- FindStat

Your data matches 68 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000297
(load all 7 compositions to match this statistic)

Mapping

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => 1 => 1
{{1,2}}
 => [2] => 10 => 1
{{1},{2}}
 => [1,1] => 11 => 2
{{1,2,3}}
 => [3] => 100 => 1
{{1,2},{3}}
 => [2,1] => 101 => 1
{{1,3},{2}}
 => [2,1] => 101 => 1
{{1},{2,3}}
 => [1,2] => 110 => 2
{{1},{2},{3}}
 => [1,1,1] => 111 => 3
{{1,2,3,4}}
 => [4] => 1000 => 1
{{1,2,3},{4}}
 => [3,1] => 1001 => 1
{{1,2,4},{3}}
 => [3,1] => 1001 => 1
{{1,2},{3,4}}
 => [2,2] => 1010 => 1
{{1,2},{3},{4}}
 => [2,1,1] => 1011 => 1
{{1,3,4},{2}}
 => [3,1] => 1001 => 1
{{1,3},{2,4}}
 => [2,2] => 1010 => 1
{{1,3},{2},{4}}
 => [2,1,1] => 1011 => 1
{{1,4},{2,3}}
 => [2,2] => 1010 => 1
{{1},{2,3,4}}
 => [1,3] => 1100 => 2
{{1},{2,3},{4}}
 => [1,2,1] => 1101 => 2
{{1,4},{2},{3}}
 => [2,1,1] => 1011 => 1
{{1},{2,4},{3}}
 => [1,2,1] => 1101 => 2
{{1},{2},{3,4}}
 => [1,1,2] => 1110 => 3
{{1},{2},{3},{4}}
 => [1,1,1,1] => 1111 => 4
{{1,2,3,4,5}}
 => [5] => 10000 => 1
{{1,2,3,4},{5}}
 => [4,1] => 10001 => 1
{{1,2,3,5},{4}}
 => [4,1] => 10001 => 1
{{1,2,3},{4,5}}
 => [3,2] => 10010 => 1
{{1,2,3},{4},{5}}
 => [3,1,1] => 10011 => 1
{{1,2,4,5},{3}}
 => [4,1] => 10001 => 1
{{1,2,4},{3,5}}
 => [3,2] => 10010 => 1
{{1,2,4},{3},{5}}
 => [3,1,1] => 10011 => 1
{{1,2,5},{3,4}}
 => [3,2] => 10010 => 1
{{1,2},{3,4,5}}
 => [2,3] => 10100 => 1
{{1,2},{3,4},{5}}
 => [2,2,1] => 10101 => 1
{{1,2,5},{3},{4}}
 => [3,1,1] => 10011 => 1
{{1,2},{3,5},{4}}
 => [2,2,1] => 10101 => 1
{{1,2},{3},{4,5}}
 => [2,1,2] => 10110 => 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => 10111 => 1
{{1,3,4,5},{2}}
 => [4,1] => 10001 => 1
{{1,3,4},{2,5}}
 => [3,2] => 10010 => 1
{{1,3,4},{2},{5}}
 => [3,1,1] => 10011 => 1
{{1,3,5},{2,4}}
 => [3,2] => 10010 => 1
{{1,3},{2,4,5}}
 => [2,3] => 10100 => 1
{{1,3},{2,4},{5}}
 => [2,2,1] => 10101 => 1
{{1,3,5},{2},{4}}
 => [3,1,1] => 10011 => 1
{{1,3},{2,5},{4}}
 => [2,2,1] => 10101 => 1
{{1,3},{2},{4,5}}
 => [2,1,2] => 10110 => 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => 10111 => 1
{{1,4,5},{2,3}}
 => [3,2] => 10010 => 1
{{1,4},{2,3,5}}
 => [2,3] => 10100 => 1

Description

The number of leading ones in a binary word.

