- FindStat

Identifier

Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001615: Lattices ⟶ ℤ

Values

[[1]] => [[1]] => ([],1) => ([],1) => 0

[[1,0],[0,1]] => [[1,1],[2]] => ([],1) => ([],1) => 0

[[0,1],[1,0]] => [[1,2],[2]] => ([(0,1)],2) => ([(0,1)],2) => 1

[[1,0,0],[0,1,0],[0,0,1]] => [[1,1,1],[2,2],[3]] => ([],1) => ([],1) => 0

[[0,1,0],[1,0,0],[0,0,1]] => [[1,1,2],[2,2],[3]] => ([(0,1)],2) => ([(0,1)],2) => 1

[[1,0,0],[0,0,1],[0,1,0]] => [[1,1,1],[2,3],[3]] => ([(0,1)],2) => ([(0,1)],2) => 1

[[0,1,0],[1,-1,1],[0,1,0]] => [[1,1,2],[2,3],[3]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 2

[[0,0,1],[1,0,0],[0,1,0]] => [[1,1,3],[2,3],[3]] => ([(0,3),(2,1),(3,2)],4) => ([(0,3),(2,1),(3,2)],4) => 3

[[0,1,0],[0,0,1],[1,0,0]] => [[1,2,2],[2,3],[3]] => ([(0,3),(2,1),(3,2)],4) => ([(0,3),(2,1),(3,2)],4) => 3

[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]] => [[1,1,1,1],[2,2,2],[3,3],[4]] => ([],1) => ([],1) => 0

[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]] => [[1,1,1,2],[2,2,2],[3,3],[4]] => ([(0,1)],2) => ([(0,1)],2) => 1

[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]] => [[1,1,1,1],[2,2,3],[3,3],[4]] => ([(0,1)],2) => ([(0,1)],2) => 1

[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]] => [[1,1,1,2],[2,2,3],[3,3],[4]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 2

[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]] => [[1,1,1,3],[2,2,3],[3,3],[4]] => ([(0,3),(2,1),(3,2)],4) => ([(0,3),(2,1),(3,2)],4) => 3

[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]] => [[1,1,2,2],[2,2,3],[3,3],[4]] => ([(0,3),(2,1),(3,2)],4) => ([(0,3),(2,1),(3,2)],4) => 3

[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]] => [[1,1,1,1],[2,2,2],[3,4],[4]] => ([(0,1)],2) => ([(0,1)],2) => 1

[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]] => [[1,1,1,2],[2,2,2],[3,4],[4]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,2),(2,1)],3) => 2

[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]] => [[1,1,1,1],[2,2,3],[3,4],[4]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 2

[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]] => [[1,1,1,2],[2,2,3],[3,4],[4]] => ([(0,3),(2,1),(3,2)],4) => ([(0,3),(2,1),(3,2)],4) => 3

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

[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]] => [[1,1,2,2],[2,2,3],[3,4],[4]] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,4),(2,3),(3,1),(4,2)],5) => 4

[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]] => [[1,1,1,1],[2,2,4],[3,4],[4]] => ([(0,3),(2,1),(3,2)],4) => ([(0,3),(2,1),(3,2)],4) => 3

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

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

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

[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]] => [[1,1,1,1],[2,3,3],[3,4],[4]] => ([(0,3),(2,1),(3,2)],4) => ([(0,3),(2,1),(3,2)],4) => 3

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

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

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

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

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

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

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

[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,4],[5]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 2

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

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

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

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

[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,4],[5]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 2

[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => [[1,1,1,1,2],[2,2,2,3],[3,3,4],[4,4],[5]] => ([(0,3),(2,1),(3,2)],4) => ([(0,3),(2,1),(3,2)],4) => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]] => [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,5],[5]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 2

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

[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]] => [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,5],[5]] => ([(0,3),(2,1),(3,2)],4) => ([(0,3),(2,1),(3,2)],4) => 3

[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]] => [[1,1,1,1,2],[2,2,2,3],[3,3,4],[4,5],[5]] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,4),(2,3),(3,1),(4,2)],5) => 4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

Description

The number of join prime elements of a lattice.
An element

$x$ of a lattice

$L$ is join-prime (or coprime) if

$x \leq a \vee b$ implies

$x \leq a$ or

$x \leq b$ for every

$a, b \in L$ .

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

to semistandard tableau via monotone triangles

Description

The semistandard tableau corresponding the monotone triangle of an alternating sign matrix.
This is obtained by interpreting each row of the monotone triangle as an integer partition, and filling the cells of the smallest partition with ones, the second smallest with twos, and so on.

Map

subcrystal

Description

The underlying poset of the subcrystal obtained by applying the raising operators to a semistandard tableau.