Processing math: 100%

Identifier
Values
[1] => [[1]] => [1] => ([],1) => 1
[2] => [[1,2]] => [2] => ([],2) => 1
[1,1] => [[1],[2]] => [1,1] => ([(0,1)],2) => 2
[3] => [[1,2,3]] => [3] => ([],3) => 1
[2,1] => [[1,2],[3]] => [2,1] => ([(0,2),(1,2)],3) => 2
[1,1,1] => [[1],[2],[3]] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 6
[4] => [[1,2,3,4]] => [4] => ([],4) => 1
[3,1] => [[1,2,3],[4]] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2
[2,2] => [[1,2],[3,4]] => [2,2] => ([(1,3),(2,3)],4) => 4
[2,1,1] => [[1,2],[3],[4]] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 6
[1,1,1,1] => [[1],[2],[3],[4]] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 24
[5] => [[1,2,3,4,5]] => [5] => ([],5) => 1
[4,1] => [[1,2,3,4],[5]] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[3,2] => [[1,2,3],[4,5]] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 4
[3,1,1] => [[1,2,3],[4],[5]] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 6
[2,2,1] => [[1,2],[3,4],[5]] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 12
[2,1,1,1] => [[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) => 24
[1,1,1,1,1] => [[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) => 120
[6] => [[1,2,3,4,5,6]] => [6] => ([],6) => 1
[5,1] => [[1,2,3,4,5],[6]] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2
[4,2] => [[1,2,3,4],[5,6]] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[4,1,1] => [[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) => 6
[3,3] => [[1,2,3],[4,5,6]] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 8
[3,2,1] => [[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) => 12
[3,1,1,1] => [[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) => 24
[2,2,2] => [[1,2],[3,4],[5,6]] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 36
[2,2,1,1] => [[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) => 48
[2,1,1,1,1] => [[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) => 120
[1,1,1,1,1,1] => [[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) => 720
[7] => [[1,2,3,4,5,6,7]] => [7] => ([],7) => 1
[6,1] => [[1,2,3,4,5,6],[7]] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[5,2] => [[1,2,3,4,5],[6,7]] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 4
[5,1,1] => [[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) => 6
[4,3] => [[1,2,3,4],[5,6,7]] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 8
[4,2,1] => [[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) => 12
[4,1,1,1] => [[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) => 24
[3,3,1] => [[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) => 24
[3,2,2] => [[1,2,3],[4,5],[6,7]] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 36
[3,2,1,1] => [[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) => 48
[3,1,1,1,1] => [[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) => 120
[2,2,2,1] => [[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) => 144
[2,2,1,1,1] => [[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) => 240
[2,1,1,1,1,1] => [[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) => 720
[1,1,1,1,1,1,1] => [[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) => 5040
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Description
The number of proper colourings of a graph with as few colours as possible.
By definition, this is the evaluation of the chromatic polynomial at the first nonnegative integer which is not a zero of the polynomial.
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
horizontal strip sizes
Description
The composition of horizontal strip sizes.
We associate to a standard Young tableau T the composition (c1,,ck), such that k is minimal and the numbers c1++ci+1,,c1++ci+1 form a horizontal strip in T for all i.
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers 1 through n row by row.