Matching statistic: St000363

Mapping

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000363: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of minimal vertex covers of a graph. A '''vertex cover''' of a graph

$G$ is a subset

$S$ of the vertices of

$G$ such that each edge of

$G$ contains at least one vertex of

$S$ . A vertex cover is minimal if it contains the least possible number of vertices. This is also the leading coefficient of the clique polynomial of the complement of

$G$ . This is also the number of independent sets of maximal cardinality of

$G$ .

Matching statistic: St000382
(load all 5 compositions to match this statistic)

Mapping

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => [1] => 1
{{1,2}}
 => [2] => [1,1] => 1
{{1},{2}}
 => [1,1] => [2] => 2
{{1,2,3}}
 => [3] => [1,1,1] => 1
{{1,2},{3}}
 => [2,1] => [1,2] => 1
{{1,3},{2}}
 => [2,1] => [1,2] => 1
{{1},{2,3}}
 => [1,2] => [2,1] => 2
{{1},{2},{3}}
 => [1,1,1] => [3] => 3
{{1,2,3,4}}
 => [4] => [1,1,1,1] => 1
{{1,2,3},{4}}
 => [3,1] => [1,1,2] => 1
{{1,2,4},{3}}
 => [3,1] => [1,1,2] => 1
{{1,2},{3,4}}
 => [2,2] => [1,2,1] => 1
{{1,2},{3},{4}}
 => [2,1,1] => [1,3] => 1
{{1,3,4},{2}}
 => [3,1] => [1,1,2] => 1
{{1,3},{2,4}}
 => [2,2] => [1,2,1] => 1
{{1,3},{2},{4}}
 => [2,1,1] => [1,3] => 1
{{1,4},{2,3}}
 => [2,2] => [1,2,1] => 1
{{1},{2,3,4}}
 => [1,3] => [2,1,1] => 2
{{1},{2,3},{4}}
 => [1,2,1] => [2,2] => 2
{{1,4},{2},{3}}
 => [2,1,1] => [1,3] => 1
{{1},{2,4},{3}}
 => [1,2,1] => [2,2] => 2
{{1},{2},{3,4}}
 => [1,1,2] => [3,1] => 3
{{1},{2},{3},{4}}
 => [1,1,1,1] => [4] => 4
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1] => 1
{{1,2,3,4},{5}}
 => [4,1] => [1,1,1,2] => 1
{{1,2,3,5},{4}}
 => [4,1] => [1,1,1,2] => 1
{{1,2,3},{4,5}}
 => [3,2] => [1,1,2,1] => 1
{{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,3] => 1
{{1,2,4,5},{3}}
 => [4,1] => [1,1,1,2] => 1
{{1,2,4},{3,5}}
 => [3,2] => [1,1,2,1] => 1
{{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,3] => 1
{{1,2,5},{3,4}}
 => [3,2] => [1,1,2,1] => 1
{{1,2},{3,4,5}}
 => [2,3] => [1,2,1,1] => 1
{{1,2},{3,4},{5}}
 => [2,2,1] => [1,2,2] => 1
{{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,3] => 1
{{1,2},{3,5},{4}}
 => [2,2,1] => [1,2,2] => 1
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,3,1] => 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,4] => 1
{{1,3,4,5},{2}}
 => [4,1] => [1,1,1,2] => 1
{{1,3,4},{2,5}}
 => [3,2] => [1,1,2,1] => 1
{{1,3,4},{2},{5}}
 => [3,1,1] => [1,1,3] => 1
{{1,3,5},{2,4}}
 => [3,2] => [1,1,2,1] => 1
{{1,3},{2,4,5}}
 => [2,3] => [1,2,1,1] => 1
{{1,3},{2,4},{5}}
 => [2,2,1] => [1,2,2] => 1
{{1,3,5},{2},{4}}
 => [3,1,1] => [1,1,3] => 1
{{1,3},{2,5},{4}}
 => [2,2,1] => [1,2,2] => 1
{{1,3},{2},{4,5}}
 => [2,1,2] => [1,3,1] => 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [1,4] => 1
{{1,4,5},{2,3}}
 => [3,2] => [1,1,2,1] => 1
{{1,4},{2,3,5}}
 => [2,3] => [1,2,1,1] => 1

Description

The first part of an integer composition.

