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Your data matches 91 different statistics following compositions of up to 3 maps.
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Matching statistic: St000010
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> [1] => [1]
=> 1
[1,2] => [[1,2]]
=> [2] => [2]
=> 1
[2,1] => [[1],[2]]
=> [2] => [2]
=> 1
[1,2,3] => [[1,2,3]]
=> [3] => [3]
=> 1
[1,3,2] => [[1,2],[3]]
=> [2,1] => [2,1]
=> 2
[2,1,3] => [[1,3],[2]]
=> [3] => [3]
=> 1
[2,3,1] => [[1,3],[2]]
=> [3] => [3]
=> 1
[3,1,2] => [[1,2],[3]]
=> [2,1] => [2,1]
=> 2
[3,2,1] => [[1],[2],[3]]
=> [3] => [3]
=> 1
[1,2,3,4] => [[1,2,3,4]]
=> [4] => [4]
=> 1
[1,2,4,3] => [[1,2,3],[4]]
=> [3,1] => [3,1]
=> 2
[1,3,2,4] => [[1,2,4],[3]]
=> [2,2] => [2,2]
=> 2
[1,3,4,2] => [[1,2,4],[3]]
=> [2,2] => [2,2]
=> 2
[1,4,2,3] => [[1,2,3],[4]]
=> [3,1] => [3,1]
=> 2
[1,4,3,2] => [[1,2],[3],[4]]
=> [2,2] => [2,2]
=> 2
[2,1,3,4] => [[1,3,4],[2]]
=> [4] => [4]
=> 1
[2,1,4,3] => [[1,3],[2,4]]
=> [3,1] => [3,1]
=> 2
[2,3,1,4] => [[1,3,4],[2]]
=> [4] => [4]
=> 1
[2,3,4,1] => [[1,3,4],[2]]
=> [4] => [4]
=> 1
[2,4,1,3] => [[1,3],[2,4]]
=> [3,1] => [3,1]
=> 2
[2,4,3,1] => [[1,3],[2],[4]]
=> [3,1] => [3,1]
=> 2
[3,1,2,4] => [[1,2,4],[3]]
=> [2,2] => [2,2]
=> 2
[3,1,4,2] => [[1,2],[3,4]]
=> [2,2] => [2,2]
=> 2
[3,2,1,4] => [[1,4],[2],[3]]
=> [4] => [4]
=> 1
[3,2,4,1] => [[1,4],[2],[3]]
=> [4] => [4]
=> 1
[3,4,1,2] => [[1,2],[3,4]]
=> [2,2] => [2,2]
=> 2
[3,4,2,1] => [[1,4],[2],[3]]
=> [4] => [4]
=> 1
[4,1,2,3] => [[1,2,3],[4]]
=> [3,1] => [3,1]
=> 2
[4,1,3,2] => [[1,2],[3],[4]]
=> [2,2] => [2,2]
=> 2
[4,2,1,3] => [[1,3],[2],[4]]
=> [3,1] => [3,1]
=> 2
[4,2,3,1] => [[1,3],[2],[4]]
=> [3,1] => [3,1]
=> 2
[4,3,1,2] => [[1,2],[3],[4]]
=> [2,2] => [2,2]
=> 2
[4,3,2,1] => [[1],[2],[3],[4]]
=> [4] => [4]
=> 1
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [5] => [5]
=> 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [4,1] => [4,1]
=> 2
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [3,2] => [3,2]
=> 2
[1,2,4,5,3] => [[1,2,3,5],[4]]
=> [3,2] => [3,2]
=> 2
[1,2,5,3,4] => [[1,2,3,4],[5]]
=> [4,1] => [4,1]
=> 2
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [3,2] => [3,2]
=> 2
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [2,3] => [3,2]
=> 2
[1,3,4,2,5] => [[1,2,4,5],[3]]
=> [2,3] => [3,2]
=> 2
[1,3,4,5,2] => [[1,2,4,5],[3]]
=> [2,3] => [3,2]
=> 2
[1,4,2,3,5] => [[1,2,3,5],[4]]
=> [3,2] => [3,2]
=> 2
[1,4,2,5,3] => [[1,2,3],[4,5]]
=> [3,2] => [3,2]
=> 2
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [2,3] => [3,2]
=> 2
[1,4,3,5,2] => [[1,2,5],[3],[4]]
=> [2,3] => [3,2]
=> 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [3,2] => [3,2]
=> 2
[1,4,5,3,2] => [[1,2,5],[3],[4]]
=> [2,3] => [3,2]
=> 2
[1,5,2,3,4] => [[1,2,3,4],[5]]
=> [4,1] => [4,1]
=> 2
[1,5,2,4,3] => [[1,2,3],[4],[5]]
=> [3,2] => [3,2]
=> 2
Description
The length of the partition.
