Identifier
-
Mp00081:
Standard tableaux
—reading word permutation⟶
Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤ
Values
[[1]] => [1] => [1] => ([],1) => 0
[[1],[2]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[1,2],[3]] => [3,1,2] => [2,3,1] => ([(0,2),(1,2)],3) => 2
[[1],[2],[3]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[1,2,3],[4]] => [4,1,2,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4) => 2
[[1,3],[2,4]] => [2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4) => 2
[[1,2],[3,4]] => [3,4,1,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 3
[[1,3],[2],[4]] => [4,2,1,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
[[1,2],[3],[4]] => [4,3,1,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
[[1],[2],[3],[4]] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
[[1,2,3,4],[5]] => [5,1,2,3,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[[1,3,4],[2,5]] => [2,5,1,3,4] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 2
[[1,2,4],[3,5]] => [3,5,1,2,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 3
[[1,2,3],[4,5]] => [4,5,1,2,3] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5) => 3
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,2,3],[4],[5]] => [5,4,1,2,3] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,4],[2,5],[3]] => [3,2,5,1,4] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 2
[[1,3],[2,5],[4]] => [4,2,5,1,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 2
[[1,2],[3,5],[4]] => [4,3,5,1,2] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 3
[[1,3],[2,4],[5]] => [5,2,4,1,3] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 3
[[1,2],[3,4],[5]] => [5,3,4,1,2] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 4
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1,2],[3],[4],[5]] => [5,4,3,1,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
[[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2
[[1,3,4,5],[2,6]] => [2,6,1,3,4,5] => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 2
[[1,2,4,5],[3,6]] => [3,6,1,2,4,5] => [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 3
[[1,2,3,5],[4,6]] => [4,6,1,2,3,5] => [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 3
[[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 3
[[1,3,4,5],[2],[6]] => [6,2,1,3,4,5] => [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => [2,4,5,6,3,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => [2,3,5,6,4,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [2,3,4,6,5,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,3,5],[2,4,6]] => [2,4,6,1,3,5] => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => 3
[[1,2,5],[3,4,6]] => [3,4,6,1,2,5] => [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 3
[[1,3,4],[2,5,6]] => [2,5,6,1,3,4] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 3
[[1,2,4],[3,5,6]] => [3,5,6,1,2,4] => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 3
[[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 3
[[1,4,5],[2,6],[3]] => [3,2,6,1,4,5] => [4,5,6,1,3,2] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,3,5],[2,6],[4]] => [4,2,6,1,3,5] => [3,5,6,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 2
[[1,2,5],[3,6],[4]] => [4,3,6,1,2,5] => [2,5,6,1,4,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 3
[[1,3,4],[2,6],[5]] => [5,2,6,1,3,4] => [3,4,6,1,5,2] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6) => 2
[[1,2,4],[3,6],[5]] => [5,3,6,1,2,4] => [2,4,6,1,5,3] => ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 3
[[1,2,3],[4,6],[5]] => [5,4,6,1,2,3] => [2,3,6,1,5,4] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 3
[[1,3,5],[2,4],[6]] => [6,2,4,1,3,5] => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 3
[[1,2,5],[3,4],[6]] => [6,3,4,1,2,5] => [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 4
[[1,3,4],[2,5],[6]] => [6,2,5,1,3,4] => [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 3
[[1,2,4],[3,5],[6]] => [6,3,5,1,2,4] => [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 4
[[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 4
[[1,4,5],[2],[3],[6]] => [6,3,2,1,4,5] => [4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,3,5],[2],[4],[6]] => [6,4,2,1,3,5] => [3,5,6,4,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,2,5],[3],[4],[6]] => [6,4,3,1,2,5] => [2,5,6,4,3,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,3,4],[2],[5],[6]] => [6,5,2,1,3,4] => [3,4,6,5,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,2,4],[3],[5],[6]] => [6,5,3,1,2,4] => [2,4,6,5,3,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [2,3,6,5,4,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,4],[2,5],[3,6]] => [3,6,2,5,1,4] => [4,5,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => 3
[[1,3],[2,5],[4,6]] => [4,6,2,5,1,3] => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6) => 3
[[1,2],[3,5],[4,6]] => [4,6,3,5,1,2] => [2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => 4
[[1,3],[2,4],[5,6]] => [5,6,2,4,1,3] => [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => 4
[[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 5
[[1,5],[2,6],[3],[4]] => [4,3,2,6,1,5] => [5,6,1,4,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 2
[[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 2