Matching statistic: St000383
(load all 7 compositions to match this statistic)

Mapping

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => [1] => 1
{{1,2}}
 => [2] => [1,1] => 1
{{1},{2}}
 => [1,1] => [2] => 2
{{1,2,3}}
 => [3] => [1,1,1] => 1
{{1,2},{3}}
 => [2,1] => [2,1] => 1
{{1,3},{2}}
 => [2,1] => [2,1] => 1
{{1},{2,3}}
 => [1,2] => [1,2] => 2
{{1},{2},{3}}
 => [1,1,1] => [3] => 3
{{1,2,3,4}}
 => [4] => [1,1,1,1] => 1
{{1,2,3},{4}}
 => [3,1] => [2,1,1] => 1
{{1,2,4},{3}}
 => [3,1] => [2,1,1] => 1
{{1,2},{3,4}}
 => [2,2] => [1,2,1] => 1
{{1,2},{3},{4}}
 => [2,1,1] => [3,1] => 1
{{1,3,4},{2}}
 => [3,1] => [2,1,1] => 1
{{1,3},{2,4}}
 => [2,2] => [1,2,1] => 1
{{1,3},{2},{4}}
 => [2,1,1] => [3,1] => 1
{{1,4},{2,3}}
 => [2,2] => [1,2,1] => 1
{{1},{2,3,4}}
 => [1,3] => [1,1,2] => 2
{{1},{2,3},{4}}
 => [1,2,1] => [2,2] => 2
{{1,4},{2},{3}}
 => [2,1,1] => [3,1] => 1
{{1},{2,4},{3}}
 => [1,2,1] => [2,2] => 2
{{1},{2},{3,4}}
 => [1,1,2] => [1,3] => 3
{{1},{2},{3},{4}}
 => [1,1,1,1] => [4] => 4
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1] => 1
{{1,2,3,4},{5}}
 => [4,1] => [2,1,1,1] => 1
{{1,2,3,5},{4}}
 => [4,1] => [2,1,1,1] => 1
{{1,2,3},{4,5}}
 => [3,2] => [1,2,1,1] => 1
{{1,2,3},{4},{5}}
 => [3,1,1] => [3,1,1] => 1
{{1,2,4,5},{3}}
 => [4,1] => [2,1,1,1] => 1
{{1,2,4},{3,5}}
 => [3,2] => [1,2,1,1] => 1
{{1,2,4},{3},{5}}
 => [3,1,1] => [3,1,1] => 1
{{1,2,5},{3,4}}
 => [3,2] => [1,2,1,1] => 1
{{1,2},{3,4,5}}
 => [2,3] => [1,1,2,1] => 1
{{1,2},{3,4},{5}}
 => [2,2,1] => [2,2,1] => 1
{{1,2,5},{3},{4}}
 => [3,1,1] => [3,1,1] => 1
{{1,2},{3,5},{4}}
 => [2,2,1] => [2,2,1] => 1
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,3,1] => 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [4,1] => 1
{{1,3,4,5},{2}}
 => [4,1] => [2,1,1,1] => 1
{{1,3,4},{2,5}}
 => [3,2] => [1,2,1,1] => 1
{{1,3,4},{2},{5}}
 => [3,1,1] => [3,1,1] => 1
{{1,3,5},{2,4}}
 => [3,2] => [1,2,1,1] => 1
{{1,3},{2,4,5}}
 => [2,3] => [1,1,2,1] => 1
{{1,3},{2,4},{5}}
 => [2,2,1] => [2,2,1] => 1
{{1,3,5},{2},{4}}
 => [3,1,1] => [3,1,1] => 1
{{1,3},{2,5},{4}}
 => [2,2,1] => [2,2,1] => 1
{{1,3},{2},{4,5}}
 => [2,1,2] => [1,3,1] => 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [4,1] => 1
{{1,4,5},{2,3}}
 => [3,2] => [1,2,1,1] => 1
{{1,4},{2,3,5}}
 => [2,3] => [1,1,2,1] => 1

Description

The last part of an integer composition.