Matching statistic: St000097
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> [1] => ([],1)
=> 1
[1,2] => [[1,2]]
=> [2] => ([],2)
=> 1
[2,1] => [[1],[2]]
=> [2] => ([],2)
=> 1
[1,2,3] => [[1,2,3]]
=> [3] => ([],3)
=> 1
[1,3,2] => [[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [[1,3],[2]]
=> [3] => ([],3)
=> 1
[2,3,1] => [[1,3],[2]]
=> [3] => ([],3)
=> 1
[3,1,2] => [[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,2,1] => [[1],[2],[3]]
=> [3] => ([],3)
=> 1
[1,2,3,4] => [[1,2,3,4]]
=> [4] => ([],4)
=> 1
[1,2,4,3] => [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,3,2,4] => [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,3,4,2] => [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [[1,2],[3],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [[1,3,4],[2]]
=> [4] => ([],4)
=> 1
[2,1,4,3] => [[1,3],[2,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,3,1,4] => [[1,3,4],[2]]
=> [4] => ([],4)
=> 1
[2,3,4,1] => [[1,3,4],[2]]
=> [4] => ([],4)
=> 1
[2,4,1,3] => [[1,3],[2,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,4,3,1] => [[1,3],[2],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [[1,4],[2],[3]]
=> [4] => ([],4)
=> 1
[3,2,4,1] => [[1,4],[2],[3]]
=> [4] => ([],4)
=> 1
[3,4,1,2] => [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,4,2,1] => [[1,4],[2],[3]]
=> [4] => ([],4)
=> 1
[4,1,2,3] => [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,1,3,2] => [[1,2],[3],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[4,2,1,3] => [[1,3],[2],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [[1,3],[2],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,3,1,2] => [[1,2],[3],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[4,3,2,1] => [[1],[2],[3],[4]]
=> [4] => ([],4)
=> 1
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [5] => ([],5)
=> 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,2,4,5,3] => [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => [[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => [[1,2,5],[3],[4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,4,5,3,2] => [[1,2,5],[3],[4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,5,2,3,4] => [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,5,2,4,3] => [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000098
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> [1] => ([],1)
=> 1
[1,2] => [[1,2]]
=> [2] => ([],2)
=> 1
[2,1] => [[1],[2]]
=> [2] => ([],2)
=> 1
[1,2,3] => [[1,2,3]]
=> [3] => ([],3)
=> 1
[1,3,2] => [[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [[1,3],[2]]
=> [3] => ([],3)
=> 1
[2,3,1] => [[1,3],[2]]
=> [3] => ([],3)
=> 1
[3,1,2] => [[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,2,1] => [[1],[2],[3]]
=> [3] => ([],3)
=> 1
[1,2,3,4] => [[1,2,3,4]]
=> [4] => ([],4)
=> 1
[1,2,4,3] => [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,3,2,4] => [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,3,4,2] => [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [[1,2],[3],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [[1,3,4],[2]]
=> [4] => ([],4)
=> 1
[2,1,4,3] => [[1,3],[2,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,3,1,4] => [[1,3,4],[2]]
=> [4] => ([],4)
=> 1
[2,3,4,1] => [[1,3,4],[2]]
=> [4] => ([],4)
=> 1
[2,4,1,3] => [[1,3],[2,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,4,3,1] => [[1,3],[2],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [[1,4],[2],[3]]
=> [4] => ([],4)
=> 1
[3,2,4,1] => [[1,4],[2],[3]]
=> [4] => ([],4)
=> 1
[3,4,1,2] => [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,4,2,1] => [[1,4],[2],[3]]
=> [4] => ([],4)
=> 1
[4,1,2,3] => [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,1,3,2] => [[1,2],[3],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[4,2,1,3] => [[1,3],[2],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [[1,3],[2],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,3,1,2] => [[1,2],[3],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[4,3,2,1] => [[1],[2],[3],[4]]
=> [4] => ([],4)
=> 1
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [5] => ([],5)
=> 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,2,4,5,3] => [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => [[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => [[1,2,5],[3],[4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,4,5,3,2] => [[1,2,5],[3],[4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,5,2,3,4] => [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,5,2,4,3] => [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St000288
