- FindStat

Identifier

Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000315: Graphs ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

$F_{2} = 2\ q^{2}$

$F_{3} = q + 3\ q^{3}$

$F_{4} = 3\ q + 3\ q^{2} + 4\ q^{4}$

$F_{5} = 8\ q + 8\ q^{2} + 5\ q^{3} + 5\ q^{5}$

$F_{6} = 25\ q + 23\ q^{2} + 15\ q^{3} + 7\ q^{4} + 6\ q^{6}$

Description

The number of isolated vertices of a graph.

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

valley composition

Description

The composition corresponding to the valley set of a standard tableau.
Let

$T$ be a standard tableau of size

$n$ .
An entry

$i$ of

$T$ is a descent if

$i+1$ is in a lower row (in English notation), otherwise

$i$ is an ascent.
An entry

$2 \leq i \leq n-1$ is a valley if

$i-1$ is a descent and

$i$ is an ascent.
This map returns the composition

$c_1,\dots,c_k$ of

$n$ such that

$\{c_1, c_1+c_2,\dots, c_1+\dots+c_k\}$ is the valley set of

$T$ .

Map

rotate back to front

Description

The back to front rotation of an integer composition.