Matching statistic: St000745

Mapping

Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [[1]]
 => [[1]]
 => 1
{{1,2}}
 => [[1,2]]
 => [[1],[2]]
 => 2
{{1},{2}}
 => [[1],[2]]
 => [[1,2]]
 => 1
{{1,2,3}}
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 3
{{1,2},{3}}
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
{{1,3},{2}}
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 1
{{1},{2,3}}
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 1
{{1},{2},{3}}
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 1
{{1,2,3,4}}
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4
{{1,2,3},{4}}
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 3
{{1,2,4},{3}}
 => [[1,2,4],[3]]
 => [[1,3],[2],[4]]
 => 2
{{1,2},{3,4}}
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
{{1,2},{3},{4}}
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 2
{{1,3,4},{2}}
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 1
{{1,3},{2,4}}
 => [[1,3],[2,4]]
 => [[1,2],[3,4]]
 => 1
{{1,3},{2},{4}}
 => [[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => 1
{{1,4},{2,3}}
 => [[1,3],[2,4]]
 => [[1,2],[3,4]]
 => 1
{{1},{2,3,4}}
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 1
{{1},{2,3},{4}}
 => [[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => 1
{{1,4},{2},{3}}
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 1
{{1},{2,4},{3}}
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 1
{{1},{2},{3,4}}
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 1
{{1},{2},{3},{4}}
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 1
{{1,2,3,4,5}}
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 5
{{1,2,3,4},{5}}
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 4
{{1,2,3,5},{4}}
 => [[1,2,3,5],[4]]
 => [[1,4],[2],[3],[5]]
 => 3
{{1,2,3},{4,5}}
 => [[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => 3
{{1,2,3},{4},{5}}
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 3
{{1,2,4,5},{3}}
 => [[1,2,4,5],[3]]
 => [[1,3],[2],[4],[5]]
 => 2
{{1,2,4},{3,5}}
 => [[1,2,4],[3,5]]
 => [[1,3],[2,5],[4]]
 => 2
{{1,2,4},{3},{5}}
 => [[1,2,4],[3],[5]]
 => [[1,3,5],[2],[4]]
 => 2
{{1,2,5},{3,4}}
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 2
{{1,2},{3,4,5}}
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 2
{{1,2},{3,4},{5}}
 => [[1,2],[3,4],[5]]
 => [[1,3,5],[2,4]]
 => 2
{{1,2,5},{3},{4}}
 => [[1,2,5],[3],[4]]
 => [[1,3,4],[2],[5]]
 => 2
{{1,2},{3,5},{4}}
 => [[1,2],[3,5],[4]]
 => [[1,3,4],[2,5]]
 => 2
{{1,2},{3},{4,5}}
 => [[1,2],[3,5],[4]]
 => [[1,3,4],[2,5]]
 => 2
{{1,2},{3},{4},{5}}
 => [[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => 2
{{1,3,4,5},{2}}
 => [[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => 1
{{1,3,4},{2,5}}
 => [[1,3,4],[2,5]]
 => [[1,2],[3,5],[4]]
 => 1
{{1,3,4},{2},{5}}
 => [[1,3,4],[2],[5]]
 => [[1,2,5],[3],[4]]
 => 1
{{1,3,5},{2,4}}
 => [[1,3,5],[2,4]]
 => [[1,2],[3,4],[5]]
 => 1
{{1,3},{2,4,5}}
 => [[1,3,5],[2,4]]
 => [[1,2],[3,4],[5]]
 => 1
{{1,3},{2,4},{5}}
 => [[1,3],[2,4],[5]]
 => [[1,2,5],[3,4]]
 => 1
{{1,3,5},{2},{4}}
 => [[1,3,5],[2],[4]]
 => [[1,2,4],[3],[5]]
 => 1
{{1,3},{2,5},{4}}
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 1
{{1,3},{2},{4,5}}
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 1
{{1,3},{2},{4},{5}}
 => [[1,3],[2],[4],[5]]
 => [[1,2,4,5],[3]]
 => 1
{{1,4,5},{2,3}}
 => [[1,3,5],[2,4]]
 => [[1,2],[3,4],[5]]
 => 1
{{1,4},{2,3,5}}
 => [[1,3,5],[2,4]]
 => [[1,2],[3,4],[5]]
 => 1

Description

The index of the last row whose first entry is the row number in a standard Young tableau.

