Identifier
Values
[[1],[2]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[1],[3]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[2],[3]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[1],[4]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[2],[4]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[3],[4]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[1],[2],[3]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[1],[5]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[2],[5]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[3],[5]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[4],[5]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[1],[2],[4]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[1],[3],[4]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[2],[3],[4]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[1,2],[2],[3]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[1],[6]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[2],[6]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[3],[6]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[4],[6]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[5],[6]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[1],[2],[5]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[1],[3],[5]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[1],[4],[5]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[2],[3],[5]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[2],[4],[5]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[3],[4],[5]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[1,2],[2],[4]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[1,3],[2],[4]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[1,3],[3],[4]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[2,3],[3],[4]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(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],[2,3],[3]] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[1],[7]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[2],[7]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[3],[7]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[4],[7]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[5],[7]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[6],[7]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[1],[2],[6]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[1],[3],[6]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[1],[4],[6]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[1],[5],[6]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[2],[3],[6]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[2],[4],[6]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[2],[5],[6]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[3],[4],[6]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[3],[5],[6]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[4],[5],[6]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[1,2],[2],[5]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[1,3],[2],[5]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[1,4],[2],[5]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[1,3],[3],[5]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[1,4],[3],[5]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[1,4],[4],[5]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[2,3],[3],[5]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[2,4],[3],[5]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[2,4],[4],[5]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[3,4],[4],[5]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[1],[2],[3],[5]] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
[[1],[2],[4],[5]] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
[[1],[3],[4],[5]] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
[[2],[3],[4],[5]] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
[[1,2],[2,4],[3]] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[1,2],[2,4],[4]] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[1,3],[2,4],[4]] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[1,3],[3,4],[4]] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[2,3],[3,4],[4]] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[1,2],[2],[3],[4]] => [5,4,2,1,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 2
[[1,3],[2],[3],[4]] => [5,3,2,1,4] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 2
[[1],[8]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[2],[8]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[3],[8]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[4],[8]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[5],[8]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[6],[8]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[7],[8]] => [2,1] => [2,1] => ([(0,1)],2) => 1
[[1],[2],[7]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[1],[3],[7]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[1],[4],[7]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[1],[5],[7]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[1],[6],[7]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[2],[3],[7]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[2],[4],[7]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[2],[5],[7]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[2],[6],[7]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[3],[4],[7]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[3],[5],[7]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[3],[6],[7]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[4],[5],[7]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[4],[6],[7]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[5],[6],[7]] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => 1
[[1,2],[2],[6]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[1,3],[2],[6]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[1,4],[2],[6]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[1,5],[2],[6]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[1,3],[3],[6]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[1,4],[3],[6]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[1,5],[3],[6]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[1,4],[4],[6]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[1,5],[4],[6]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[1,5],[5],[6]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
>>> Load all 170 entries. <<<
[[2,3],[3],[6]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[2,4],[3],[6]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[2,5],[3],[6]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[2,4],[4],[6]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[2,5],[4],[6]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[2,5],[5],[6]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[3,4],[4],[6]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[3,5],[4],[6]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[3,5],[5],[6]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[4,5],[5],[6]] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 2
[[1],[2],[3],[6]] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
[[1],[2],[4],[6]] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
[[1],[2],[5],[6]] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
[[1],[3],[4],[6]] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
[[1],[3],[5],[6]] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
[[1],[4],[5],[6]] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
[[2],[3],[4],[6]] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
[[2],[3],[5],[6]] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
[[2],[4],[5],[6]] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
[[3],[4],[5],[6]] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
[[1,2],[2,5],[3]] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[1,2],[2,5],[4]] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[1,2],[2,5],[5]] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[1,3],[2,5],[4]] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[1,3],[2,5],[5]] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[1,4],[2,5],[5]] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[1,3],[3,5],[4]] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[1,3],[3,5],[5]] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[1,4],[3,5],[5]] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[1,4],[4,5],[5]] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[2,3],[3,5],[4]] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[2,3],[3,5],[5]] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[2,4],[3,5],[5]] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[2,4],[4,5],[5]] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[3,4],[4,5],[5]] => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[1,2],[2],[3],[5]] => [5,4,2,1,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 2
[[1,2],[2],[4],[5]] => [5,4,2,1,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 2
[[1,3],[2],[3],[5]] => [5,3,2,1,4] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 2
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 2
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 2
[[1,4],[2],[4],[5]] => [5,3,2,1,4] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 2
[[1,3],[3],[4],[5]] => [5,4,2,1,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 2
[[1,4],[3],[4],[5]] => [5,3,2,1,4] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 2
[[2,3],[3],[4],[5]] => [5,4,2,1,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 2
[[2,4],[3],[4],[5]] => [5,3,2,1,4] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(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],[2],[3],[4]] => [6,4,2,1,3,5] => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2
[[1,3,3],[2],[3],[4]] => [6,3,2,1,4,5] => [3,2,6,1,4,5] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 2
[[1,2],[2,3],[3,4]] => [4,6,2,5,1,3] => [2,4,1,6,5,3] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => 2
[[1,2],[2,3],[3],[4]] => [6,4,2,5,1,3] => [2,4,6,1,5,3] => ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 2
[[1,2],[2,4],[3],[4]] => [5,4,2,6,1,3] => [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 2
[[1,3],[2,4],[3],[4]] => [5,3,2,6,1,4] => [3,5,2,1,6,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6) => 2
[[1]] => [1] => [1] => ([],1) => 0
[[2]] => [1] => [1] => ([],1) => 0
[[3]] => [1] => [1] => ([],1) => 0
[[4]] => [1] => [1] => ([],1) => 0
[[5]] => [1] => [1] => ([],1) => 0
[[6]] => [1] => [1] => ([],1) => 0
[[1,3,5],[2],[4],[6]] => [6,4,2,1,3,5] => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2
[[1,3],[2],[4],[5],[6]] => [6,5,4,2,1,3] => [2,6,5,4,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,4],[2,5],[3],[6]] => [6,3,2,5,1,4] => [3,2,6,1,5,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6) => 2
[[1,4,5],[2],[3],[6]] => [6,3,2,1,4,5] => [3,2,6,1,4,5] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 2
[[1,4],[2],[3],[5],[6]] => [6,5,3,2,1,4] => [3,6,5,2,1,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,3],[2,5],[4,6]] => [4,6,2,5,1,3] => [2,4,1,6,5,3] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => 2
[[1,3],[2,5],[4],[6]] => [6,4,2,5,1,3] => [2,4,6,1,5,3] => ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 2
[[1,5],[2],[3],[4],[6]] => [6,4,3,2,1,5] => [4,6,3,2,1,5] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[[1,3],[2,6],[4],[5]] => [5,4,2,6,1,3] => [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 2
[[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => [3,5,2,1,6,4] => ([(0,2),(1,4),(1,5),(2,3),(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
search for individual values
searching the database for the individual values of this statistic
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottommost row (in English notation).
Map
Foata bijection
Description
Sends a permutation to its image under the Foata bijection.
The Foata bijection $\phi$ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$.
At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
  • If $w_{i+1} \geq v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$.
  • If $w_{i+1} < v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$.
In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
  • $1$
  • $|1|4 \to 14$
  • $|14|2 \to 412$
  • $|4|1|2|5 \to 4125$
  • $|4|125|3 \to 45123.$
In total, this gives $\phi([1,4,2,5,3]) = [4,5,1,2,3]$.
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).
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.