secondaryInvariants(List,PolynomialRing) -- the set of secondaryInvariants of a permutation group

Synopsis

Function: secondaryInvariants
Usage:

secondaryInvariants(gens,R)
Inputs:
- gens, a list, a list of generators of a permutation group H
- R, a polynomial ring, a polynomial ring
Optional inputs:
- AsExpression => ..., default value false, an optional argument that returns polynomials as expressions
- Robust => ..., default value false, an optional argument for specifying the algorithm for calculating higherSpechtPolynomials
Outputs:
- a hash table, the set of secondary invariants indexed by the representation in which they are found

Description

Let $R$ be a polynomial ring with $n$ generators. The secondary invariants of a group $H$ in $GL(n)$ are the set of generators of the ring of invariants $R^H$ as a $K[\theta_1,\ldots,\theta_n]$-module. For this algorithm we always take the primary invariants $\theta_1,\ldots,\theta_n$ to be the elementary symmetric polynomials $e_1,\ldots e_n$ so that the ring $K[\theta_1,\ldots,\theta_n]$ is the ring of symmetric polynomials.

The secondary invariants are obtained by considering the quotient ring $R/(e_1,\ldots,e_n)$. This quotient ring is called the coinvariant algebra of $S_n$. This quotient is isomorphic to the regular representation of $S_n$. In particular as a K-vector space it is finite dimensional. In this space we find the subspace that is invariant under the action of $H$. The secondary invariants correspond to a basis for this space.

The advantage of this algorithm is that it decomposes the regular representation into its irreducible representation by means of the higher Specht polynomials basis. This reduces significantly the dimension of the vector spaces in which the invariant spaces must be found.

To illustrate we calculate the secondary invariants for a subgroup of cardinality 24 in $S_6$.

i1 : R = QQ[x_1..x_6]

o1 = R

o1 : PolynomialRing

i2 : genList = {{1,2,3,0,5,4},{0,4,2,5,1,3}}

o2 = {{1, 2, 3, 0, 5, 4}, {0, 4, 2, 5, 1, 3}}

o2 : List

i3 : time seco = secondaryInvariants(genList,R);
 -- used 1.02706s (cpu); 0.815718s (thread); 0s (gc)
(Partition{6}, Ambient_Dimension, 1, Rank, 1)
(Partition{5, 1}, Ambient_Dimension, 5, Rank, 0)
(Partition{4, 2}, Ambient_Dimension, 9, Rank, 1)
(Partition{4, 1, 1}, Ambient_Dimension, 10, Rank, 0)
(Partition{3, 3}, Ambient_Dimension, 5, Rank, 1)
(Partition{3, 2, 1}, Ambient_Dimension, 16, Rank, 0)
(Partition{3, 1, 1, 1}, Ambient_Dimension, 10, Rank, 0)
(Partition{2, 2, 2}, Ambient_Dimension, 5, Rank, 1)
(Partition{2, 2, 1, 1}, Ambient_Dimension, 9, Rank, 1)
(Partition{2, 1, 1, 1, 1}, Ambient_Dimension, 5, Rank, 0)
(Partition{1, 1, 1, 1, 1, 1}, Ambient_Dimension, 1, Rank, 1)

i4 : seco#(new Partition from {2,2,2})