Matching statistic: St000025

Mapping

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => [1] => [1,0]
 => 1
{{1,2}}
 => [2] => [1,1] => [1,0,1,0]
 => 1
{{1},{2}}
 => [1,1] => [2] => [1,1,0,0]
 => 2
{{1,2,3}}
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 1
{{1,2},{3}}
 => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 1
{{1,3},{2}}
 => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 1
{{1},{2,3}}
 => [1,2] => [2,1] => [1,1,0,0,1,0]
 => 2
{{1},{2},{3}}
 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 3
{{1,2,3,4}}
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
{{1,2,3},{4}}
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
{{1,2,4},{3}}
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
{{1,2},{3,4}}
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
{{1,2},{3},{4}}
 => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
{{1,3,4},{2}}
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
{{1,3},{2,4}}
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
{{1,3},{2},{4}}
 => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
{{1,4},{2,3}}
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
{{1},{2,3,4}}
 => [1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
{{1},{2,3},{4}}
 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
{{1,4},{2},{3}}
 => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
{{1},{2,4},{3}}
 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
{{1},{2},{3,4}}
 => [1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
{{1},{2},{3},{4}}
 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 4
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
{{1,2,3,4},{5}}
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,2,3,5},{4}}
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,2,3},{4,5}}
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
{{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2,4,5},{3}}
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,2,4},{3,5}}
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
{{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2,5},{3,4}}
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
{{1,2},{3,4,5}}
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
{{1,2},{3,4},{5}}
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
{{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2},{3,5},{4}}
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
{{1,3,4,5},{2}}
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,3,4},{2,5}}
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
{{1,3,4},{2},{5}}
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3,5},{2,4}}
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
{{1,3},{2,4,5}}
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
{{1,3},{2,4},{5}}
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
{{1,3,5},{2},{4}}
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2,5},{4}}
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
{{1,3},{2},{4,5}}
 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
{{1,4,5},{2,3}}
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
{{1,4},{2,3,5}}
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1

Description

The number of initial rises of a Dyck path. In other words, this is the height of the first peak of

$D$ .

Matching statistic: St000026
(load all 2 compositions to match this statistic)

Mapping

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => [1] => [1,0]
 => 1
{{1,2}}
 => [2] => [1,1] => [1,0,1,0]
 => 1
{{1},{2}}
 => [1,1] => [2] => [1,1,0,0]
 => 2
{{1,2,3}}
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 1
{{1,2},{3}}
 => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 1
{{1,3},{2}}
 => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 1
{{1},{2,3}}
 => [1,2] => [2,1] => [1,1,0,0,1,0]
 => 2
{{1},{2},{3}}
 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 3
{{1,2,3,4}}
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
{{1,2,3},{4}}
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
{{1,2,4},{3}}
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
{{1,2},{3,4}}
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
{{1,2},{3},{4}}
 => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
{{1,3,4},{2}}
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
{{1,3},{2,4}}
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
{{1,3},{2},{4}}
 => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
{{1,4},{2,3}}
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
{{1},{2,3,4}}
 => [1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
{{1},{2,3},{4}}
 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
{{1,4},{2},{3}}
 => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
{{1},{2,4},{3}}
 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
{{1},{2},{3,4}}
 => [1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
{{1},{2},{3},{4}}
 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 4
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
{{1,2,3,4},{5}}
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,2,3,5},{4}}
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,2,3},{4,5}}
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
{{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2,4,5},{3}}
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,2,4},{3,5}}
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
{{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2,5},{3,4}}
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
{{1,2},{3,4,5}}
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
{{1,2},{3,4},{5}}
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
{{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2},{3,5},{4}}
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
{{1,3,4,5},{2}}
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,3,4},{2,5}}
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
{{1,3,4},{2},{5}}
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3,5},{2,4}}
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
{{1,3},{2,4,5}}
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
{{1,3},{2,4},{5}}
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
{{1,3,5},{2},{4}}
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2,5},{4}}
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
{{1,3},{2},{4,5}}
 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
{{1,4,5},{2,3}}
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
{{1,4},{2,3,5}}
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1

Description

The position of the first return of a Dyck path.

