Identifier
Values
([],1) => [1] => [[1]] => [[1]] => 1
([],2) => [1,1] => [[1],[2]] => [[1,2]] => 2
([(0,1)],2) => [2] => [[1,2]] => [[1],[2]] => 1
([],3) => [1,1,1] => [[1],[2],[3]] => [[1,2,3]] => 3
([(1,2)],3) => [2,1] => [[1,2],[3]] => [[1,3],[2]] => 2
([(0,2),(1,2)],3) => [2,2] => [[1,2],[3,4]] => [[1,3],[2,4]] => 2
([(0,1),(0,2),(1,2)],3) => [3] => [[1,2,3]] => [[1],[2],[3]] => 1
([],4) => [1,1,1,1] => [[1],[2],[3],[4]] => [[1,2,3,4]] => 4
([(2,3)],4) => [2,1,1] => [[1,2],[3],[4]] => [[1,3,4],[2]] => 3
([(1,3),(2,3)],4) => [2,2,1] => [[1,2],[3,4],[5]] => [[1,3,5],[2,4]] => 3
([(0,3),(1,3),(2,3)],4) => [2,2,2] => [[1,2],[3,4],[5,6]] => [[1,3,5],[2,4,6]] => 3
([(0,3),(1,2)],4) => [2,2] => [[1,2],[3,4]] => [[1,3],[2,4]] => 2
([(0,3),(1,2),(2,3)],4) => [2,2,2] => [[1,2],[3,4],[5,6]] => [[1,3,5],[2,4,6]] => 3
([(1,2),(1,3),(2,3)],4) => [3,1] => [[1,2,3],[4]] => [[1,4],[2],[3]] => 2
([(0,3),(1,2),(1,3),(2,3)],4) => [3,2] => [[1,2,3],[4,5]] => [[1,4],[2,5],[3]] => 2
([(0,2),(0,3),(1,2),(1,3)],4) => [2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [[1,3,5,7],[2,4,6,8]] => 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [3,3] => [[1,2,3],[4,5,6]] => [[1,4],[2,5],[3,6]] => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [4] => [[1,2,3,4]] => [[1],[2],[3],[4]] => 1
([],5) => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [[1,2,3,4,5]] => 5
([(3,4)],5) => [2,1,1,1] => [[1,2],[3],[4],[5]] => [[1,3,4,5],[2]] => 4
([(2,4),(3,4)],5) => [2,2,1,1] => [[1,2],[3,4],[5],[6]] => [[1,3,5,6],[2,4]] => 4
([(1,4),(2,4),(3,4)],5) => [2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [[1,3,5,7],[2,4,6]] => 4
([(0,4),(1,4),(2,4),(3,4)],5) => [2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [[1,3,5,7],[2,4,6,8]] => 4
([(1,4),(2,3)],5) => [2,2,1] => [[1,2],[3,4],[5]] => [[1,3,5],[2,4]] => 3
([(1,4),(2,3),(3,4)],5) => [2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [[1,3,5,7],[2,4,6]] => 4
([(0,1),(2,4),(3,4)],5) => [2,2,2] => [[1,2],[3,4],[5,6]] => [[1,3,5],[2,4,6]] => 3
([(2,3),(2,4),(3,4)],5) => [3,1,1] => [[1,2,3],[4],[5]] => [[1,4,5],[2],[3]] => 3
([(0,4),(1,4),(2,3),(3,4)],5) => [2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [[1,3,5,7],[2,4,6,8]] => 4
([(1,4),(2,3),(2,4),(3,4)],5) => [3,2,1] => [[1,2,3],[4,5],[6]] => [[1,4,6],[2,5],[3]] => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [3,2,2] => [[1,2,3],[4,5],[6,7]] => [[1,4,6],[2,5,7],[3]] => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,3,1] => [[1,2,3],[4,5,6],[7]] => [[1,4,7],[2,5],[3,6]] => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [3,2,2] => [[1,2,3],[4,5],[6,7]] => [[1,4,6],[2,5,7],[3]] => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,3,2] => [[1,2,3],[4,5,6],[7,8]] => [[1,4,7],[2,5,8],[3,6]] => 3
([(0,4),(1,3),(2,3),(2,4)],5) => [2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [[1,3,5,7],[2,4,6,8]] => 4
([(0,1),(2,3),(2,4),(3,4)],5) => [3,2] => [[1,2,3],[4,5]] => [[1,4],[2,5],[3]] => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [3,2,2] => [[1,2,3],[4,5],[6,7]] => [[1,4,6],[2,5,7],[3]] => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => [3,3] => [[1,2,3],[4,5,6]] => [[1,4],[2,5],[3,6]] => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => [3,3,2] => [[1,2,3],[4,5,6],[7,8]] => [[1,4,7],[2,5,8],[3,6]] => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [[1,2,3,4],[5]] => [[1,5],[2],[3],[4]] => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,2] => [[1,2,3,4],[5,6]] => [[1,5],[2,6],[3],[4]] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,3] => [[1,2,3,4],[5,6,7]] => [[1,5],[2,6],[3,7],[4]] => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,4] => [[1,2,3,4],[5,6,7,8]] => [[1,5],[2,6],[3,7],[4,8]] => 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [[1,2,3,4,5]] => [[1],[2],[3],[4],[5]] => 1
([],6) => [1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [[1,2,3,4,5,6]] => 6
([(4,5)],6) => [2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [[1,3,4,5,6],[2]] => 5
([(3,5),(4,5)],6) => [2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [[1,3,5,6,7],[2,4]] => 5
