- FindStat

Identifier

Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000262: Graphs ⟶ ℤ

Values

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 5$

$F_{4} = 12 + 2\ q$

Description

The vertex connectivity of a graph.
For non-complete graphs, this is the minimum number of vertices that has to be removed to make the graph disconnected.

Map

incomparability graph

Description

The incomparability graph of a poset.

Map

skew partition

Description

The parallelogram polyomino corresponding to a Dyck path, interpreted as a skew partition.
Let

$D$ be a Dyck path of semilength

$n$ . The parallelogram polyomino

$\gamma(D)$ is defined as follows: let

$\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to

$D$ . Then, the upper path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with even indices, and the lower path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with odd indices.
This map returns the skew partition definded by the diagram of

$\gamma(D)$ .

Map

cell poset

Description

The Young diagram of a skew partition regarded as a poset.
This is the poset on the cells of the Young diagram, such that a cell

$d$ is greater than a cell

$c$ if the entry in

$d$ must be larger than the entry of

$c$ in any standard Young tableau on the skew partition.