                                                      2 2 2       4 2   2     2   2 2     2 2     2   4   2   2   2     2 2   1 2 2       2 2   2     1   2 2     1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2     1   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2   2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1     2 2   1 2 2       2 2   2     1   2 2     1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2     1   2 2     1   2 2     2 2 2       4 2 2       2 2 2       2 2 2       4 2 2       2 2 2       1 2   2     1   2 2     2 2   2     1 2   2     2   2 2     1   2 2     1 2   2     2 2   2     1 2   2     1   2 2     2   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2   2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1     2 2   1 2     2   1   2   2   2 2     2   1 2     2   2   2   2   1   2   2   1 2     2   2 2     2   1 2     2   1   2   2   2   2   2   1   2   2   2     2 2   4     2 2   2     2 2   2     2 2   4     2 2   2     2 2
o4 = HashTable{{0, 1, 2, 3, 4, 5} => HashTable{0 => - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x }                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        }
                                                      3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 5   3 1 2 3 5   3 1 2 3 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 3 5   3 1 2 3 5   3 1 2 3 5   3 1 2 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 3 6   3 1 2 3 6   3 1 2 3 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 3 6   3 1 2 3 6   3 1 2 3 6   3 1 2 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6
                                                    2 3 2 2     4 2 3 2     2 2 2 3     4 3 2   2   2 2 3   2   2 3   2 2   2   3 2 2   2 2   3 2   4   2 3 2   2 2 2   3   4 2   2 3   2   2 2 3   1 3 2 2     2 2 3 2     1 2 2 3     2 3 2 2     1 2 3 2     1 3 2 2     1 3 2 2     1 2 3 2     2 2 3 2     1 2 2 3     2 2 2 3     1 2 2 3     2 3 2   2   1 2 3   2   1 3   2 2   1   3 2 2   1 2   3 2   2   2 3 2   1 3 2   2   2 2 3   2   1 3 2   2   2 3 2   2   1 2 3   2   1 2 3   2   1 3   2 2   1   3 2 2   2 3   2 2   1 3   2 2   2   3 2 2   1   3 2 2   2 2   3 2   1   2 3 2   1 2   3 2   1 2   3 2   1   2 3 2   2   2 3 2   1 2 2   3   2 2   2 3   1   2 2 3   1 2 2   3   2 2 2   3   1 2 2   3   1 2   2 3   2   2 2 3   1 2   2 3   2 2   2 3   1   2 2 3   1   2 2 3   1 3 2 2     2 2 3 2     1 2 2 3     2 3 2 2     1 2 3 2     1 3 2 2     1 3 2 2     1 2 3 2     2 2 3 2     1 2 2 3     2 2 2 3     1 2 2 3     1 3 2 2     1 2 3 2     2 3 2 2     1 3 2 2     2 2 3 2     1 2 3 2     1 3 2 2     2 3 2 2     1 3 2 2     1 2 3 2     2 2 3 2     1 2 3 2     2 2 2 3     4 2 2 3     2 2 2 3     2 2 2 3     4 2 2 3     2 2 2 3     2 3 2   2   1 2 3   2   1 3   2 2   1   3 2 2   1 2   3 2   2   2 3 2   1 3 2   2   2 2 3   2   1 3 2   2   2 3 2   2   1 2 3   2   1 2 3   2   1 3   2 2   1   3 2 2   2 3   2 2   1 3   2 2   2   3 2 2   1   3 2 2   2 2   3 2   1   2 3 