Identifier
Values
[1,1] => [1,1,0,0] => [2,1] => ([(0,1)],2) => 1
[1,1,1] => [1,1,0,1,0,0] => [2,3,1] => ([(0,2),(1,2)],3) => 1
[2,2] => [1,1,1,0,0,0] => [3,1,2] => ([(0,2),(1,2)],3) => 1
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4) => 1
[2,2,1] => [1,1,1,0,0,1,0,0] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4) => 1
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 1
[3,3] => [1,1,1,0,1,0,0,0] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4) => 2
[2,2,2] => [1,1,1,1,0,0,0,0] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4) => 1
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 1
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 2
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 1
[1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 1
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 3
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 2
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 2
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 3
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 1
[2,2,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [3,1,4,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 1
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 3
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5) => 1
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => 2
[3,3,1,1,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0] => [3,4,1,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 2
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [4,5,1,2,6,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 3
[2,2,2,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [4,1,2,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 1
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 4
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 3
[4,4,1,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0] => [3,4,5,1,6,7,2] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 3
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 1
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => 4
[3,3,2,1,1] => [1,1,1,0,1,1,0,0,0,1,0,1,0,0] => [3,5,1,2,6,7,4] => ([(0,6),(1,6),(2,3),(2,4),(3,5),(4,5),(5,6)],7) => 2
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 5
[2,2,2,2,1,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0] => [4,5,1,2,6,7,3] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 3
[5,5,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0] => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 4
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 2
[4,4,2,1] => [1,1,1,0,1,0,1,1,0,0,0,1,0,0] => [3,4,6,1,2,7,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,6),(4,6),(5,6)],7) => 3
[3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [5,1,6,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 3
[3,3,3,1,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0] => [5,1,2,3,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 1
[3,3,2,2,1] => [1,1,1,0,1,1,0,1,0,0,0,1,0,0] => [3,5,6,1,2,7,4] => ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => 4
[2,2,2,2,2,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0] => [4,5,6,1,2,7,3] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7) => 5
[6,6] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [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) => 5
[5,5,2] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => [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) => 4
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 4
[4,4,3,1] => [1,1,1,0,1,1,1,0,0,0,0,1,0,0] => [3,6,1,2,4,7,5] => ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7) => 2
[4,4,2,2] => [1,1,1,0,1,0,1,1,0,1,0,0,0,0] => [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) => 5
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 1
[3,3,3,2,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0] => [5,1,6,2,3,7,4] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 3
[3,3,2,2,2] => [1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [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) => 6
[2,2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => 7
[5,5,3] => [1,1,1,0,1,0,1,1,1,0,0,0,0,0] => [3,4,7,1,2,5,6] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 3
[4,4,4,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0] => [5,6,1,2,3,7,4] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 4
[4,4,3,2] => [1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [3,6,1,7,2,4,5] => ([(0,1),(0,6),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7) => 4
[3,3,3,3,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [6,1,2,3,4,7,5] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 1
[3,3,3,2,2] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [5,1,6,7,2,3,4] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7) => 5
[5,5,4] => [1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [3,6,7,1,2,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7) => 5
[4,4,4,2] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [5,6,1,7,2,3,4] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 6
[4,4,3,3] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [3,7,1,2,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 2
[3,3,3,3,2] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [6,1,2,7,3,4,5] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 3
[5,5,5] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => 7
[4,4,4,3] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [5,7,1,2,3,4,6] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 4
[3,3,3,3,3] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [6,7,1,2,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 5
[4,4,4,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 1
search for individual values
searching the database for the individual values of this statistic
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
to 321-avoiding permutation (Krattenthaler)
Description
Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
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.