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> [1] => 1 => 1
[1,2] => [[1,2]]
=> [2] => 10 => 1
[2,1] => [[1],[2]]
=> [2] => 10 => 1
[1,2,3] => [[1,2,3]]
=> [3] => 100 => 1
[1,3,2] => [[1,2],[3]]
=> [2,1] => 101 => 2
[2,1,3] => [[1,3],[2]]
=> [3] => 100 => 1
[2,3,1] => [[1,3],[2]]
=> [3] => 100 => 1
[3,1,2] => [[1,2],[3]]
=> [2,1] => 101 => 2
[3,2,1] => [[1],[2],[3]]
=> [3] => 100 => 1
[1,2,3,4] => [[1,2,3,4]]
=> [4] => 1000 => 1
[1,2,4,3] => [[1,2,3],[4]]
=> [3,1] => 1001 => 2
[1,3,2,4] => [[1,2,4],[3]]
=> [2,2] => 1010 => 2
[1,3,4,2] => [[1,2,4],[3]]
=> [2,2] => 1010 => 2
[1,4,2,3] => [[1,2,3],[4]]
=> [3,1] => 1001 => 2
[1,4,3,2] => [[1,2],[3],[4]]
=> [2,2] => 1010 => 2
[2,1,3,4] => [[1,3,4],[2]]
=> [4] => 1000 => 1
[2,1,4,3] => [[1,3],[2,4]]
=> [3,1] => 1001 => 2
[2,3,1,4] => [[1,3,4],[2]]
=> [4] => 1000 => 1
[2,3,4,1] => [[1,3,4],[2]]
=> [4] => 1000 => 1
[2,4,1,3] => [[1,3],[2,4]]
=> [3,1] => 1001 => 2
[2,4,3,1] => [[1,3],[2],[4]]
=> [3,1] => 1001 => 2
[3,1,2,4] => [[1,2,4],[3]]
=> [2,2] => 1010 => 2
[3,1,4,2] => [[1,2],[3,4]]
=> [2,2] => 1010 => 2
[3,2,1,4] => [[1,4],[2],[3]]
=> [4] => 1000 => 1
[3,2,4,1] => [[1,4],[2],[3]]
=> [4] => 1000 => 1
[3,4,1,2] => [[1,2],[3,4]]
=> [2,2] => 1010 => 2
[3,4,2,1] => [[1,4],[2],[3]]
=> [4] => 1000 => 1
[4,1,2,3] => [[1,2,3],[4]]
=> [3,1] => 1001 => 2
[4,1,3,2] => [[1,2],[3],[4]]
=> [2,2] => 1010 => 2
[4,2,1,3] => [[1,3],[2],[4]]
=> [3,1] => 1001 => 2
[4,2,3,1] => [[1,3],[2],[4]]
=> [3,1] => 1001 => 2
[4,3,1,2] => [[1,2],[3],[4]]
=> [2,2] => 1010 => 2
[4,3,2,1] => [[1],[2],[3],[4]]
=> [4] => 1000 => 1
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [5] => 10000 => 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [4,1] => 10001 => 2
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [3,2] => 10010 => 2
[1,2,4,5,3] => [[1,2,3,5],[4]]
=> [3,2] => 10010 => 2
[1,2,5,3,4] => [[1,2,3,4],[5]]
=> [4,1] => 10001 => 2
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [3,2] => 10010 => 2
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [2,3] => 10100 => 2
[1,3,4,2,5] => [[1,2,4,5],[3]]
=> [2,3] => 10100 => 2
[1,3,4,5,2] => [[1,2,4,5],[3]]
=> [2,3] => 10100 => 2
[1,4,2,3,5] => [[1,2,3,5],[4]]
=> [3,2] => 10010 => 2
[1,4,2,5,3] => [[1,2,3],[4,5]]
=> [3,2] => 10010 => 2
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [2,3] => 10100 => 2
[1,4,3,5,2] => [[1,2,5],[3],[4]]
=> [2,3] => 10100 => 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [3,2] => 10010 => 2
[1,4,5,3,2] => [[1,2,5],[3],[4]]
=> [2,3] => 10100 => 2
[1,5,2,3,4] => [[1,2,3,4],[5]]
=> [4,1] => 10001 => 2
[1,5,2,4,3] => [[1,2,3],[4],[5]]
=> [3,2] => 10010 => 2
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000346
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000346: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000346: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> [1] => [1]
=> 1
[1,2] => [[1,2]]
=> [2] => [2]
=> 1
[2,1] => [[1],[2]]
=> [2] => [2]
=> 1
[1,2,3] => [[1,2,3]]
=> [3] => [3]
=> 1
[1,3,2] => [[1,2],[3]]
=> [2,1] => [2,1]
=> 2
[2,1,3] => [[1,3],[2]]
=> [3] => [3]
=> 1
[2,3,1] => [[1,3],[2]]
=> [3] => [3]
=> 1
[3,1,2] => [[1,2],[3]]
=> [2,1] => [2,1]
=> 2
[3,2,1] => [[1],[2],[3]]
=> [3] => [3]
=> 1
[1,2,3,4] => [[1,2,3,4]]
=> [4] => [4]
=> 1
[1,2,4,3] => [[1,2,3],[4]]
=> [3,1] => [3,1]
=> 2
[1,3,2,4] => [[1,2,4],[3]]
=> [2,2] => [2,2]
=> 2
[1,3,4,2] => [[1,2,4],[3]]
=> [2,2] => [2,2]
=> 2
[1,4,2,3] => [[1,2,3],[4]]
=> [3,1] => [3,1]
=> 2
[1,4,3,2] => [[1,2],[3],[4]]
=> [2,2] => [2,2]
=> 2
[2,1,3,4] => [[1,3,4],[2]]
=> [4] => [4]
=> 1
[2,1,4,3] => [[1,3],[2,4]]
=> [3,1] => [3,1]
=> 2
[2,3,1,4] => [[1,3,4],[2]]
=> [4] => [4]
=> 1
[2,3,4,1] => [[1,3,4],[2]]
=> [4] => [4]
=> 1
[2,4,1,3] => [[1,3],[2,4]]
=> [3,1] => [3,1]
=> 2
[2,4,3,1] => [[1,3],[2],[4]]
=> [3,1] => [3,1]
=> 2
[3,1,2,4] => [[1,2,4],[3]]
=> [2,2] => [2,2]
=> 2
[3,1,4,2] => [[1,2],[3,4]]
=> [2,2] => [2,2]
=> 2
[3,2,1,4] => [[1,4],[2],[3]]
=> [4] => [4]
=> 1
[3,2,4,1] => [[1,4],[2],[3]]
=> [4] => [4]
=> 1
[3,4,1,2] => [[1,2],[3,4]]
=> [2,2] => [2,2]
=> 2
[3,4,2,1] => [[1,4],[2],[3]]
=> [4] => [4]
=> 1
[4,1,2,3] => [[1,2,3],[4]]
=> [3,1] => [3,1]
=> 2
