SolvRun
[2, 3, 2, 3], [4, 4, 1, 1]
=
[1, 1, 1, 1]
=
[2, 2, 2, 2]
POSSIBLE RANKS
1 x 4
2 x 2
BASE DETERMINANT
117/512, .2285156250
NullSpace of Δ
{1, 2, 3, 4}
Nullspace of A
[{1, 4},{2, 3}]
STRATIFIED CYCLE COVERS
1
0
v[2] v[3] + v[1] v[4]
v[2] v[4] v[3] + v[1] v[4] v[3] + v[1] v[2] v[4] + v[1] v[2] v[3]
2 v[1] v[2] v[4] v[3]
============================================================================
1, [1, 1, 1, 1]
"Coloring", {}
"R", [2, 3, 2, 3]
"B", [4, 4, 1, 1]
NOT SYNC'D
Kernel has RANK, 2,
N=
"R CYCLES", 1 + v[2] v[3]
"B CYCLES", 1 + v[1] v[4]
"Char Poly V"
(
)
CHECKS
"OM-R"
"CESARO CHECK", true
"OM-B"
"CESARO CHECK", true
"CHECKING OMEGA-R", true
"Kernel Check OMEGA-R", true
"RANKING B from RC"
1, "vs", 2
"RANKING B-NAT from RC"
1, "vs", 2
"RANKING B from FIRST ROW of RC"
1, "vs", 2
"CHECKING OMEGA-B", true
"Kernel Check OMEGA-B", true
"RANKING R from BC"
1, "vs", 2
"RANKING R-NAT from BC"
1, "vs", 2
"RANKING R from FIRST ROW of BC"
1, "vs", 2
NOT SYNC'D
"RANK of R is ", 2
"R ranking is ", 1, "vs", 2
=
[3, 2, 3, 2]
"RBAR ranking", 1, "vs", 2
"RANK of B is ", 2
"B ranking is ", 1, "vs", 2
=
[1, 1, 4, 4]
"BBAR ranking", 1, "vs", 2
=
[4, 1, 4, 1]
=
[3, 3, 2, 2]
"RNATBNAT ranking", 1, "vs", 2
"BNATRNAT ranking", 1, "vs", 2
"Centralizer "
[4, 3, 2, 1]
[1, 2, 3, 4]
"Char Poly Commutator"
"Min Poly Commutator"
CYCLES , 1 + v[1] v[4]
CYCLES , 1 + v[2] v[3]
=
[1, 4, 1, 4], (v[1] + 1) (v[4] + 1)
=
[2, 2, 3, 3], (v[2] + 1) (v[3] + 1)
"RNATB NAT RANK", 2
"BNATR NAT RANK", 2
"IDEMSOLVRANK", 2, "LOCAL TRACE", 0
"IDEMSOLVRANK", 2, "LOCAL TRACE", 0
"IDEMSOLVER", [1, 4, 1, 4]
"IDEMSOLVER", [2, 2, 3, 3]
"IDEMSOLVABLE?", false
"ABELIAN? "
false
"CMM = "
"ISIDEM?", false
RBSOLVRANK, 2, with index , 2
NOT SOLVABLE
"COLORING IS CC"
============================================================================
2, [1, -1, 1, 1]
"Coloring", {2}
"R", [2, 4, 2, 3]
"B", [4, 3, 1, 1]
SYNC'D
"R CYCLES", 1 + v[2] v[4] v[3]
"B CYCLES", 1 + v[1] v[4]
"Char Poly V"
(
)
CHECKS
"OM-R"
"CESARO CHECK", true
"OM-B"
"CESARO CHECK", true
"CHECKING OMEGA-R", true
"RANKING B from RC"
3, "vs", 3
"RANKING B-NAT from RC"
1, "vs", 2
"RANKING B from FIRST ROW of RC"
