Identifier
-
Mp00209:
Permutations
—pattern poset⟶
Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001571: Integer partitions ⟶ ℤ
Values
[1] => ([],1) => [1] => [1] => 1
[1,2] => ([(0,1)],2) => [2] => [1,1] => 1
[2,1] => ([(0,1)],2) => [2] => [1,1] => 1
[1,2,3] => ([(0,2),(2,1)],3) => [3] => [1,1,1] => 1
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => [3,1] => [2,1,1] => 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => [3,1] => [2,1,1] => 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => [3,1] => [2,1,1] => 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => [3,1] => [2,1,1] => 1
[3,2,1] => ([(0,2),(2,1)],3) => [3] => [1,1,1] => 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => [4] => [1,1,1,1] => 1
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [4,2] => [2,2,1,1] => 1
[1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => [4,2,1] => [3,2,1,1] => 1
[1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => [4,2,1] => [3,2,1,1] => 1
[1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => [4,2,1] => [3,2,1,1] => 1
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [4,2] => [2,2,1,1] => 1
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [4,2] => [2,2,1,1] => 1
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => [4,2] => [2,2,1,1] => 1
[2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => [4,2,1] => [3,2,1,1] => 1
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [4,2] => [2,2,1,1] => 1
[2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8) => [4,2,1,1] => [4,2,1,1] => 2
[2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => [4,2,1] => [3,2,1,1] => 1
[3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => [4,2,1] => [3,2,1,1] => 1
[3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8) => [4,2,1,1] => [4,2,1,1] => 2
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [4,2] => [2,2,1,1] => 1
[3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => [4,2,1] => [3,2,1,1] => 1
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => [4,2] => [2,2,1,1] => 1
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [4,2] => [2,2,1,1] => 1
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [4,2] => [2,2,1,1] => 1
[4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => [4,2,1] => [3,2,1,1] => 1
[4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => [4,2,1] => [3,2,1,1] => 1
[4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => [4,2,1] => [3,2,1,1] => 1
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [4,2] => [2,2,1,1] => 1
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => [4] => [1,1,1,1] => 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,1,1,1,1] => 1
[1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => [5,3] => [2,2,2,1,1] => 1
[1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10) => [5,3,2] => [3,3,2,1,1] => 1
[1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10) => [5,3,2] => [3,3,2,1,1] => 1
[1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10) => [5,3,2] => [3,3,2,1,1] => 1
[1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => [5,3,1] => [3,2,2,1,1] => 1
[1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10) => [5,3,2] => [3,3,2,1,1] => 1
[1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10) => [5,3,2] => [3,3,2,1,1] => 1
[1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10) => [5,3,2] => [3,3,2,1,1] => 1
[1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11) => [5,3,2,1] => [4,3,2,1,1] => 1
[1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11) => [5,3,2,1] => [4,3,2,1,1] => 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11) => [5,3,2,1] => [4,3,2,1,1] => 1
[1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10) => [5,3,2] => [3,3,2,1,1] => 1
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11) => [5,3,2,1] => [4,3,2,1,1] => 1
[1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => [5,3] => [2,2,2,1,1] => 1
[2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => [5,3] => [2,2,2,1,1] => 1
[2,1,3,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10) => [5,3,2] => [3,3,2,1,1] => 1
[2,1,4,3,5] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10) => [5,3,2] => [3,3,2,1,1] => 1
[2,1,4,5,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11) => [5,3,2,1] => [4,3,2,1,1] => 1
[2,1,5,3,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11) => [5,3,2,1] => [4,3,2,1,1] => 1
[2,1,5,4,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9) => [5,3,1] => [3,2,2,1,1] => 1
[2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10) => [5,3,2] => [3,3,2,1,1] => 1
[2,3,1,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11) => [5,3,2,1] => [4,3,2,1,1] => 1
[2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10) => [5,3,2] => [3,3,2,1,1] => 1
[2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => [5,3] => [2,2,2,1,1] => 1
[2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11) => [5,3,2,1] => [4,3,2,1,1] => 1
[2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10) => [5,3,2] => [3,3,2,1,1] => 1
[3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10) => [5,3,2] => [3,3,2,1,1] => 1
[3,1,2,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11) => [5,3,2,1] => [4,3,2,1,1] => 1
[3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => [5,3,1] => [3,2,2,1,1] => 1
[3,2,1,5,4] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9) => [5,3,1] => [3,2,2,1,1] => 1
[3,2,4,5,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11) => [5,3,2,1] => [4,3,2,1,1] => 1
[3,2,5,4,1] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11) => [5,3,2,1] => [4,3,2,1,1] => 1