[4,1,3,2] => [[1,2],[3],[4]]
=> [2,2] => [2,2]
=> 2
[4,2,1,3] => [[1,3],[2],[4]]
=> [3,1] => [3,1]
=> 2
[4,2,3,1] => [[1,3],[2],[4]]
=> [3,1] => [3,1]
=> 2
[4,3,1,2] => [[1,2],[3],[4]]
=> [2,2] => [2,2]
=> 2
[4,3,2,1] => [[1],[2],[3],[4]]
=> [4] => [4]
=> 1
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [5] => [5]
=> 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [4,1] => [4,1]
=> 2
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [3,2] => [3,2]
=> 2
[1,2,4,5,3] => [[1,2,3,5],[4]]
=> [3,2] => [3,2]
=> 2
[1,2,5,3,4] => [[1,2,3,4],[5]]
=> [4,1] => [4,1]
=> 2
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [3,2] => [3,2]
=> 2
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [2,3] => [3,2]
=> 2
[1,3,4,2,5] => [[1,2,4,5],[3]]
=> [2,3] => [3,2]
=> 2
[1,3,4,5,2] => [[1,2,4,5],[3]]
=> [2,3] => [3,2]
=> 2
[1,4,2,3,5] => [[1,2,3,5],[4]]
=> [3,2] => [3,2]
=> 2
[1,4,2,5,3] => [[1,2,3],[4,5]]
=> [3,2] => [3,2]
=> 2
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [2,3] => [3,2]
=> 2
[1,4,3,5,2] => [[1,2,5],[3],[4]]
=> [2,3] => [3,2]
=> 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [3,2] => [3,2]
=> 2
[1,4,5,3,2] => [[1,2,5],[3],[4]]
=> [2,3] => [3,2]
=> 2
[1,5,2,3,4] => [[1,2,3,4],[5]]
=> [4,1] => [4,1]
=> 2
[1,5,2,4,3] => [[1,2,3],[4],[5]]
=> [3,2] => [3,2]
=> 2
Description
The number of coarsenings of a partition.
A partition $\mu$ coarsens a partition $\lambda$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$.
Matching statistic: St000381
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> [1] => [1] => 1
[1,2] => [[1,2]]
=> [2] => [1,1] => 1
[2,1] => [[1],[2]]
=> [2] => [1,1] => 1
[1,2,3] => [[1,2,3]]
=> [3] => [1,1,1] => 1
[1,3,2] => [[1,2],[3]]
=> [2,1] => [2,1] => 2
[2,1,3] => [[1,3],[2]]
=> [3] => [1,1,1] => 1
[2,3,1] => [[1,3],[2]]
=> [3] => [1,1,1] => 1
[3,1,2] => [[1,2],[3]]
=> [2,1] => [2,1] => 2
[3,2,1] => [[1],[2],[3]]
=> [3] => [1,1,1] => 1
[1,2,3,4] => [[1,2,3,4]]
=> [4] => [1,1,1,1] => 1
[1,2,4,3] => [[1,2,3],[4]]
=> [3,1] => [2,1,1] => 2
[1,3,2,4] => [[1,2,4],[3]]
=> [2,2] => [1,2,1] => 2
[1,3,4,2] => [[1,2,4],[3]]
=> [2,2] => [1,2,1] => 2
[1,4,2,3] => [[1,2,3],[4]]
=> [3,1] => [2,1,1] => 2
[1,4,3,2] => [[1,2],[3],[4]]
=> [2,2] => [1,2,1] => 2
[2,1,3,4] => [[1,3,4],[2]]
=> [4] => [1,1,1,1] => 1
[2,1,4,3] => [[1,3],[2,4]]
=> [3,1] => [2,1,1] => 2
[2,3,1,4] => [[1,3,4],[2]]
=> [4] => [1,1,1,1] => 1
[2,3,4,1] => [[1,3,4],[2]]
=> [4] => [1,1,1,1] => 1
[2,4,1,3] => [[1,3],[2,4]]
=> [3,1] => [2,1,1] => 2
[2,4,3,1] => [[1,3],[2],[4]]
=> [3,1] => [2,1,1] => 2
[3,1,2,4] => [[1,2,4],[3]]
=> [2,2] => [1,2,1] => 2
[3,1,4,2] => [[1,2],[3,4]]
=> [2,2] => [1,2,1] => 2
[3,2,1,4] => [[1,4],[2],[3]]
=> [4] => [1,1,1,1] => 1
[3,2,4,1] => [[1,4],[2],[3]]
=> [4] => [1,1,1,1] => 1
[3,4,1,2] => [[1,2],[3,4]]
=> [2,2] => [1,2,1] => 2
[3,4,2,1] => [[1,4],[2],[3]]
=> [4] => [1,1,1,1] => 1
[4,1,2,3] => [[1,2,3],[4]]
=> [3,1] => [2,1,1] => 2
[4,1,3,2] => [[1,2],[3],[4]]
=> [2,2] => [1,2,1] => 2
[4,2,1,3] => [[1,3],[2],[4]]
=> [3,1] => [2,1,1] => 2
[4,2,3,1] => [[1,3],[2],[4]]
=> [3,1] => [2,1,1] => 2
[4,3,1,2] => [[1,2],[3],[4]]
=> [2,2] => [1,2,1] => 2
[4,3,2,1] => [[1],[2],[3],[4]]
=> [4] => [1,1,1,1] => 1
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [5] => [1,1,1,1,1] => 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [4,1] => [2,1,1,1] => 2
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [3,2] => [1,2,1,1] => 2
[1,2,4,5,3] => [[1,2,3,5],[4]]
=> [3,2] => [1,2,1,1] => 2
[1,2,5,3,4] => [[1,2,3,4],[5]]
=> [4,1] => [2,1,1,1] => 2
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [3,2] => [1,2,1,1] => 2
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [2,3] => [1,1,2,1] => 2
[1,3,4,2,5] => [[1,2,4,5],[3]]
=> [2,3] => [1,1,2,1] => 2
[1,3,4,5,2] => [[1,2,4,5],[3]]
=> [2,3] => [1,1,2,1] => 2
[1,4,2,3,5] => [[1,2,3,5],[4]]
=> [3,2] => [1,2,1,1] => 2
[1,4,2,5,3] => [[1,2,3],[4,5]]
=> [3,2] => [1,2,1,1] => 2
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [2,3] => [1,1,2,1] => 2
[1,4,3,5,2] => [[1,2,5],[3],[4]]
=> [2,3] => [1,1,2,1] => 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [3,2] => [1,2,1,1] => 2
[1,4,5,3,2] => [[1,2,5],[3],[4]]
=> [2,3] => [1,1,2,1] => 2
[1,5,2,3,4] => [[1,2,3,4],[5]]
=> [4,1] => [2,1,1,1] => 2
[1,5,2,4,3] => [[1,2,3],[4],[5]]
=> [3,2] => [1,2,1,1] => 2
Description
The largest part of an integer composition.