2   1 2   3 2   1 2   3 2   1   2 3 2   2   2 3 2   1 3 2   2   1 2 3   2   2 3 2   2   1 3 2   2   2 2 3   2   1 2 3   2   1 3 2   2   2 3 2   2   1 3 2   2   1 2 3   2   2 2 3   2   1 2 3   2   2 3   2 2   2   3 2 2   4 3   2 2   2 3   2 2   4   3 2 2   2   3 2 2   2 3   2 2   4 3   2 2   2 3   2 2   2   3 2 2   4   3 2 2   2   3 2 2   1 2   3 2   1   2 3 2   2 2   3 2   1 2   3 2   2   2 3 2   1   2 3 2   1 2   3 2   2 2   3 2   1 2   3 2   1   2 3 2   2   2 3 2   1   2 3 2   1 2 2   3   2 2   2 3   1   2 2 3   1 2 2   3   2 2 2   3   1 2 2   3   1 2   2 3   2   2 2 3   1 2   2 3   2 2   2 3   1   2 2 3   1   2 2 3   2 2 2   3   4 2 2   3   2 2 2   3   2 2 2   3   4 2 2   3   2 2 2   3   1 2   2 3   1   2 2 3   2 2   2 3   1 2   2 3   2   2 2 3   1   2 2 3   1 2   2 3   2 2   2 3   1 2   2 3   1   2 2 3   2   2 2 3   1   2 2 3
               {0, 1, 2, 4, 3, 5} => HashTable{0 => -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  - -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x }
                                                    3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 5   3 1 2 3 5   3 1 2 3 5   3 1 2 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 3 5   3 1 2 3 5   3 1 2 3 5   3 1 2 3 5   3 1 2 3 5   3 1 2 3 5   3 1 2 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 3 5   3 1 2 3 5   3 1 2 3 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 3 6   3 1 2 3 6   3 1 2 3 6   3 1 2 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 3 6   3 1 2 3 6   3 1 2 3 6   3 1 2 3 6   3 1 2 3 6   3 1 2 3 6   3 1 2 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 3 6   3 1 2 3 6   3 1 2 3 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6
                                                      1 3 3 2       2 3 2 3       1 2 3 3       1 3 3   2     2 3   3 2     1   3 3 2     1 3 2   3     2 2 3   3     1 3   2 3     2   3 2 3     1 2   3 3     1   2 3 3     2 3 3     2   4 3   3   2   2   3 3   2   2 3     3 2   4   3   3 2   2     3 3 2   1 3 2     3   1 2 3     3   2 3   2   3   1   3 2   3   2 2   3   3   1   2 3   3   1 3     2 3   2   3   2 3   1     3 2 3   1 2     3 3   2   2   3 3   1     2 3 3   1 3 3 2       2 3 2 3       1 2 3 3       1 3 3   2     2 3   3 2     1   3 3 2     1 3 2   3     2 2 3   3     1 3   2 3     2   3 2 3     1 2   3 3     1   2 3 3     2 3 3 2       4 3 2 3       2 2 3 3       2 3 3 2       4 3 3 2       2 3 3 2       2 3 2 3       4 2 3 3       2 3 2 3       4 3 2 3       2 2 3 3       2 2 3 3       1 3 3   2     2 3   3 2     1   3 3 2     1 3 3   2     2 3 3   2     1 3 3   2     1 3   3 2     2   3 3 2     1 3   3 2     2 3   3 2     1   3 3 2     1   3 3 2     2 3 2   3     1 2 3   3     1 3   2 3     1   3 2 3     1 2   3 3     2   2 3 3     1 3 2   3     2 2 3   3     1 3 2   3     2 3 2   3     1 2 3   3     1 2 3   3     1 3   2 3     1   3 2 3     2 3   2 3     1 3   2 3     2   3 2 3     1   3 2 3     2 2   3 3     1   2 3 3     1 2   3 3     1 2   3 3     1   2 3 3     