Matching statistic: St000056
(load all 5 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => [1] => [1] => 1
{{1,2}}
 => [2,1] => [2,1] => [2,1] => 1
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 2
{{1,2,3}}
 => [2,3,1] => [3,2,1] => [3,2,1] => 1
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1,3] => 2
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => [2,3,1] => 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [3,1,2] => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,2,3] => 3
{{1,2,3,4}}
 => [2,3,4,1] => [4,3,2,1] => [4,3,2,1] => 1
{{1,2,3},{4}}
 => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 2
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,2,1] => [3,4,2,1] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [4,2,1,3] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 3
{{1,3,4},{2}}
 => [3,2,4,1] => [2,4,3,1] => [4,2,3,1] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => [3,2,4,1] => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,3,2] => [4,3,1,2] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 2
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => [2,3,4,1] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => [3,1,4,2] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,4,3,2,1] => [5,4,3,2,1] => 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 2
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,5,3,2,1] => [4,5,3,2,1] => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,2,1,5,4] => [5,3,2,1,4] => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 3
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [3,5,4,2,1] => [5,3,4,2,1] => 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [5,2,1,4,3] => [5,2,4,1,3] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [3,4,2,1,5] => [3,4,2,1,5] => 2
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [4,3,5,2,1] => [4,3,5,2,1] => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,4,3] => [5,4,2,1,3] => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => [4,2,1,3,5] => 2
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [3,4,5,2,1] => [3,4,5,2,1] => 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,4,5,3] => [4,2,1,5,3] => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => [5,2,1,3,4] => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 4
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,5,4,3,1] => [5,4,2,3,1] => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [4,1,5,3,2] => [4,5,3,1,2] => 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,4,3,1,5] => [4,2,3,1,5] => 2
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [5,2,4,3,1] => [5,2,4,3,1] => 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [5,4,1,3,2] => [5,1,4,3,2] => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => 2
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,4,5,3,1] => [4,2,5,3,1] => 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [4,5,1,3,2] => [4,1,5,3,2] => 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,3,1,5,4] => [5,2,3,1,4] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => 3
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,5,4,1] => [5,3,2,4,1] => 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [5,3,1,4,2] => [3,5,4,1,2] => 1

Description

The decomposition (or block) number of a permutation. For

$\pi \in \mathcal{S}_n$ , this is given by

$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$ This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum. This is one plus [[St000234]].

Matching statistic: St000273

Mapping

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000273: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The domination number of a graph. The domination number of a graph is given by the minimum size of a dominating set of vertices. A dominating set of vertices is a subset of the vertex set of such that every vertex is either in this subset or adjacent to an element of this subset.

Matching statistic: St000286

Mapping

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000286: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of connected components of the complement of a graph. The complement of a graph is the graph on the same vertex set with complementary edges.

The following 58 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000287The number of connected components of a graph. St000335The difference of lower and upper interactions. St000501The size of the first part in the decomposition of a permutation. St000544The cop number of a graph. St000617The number of global maxima of a Dyck path. St000759The smallest missing part in an integer partition. St000916The packing number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001316The domatic number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001481The minimal height of a peak of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St001829The common independence number of a graph. St000234The number of global ascents of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000439The position of the first down step of a Dyck path. St000546The number of global descents of a permutation. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000729The minimal arc length of a set partition. St000989The number of final rises of a permutation. St000654The first descent of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000990The first ascent of a permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000054The first entry of the permutation. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001889The size of the connectivity set of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function.