Matching statistic: St000389
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> [1] => 1 => 1
[1,2] => [[1,2]]
=> [2] => 10 => 1
[2,1] => [[1],[2]]
=> [2] => 10 => 1
[1,2,3] => [[1,2,3]]
=> [3] => 100 => 1
[1,3,2] => [[1,2],[3]]
=> [2,1] => 101 => 2
[2,1,3] => [[1,3],[2]]
=> [3] => 100 => 1
[2,3,1] => [[1,3],[2]]
=> [3] => 100 => 1
[3,1,2] => [[1,2],[3]]
=> [2,1] => 101 => 2
[3,2,1] => [[1],[2],[3]]
=> [3] => 100 => 1
[1,2,3,4] => [[1,2,3,4]]
=> [4] => 1000 => 1
[1,2,4,3] => [[1,2,3],[4]]
=> [3,1] => 1001 => 2
[1,3,2,4] => [[1,2,4],[3]]
=> [2,2] => 1010 => 2
[1,3,4,2] => [[1,2,4],[3]]
=> [2,2] => 1010 => 2
[1,4,2,3] => [[1,2,3],[4]]
=> [3,1] => 1001 => 2
[1,4,3,2] => [[1,2],[3],[4]]
=> [2,2] => 1010 => 2
[2,1,3,4] => [[1,3,4],[2]]
=> [4] => 1000 => 1
[2,1,4,3] => [[1,3],[2,4]]
=> [3,1] => 1001 => 2
[2,3,1,4] => [[1,3,4],[2]]
=> [4] => 1000 => 1
[2,3,4,1] => [[1,3,4],[2]]
=> [4] => 1000 => 1
[2,4,1,3] => [[1,3],[2,4]]
=> [3,1] => 1001 => 2
[2,4,3,1] => [[1,3],[2],[4]]
=> [3,1] => 1001 => 2
[3,1,2,4] => [[1,2,4],[3]]
=> [2,2] => 1010 => 2
[3,1,4,2] => [[1,2],[3,4]]
=> [2,2] => 1010 => 2
[3,2,1,4] => [[1,4],[2],[3]]
=> [4] => 1000 => 1
[3,2,4,1] => [[1,4],[2],[3]]
=> [4] => 1000 => 1
[3,4,1,2] => [[1,2],[3,4]]
=> [2,2] => 1010 => 2
[3,4,2,1] => [[1,4],[2],[3]]
=> [4] => 1000 => 1
[4,1,2,3] => [[1,2,3],[4]]
=> [3,1] => 1001 => 2
[4,1,3,2] => [[1,2],[3],[4]]
=> [2,2] => 1010 => 2
[4,2,1,3] => [[1,3],[2],[4]]
=> [3,1] => 1001 => 2
[4,2,3,1] => [[1,3],[2],[4]]
=> [3,1] => 1001 => 2
[4,3,1,2] => [[1,2],[3],[4]]
=> [2,2] => 1010 => 2
[4,3,2,1] => [[1],[2],[3],[4]]
=> [4] => 1000 => 1
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [5] => 10000 => 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [4,1] => 10001 => 2
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [3,2] => 10010 => 2
[1,2,4,5,3] => [[1,2,3,5],[4]]
=> [3,2] => 10010 => 2
[1,2,5,3,4] => [[1,2,3,4],[5]]
=> [4,1] => 10001 => 2
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [3,2] => 10010 => 2
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [2,3] => 10100 => 2
[1,3,4,2,5] => [[1,2,4,5],[3]]
=> [2,3] => 10100 => 2
[1,3,4,5,2] => [[1,2,4,5],[3]]
=> [2,3] => 10100 => 2
[1,4,2,3,5] => [[1,2,3,5],[4]]
=> [3,2] => 10010 => 2
[1,4,2,5,3] => [[1,2,3],[4,5]]
=> [3,2] => 10010 => 2
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [2,3] => 10100 => 2
[1,4,3,5,2] => [[1,2,5],[3],[4]]
=> [2,3] => 10100 => 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [3,2] => 10010 => 2
[1,4,5,3,2] => [[1,2,5],[3],[4]]
=> [2,3] => 10100 => 2
[1,5,2,3,4] => [[1,2,3,4],[5]]
=> [4,1] => 10001 => 2
[1,5,2,4,3] => [[1,2,3],[4],[5]]
=> [3,2] => 10010 => 2
Description
The number of runs of ones of odd length in a binary word.