3, "vs", 3
"CHECKING OMEGA-B", true
"RANKING R from BC"
3, "vs", 3
"RANKING R-NAT from BC"
1, "vs", 3
"RANKING R from FIRST ROW of BC"
3, "vs", 3
SYNC'D
"RANK of R is ", 3
"R ranking is ", 3, "vs", 3
=
[3, 2, 3, 4]
"RBAR ranking", 3, "vs", 3
"RANK of B is ", 3
"B ranking is ", 2, "vs", 3
=
[1, 1, 4, 4]
"BBAR ranking", 1, "vs", 2
=
[4, 1, 4, 4]
=
[3, 3, 4, 4]
"RNATBNAT ranking", 2, "vs", 2
"BNATRNAT ranking", 2, "vs", 2
"Centralizer "
"Char Poly Commutator"
"Min Poly Commutator"
CYCLES , v[4] + 1
CYCLES , v[4] + 1
=
[1, 3, 1, 1], v[1] + 1
=
[2, 2, 3, 3], (v[2] + 1) (v[3] + 1)
"RNATB NAT RANK", 1
"BNATR NAT RANK", 2
"IDEMSOLVRANK", 1, "LOCAL TRACE", 1
"IDEMSOLVRANK", 1, "LOCAL TRACE", 1
"IDEMSOLVER", [4, 4, 4, 4]
"IDEMSOLVER", [4, 4, 4, 4]
"IDEMSOLVABLE?", true
"ABELIAN? "
false
"CMM = "
"ISIDEM?", false
RB-RANKED. KERNEL IS RANK ONE.
RBSOLVRANK, 1, with index , 2
RBSOLVABLE
============================================================================
3, [1, 1, -1, 1]
"Coloring", {3}
"R", [2, 3, 1, 3]
"B", [4, 4, 2, 1]
SYNC'D
"R CYCLES", 1 + v[1] v[2] v[3]
"B CYCLES", 1 + v[1] v[4]
"Char Poly V"
(
)
CHECKS
"OM-R"
"CESARO CHECK", true
"OM-B"
"CESARO CHECK", true
"CHECKING OMEGA-R", true
"RANKING B from RC"
3, "vs", 3
"RANKING B-NAT from RC"
1, "vs", 2
"RANKING B from FIRST ROW of RC"
3, "vs", 3
"CHECKING OMEGA-B", true
"RANKING R from BC"
3, "vs", 3
"RANKING R-NAT from BC"
1, "vs", 3
"RANKING R from FIRST ROW of BC"
3, "vs", 3
SYNC'D
"RANK of R is ", 3
"R ranking is ", 3, "vs", 3
=
[1, 2, 3, 2]
"RBAR ranking", 3, "vs", 3
"RANK of B is ", 3
"B ranking is ", 2, "vs", 3
=
[1, 1, 4, 4]
"BBAR ranking", 1, "vs", 2
=
[1, 1, 4, 1]
=
[1, 1, 2, 2]
"RNATBNAT ranking", 2, "vs", 2
"BNATRNAT ranking", 2, "vs", 2
"Centralizer "
"Char Poly Commutator"
"Min Poly Commutator"
CYCLES , v[1] + 1
CYCLES , v[1] + 1
=
[4, 4, 2, 4], v[4] + 1
=
[2, 2, 3, 3], (v[2] + 1) (v[3] + 1)
"RNATB NAT RANK", 1
"BNATR NAT RANK", 2
"IDEMSOLVRANK", 1, "LOCAL TRACE", 1
"IDEMSOLVRANK", 1, "LOCAL TRACE", 1
"IDEMSOLVER", [1, 1, 1, 1]
"IDEMSOLVER", [1, 1, 1, 1]
"IDEMSOLVABLE?", true
"ABELIAN? "
false
"CMM = "
"ISIDEM?", false
RB-RANKED. KERNEL IS RANK ONE.