2   2 3 3     2 3 3     2   4 3   3   2   2   3 3   2   2 3     3 2   4   3   3 2   2     3 3 2   1 3 3     2   2 3   3   2   1   3 3   2   1 3 3     2   2 3 3     2   1 3 3     2   1 3   3   2   2   3 3   2   1 3   3   2   2 3   3   2   1   3 3   2   1   3 3   2   1 3     3 2   2   3   3 2   1     3 3 2   2 3     3 2   1   3   3 2   1 3     3 2   1 3     3 2   1   3   3 2   2   3   3 2   1     3 3 2   2     3 3 2   1     3 3 2   1 3 2     3   1 2 3     3   2 3   2   3   1   3 2   3   2 2   3   3   1   2 3   3   1 3     2 3   2   3   2 3   1     3 2 3   1 2     3 3   2   2   3 3   1     2 3 3   2 3 2     3   1 2 3     3   1 3   2   3   1   3 2   3   1 2   3   3   2   2 3   3   1 3 2     3   2 2 3     3   1 3 2     3   2 3 2     3   1 2 3     3   1 2 3     3   1 3   2   3   1   3 2   3   2 3   2   3   1 3   2   3   2   3 2   3   1   3 2   3   2 2   3   3   1   2 3   3   1 2   3   3   1 2   3   3   1   2 3   3   2   2 3   3   1 3     2 3   2   3   2 3   1     3 2 3   2 3     2 3   1   3   2 3   1 3     2 3   1 3     2 3   1   3   2 3   2   3   2 3   1     3 2 3   2     3 2 3   1     3 2 3   2 2     3 3   4   2   3 3   2     2 3 3   4 2     3 3   2   2   3 3   2 2     3 3   2 2     3 3   2   2   3 3   4   2   3 3   2     2 3 3   4     2 3 3   2     2 3 3
               {0, 2, 1, 3, 4, 5} => HashTable{0 => - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x }
                                                      3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6
                                                    1 3 2 2       2 2 3 2       1 2 2 3       2 3 2   2     1 2 3   2     1 3   2 2     1   3 2 2     1 2   3 2     2   2 3 2     1 2 2   3     2 2   2 3     1   2 2 3     1 3 2     2   1 2 3     2   2 3   2   2   1   3 2   2   2 2   3   2   1   2 3   2   1 3     2 2   2   3   2 2   1     3 2 2   1 2     3 2   2   2   3 2   1     2 3 2   2 2 2     3   4 2   2   3   2   2 2   3   2 2     2 3   4   2   2 3   2     2 2 3   1 3 2 2       2 2 3 2       1 2 2 3       2 3 2   2     1 2 3   2     1 3   2 2     1   3 2 2     1 2   3 2     2   2 3 2     1 2 2   3     2 2   2 3     1   2 2 3     2 3 2 2       4 2 3 2       2 2 2 3       4 3 2 2       2 2 3 2       2 3 2 2       2 3 2 2       2 2 3 2       4 2 3 2       2 2 2 3       4 2 2 3       2 2 2 3       1 3 2   2     2 2 3   2     1 3   2 2     2   3 2 2     1 2   3 2     1   2 3 2     2 3 2   2     1 2 3   2     1 3 2   2     1 3 2   2     1 2 3   2     2 2 3   2     2 3   2 2     1   3 2 2     1 3   2 2     1 3   2 2     1   3 2 2     2   3 2 2     1 2   3 2     1   2 3 2     2 2   3 2     1 2   3 2     2   2 3 2     1   2 3 2     1 2 2   3     2 2   2 3     1   2 2 3     1 2 2   3     2 2 2   3     1 2 2   3     1 2   2 3     2   2 2 3     1 2   2 3     2 2   2 3     1   2 2 3     1   2 2 3     1 3 2     2   1 2 3     2   2 3   2   2   1   3 2   2   2 2   3   2   1   2 3   2   1 3     2 2   2   3   2 2   1     3 2 2   1 2     3 2   2   2   3 2   1     2 3 2   1 3 2     2   2 2 3     2   1 3   2   