Matching statistic: St000390
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> [1] => 1 => 1
[1,2] => [[1,2]]
=> [2] => 10 => 1
[2,1] => [[1],[2]]
=> [2] => 10 => 1
[1,2,3] => [[1,2,3]]
=> [3] => 100 => 1
[1,3,2] => [[1,2],[3]]
=> [2,1] => 101 => 2
[2,1,3] => [[1,3],[2]]
=> [3] => 100 => 1
[2,3,1] => [[1,3],[2]]
=> [3] => 100 => 1
[3,1,2] => [[1,2],[3]]
=> [2,1] => 101 => 2
[3,2,1] => [[1],[2],[3]]
=> [3] => 100 => 1
[1,2,3,4] => [[1,2,3,4]]
=> [4] => 1000 => 1
[1,2,4,3] => [[1,2,3],[4]]
=> [3,1] => 1001 => 2
[1,3,2,4] => [[1,2,4],[3]]
=> [2,2] => 1010 => 2
[1,3,4,2] => [[1,2,4],[3]]
=> [2,2] => 1010 => 2
[1,4,2,3] => [[1,2,3],[4]]
=> [3,1] => 1001 => 2
[1,4,3,2] => [[1,2],[3],[4]]
=> [2,2] => 1010 => 2
[2,1,3,4] => [[1,3,4],[2]]
=> [4] => 1000 => 1
[2,1,4,3] => [[1,3],[2,4]]
=> [3,1] => 1001 => 2
[2,3,1,4] => [[1,3,4],[2]]
=> [4] => 1000 => 1
[2,3,4,1] => [[1,3,4],[2]]
=> [4] => 1000 => 1
[2,4,1,3] => [[1,3],[2,4]]
=> [3,1] => 1001 => 2
[2,4,3,1] => [[1,3],[2],[4]]
=> [3,1] => 1001 => 2
[3,1,2,4] => [[1,2,4],[3]]
=> [2,2] => 1010 => 2
[3,1,4,2] => [[1,2],[3,4]]
=> [2,2] => 1010 => 2
[3,2,1,4] => [[1,4],[2],[3]]
=> [4] => 1000 => 1
[3,2,4,1] => [[1,4],[2],[3]]
=> [4] => 1000 => 1
[3,4,1,2] => [[1,2],[3,4]]
=> [2,2] => 1010 => 2
[3,4,2,1] => [[1,4],[2],[3]]
=> [4] => 1000 => 1
[4,1,2,3] => [[1,2,3],[4]]
=> [3,1] => 1001 => 2
[4,1,3,2] => [[1,2],[3],[4]]
=> [2,2] => 1010 => 2
[4,2,1,3] => [[1,3],[2],[4]]
=> [3,1] => 1001 => 2
[4,2,3,1] => [[1,3],[2],[4]]
=> [3,1] => 1001 => 2
[4,3,1,2] => [[1,2],[3],[4]]
=> [2,2] => 1010 => 2
[4,3,2,1] => [[1],[2],[3],[4]]
=> [4] => 1000 => 1
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [5] => 10000 => 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [4,1] => 10001 => 2
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [3,2] => 10010 => 2
[1,2,4,5,3] => [[1,2,3,5],[4]]
=> [3,2] => 10010 => 2
[1,2,5,3,4] => [[1,2,3,4],[5]]
=> [4,1] => 10001 => 2
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [3,2] => 10010 => 2
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [2,3] => 10100 => 2
[1,3,4,2,5] => [[1,2,4,5],[3]]
=> [2,3] => 10100 => 2
[1,3,4,5,2] => [[1,2,4,5],[3]]
=> [2,3] => 10100 => 2
[1,4,2,3,5] => [[1,2,3,5],[4]]
=> [3,2] => 10010 => 2
[1,4,2,5,3] => [[1,2,3],[4,5]]
=> [3,2] => 10010 => 2
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [2,3] => 10100 => 2
[1,4,3,5,2] => [[1,2,5],[3],[4]]
=> [2,3] => 10100 => 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [3,2] => 10010 => 2
[1,4,5,3,2] => [[1,2,5],[3],[4]]
=> [2,3] => 10100 => 2
[1,5,2,3,4] => [[1,2,3,4],[5]]
=> [4,1] => 10001 => 2
[1,5,2,4,3] => [[1,2,3],[4],[5]]
=> [3,2] => 10010 => 2
Description
The number of runs of ones in a binary word.
