- FindStat

Identifier

Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001624: Lattices ⟶ ℤ

Values

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

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$F_{2} = q$

$F_{4} = 2\ q + q^{2}$

Description

The breadth of a lattice.
The breadth of a lattice is the least integer

$b$ such that any join

$x_1\vee x_2\vee\cdots\vee x_n$ , with

$n > b$ , can be expressed as a join over a proper subset of

$\{x_1,x_2,\ldots,x_n\}$ .

Map

maximal antichains

Description

The lattice of maximal antichains in a poset.
An antichain

$A$ in a poset is maximal if there is no antichain of larger cardinality which contains all elements of

$A$ .
The set of maximal antichains can be ordered by setting

$A \leq B \Leftrightarrow \mathop{\downarrow} A \subseteq \mathop{\downarrow}B$ , where

$\mathop{\downarrow}A$ is the order ideal generated by

$A$ .

Map

permutation poset

Description

Sends a permutation to its permutation poset.
For a permutation

$\pi$ of length

$n$ , this poset has vertices

$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$
and the cover relation is given by

$(w, x) \leq (y, z)$ if

$w \leq y$ and

$x \leq z$ .
For example, the permutation

$[3,1,5,4,2]$ is mapped to the poset with cover relations

$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$

Map

to permutation

Description

Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.