Identifier
-
Mp00001:
Alternating sign matrices
—to semistandard tableau via monotone triangles⟶
Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000384: Integer partitions ⟶ ℤ
Values
[[1]] => [[1]] => ([],1) => [1] => 1
[[1,0],[0,1]] => [[1,1],[2]] => ([],1) => [1] => 1
[[0,1],[1,0]] => [[1,2],[2]] => ([(0,1)],2) => [2] => 2
[[1,0,0],[0,1,0],[0,0,1]] => [[1,1,1],[2,2],[3]] => ([],1) => [1] => 1
[[0,1,0],[1,0,0],[0,0,1]] => [[1,1,2],[2,2],[3]] => ([(0,1)],2) => [2] => 2
[[1,0,0],[0,0,1],[0,1,0]] => [[1,1,1],[2,3],[3]] => ([(0,1)],2) => [2] => 2
[[0,1,0],[1,-1,1],[0,1,0]] => [[1,1,2],[2,3],[3]] => ([(0,2),(2,1)],3) => [3] => 3
[[0,0,1],[1,0,0],[0,1,0]] => [[1,1,3],[2,3],[3]] => ([(0,3),(2,1),(3,2)],4) => [4] => 4
[[0,1,0],[0,0,1],[1,0,0]] => [[1,2,2],[2,3],[3]] => ([(0,3),(2,1),(3,2)],4) => [4] => 4
[[0,0,1],[0,1,0],[1,0,0]] => [[1,2,3],[2,3],[3]] => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8) => [5,3] => 5
[[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] => 1
[[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) => [2] => 2
[[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) => [2] => 2
[[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) => [3] => 3
[[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) => [4] => 4
[[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) => [4] => 4
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]] => [[1,1,2,3],[2,2,3],[3,3],[4]] => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8) => [5,3] => 5
[[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) => [2] => 2
[[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) => [3,1] => 3
[[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) => [3] => 3
[[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) => [4] => 4
[[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) => [5,1] => 5
[[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) => [5] => 5
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]] => [[1,1,2,3],[2,2,3],[3,4],[4]] => ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10) => [6,4] => 6
[[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) => [4] => 4
[[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) => [5,2] => 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) => [6,3] => 6
[[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) => [7,1] => 7
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]] => [[1,1,2,2],[2,2,4],[3,4],[4]] => ([(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
[[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) => [4] => 4
[[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) => [5,1] => 5
[[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) => [6,1] => 6
[[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) => [6,1] => 6
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]] => [[1,1,2,3],[2,3,3],[3,4],[4]] => ([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12) => [7,5] => 7
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]] => [[1,1,1,1],[2,3,4],[3,4],[4]] => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8) => [5,3] => 5
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]] => [[1,1,1,2],[2,3,4],[3,4],[4]] => ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11) => [6,4,1] => 6
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]] => [[1,1,1,3],[2,3,4],[3,4],[4]] => ([(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
[[0,1,0,0],[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) => [7,1] => 7
[[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] => 1
[[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) => [2] => 2
[[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) => [2] => 2
[[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) => [3] => 3
[[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) => [4] => 4
[[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) => [4] => 4
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]] => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8) => [5,3] => 5
[[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) => [2] => 2
[[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) => [3,1] => 3
[[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) => [3] => 3
[[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) => [4] => 4
[[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) => [5,1] => 5
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => [[1,1,1,2,3],[2,2,2,3],[3,3,4],[4,4],[5]] => ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10) => [6,4] => 6
[[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) => [4] => 4
[[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) => [5,2] => 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) => [6,3] => 6
[[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) => [7,1] => 7
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,4],[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) => [6,3,1] => 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) => [4] => 4
[[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) => [6,1] => 6
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]] => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8) => [5,3] => 5
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [[1,1,1,1,2],[2,2,3,4],[3,3,4],[4,4],[5]] => ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11) => [6,4,1] => 6
[[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) => [7,1] => 7
[[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) => [2] => 2
[[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) => [3,1] => 3
[[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) => [3,1] => 3
[[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) => [4,2] => 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) => [5,3] => 5
[[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) => [5,3] => 5
[[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) => [3] => 3
[[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) => [4,2] => 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) => [4] => 4
[[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) => [5] => 5
[[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) => [6,2] => 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) => [5,1] => 5
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]] => [[1,1,1,1,2],[2,2,2,4],[3,3,4],[4,5],[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) => [6,4] => 6
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]] => [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,5],[5]] => ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10) => [6,4] => 6
[[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) => [4] => 4
[[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) => [5,3] => 5
[[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) => [5,2] => 5
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]] => [[1,1,1,1,2],[2,2,2,3],[3,3,5],[4,5],[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) => [6,3,1] => 6
[[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) => [6,3] => 6
[[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) => [7,1] => 7
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]] => [[1,1,1,1,1],[2,2,3,3],[3,3,5],[4,5],[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) => [6,3,1] => 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) => [4] => 4
[[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) => [5,3] => 5
[[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) => [6,1] => 6
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]] => [[1,1,1,1,1],[2,2,2,2],[3,4,5],[4,5],[5]] => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8) => [5,3] => 5
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]] => [[1,1,1,1,1],[2,2,2,3],[3,4,5],[4,5],[5]] => ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11) => [6,4,1] => 6
[[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) => [7,1] => 7
[[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] => 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],[0,0,0,0,1,0],[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) => [2] => 2
[[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) => [2] => 2
[[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) => [3] => 3
[[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) => [4] => 4
[[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]] => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8) => [5,3] => 5
[[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) => [2] => 2
[[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) => [3,1] => 3
[[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) => [3] => 3
[[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) => [4] => 4
[[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) => [5,1] => 5
[[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,5],[6]] => ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10) => [6,4] => 6
[[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) => [4] => 4
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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..
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
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.
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.
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.
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