- FindStat

Identifier

Mp00226: Standard tableaux —row-to-column-descents⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 171 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

click to show known generating functions

$F_{1} = 1$

Description

The diameter of a connected graph.
This is the greatest distance between any pair of vertices.

Map

to threshold graph

Description

The threshold graph corresponding to the composition.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.

Map

row-to-column-descents

Description

Return a standard tableau whose column descent set equals the row descent set of the original tableau.
A column descent in a standard tableau is an entry

$i$ such that

$i+1$ appears in a column to the left of the cell containing

$i$ , in English notation.
A row descent is an entry

$i$ such that

$i+1$ appears in a row above of the cell containing

$i$ .

Map

horizontal strip sizes

Description

The composition of horizontal strip sizes.
We associate to a standard Young tableau

$T$ the composition

$(c_1,\dots,c_k)$ , such that

$k$ is minimal and the numbers

$c_1+\dots+c_i + 1,\dots,c_1+\dots+c_{i+1}$ form a horizontal strip in

$T$ for all

$i$ .