([(2,5),(3,5),(4,5)],6) => [2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => [[1,3,5,7,8],[2,4,6]] => 5
([(2,5),(3,4)],6) => [2,2,1,1] => [[1,2],[3,4],[5],[6]] => [[1,3,5,6],[2,4]] => 4
([(2,5),(3,4),(4,5)],6) => [2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => [[1,3,5,7,8],[2,4,6]] => 5
([(1,2),(3,5),(4,5)],6) => [2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [[1,3,5,7],[2,4,6]] => 4
([(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => [[1,4,5,6],[2],[3]] => 4
([(0,1),(2,5),(3,5),(4,5)],6) => [2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [[1,3,5,7],[2,4,6,8]] => 4
([(2,5),(3,4),(3,5),(4,5)],6) => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => [[1,4,6,7],[2,5],[3]] => 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => [[1,4,6,8],[2,5,7],[3]] => 4
([(0,5),(1,5),(2,4),(3,4)],6) => [2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [[1,3,5,7],[2,4,6,8]] => 4
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => [[1,4,7,8],[2,5],[3,6]] => 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => [[1,4,6,8],[2,5,7],[3]] => 4
([(0,5),(1,4),(2,3)],6) => [2,2,2] => [[1,2],[3,4],[5,6]] => [[1,3,5],[2,4,6]] => 3
([(0,1),(2,5),(3,4),(4,5)],6) => [2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [[1,3,5,7],[2,4,6,8]] => 4
([(1,2),(3,4),(3,5),(4,5)],6) => [3,2,1] => [[1,2,3],[4,5],[6]] => [[1,4,6],[2,5],[3]] => 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => [[1,4,6,8],[2,5,7],[3]] => 4
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [3,2,2] => [[1,2,3],[4,5],[6,7]] => [[1,4,6],[2,5,7],[3]] => 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,3,1] => [[1,2,3],[4,5,6],[7]] => [[1,4,7],[2,5],[3,6]] => 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,3,2] => [[1,2,3],[4,5,6],[7,8]] => [[1,4,7],[2,5,8],[3,6]] => 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => [3,2,2] => [[1,2,3],[4,5],[6,7]] => [[1,4,6],[2,5,7],[3]] => 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [3,3,2] => [[1,2,3],[4,5,6],[7,8]] => [[1,4,7],[2,5,8],[3,6]] => 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,3,2] => [[1,2,3],[4,5,6],[7,8]] => [[1,4,7],[2,5,8],[3,6]] => 3
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [[1,2,3,4],[5],[6]] => [[1,5,6],[2],[3],[4]] => 3
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,2,1] => [[1,2,3,4],[5,6],[7]] => [[1,5,7],[2,6],[3],[4]] => 3
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,2,2] => [[1,2,3,4],[5,6],[7,8]] => [[1,5,7],[2,6,8],[3],[4]] => 3
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,3,1] => [[1,2,3,4],[5,6,7],[8]] => [[1,5,8],[2,6],[3,7],[4]] => 3
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,2,2] => [[1,2,3,4],[5,6],[7,8]] => [[1,5,7],[2,6,8],[3],[4]] => 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => [3,3] => [[1,2,3],[4,5,6]] => [[1,4],[2,5],[3,6]] => 2
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6) => [3,3,2] => [[1,2,3],[4,5,6],[7,8]] => [[1,4,7],[2,5,8],[3,6]] => 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,2] => [[1,2,3,4],[5,6]] => [[1,5],[2,6],[3],[4]] => 2
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,2,2] => [[1,2,3,4],[5,6],[7,8]] => [[1,5,7],[2,6,8],[3],[4]] => 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,3] => [[1,2,3,4],[5,6,7]] => [[1,5],[2,6],[3,7],[4]] => 2
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,4] => [[1,2,3,4],[5,6,7,8]] => [[1,5],[2,6],[3,7],[4,8]] => 2
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [[1,2,3,4,5],[6]] => [[1,6],[2],[3],[4],[5]] => 2
([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,2] => [[1,2,3,4,5],[6,7]] => [[1,6],[2,7],[3],[4],[5]] => 2
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,3] => [[1,2,3,4,5],[6,7,8]] => [[1,6],[2,7],[3,8],[4],[5]] => 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [[1,2,3,4,5,6]] => [[1],[2],[3],[4],[5],[6]] => 1
([],7) => [1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [[1,2,3,4,5,6,7]] => 7
([(5,6)],7) => [2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [[1,3,4,5,6,7],[2]] => 6
