2,3,3,4}\,\left(3\,4\,2\,4\right)={\mathrm{`?`}}_{2,4,3,4}\,\left(3\,4\,3\,1\right)=-{\mathrm{`?`}}_{1,3,3,4}\,\left(3\,4\,3\,2\right)=-{\mathrm{`?`}}_{2,3,3,4}\,\left(3\,4\,3\,3\right)=0\,\left(3\,4\,3\,4\right)={\mathrm{`?`}}_{3,4,3,4}\,\left(3\,4\,4\,1\right)=-{\mathrm{`?`}}_{1,4,3,4}\,\left(3\,4\,4\,2\right)=-{\mathrm{`?`}}_{2,4,3,4}\,\left(3\,4\,4\,3\right)=-{\mathrm{`?`}}_{3,4,3,4}\,\left(3\,4\,4\,4\right)=0\,\left(4\,1\,1\,1\right)=0\,\left(4\,1\,1\,2\right)=-{\mathrm{`?`}}_{1,2,1,4}\,\left(4\,1\,1\,3\right)=-{\mathrm{`?`}}_{1,3,1,4}\,\left(4\,1\,1\,4\right)=-{\mathrm{`?`}}_{1,4,1,4}\,\left(4\,1\,2\,1\right)={\mathrm{`?`}}_{1,2,1,4}\,\left(4\,1\,2\,2\right)=0\,\left(4\,1\,2\,3\right)=-{\mathrm{`?`}}_{1,4,2,3}\,\left(4\,1\,2\,4\right)=-{\mathrm{`?`}}_{1,4,2,4}\,\left(4\,1\,3\,1\right)={\mathrm{`?`}}_{1,3,1,4}\,\left(4\,1\,3\,2\right)={\mathrm{`?`}}_{1,4,2,3}\,\left(4\,1\,3\,3\right)=0\,\left(4\,1\,3\,4\right)=-{\mathrm{`?`}}_{1,4,3,4}\,\left(4\,1\,4\,1\right)={\mathrm{`?`}}_{1,4,1,4}\,\left(4\,1\,4\,2\right)={\mathrm{`?`}}_{1,4,2,4}\,\left(4\,1\,4\,3\right)={\mathrm{`?`}}_{1,4,3,4}\,\left(4\,1\,4\,4\right)=0\,\left(4\,2\,1\,1\right)=0\,\left(4\,2\,1\,2\right)=-{\mathrm{`?`}}_{1,2,2,4}\,\left(4\,2\,1\,3\right)=-{\mathrm{`?`}}_{1,3,2,4}\,\left(4\,2\,1\,4\right)=-{\mathrm{`?`}}_{1,4,2,4}\,\left(4\,2\,2\,1\right)={\mathrm{`?`}}_{1,2,2,4}\,\left(4\,2\,2\,2\right)=0\,\left(4\,2\,2\,3\right)=-{\mathrm{`?`}}_{2,3,2,4}\,\left(4\,2\,2\,4\right)=-{\mathrm{`?`}}_{2,4,2,4}\,\left(4\,2\,3\,1\right)={\mathrm{`?`}}_{1,3,2,4}\,\left(4\,2\,3\,2\right)={\mathrm{`?`}}_{2,3,2,4}\,\left(4\,2\,3\,3\right)=0\,\left(4\,2\,3\,4\right)=-{\mathrm{`?`}}_{2,4,3,4}\,\left(4\,2\,4\,1\right)={\mathrm{`?`}}_{1,4,2,4}\,\left(4\,2\,4\,2\right)={\mathrm{`?`}}_{2,4,2,4}\,\left(4\,2\,4\,3\right)={\mathrm{`?`}}_{2,4,3,4}\,\left(4\,2\,4\,4\right)=0\,\left(4\,3\,1\,1\right)=0\,\left(4\,3\,1\,2\right)=-{\mathrm{`?`}}_{1,2,3,4}\,\left(4\,3\,1\,3\right)=-{\mathrm{`?`}}_{1,3,3,4}\,\left(4\,3\,1\,4\right)=-{\mathrm{`?`}}_{1,4,3,4}\,\left(4\,3\,2\,1\right)={\mathrm{`?`}}_{1,2,3,4}\,\left(4\,3\,2\,2\right)=0\,\left(4\,3\,2\,3\right)=-{\mathrm{`?`}}_{2,3,3,4}\,\left(4\,3\,2\,4\right)=-{\mathrm{`?`}}_{2,4,3,4}\,\left(4\,3\,3\,1\right)={\mathrm{`?`}}_{1,3,3,4}\,\left(4\,3\,3\,2\right)={\mathrm{`?`}}_{2,3,3,4}\,\left(4\,3\,3\,3\right)=0\,\left(4\,3\,3\,4\right)=-{\mathrm{`?`}}_{3,4,3,4}\,\left(4\,3\,4\,1\right)={\mathrm{`?`}}_{1,4,3,4}\,\left(4\,3\,4\,2\right)={\mathrm{`?`}}_{2,4,3,4}\,\left(4\,3\,4\,3\right)={\mathrm{`?`}}_{3,4,3,4}\,\left(4\,3\,4\,4\right)=0\,\left(4\,4\,1\,1\right)=0\,\left(4\,4\,1\,2\right)=0\,\left(4\,4\,1\,3\right)=0\,\left(4\,4\,1\,4\right)=0\,\left(4\,4\,2\,1\right)=0\,\left(4\,4\,2\,2\right)=0\,\left(4\,4\,2\,3\right)=0\,\left(4\,4\,2\,4\right)=0\,\left(4\,4\,3\,1\right)=0\,\left(4\,4\,3\,2\right)=0\,\left(4\,4\,3\,3\right)=0\,\left(4\,4\,3\,4\right)=0\,\left(4\,4\,4\,1\right)=0\,\left(4\,4\,4\,2\right)=0\,\left(4\,4\,4\,3\right)=0\,\left(4\,4\,4\,4\right)=0\right]\right)$