[[1,3],[2,6],[4],[5]] => [5,4,2,6,1,3] => [3,6,1,5,4,2] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,2],[3,6],[4],[5]] => [5,4,3,6,1,2] => [2,6,1,5,4,3] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,4],[2,5],[3],[6]] => [6,3,2,5,1,4] => [4,5,1,6,3,2] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,3],[2,5],[4],[6]] => [6,4,2,5,1,3] => [3,5,1,6,4,2] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[[1,2],[3,5],[4],[6]] => [6,4,3,5,1,2] => [2,5,1,6,4,3] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6) => 4
[[1,3],[2,4],[5],[6]] => [6,5,2,4,1,3] => [3,4,1,6,5,2] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => 3
[[1,2],[3,4],[5],[6]] => [6,5,3,4,1,2] => [2,4,1,6,5,3] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => 4
[[1,5],[2],[3],[4],[6]] => [6,4,3,2,1,5] => [5,6,4,3,2,1] => ([(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) => 2
[[1,4],[2],[3],[5],[6]] => [6,5,3,2,1,4] => [4,6,5,3,2,1] => ([(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) => 2
[[1,3],[2],[4],[5],[6]] => [6,5,4,2,1,3] => [3,6,5,4,2,1] => ([(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) => 2
[[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(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) => 1
[[1,2,3,4,5,6],[7]] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[[1,3,4,5,6],[2,7]] => [2,7,1,3,4,5,6] => [3,4,5,6,7,1,2] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 2
[[1,2,4,5,6],[3,7]] => [3,7,1,2,4,5,6] => [2,4,5,6,7,1,3] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 3
[[1,2,3,5,6],[4,7]] => [4,7,1,2,3,5,6] => [2,3,5,6,7,1,4] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 3
[[1,2,3,4,6],[5,7]] => [5,7,1,2,3,4,6] => [2,3,4,6,7,1,5] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 3
[[1,2,3,4,5],[6,7]] => [6,7,1,2,3,4,5] => [2,3,4,5,7,1,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 3
[[1,3,4,5,6],[2],[7]] => [7,2,1,3,4,5,6] => [3,4,5,6,7,2,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 2
[[1,2,4,5,6],[3],[7]] => [7,3,1,2,4,5,6] => [2,4,5,6,7,3,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 2
[[1,2,3,5,6],[4],[7]] => [7,4,1,2,3,5,6] => [2,3,5,6,7,4,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 2
[[1,2,3,4,6],[5],[7]] => [7,5,1,2,3,4,6] => [2,3,4,6,7,5,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 2
[[1,2,3,4,5],[6],[7]] => [7,6,1,2,3,4,5] => [2,3,4,5,7,6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 2
[[1,3,5,6],[2,4,7]] => [2,4,7,1,3,5,6] => [3,5,6,7,1,2,4] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 3
[[1,2,5,6],[3,4,7]] => [3,4,7,1,2,5,6] => [2,5,6,7,1,3,4] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7) => 3
[[1,3,4,6],[2,5,7]] => [2,5,7,1,3,4,6] => [3,4,6,7,1,2,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7) => 3
[[1,2,4,6],[3,5,7]] => [3,5,7,1,2,4,6] => [2,4,6,7,1,3,5] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 3
[[1,2,3,6],[4,5,7]] => [4,5,7,1,2,3,6] => [2,3,6,7,1,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 3
[[1,3,4,5],[2,6,7]] => [2,6,7,1,3,4,5] => [3,4,5,7,1,2,6] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 3
[[1,2,4,5],[3,6,7]] => [3,6,7,1,2,4,5] => [2,4,5,7,1,3,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 3
[[1,2,3,5],[4,6,7]] => [4,6,7,1,2,3,5] => [2,3,5,7,1,4,6] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7) => 3
[[1,2,3,4],[5,6,7]] => [5,6,7,1,2,3,4] => [2,3,4,7,1,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 3
[[1,4,5,6],[2,7],[3]] => [3,2,7,1,4,5,6] => [4,5,6,7,1,3,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7) => 2
[[1,3,5,6],[2,7],[4]] => [4,2,7,1,3,5,6] => [3,5,6,7,1,4,2] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6)],7) => 2
[[1,2,5,6],[3,7],[4]] => [4,3,7,1,2,5,6] => [2,5,6,7,1,4,3] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5)],7) => 3
[[1,3,4,6],[2,7],[5]] => [5,2,7,1,3,4,6] => [3,4,6,7,1,5,2] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 2
[[1,2,4,6],[3,7],[5]] => [5,3,7,1,2,4,6] => [2,4,6,7,1,5,3] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6)],7) => 3
>>> Load all 232 entries. <<<
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
This is the greatest distance between any pair of vertices.
Map
invert Laguerre heap
Description
The permutation obtained by inverting the corresponding Laguerre heap, according to Viennot.
Let $\pi$ be a permutation. Following Viennot [1], we associate to $\pi$ a heap of pieces, by considering each decreasing run $(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of $\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.
Let $\pi$ be a permutation. Following Viennot [1], we associate to $\pi$ a heap of pieces, by considering each decreasing run $(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of $\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.
Map
graph of inversions
Description
The graph of inversions of a permutation.
For a permutation of $\{1,\dots,n\}$, this is the graph with vertices $\{1,\dots,n\}$, where $(i,j)$ is an edge if and only if it is an inversion of the permutation.
For a permutation of $\{1,\dots,n\}$, this is the graph with vertices $\{1,\dots,n\}$, where $(i,j)$ is an edge if and only if it is an inversion of the permutation.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.
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