([(4,6),(5,6)],7) => [2,2,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8]] => [[1,3,5,6,7,8],[2,4]] => 6
([(3,6),(4,5)],7) => [2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [[1,3,5,6,7],[2,4]] => 5
([(2,3),(4,6),(5,6)],7) => [2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => [[1,3,5,7,8],[2,4,6]] => 5
([(4,5),(4,6),(5,6)],7) => [3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [[1,4,5,6,7],[2],[3]] => 5
([(3,6),(4,5),(4,6),(5,6)],7) => [3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => [[1,4,6,7,8],[2,5],[3]] => 5
([(1,6),(2,5),(3,4)],7) => [2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [[1,3,5,7],[2,4,6]] => 4
([(0,3),(1,2),(4,6),(5,6)],7) => [2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [[1,3,5,7],[2,4,6,8]] => 4
([(2,3),(4,5),(4,6),(5,6)],7) => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => [[1,4,6,7],[2,5],[3]] => 4
([(1,2),(3,6),(4,5),(4,6),(5,6)],7) => [3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => [[1,4,6,8],[2,5,7],[3]] => 4
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => [[1,4,7,8],[2,5],[3,6]] => 4
([(1,6),(2,6),(3,4),(3,5),(4,5)],7) => [3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => [[1,4,6,8],[2,5,7],[3]] => 4
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => [[1,5,6,7],[2],[3],[4]] => 4
([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => [[1,5,7,8],[2,6],[3],[4]] => 4
([(0,3),(1,2),(4,5),(4,6),(5,6)],7) => [3,2,2] => [[1,2,3],[4,5],[6,7]] => [[1,4,6],[2,5,7],[3]] => 3
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [3,3,2] => [[1,2,3],[4,5,6],[7,8]] => [[1,4,7],[2,5,8],[3,6]] => 3
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7) => [3,3,1] => [[1,2,3],[4,5,6],[7]] => [[1,4,7],[2,5],[3,6]] => 3
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7) => [3,3,2] => [[1,2,3],[4,5,6],[7,8]] => [[1,4,7],[2,5,8],[3,6]] => 3
>>> Load all 114 entries. <<<
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,2,1] => [[1,2,3,4],[5,6],[7]] => [[1,5,7],[2,6],[3],[4]] => 3
([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,2,2] => [[1,2,3,4],[5,6],[7,8]] => [[1,5,7],[2,6,8],[3],[4]] => 3
([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,3,1] => [[1,2,3,4],[5,6,7],[8]] => [[1,5,8],[2,6],[3,7],[4]] => 3
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,2,2] => [[1,2,3,4],[5,6],[7,8]] => [[1,5,7],[2,6,8],[3],[4]] => 3
([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => [[1,6,7],[2],[3],[4],[5]] => 3
([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,2,1] => [[1,2,3,4,5],[6,7],[8]] => [[1,6,8],[2,7],[3],[4],[5]] => 3
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,3] => [[1,2,3,4],[5,6,7]] => [[1,5],[2,6],[3,7],[4]] => 2
([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [4,4] => [[1,2,3,4],[5,6,7,8]] => [[1,5],[2,6],[3,7],[4,8]] => 2
([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,2] => [[1,2,3,4,5],[6,7]] => [[1,6],[2,7],[3],[4],[5]] => 2
([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,3] => [[1,2,3,4,5],[6,7,8]] => [[1,6],[2,7],[3,8],[4],[5]] => 2
([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6,1] => [[1,2,3,4,5,6],[7]] => [[1,7],[2],[3],[4],[5],[6]] => 2
([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6,2] => [[1,2,3,4,5,6],[7,8]] => [[1,7],[2,8],[3],[4],[5],[6]] => 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [7] => [[1,2,3,4,5,6,7]] => [[1],[2],[3],[4],[5],[6],[7]] => 1
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Description
The number of factors of a standard tableaux under concatenation.
The concatenation of two standard Young tableaux $T_1$ and $T_2$ is obtained by adding the largest entry of $T_1$ to each entry of $T_2$, and then appending the rows of the result to $T_1$, see [1, dfn 2.10].
This statistic returns the maximal number of standard tableaux such that their concatenation is the given tableau.
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.
Map
clique sizes
Description
The integer partition of the sizes of the maximal cliques of a graph.
Map
conjugate
Description
Sends a standard tableau to its conjugate tableau.