2   2   3 2   2   1 2   3   2   1   2 3   2   2 3 2     2   1 2 3     2   1 3 2     2   1 3 2     2   1 2 3     2   2 2 3     2   2 3   2   2   1   3 2   2   1 3   2   2   1 3   2   2   1   3 2   2   2   3 2   2   1 2   3   2   1   2 3   2   2 2   3   2   1 2   3   2   2   2 3   2   1   2 3   2   2 3     2 2   4   3   2 2   2     3 2 2   4 3     2 2   2   3   2 2   2 3     2 2   2 3     2 2   2   3   2 2   4   3   2 2   2     3 2 2   4     3 2 2   2     3 2 2   1 2     3 2   2   2   3 2   1     2 3 2   2 2     3 2   1   2   3 2   1 2     3 2   1 2     3 2   1   2   3 2   2   2   3 2   1     2 3 2   2     2 3 2   1     2 3 2   2 2 2     3   4 2   2   3   2   2 2   3   2 2     2 3   4   2   2 3   2     2 2 3   1 2 2     3   2 2   2   3   1   2 2   3   1 2 2     3   2 2 2     3   1 2 2     3   1 2   2   3   2   2 2   3   1 2   2   3   2 2   2   3   1   2 2   3   1   2 2   3   1 2     2 3   2   2   2 3   1     2 2 3   2 2     2 3   1   2   2 3   1 2     2 3   1 2     2 3   1   2   2 3   2   2   2 3   1     2 2 3   2     2 2 3   1     2 2 3
               {0, 2, 1, 4, 3, 5} => HashTable{0 => -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  + -x x x x x  + -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x  - -x x x x x  + -x x x x x }
                                                    3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6
                                                      1 4 3 2 2     1 3 4 2 2     2 4 2 3 2     1 2 4 3 2     2 3 2 4 2     1 2 3 4 2     1 4 2 2 3     2 2 4 2 3     1 2 2 4 3     1 3 2 2 4     2 2 3 2 4     1 2 2 3 4     2 4 3 2   2   1 3 4 2   2   1 4 2 3   2   1 2 4 3   2   1 3 2 4   2   2 2 3 4   2   1 4 3   2 2   2 3 4   2 2   1 4   3 2 2   2   4 3 2 2   1 3   4 2 2   1   3 4 2 2   1 4 2   3 2   1 2 4   3 2   2 4   2 3 2   1   4 2 3 2   2 2   4 3 2   1   2 4 3 2   2 3 2   4 2   1 2 3   4 2   1 3   2 4 2   1   3 2 4 2   1 2   3 4 2   2   2 3 4 2   1 4 2 2   3   2 2 4 2   3   1 2 2 4   3   2 4 2   2 3   1 2 4   2 3   1 4   2 2 3   1   4 2 2 3   1 2   4 2 3   2   2 4 2 3   1 2 2   4 3   2 2   2 4 3   1   2 2 4 3   2 3 2 2   4   4 2 3 2   4   2 2 2 3   4   4 3 2   2 4   2 2 3   2 4   2 3   2 2 4   2   3 2 2 4   2 2   3 2 4   4   2 3 2 4   2 2 2   3 4   4 2   2 3 4   2   2 2 3 4   1 4 3 2 2     1 3 4 2 2     2 4 2 3 2     1 2 4 3 2     2 3 2 4 2     1 2 3 4 2     1 4 2 2 3     2 2 4 2 3     1 2 2 4 3     1 3 2 2 4     2 2 3 2 4     1 2 2 3 4     1 4 3 2 2     2 3 4 2 2     1 4 2 3 2     2 2 4 3 2     1 3 2 4 2     1 2 3 4 2     2 4 3 2 2     1 3 4 2 2     1 4 3 2 2     1 4 3 2 2     1 3 4 2 2     2 3 4 2 2     2 4 2 3 2     1 2 4 3 2     1 4 2 3 2     1 4 2 3 2     1 2 4 3 2     2 2 4 3 2     1 3 2 4 2     1 2 3 4 2     2 3 2 4 2     1 3 2 4 2     2 2 3 4 2     1 2 3 4 2     2 4 2 2 3     4 2 4 2 3     2 2 2 4 3     4 4 2 2 3     2 2 4 2 3     2 4 2 2 3     2 4 2 2 3     2 2 4 2 3     4 2 4 2 3     2 2 2 4 3     4 2 2 4 3     2 2 2 4 3     1 3 2 2 4     2 2 3 2 4     1 2 2 3 4     2 3 2 2 4     1 2 3 2 4     