- FindStat

Identifier

Mp00195: Posets —order ideals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000384: Integer partitions ⟶ ℤ

Values

([],1) => ([(0,1)],2) => ([(0,1)],2) => [2] => 2
([],2) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => [3,1] => 3
([(0,1)],2) => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => [3] => 3
([],3) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => [4,2,2] => 4
([(1,2)],3) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [4,2] => 4
([(0,1),(0,2)],3) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => [4,1] => 4
([(0,2),(2,1)],3) => ([(0,3),(2,1),(3,2)],4) => ([(0,3),(2,1),(3,2)],4) => [4] => 4
([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => [4,1] => 4
([(2,3)],4) => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12) => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12) => [5,3,3,1] => 5
([(1,2),(1,3)],4) => ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10) => ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10) => [5,3,2] => 5
([(0,1),(0,2),(0,3)],4) => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9) => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9) => [5,2,2] => 5
([(0,2),(0,3),(3,1)],4) => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => [5,2] => 5
([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => [5,1] => 5
([(1,2),(2,3)],4) => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => [5,3] => 5
([(0,3),(3,1),(3,2)],4) => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => [5,1] => 5
([(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10) => ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10) => [5,3,2] => 5
([(0,3),(1,3),(3,2)],4) => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => [5,1] => 5
([(0,3),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9) => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9) => [5,2,2] => 5
([(0,3),(1,2)],4) => ([(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) => ([(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] => 5
([(0,3),(1,2),(1,3)],4) => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8) => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8) => [5,3] => 5
([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => [5,2] => 5
([(0,3),(2,1),(3,2)],4) => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => 5
([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => [5,2] => 5
([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11) => ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11) => [6,3,2] => 6
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10) => ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10) => [6,2,2] => 6
([(0,3),(0,4),(4,1),(4,2)],5) => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11) => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11) => [6,3,2] => 6
([(1,2),(1,3),(2,4),(3,4)],5) => ([(0,4),(0,5),(1,9),(1,10),(2,8),(2,10),(3,7),(4,6),(5,1),(5,2),(5,6),(6,8),(6,9),(8,11),(9,11),(10,3),(10,11),(11,7)],12) => ([(0,4),(0,5),(1,9),(1,10),(2,8),(2,10),(3,7),(4,6),(5,1),(5,2),(5,6),(6,8),(6,9),(8,11),(9,11),(10,3),(10,11),(11,7)],12) => [6,4,2] => 6
([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => [6,1] => 6
([(0,3),(0,4),(3,2),(4,1)],5) => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10) => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10) => [6,3,1] => 6
([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9) => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9) => [6,3] => 6
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8) => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8) => [6,2] => 6
([(1,4),(4,2),(4,3)],5) => ([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12) => ([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12) => [6,4,2] => 6
([(0,4),(4,1),(4,2),(4,3)],5) => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10) => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10) => [6,2,2] => 6
([(1,4),(2,4),(4,3)],5) => ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12) => ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12) => [6,4,2] => 6
([(0,4),(1,4),(4,2),(4,3)],5) => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8) => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8) => [6,2] => 6
([(0,4),(1,4),(2,4),(4,3)],5) => ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10) => ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10) => [6,2,2] => 6
([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12) => ([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12) => [6,4,2] => 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11) => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11) => [6,3,2] => 6
([(0,4),(1,4),(2,3),(4,2)],5) => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7) => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7) => [6,1] => 6
([(0,4),(1,3),(2,3),(3,4)],5) => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11) => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11) => [6,3,2] => 6
([(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11) => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11) => [6,4,1] => 6
([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9) => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9) => [6,3] => 6
([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9) => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9) => [6,3] => 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8) => ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8) => [6,2] => 6
([(0,4),(1,2),(1,4),(4,3)],5) => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10) => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10) => [6,4] => 6
([(0,2),(0,4),(3,1),(4,3)],5) => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9) => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9) => [6,3] => 6
([(0,4),(1,2),(1,3),(3,4)],5) => ([(0,4),(0,5),(1,8),(2,7),(2,9),(3,7),(3,10),(4,6),(5,2),(5,3),(5,6),(6,9),(6,10),(7,11),(9,11),(10,1),(10,11),(11,8)],12) => ([(0,4),(0,5),(1,8),(2,7),(2,9),(3,7),(3,10),(4,6),(5,2),(5,3),(5,6),(6,9),(6,10),(7,11),(9,11),(10,1),(10,11),(11,8)],12) => [6,4,2] => 6
([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8) => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8) => [6,2] => 6
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11) => ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11) => [6,3,2] => 6
([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => ([(0,4),(0,5),(1,7),(1,9),(2,7),(2,8),(3,6),(4,10),(5,3),(5,10),(6,8),(6,9),(7,11),(8,11),(9,11),(10,1),(10,2),(10,6)],12) => ([(0,4),(0,5),(1,7),(1,9),(2,7),(2,8),(3,6),(4,10),(5,3),(5,10),(6,8),(6,9),(7,11),(8,11),(9,11),(10,1),(10,2),(10,6)],12) => [6,4,2] => 6
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11) => ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11) => [6,3,2] => 6
([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10) => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10) => [6,4] => 6
([(0,3),(1,2),(1,4),(3,4)],5) => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11) => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11) => [6,4,1] => 6
([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9) => ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9) => [6,3] => 6
([(1,4),(3,2),(4,3)],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) => ([(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] => 6
([(0,3),(3,4),(4,1),(4,2)],5) => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => [6,1] => 6
([(0,4),(1,2),(2,4),(4,3)],5) => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8) => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8) => [6,2] => 6
([(0,3),(1,4),(4,2)],5) => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12) => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12) => [6,4,2] => 6
([(0,4),(3,2),(4,1),(4,3)],5) => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8) => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8) => [6,2] => 6
([(0,4),(1,2),(2,3),(2,4)],5) => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10) => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10) => [6,4] => 6
([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => 6
([(0,3),(1,2),(2,4),(3,4)],5) => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10) => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10) => [6,3,1] => 6
([(0,4),(1,2),(2,3),(3,4)],5) => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9) => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9) => [6,3] => 6
([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => [6,1] => 6
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10) => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10) => [7,3] => 7
([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11) => ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11) => [7,4] => 7
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8) => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8) => [7,1] => 7
([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => [7] => 7
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11) => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11) => [8,3] => 8
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11) => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11) => [8,3] => 8
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) => [8] => 8
([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9) => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9) => [9] => 9
([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9) => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => [10] => 10

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$F_{1} = q^{2}$

$F_{2} = 2\ q^{3}$

$F_{3} = 5\ q^{4}$

Description

The maximal part of the shifted composition of an integer partition.
A partition

$\lambda = (\lambda_1,\ldots,\lambda_k)$ is shifted into a composition by adding

$i-1$ to the

$i$ -th part.
The statistic is then

$\operatorname{max}_i\{ \lambda_i + i - 1 \}$ .
See also St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition..

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.

Map

order ideals

Description

The lattice of order ideals of a poset.
An order ideal

$\mathcal I$ in a poset

$P$ is a downward closed set, i.e.,

$a \in \mathcal I$ and

$b \leq a$ implies

$b \in \mathcal I$ . This map sends a poset to the lattice of all order ideals sorted by inclusion with meet being intersection and join being union.

Map

to poset

Description

Return the poset corresponding to the lattice.