Matching statistic: St000533
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000533: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000533: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> [1] => [1]
=> 1
[1,2] => [[1,2]]
=> [2] => [2]
=> 1
[2,1] => [[1],[2]]
=> [2] => [2]
=> 1
[1,2,3] => [[1,2,3]]
=> [3] => [3]
=> 1
[1,3,2] => [[1,2],[3]]
=> [2,1] => [2,1]
=> 2
[2,1,3] => [[1,3],[2]]
=> [3] => [3]
=> 1
[2,3,1] => [[1,3],[2]]
=> [3] => [3]
=> 1
[3,1,2] => [[1,2],[3]]
=> [2,1] => [2,1]
=> 2
[3,2,1] => [[1],[2],[3]]
=> [3] => [3]
=> 1
[1,2,3,4] => [[1,2,3,4]]
=> [4] => [4]
=> 1
[1,2,4,3] => [[1,2,3],[4]]
=> [3,1] => [3,1]
=> 2
[1,3,2,4] => [[1,2,4],[3]]
=> [2,2] => [2,2]
=> 2
[1,3,4,2] => [[1,2,4],[3]]
=> [2,2] => [2,2]
=> 2
[1,4,2,3] => [[1,2,3],[4]]
=> [3,1] => [3,1]
=> 2
[1,4,3,2] => [[1,2],[3],[4]]
=> [2,2] => [2,2]
=> 2
[2,1,3,4] => [[1,3,4],[2]]
=> [4] => [4]
=> 1
[2,1,4,3] => [[1,3],[2,4]]
=> [3,1] => [3,1]
=> 2
[2,3,1,4] => [[1,3,4],[2]]
=> [4] => [4]
=> 1
[2,3,4,1] => [[1,3,4],[2]]
=> [4] => [4]
=> 1
[2,4,1,3] => [[1,3],[2,4]]
=> [3,1] => [3,1]
=> 2
[2,4,3,1] => [[1,3],[2],[4]]
=> [3,1] => [3,1]
=> 2
[3,1,2,4] => [[1,2,4],[3]]
=> [2,2] => [2,2]
=> 2
[3,1,4,2] => [[1,2],[3,4]]
=> [2,2] => [2,2]
=> 2
[3,2,1,4] => [[1,4],[2],[3]]
=> [4] => [4]
=> 1
[3,2,4,1] => [[1,4],[2],[3]]
=> [4] => [4]
=> 1
[3,4,1,2] => [[1,2],[3,4]]
=> [2,2] => [2,2]
=> 2
[3,4,2,1] => [[1,4],[2],[3]]
=> [4] => [4]
=> 1
[4,1,2,3] => [[1,2,3],[4]]
=> [3,1] => [3,1]
=> 2
[4,1,3,2] => [[1,2],[3],[4]]
=> [2,2] => [2,2]
=> 2
[4,2,1,3] => [[1,3],[2],[4]]
=> [3,1] => [3,1]
=> 2
[4,2,3,1] => [[1,3],[2],[4]]
=> [3,1] => [3,1]
=> 2
[4,3,1,2] => [[1,2],[3],[4]]
=> [2,2] => [2,2]
=> 2
[4,3,2,1] => [[1],[2],[3],[4]]
=> [4] => [4]
=> 1
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [5] => [5]
=> 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [4,1] => [4,1]
=> 2
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [3,2] => [3,2]
=> 2
[1,2,4,5,3] => [[1,2,3,5],[4]]
=> [3,2] => [3,2]
=> 2
[1,2,5,3,4] => [[1,2,3,4],[5]]
=> [4,1] => [4,1]
=> 2
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [3,2] => [3,2]
=> 2
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [2,3] => [3,2]
=> 2
[1,3,4,2,5] => [[1,2,4,5],[3]]
=> [2,3] => [3,2]
=> 2
[1,3,4,5,2] => [[1,2,4,5],[3]]
=> [2,3] => [3,2]
=> 2
[1,4,2,3,5] => [[1,2,3,5],[4]]
=> [3,2] => [3,2]
=> 2
[1,4,2,5,3] => [[1,2,3],[4,5]]
=> [3,2] => [3,2]
=> 2
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [2,3] => [3,2]
=> 2
[1,4,3,5,2] => [[1,2,5],[3],[4]]
=> [2,3] => [3,2]
=> 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [3,2] => [3,2]
=> 2
[1,4,5,3,2] => [[1,2,5],[3],[4]]
=> [2,3] => [3,2]
=> 2
[1,5,2,3,4] => [[1,2,3,4],[5]]
=> [4,1] => [4,1]
=> 2
[1,5,2,4,3] => [[1,2,3],[4],[5]]
=> [3,2] => [3,2]
=> 2
Description
The minimum of the number of parts and the size of the first part of an integer partition.
This is also an upper bound on the maximal number of non-attacking rooks that can be placed on the Ferrers board.