RBSOLVRANK, 1, with index , 2
RBSOLVABLE
============================================================================
4, [1, 1, 1, -1]
"Coloring", {4}
"R", [2, 3, 2, 1]
"B", [4, 4, 1, 3]
SYNC'D
"R CYCLES", 1 + v[2] v[3]
"B CYCLES", 1 + v[1] v[4] v[3]
"Char Poly V"
(
)
CHECKS
"OM-R"
"CESARO CHECK", true
"OM-B"
"CESARO CHECK", true
"CHECKING OMEGA-R", true
"RANKING B from RC"
3, "vs", 3
"RANKING B-NAT from RC"
1, "vs", 3
"RANKING B from FIRST ROW of RC"
3, "vs", 3
"CHECKING OMEGA-B", true
"RANKING R from BC"
3, "vs", 3
"RANKING R-NAT from BC"
1, "vs", 2
"RANKING R from FIRST ROW of BC"
3, "vs", 3
SYNC'D
"RANK of R is ", 3
"R ranking is ", 2, "vs", 3
=
[3, 2, 3, 2]
"RBAR ranking", 1, "vs", 2
"RANK of B is ", 3
"B ranking is ", 3, "vs", 3
=
[1, 1, 3, 4]
"BBAR ranking", 3, "vs", 3
=
[3, 1, 3, 1]
=
[3, 3, 3, 2]
"RNATBNAT ranking", 2, "vs", 2
"BNATRNAT ranking", 2, "vs", 2
"Centralizer "
"Char Poly Commutator"
"Min Poly Commutator"
CYCLES , v[3] + 1
CYCLES , v[3] + 1
=
[1, 4, 1, 4], (v[1] + 1) (v[4] + 1)
=
[2, 2, 2, 1], v[2] + 1
"RNATB NAT RANK", 2
"BNATR NAT RANK", 1
"IDEMSOLVRANK", 1, "LOCAL TRACE", 1
"IDEMSOLVRANK", 1, "LOCAL TRACE", 1
"IDEMSOLVER", [3, 3, 3, 3]
"IDEMSOLVER", [3, 3, 3, 3]
"IDEMSOLVABLE?", true
"ABELIAN? "
false
"CMM = "
"ISIDEM?", false
RB-RANKED. KERNEL IS RANK ONE.
RBSOLVRANK, 1, with index , 2
RBSOLVABLE
============================================================================
5, [1, -1, -1, 1]
"Coloring", {2, 3}
"R", [2, 4, 1, 3]
"B", [4, 3, 2, 1]
NOT SYNC'D
Kernel has RANK, 4,
N=
"R CYCLES", 1 + v[1] v[2] v[4] v[3]
"B CYCLES", (1 + v[2] v[3]) (1 + v[1] v[4])
"Char Poly V"
(
)
(
)
"CYCLE CHECK B"
CHECKS
"OM-R"
"CESARO CHECK", true
"OM-B"
"CESARO CHECK", true
"CHECKING OMEGA-R", true
"Kernel Check OMEGA-R", true
"RANKING B from RC"
1, "vs", 4
"RANKING B-NAT from RC"
1, "vs", 4
"RANKING B from FIRST ROW of RC"
1, "vs", 4
"CHECKING OMEGA-B", true
"Kernel Check OMEGA-B", false
"RANKING R from BC"
2, "vs", 4
"RANKING R-NAT from BC"
2, "vs", 4
"RANKING R from FIRST ROW of BC"
2, "vs", 4
NOT SYNC'D
"RANK of R is ", 4
"R ranking is ", 1, "vs", 4
=
[1, 2, 3, 4]
"RBAR ranking", 1, "vs", 4
"RANK of B is ", 4
"B ranking is ", 1, "vs", 4
=
[1, 2, 3, 4]
"BBAR ranking", 1, "vs", 4
=
[1, 2, 3, 4]
=
[1, 2, 3, 4]
"RNATBNAT ranking", 1, "vs", 4
"BNATRNAT ranking", 1, "vs", 4
"Centralizer "
[1, 2, 3, 4]
[4, 3, 2, 1]
[3, 1, 4, 2]
[2, 4, 1, 3]
"Char Poly Commutator"
"Min Poly Commutator"
CYCLES ,
(v[1] + 1) (v[2] + 1) (v[3] + 1) (v[4] + 1)
CYCLES ,
(v[1] + 1) (v[2] + 1) (v[3] + 1) (v[4] + 1)
=
[4, 3, 2, 1], (1 + v[2] v[3]) (1 + v[1] v[4])
=
[2, 4, 1, 3], 1 + v[1] v[2] v[4] v[3]
"RNATB NAT RANK", 4
"BNATR NAT RANK", 4
"IDEMSOLVRANK", 4, "LOCAL TRACE", 4
"IDEMSOLVRANK", 4, "LOCAL TRACE", 4
"IDEMSOLVER", [1, 2, 3, 4]
"IDEMSOLVER", [1, 2, 3, 4]
"IDEMSOLVABLE?", true
"ABELIAN? "
true
"CMM = "
"ISIDEM?", true
RBSOLVRANK, 4, with index , 4
NOT SOLVABLE
============================================================================
6, [1, -1, 1, -1]
"Coloring", {2, 4}
"R", [2, 4, 2, 1]
"B", [4, 3, 1, 3]
SYNC'D
"R CYCLES", 1 + v[1] v[2] v[4]
"B CYCLES", 1 + v[1] v[4] v[3]
"Char Poly V"
(
)
CHECKS
"OM-R"
"CESARO CHECK", true
"OM-B"
"CESARO CHECK", true
"CHECKING OMEGA-R", true
"RANKING B from RC"
3, "vs", 3
"RANKING B-NAT from RC"
1, "vs", 3
"RANKING B from FIRST ROW of RC"
3, "vs", 3
"CHECKING OMEGA-B", true
"RANKING R from BC"
3, "vs", 3
"RANKING R-NAT from BC"
1, "vs", 3
"RANKING R from FIRST ROW of BC"
3, "vs", 3
SYNC'D
"RANK of R is ", 3
"R ranking is ", 3, "vs", 3
=
[1, 2, 1, 4]
"RBAR ranking", 3, "vs", 3
"RANK of B is ", 3
"B ranking is ", 3, "vs", 3
=
[1, 4, 3, 4]
"BBAR ranking", 3, "vs", 3
=
[1, 4, 1, 4]
=
[1, 4, 1, 4]
"RNATBNAT ranking", 1, "vs", 2
"BNATRNAT ranking", 1, "vs", 2
"Centralizer "
"Char Poly Commutator"
"Min Poly Commutator"
CYCLES , (v[1] + 1) (v[4] + 1)
CYCLES , (v[1] + 1) (v[4] + 1)
=
[4, 3, 4, 3], 1 + v[4] v[3]
=
[2, 1, 2, 1], 1 + v[1] v[2]
"RNATB NAT RANK", 2
"BNATR NAT RANK", 2
"IDEMSOLVRANK", 2, "LOCAL TRACE", 2
"IDEMSOLVRANK", 2, "LOCAL TRACE", 2
"IDEMSOLVER", [1, 4, 1, 4]
"IDEMSOLVER", [1, 4, 1, 4]
"IDEMSOLVABLE?", true
"ABELIAN? "
true
"CMM = "
"ISIDEM?", true
RB-RANKED. KERNEL IS RANK ONE.
RBSOLVRANK, 2, with index , 1
NOT SOLVABLE
============================================================================
7, [1, 1, -1, -1]
"Coloring", {3, 4}
"R", [2, 3, 1, 1]
"B", [4, 4, 2, 3]
SYNC'D
"R CYCLES", 1 + v[1] v[2] v[3]
"B CYCLES", 1 + v[2] v[4] v[3]
"Char Poly V"
(
)
CHECKS
"OM-R"
"CESARO CHECK", true
"OM-B"
"CESARO CHECK", true
"CHECKING OMEGA-R", true
"RANKING B from RC"
3, "vs", 3
"RANKING B-NAT from RC"
1, "vs", 3
"RANKING B from FIRST ROW of RC"
3, "vs", 3
"CHECKING OMEGA-B", true
"RANKING R from BC"
3, "vs", 3
"RANKING R-NAT from BC"
1, "vs", 3
"RANKING R from FIRST ROW of BC"
3, "vs", 3
SYNC'D
"RANK of R is ", 3
"R ranking is ", 3, "vs", 3
=
[1, 2, 3, 3]
"RBAR ranking", 3, "vs", 3
"RANK of B is ", 3
"B ranking is ", 3, "vs", 3
=
[2, 2, 3, 4]
"BBAR ranking", 3, "vs", 3
=
[2, 2, 3, 3]
=
[2, 2, 3, 3]
"RNATBNAT ranking", 1, "vs", 2
"BNATRNAT ranking", 1, "vs", 2
"Centralizer "
"Char Poly Commutator"
"Min Poly Commutator"
CYCLES , (v[2] + 1) (v[3] + 1)
CYCLES , (v[2] + 1) (v[3] + 1)
=
[4, 4, 2, 2], 1 + v[2] v[4]
=
[3, 3, 1, 1], 1 + v[1] v[3]
"RNATB NAT RANK", 2
"BNATR NAT RANK", 2
"IDEMSOLVRANK", 2, "LOCAL TRACE", 2
"IDEMSOLVRANK", 2, "LOCAL TRACE", 2
"IDEMSOLVER", [2, 2, 3, 3]
"IDEMSOLVER", [2, 2, 3, 3]
"IDEMSOLVABLE?", true
"ABELIAN? "
true
"CMM = "
"ISIDEM?", true
RB-RANKED. KERNEL IS RANK ONE.
RBSOLVRANK, 2, with index , 1
NOT SOLVABLE
============================================================================
8, [1, -1, -1, -1]
"Coloring", {2, 3, 4}
"R", [2, 4, 1, 1]
"B", [4, 3, 2, 3]
SYNC'D
"R CYCLES", 1 + v[1] v[2] v[4]
"B CYCLES", 1 + v[2] v[3]
"Char Poly V"
(
)
CHECKS
"OM-R"
"CESARO CHECK", true
"OM-B"
"CESARO CHECK", true
"CHECKING OMEGA-R", true
"RANKING B from RC"
3, "vs", 3
"RANKING B-NAT from RC"
1, "vs", 2
"RANKING B from FIRST ROW of RC"
3, "vs", 3
"CHECKING OMEGA-B", true
"RANKING R from BC"
3, "vs", 3
"RANKING R-NAT from BC"
1, "vs", 3
"RANKING R from FIRST ROW of BC"
3, "vs", 3
SYNC'D
"RANK of R is ", 3
"R ranking is ", 3, "vs", 3
=
[1, 2, 4, 4]
"RBAR ranking", 3, "vs", 3
"RANK of B is ", 3
"B ranking is ", 2, "vs", 3
=
[3, 2, 3, 2]
"BBAR ranking", 1, "vs", 2
=
[3, 2, 2, 2]
=
[4, 2, 4, 2]
"RNATBNAT ranking", 2, "vs", 2
"BNATRNAT ranking", 2, "vs", 2
"Centralizer "
"Char Poly Commutator"
"Min Poly Commutator"
CYCLES , v[2] + 1
CYCLES , v[2] + 1
=
[4, 3, 3, 3], v[3] + 1
=
[1, 4, 1, 4], (v[1] + 1) (v[4] + 1)
"RNATB NAT RANK", 1
"BNATR NAT RANK", 2
"IDEMSOLVRANK", 1, "LOCAL TRACE", 1
"IDEMSOLVRANK", 1, "LOCAL TRACE", 1
"IDEMSOLVER", [2, 2, 2, 2]
"IDEMSOLVER", [2, 2, 2, 2]
"IDEMSOLVABLE?", true
"ABELIAN? "
false
"CMM = "
"ISIDEM?", false
RB-RANKED. KERNEL IS RANK ONE.
RBSOLVRANK, 1, with index , 2
RBSOLVABLE
============================================================================
"SANDWICH SUMMARY", 0
"RG SUMMARY", 0
"SOLVSUMMARY", 4
2
.
{2}, rank:
1/1
3
.
{3}, rank:
1/1
4
.
{4}, rank:
1/1
8
.
{2, 3, 4}, rank:
1/1
"IDEMSOLVSUMMARY", 7
2
.
{2}, rank:
1/1
3
.
{3}, rank:
1/1
4
.
{4}, rank:
1/1
5
.
{2, 3}, rank:
4/1
6
.
{2, 4}, rank:
2/1
7
.
{3, 4}, rank:
2/1
8
.
{2, 3, 4}, rank:
1/1
"SUMMARY: NON-SYNCED CC", 1
{[1, {}, sw, 2]}