[3,4,1,2,5] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11) => [5,3,2,1] => [4,3,2,1,1] => 1
[3,4,2,1,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11) => [5,3,2,1] => [4,3,2,1,1] => 1
[3,4,5,1,2] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9) => [5,3,1] => [3,2,2,1,1] => 1
[3,4,5,2,1] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => [5,3,1] => [3,2,2,1,1] => 1
[3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11) => [5,3,2,1] => [4,3,2,1,1] => 1
[3,5,4,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10) => [5,3,2] => [3,3,2,1,1] => 1
[4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10) => [5,3,2] => [3,3,2,1,1] => 1
[4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11) => [5,3,2,1] => [4,3,2,1,1] => 1
[4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => [5,3] => [2,2,2,1,1] => 1
[4,3,2,5,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10) => [5,3,2] => [3,3,2,1,1] => 1
[4,3,5,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11) => [5,3,2,1] => [4,3,2,1,1] => 1
[4,3,5,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10) => [5,3,2] => [3,3,2,1,1] => 1
[4,5,1,2,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9) => [5,3,1] => [3,2,2,1,1] => 1
[4,5,1,3,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11) => [5,3,2,1] => [4,3,2,1,1] => 1
[4,5,2,1,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11) => [5,3,2,1] => [4,3,2,1,1] => 1
[4,5,2,3,1] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10) => [5,3,2] => [3,3,2,1,1] => 1
[4,5,3,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10) => [5,3,2] => [3,3,2,1,1] => 1
[4,5,3,2,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => [5,3] => [2,2,2,1,1] => 1
[5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => [5,3] => [2,2,2,1,1] => 1
[5,1,2,4,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11) => [5,3,2,1] => [4,3,2,1,1] => 1
[5,1,4,3,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10) => [5,3,2] => [3,3,2,1,1] => 1
[5,2,1,3,4] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11) => [5,3,2,1] => [4,3,2,1,1] => 1
[5,2,1,4,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11) => [5,3,2,1] => [4,3,2,1,1] => 1
[5,2,3,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11) => [5,3,2,1] => [4,3,2,1,1] => 1
[5,3,2,1,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10) => [5,3,2] => [3,3,2,1,1] => 1
[5,3,4,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10) => [5,3,2] => [3,3,2,1,1] => 1
[5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10) => [5,3,2] => [3,3,2,1,1] => 1
[5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => [5,3,1] => [3,2,2,1,1] => 1
[5,4,1,3,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10) => [5,3,2] => [3,3,2,1,1] => 1
[5,4,2,1,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10) => [5,3,2] => [3,3,2,1,1] => 1
[5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10) => [5,3,2] => [3,3,2,1,1] => 1
[5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => [5,3] => [2,2,2,1,1] => 1
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,1,1,1,1] => 1
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1] => 1
[1,2,3,4,6,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10) => [6,4] => [2,2,2,2,1,1] => 1
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Description
The Cartan determinant of the integer partition.
Let $p=[p_1,...,p_r]$ be a given integer partition with highest part t. Let $A=K[x]/(x^t)$ be the finite dimensional algebra over the field $K$ and $M$ the direct sum of the indecomposable $A$-modules of vector space dimension $p_i$ for each $i$. Then the Cartan determinant of $p$ is the Cartan determinant of the endomorphism algebra of $M$ over $A$.
Explicitly, this is the determinant of the matrix $\left(\min(\bar p_i, \bar p_j)\right)_{i,j}$, where $\bar p$ is the set of distinct parts of the partition.
Let $p=[p_1,...,p_r]$ be a given integer partition with highest part t. Let $A=K[x]/(x^t)$ be the finite dimensional algebra over the field $K$ and $M$ the direct sum of the indecomposable $A$-modules of vector space dimension $p_i$ for each $i$. Then the Cartan determinant of $p$ is the Cartan determinant of the endomorphism algebra of $M$ over $A$.
Explicitly, this is the determinant of the matrix $\left(\min(\bar p_i, \bar p_j)\right)_{i,j}$, where $\bar p$ is the set of distinct parts of the partition.
Map
pattern poset
Description
The pattern poset of a permutation.
This is the poset of all non-empty permutations that occur in the given permutation as a pattern, ordered by pattern containment.
This is the poset of all non-empty permutations that occur in the given permutation as a pattern, ordered by pattern containment.
Map
Greene-Kleitman invariant
Description
The Greene-Kleitman invariant of a poset.
This is the partition $(c_1 - c_0, c_2 - c_1, c_3 - c_2, \ldots)$, where $c_k$ is the maximum cardinality of a union of $k$ chains of the poset. Equivalently, this is the conjugate of the partition $(a_1 - a_0, a_2 - a_1, a_3 - a_2, \ldots)$, where $a_k$ is the maximum cardinality of a union of $k$ antichains of the poset.
This is the partition $(c_1 - c_0, c_2 - c_1, c_3 - c_2, \ldots)$, where $c_k$ is the maximum cardinality of a union of $k$ chains of the poset. Equivalently, this is the conjugate of the partition $(a_1 - a_0, a_2 - a_1, a_3 - a_2, \ldots)$, where $a_k$ is the maximum cardinality of a union of $k$ antichains of the poset.
Map
conjugate
Description
Return the conjugate partition of the partition.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
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