Matching statistic: St000758
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000758: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000758: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> [1] => [1] => 1
[1,2] => [[1,2]]
=> [2] => [1,1] => 1
[2,1] => [[1],[2]]
=> [2] => [1,1] => 1
[1,2,3] => [[1,2,3]]
=> [3] => [1,1,1] => 1
[1,3,2] => [[1,2],[3]]
=> [2,1] => [1,2] => 2
[2,1,3] => [[1,3],[2]]
=> [3] => [1,1,1] => 1
[2,3,1] => [[1,3],[2]]
=> [3] => [1,1,1] => 1
[3,1,2] => [[1,2],[3]]
=> [2,1] => [1,2] => 2
[3,2,1] => [[1],[2],[3]]
=> [3] => [1,1,1] => 1
[1,2,3,4] => [[1,2,3,4]]
=> [4] => [1,1,1,1] => 1
[1,2,4,3] => [[1,2,3],[4]]
=> [3,1] => [1,1,2] => 2
[1,3,2,4] => [[1,2,4],[3]]
=> [2,2] => [1,2,1] => 2
[1,3,4,2] => [[1,2,4],[3]]
=> [2,2] => [1,2,1] => 2
[1,4,2,3] => [[1,2,3],[4]]
=> [3,1] => [1,1,2] => 2
[1,4,3,2] => [[1,2],[3],[4]]
=> [2,2] => [1,2,1] => 2
[2,1,3,4] => [[1,3,4],[2]]
=> [4] => [1,1,1,1] => 1
[2,1,4,3] => [[1,3],[2,4]]
=> [3,1] => [1,1,2] => 2
[2,3,1,4] => [[1,3,4],[2]]
=> [4] => [1,1,1,1] => 1
[2,3,4,1] => [[1,3,4],[2]]
=> [4] => [1,1,1,1] => 1
[2,4,1,3] => [[1,3],[2,4]]
=> [3,1] => [1,1,2] => 2
[2,4,3,1] => [[1,3],[2],[4]]
=> [3,1] => [1,1,2] => 2
[3,1,2,4] => [[1,2,4],[3]]
=> [2,2] => [1,2,1] => 2
[3,1,4,2] => [[1,2],[3,4]]
=> [2,2] => [1,2,1] => 2
[3,2,1,4] => [[1,4],[2],[3]]
=> [4] => [1,1,1,1] => 1
[3,2,4,1] => [[1,4],[2],[3]]
=> [4] => [1,1,1,1] => 1
[3,4,1,2] => [[1,2],[3,4]]
=> [2,2] => [1,2,1] => 2
[3,4,2,1] => [[1,4],[2],[3]]
=> [4] => [1,1,1,1] => 1
[4,1,2,3] => [[1,2,3],[4]]
=> [3,1] => [1,1,2] => 2
[4,1,3,2] => [[1,2],[3],[4]]
=> [2,2] => [1,2,1] => 2
[4,2,1,3] => [[1,3],[2],[4]]
=> [3,1] => [1,1,2] => 2
[4,2,3,1] => [[1,3],[2],[4]]
=> [3,1] => [1,1,2] => 2
[4,3,1,2] => [[1,2],[3],[4]]
=> [2,2] => [1,2,1] => 2
[4,3,2,1] => [[1],[2],[3],[4]]
=> [4] => [1,1,1,1] => 1
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [5] => [1,1,1,1,1] => 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [4,1] => [1,1,1,2] => 2
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [3,2] => [1,1,2,1] => 2
[1,2,4,5,3] => [[1,2,3,5],[4]]
=> [3,2] => [1,1,2,1] => 2
[1,2,5,3,4] => [[1,2,3,4],[5]]
=> [4,1] => [1,1,1,2] => 2
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [3,2] => [1,1,2,1] => 2
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [2,3] => [1,2,1,1] => 2
[1,3,4,2,5] => [[1,2,4,5],[3]]
=> [2,3] => [1,2,1,1] => 2
[1,3,4,5,2] => [[1,2,4,5],[3]]
=> [2,3] => [1,2,1,1] => 2
[1,4,2,3,5] => [[1,2,3,5],[4]]
=> [3,2] => [1,1,2,1] => 2
[1,4,2,5,3] => [[1,2,3],[4,5]]
=> [3,2] => [1,1,2,1] => 2
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [2,3] => [1,2,1,1] => 2
[1,4,3,5,2] => [[1,2,5],[3],[4]]
=> [2,3] => [1,2,1,1] => 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [3,2] => [1,1,2,1] => 2
[1,4,5,3,2] => [[1,2,5],[3],[4]]
=> [2,3] => [1,2,1,1] => 2
[1,5,2,3,4] => [[1,2,3,4],[5]]
=> [4,1] => [1,1,1,2] => 2
[1,5,2,4,3] => [[1,2,3],[4],[5]]
=> [3,2] => [1,1,2,1] => 2
Description
The length of the longest staircase fitting into an integer composition.
For a given composition $c_1,\dots,c_n$, this is the maximal number $\ell$ such that there are indices $i_1 < \dots < i_\ell$ with $c_{i_k} \geq k$, see [def.3.1, 1]
The following 81 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000808The number of up steps of the associated bargraph. St000903The number of different parts of an integer composition. St001330The hat guessing number of a graph. St001432The order dimension of the partition. St001486The number of corners of the ribbon associated with an integer composition. St001581The achromatic number of a graph. St000292The number of ascents of a binary word. St000386The number of factors DDU in a Dyck path. St000481The number of upper covers of a partition in dominance order. St000761The number of ascents in an integer composition. St001712The number of natural descents of a standard Young tableau. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000396The register function (or Horton-Strahler number) of a binary tree. St000291The number of descents of a binary word. St000920The logarithmic height of a Dyck path. St000806The semiperimeter of the associated bargraph. St000201The number of leaf nodes in a binary tree. St000862The number of parts of the shifted shape of a permutation. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000172The Grundy number of a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000527The width of the poset. St001029The size of the core of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001716The 1-improper chromatic number of a graph. St001963The tree-depth of a graph. St000091The descent variation of a composition. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000552The number of cut vertices of a graph. St000632The jump number of the poset. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001277The degeneracy of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001393The induced matching number of a graph. St001459The number of zero columns in the nullspace of a graph. St001689The number of celebrities in a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000353The number of inner valleys of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000099The number of valleys of a permutation, including the boundary. St000023The number of inner peaks of a permutation. St000092The number of outer peaks of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000822The Hadwiger number of the graph. St001734The lettericity of a graph. St000397The Strahler number of a rooted tree. St000523The number of 2-protected nodes of a rooted tree. St001812The biclique partition number of a graph. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000779The tier of a permutation. St000264The girth of a graph, which is not a tree. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St001487The number of inner corners of a skew partition. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001060The distinguishing index of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000699The toughness times the least common multiple of 1,.
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