1 3 2 2 4     1 3 2 2 4     1 2 3 2 4     2 2 3 2 4     1 2 2 3 4     2 2 2 3 4     1 2 2 3 4     2 4 3 2   2   1 3 4 2   2   1 4 2 3   2   1 2 4 3   2   1 3 2 4   2   2 2 3 4   2   1 4 3   2 2   2 3 4   2 2   1 4   3 2 2   2   4 3 2 2   1 3   4 2 2   1   3 4 2 2   1 4 2   3 2   1 2 4   3 2   2 4   2 3 2   1   4 2 3 2   2 2   4 3 2   1   2 4 3 2   2 3 2   4 2   1 2 3   4 2   1 3   2 4 2   1   3 2 4 2   1 2   3 4 2   2   2 3 4 2   1 4 3 2   2   2 3 4 2   2   1 4 2 3   2   2 2 4 3   2   1 3 2 4   2   1 2 3 4   2   2 4 3 2   2   1 3 4 2   2   1 4 3 2   2   1 4 3 2   2   1 3 4 2   2   2 3 4 2   2   2 4 2 3   2   1 2 4 3   2   1 4 2 3   2   1 4 2 3   2   1 2 4 3   2   2 2 4 3   2   1 3 2 4   2   1 2 3 4   2   2 3 2 4   2   1 3 2 4   2   2 2 3 4   2   1 2 3 4   2   2 4 3   2 2   4 3 4   2 2   2 4   3 2 2   4   4 3 2 2   2 3   4 2 2   2   3 4 2 2   4 4 3   2 2   2 3 4   2 2   2 4 3   2 2   2 4 3   2 2   2 3 4   2 2   4 3 4   2 2   4 4   3 2 2   2   4 3 2 2   2 4   3 2 2   2 4   3 2 2   2   4 3 2 2   4   4 3 2 2   2 3   4 2 2   2   3 4 2 2   4 3   4 2 2   2 3   4 2 2   4   3 4 2 2   2   3 4 2 2   1 4 2   3 2   2 2 4   3 2   1 4   2 3 2   2   4 2 3 2   1 2   4 3 2   1   2 4 3 2   2 4 2   3 2   1 2 4   3 2   1 4 2   3 2   1 4 2   3 2   1 2 4   3 2   2 2 4   3 2   2 4   2 3 2   1   4 2 3 2   1 4   2 3 2   1 4   2 3 2   1   4 2 3 2   2   4 2 3 2   1 2   4 3 2   1   2 4 3 2   2 2   4 3 2   1 2   4 3 2   2   2 4 3 2   1   2 4 3 2   2 3 2   4 2   1 2 3   4 2   1 3   2 4 2   1   3 2 4 2   1 2   3 4 2   2   2 3 4 2   1 3 2   4 2   2 2 3   4 2   1 3 2   4 2   2 3 2   4 2   1 2 3   4 2   1 2 3   4 2   1 3   2 4 2   1   3 2 4 2   2 3   2 4 2   1 3   2 4 2   2   3 2 4 2   1   3 2 4 2   2 2   3 4 2   1   2 3 4 2   1 2   3 4 2   1 2   3 4 2   1   2 3 4 2   2   2 3 4 2   1 4 2 2   3   2 2 4 2   3   1 2 2 4   3   2 4 2   2 3   1 2 4   2 3   1 4   2 2 3   1   4 2 2 3   1 2   4 2 3   2   2 4 2 3   1 2 2   4 3   2 2   2 4 3   1   2 2 4 3   2 4 2 2   3   4 2 4 2   3   2 2 2 4   3   4 4 2 2   3   2 2 4 2   3   2 4 2 2   3   2 4 2 2   3   2 2 4 2   3   4 2 4 2   3   2 2 2 4   3   4 2 2 4   3   2 2 2 4   3   1 4 2   2 3   2 2 4   2 3   1 4   2 2 3   2   4 2 2 3   1 2   4 2 3   1   2 4 2 3   2 4 2   2 3   1 2 4   2 3   1 4 2   2 3   1 4 2   2 3   1 2 4   2 3   2 2 4   2 3   2 4   2 2 3   1   4 2 2 3   1 4   2 2 3   1 4   2 2 3   1   4 2 2 3   2   4 2 2 3  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  3 4   1 2   2 3 4   2   2 2 3 4   1 2   2 3 4   2 2   2 3 4   1   2 2 3 4   1   2 2 3 4
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-x x x x x  - -x x x x x }
                                                      3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 5   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 4 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 3 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 2 4 5 6   3 1 2 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6   3 1 3 4 5 6   3 2 3 4 5 6

o4 : HashTable

Ways to use this method: