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Your data matches 58 different statistics following compositions of up to 3 maps.
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Matching statistic: St000470
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(load all 8 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => 1
[2,1] => [1] => 1
[1,2,3] => [1,2] => 1
[1,3,2] => [1,2] => 1
[2,1,3] => [2,1] => 2
[2,3,1] => [2,1] => 2
[3,1,2] => [1,2] => 1
[3,2,1] => [2,1] => 2
[1,2,3,4] => [1,2,3] => 1
[1,2,4,3] => [1,2,3] => 1
[1,3,2,4] => [1,3,2] => 2
[1,3,4,2] => [1,3,2] => 2
[1,4,2,3] => [1,2,3] => 1
[1,4,3,2] => [1,3,2] => 2
[2,1,3,4] => [2,1,3] => 2
[2,1,4,3] => [2,1,3] => 2
[2,3,1,4] => [2,3,1] => 2
[2,3,4,1] => [2,3,1] => 2
[2,4,1,3] => [2,1,3] => 2
[2,4,3,1] => [2,3,1] => 2
[3,1,2,4] => [3,1,2] => 2
[3,1,4,2] => [3,1,2] => 2
[3,2,1,4] => [3,2,1] => 3
[3,2,4,1] => [3,2,1] => 3
[3,4,1,2] => [3,1,2] => 2
[3,4,2,1] => [3,2,1] => 3
[4,1,2,3] => [1,2,3] => 1
[4,1,3,2] => [1,3,2] => 2
[4,2,1,3] => [2,1,3] => 2
[4,2,3,1] => [2,3,1] => 2
[4,3,1,2] => [3,1,2] => 2
[4,3,2,1] => [3,2,1] => 3
[1,2,3,4,5] => [1,2,3,4] => 1
[1,2,3,5,4] => [1,2,3,4] => 1
[1,2,4,3,5] => [1,2,4,3] => 2
[1,2,4,5,3] => [1,2,4,3] => 2
[1,2,5,3,4] => [1,2,3,4] => 1
[1,2,5,4,3] => [1,2,4,3] => 2
[1,3,2,4,5] => [1,3,2,4] => 2
[1,3,2,5,4] => [1,3,2,4] => 2
[1,3,4,2,5] => [1,3,4,2] => 2
[1,3,4,5,2] => [1,3,4,2] => 2
[1,3,5,2,4] => [1,3,2,4] => 2
[1,3,5,4,2] => [1,3,4,2] => 2
[1,4,2,3,5] => [1,4,2,3] => 2
[1,4,2,5,3] => [1,4,2,3] => 2
[1,4,3,2,5] => [1,4,3,2] => 3
[1,4,3,5,2] => [1,4,3,2] => 3
[1,4,5,2,3] => [1,4,2,3] => 2
[1,4,5,3,2] => [1,4,3,2] => 3
Description
The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.
Matching statistic: St000157
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(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [[1]]
=> 0 = 1 - 1
[2,1] => [1] => [[1]]
=> 0 = 1 - 1
[1,2,3] => [1,2] => [[1,2]]
=> 0 = 1 - 1
[1,3,2] => [1,2] => [[1,2]]
=> 0 = 1 - 1
[2,1,3] => [2,1] => [[1],[2]]
=> 1 = 2 - 1
[2,3,1] => [2,1] => [[1],[2]]
=> 1 = 2 - 1
[3,1,2] => [1,2] => [[1,2]]
=> 0 = 1 - 1
[3,2,1] => [2,1] => [[1],[2]]
=> 1 = 2 - 1
[1,2,3,4] => [1,2,3] => [[1,2,3]]
=> 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [[1,2,3]]
=> 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [[1,2],[3]]
=> 1 = 2 - 1
[1,3,4,2] => [1,3,2] => [[1,2],[3]]
=> 1 = 2 - 1
[1,4,2,3] => [1,2,3] => [[1,2,3]]
=> 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [[1,2],[3]]
=> 1 = 2 - 1
[2,1,3,4] => [2,1,3] => [[1,3],[2]]
=> 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [[1,3],[2]]
=> 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [[1,2],[3]]
=> 1 = 2 - 1
[2,3,4,1] => [2,3,1] => [[1,2],[3]]
=> 1 = 2 - 1
[2,4,1,3] => [2,1,3] => [[1,3],[2]]
=> 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [[1,2],[3]]
=> 1 = 2 - 1
[3,1,2,4] => [3,1,2] => [[1,3],[2]]
=> 1 = 2 - 1
[3,1,4,2] => [3,1,2] => [[1,3],[2]]
=> 1 = 2 - 1
[3,2,1,4] => [3,2,1] => [[1],[2],[3]]
=> 2 = 3 - 1
[3,2,4,1] => [3,2,1] => [[1],[2],[3]]
=> 2 = 3 - 1
[3,4,1,2] => [3,1,2] => [[1,3],[2]]
=> 1 = 2 - 1
[3,4,2,1] => [3,2,1] => [[1],[2],[3]]
=> 2 = 3 - 1
[4,1,2,3] => [1,2,3] => [[1,2,3]]
=> 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [[1,2],[3]]
=> 1 = 2 - 1
[4,2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [[1,2],[3]]
=> 1 = 2 - 1
[4,3,1,2] => [3,1,2] => [[1,3],[2]]
=> 1 = 2 - 1
[4,3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 2 = 3 - 1
[1,2,3,4,5] => [1,2,3,4] => [[1,2,3,4]]
=> 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [[1,2,3],[4]]
=> 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => [[1,2,3],[4]]
=> 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,2] => [[1,2,3],[4]]
=> 1 = 2 - 1
[1,3,5,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => [[1,2,3],[4]]
=> 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3] => [[1,2,4],[3]]
=> 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [[1,2,4],[3]]
=> 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 3 - 1
[1,4,3,5,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 3 - 1
[1,4,5,2,3] => [1,4,2,3] => [[1,2,4],[3]]
=> 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 3 - 1
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000010
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(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00071: Permutations —descent 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%
Mp00071: Permutations —descent 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
[1,2] => [1] => [1] => [1]
=> 1
[2,1] => [1] => [1] => [1]
=> 1
[1,2,3] => [1,2] => [2] => [2]
=> 1
[1,3,2] => [1,2] => [2] => [2]
=> 1
[2,1,3] => [2,1] => [1,1] => [1,1]
=> 2
[2,3,1] => [2,1] => [1,1] => [1,1]
=> 2
[3,1,2] => [1,2] => [2] => [2]
=> 1
[3,2,1] => [2,1] => [1,1] => [1,1]
=> 2
[1,2,3,4] => [1,2,3] => [3] => [3]
=> 1
[1,2,4,3] => [1,2,3] => [3] => [3]
=> 1
[1,3,2,4] => [1,3,2] => [2,1] => [2,1]
=> 2
[1,3,4,2] => [1,3,2] => [2,1] => [2,1]
=> 2
[1,4,2,3] => [1,2,3] => [3] => [3]
=> 1
[1,4,3,2] => [1,3,2] => [2,1] => [2,1]
=> 2
[2,1,3,4] => [2,1,3] => [1,2] => [2,1]
=> 2
[2,1,4,3] => [2,1,3] => [1,2] => [2,1]
=> 2
[2,3,1,4] => [2,3,1] => [2,1] => [2,1]
=> 2
[2,3,4,1] => [2,3,1] => [2,1] => [2,1]
=> 2
[2,4,1,3] => [2,1,3] => [1,2] => [2,1]
=> 2
[2,4,3,1] => [2,3,1] => [2,1] => [2,1]
=> 2
[3,1,2,4] => [3,1,2] => [1,2] => [2,1]
=> 2
[3,1,4,2] => [3,1,2] => [1,2] => [2,1]
=> 2
[3,2,1,4] => [3,2,1] => [1,1,1] => [1,1,1]
=> 3
[3,2,4,1] => [3,2,1] => [1,1,1] => [1,1,1]
=> 3
[3,4,1,2] => [3,1,2] => [1,2] => [2,1]
=> 2
[3,4,2,1] => [3,2,1] => [1,1,1] => [1,1,1]
=> 3
[4,1,2,3] => [1,2,3] => [3] => [3]
=> 1
[4,1,3,2] => [1,3,2] => [2,1] => [2,1]
=> 2
[4,2,1,3] => [2,1,3] => [1,2] => [2,1]
=> 2
[4,2,3,1] => [2,3,1] => [2,1] => [2,1]
=> 2
[4,3,1,2] => [3,1,2] => [1,2] => [2,1]
=> 2
[4,3,2,1] => [3,2,1] => [1,1,1] => [1,1,1]
=> 3
[1,2,3,4,5] => [1,2,3,4] => [4] => [4]
=> 1
[1,2,3,5,4] => [1,2,3,4] => [4] => [4]
=> 1
[1,2,4,3,5] => [1,2,4,3] => [3,1] => [3,1]
=> 2
[1,2,4,5,3] => [1,2,4,3] => [3,1] => [3,1]
=> 2
[1,2,5,3,4] => [1,2,3,4] => [4] => [4]
=> 1
[1,2,5,4,3] => [1,2,4,3] => [3,1] => [3,1]
=> 2
[1,3,2,4,5] => [1,3,2,4] => [2,2] => [2,2]
=> 2
[1,3,2,5,4] => [1,3,2,4] => [2,2] => [2,2]
=> 2
[1,3,4,2,5] => [1,3,4,2] => [3,1] => [3,1]
=> 2
[1,3,4,5,2] => [1,3,4,2] => [3,1] => [3,1]
=> 2
[1,3,5,2,4] => [1,3,2,4] => [2,2] => [2,2]
=> 2
[1,3,5,4,2] => [1,3,4,2] => [3,1] => [3,1]
=> 2
[1,4,2,3,5] => [1,4,2,3] => [2,2] => [2,2]
=> 2
[1,4,2,5,3] => [1,4,2,3] => [2,2] => [2,2]
=> 2
[1,4,3,2,5] => [1,4,3,2] => [2,1,1] => [2,1,1]
=> 3
[1,4,3,5,2] => [1,4,3,2] => [2,1,1] => [2,1,1]
=> 3
[1,4,5,2,3] => [1,4,2,3] => [2,2] => [2,2]
=> 2
[1,4,5,3,2] => [1,4,3,2] => [2,1,1] => [2,1,1]
=> 3
Description
The length of the partition.
Matching statistic: St000097
Mp00252: Permutations —restriction⟶ Permutations
Mp00071: Permutations —descent 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%
Mp00071: Permutations —descent 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
[1,2] => [1] => [1] => ([],1)
=> 1
[2,1] => [1] => [1] => ([],1)
=> 1
[1,2,3] => [1,2] => [2] => ([],2)
=> 1
[1,3,2] => [1,2] => [2] => ([],2)
=> 1
[2,1,3] => [2,1] => [1,1] => ([(0,1)],2)
=> 2
[2,3,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 2
[3,1,2] => [1,2] => [2] => ([],2)
=> 1
[3,2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 2
[1,2,3,4] => [1,2,3] => [3] => ([],3)
=> 1
[1,2,4,3] => [1,2,3] => [3] => ([],3)
=> 1
[1,3,2,4] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,3,4,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,4,2,3] => [1,2,3] => [3] => ([],3)
=> 1
[1,4,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,3,4] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 2
[2,1,4,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 2
[2,3,1,4] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,3,4,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,4,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 2
[2,4,3,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,1,2,4] => [3,1,2] => [1,2] => ([(1,2)],3)
=> 2
[3,1,4,2] => [3,1,2] => [1,2] => ([(1,2)],3)
=> 2
[3,2,1,4] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[3,2,4,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[3,4,1,2] => [3,1,2] => [1,2] => ([(1,2)],3)
=> 2
[3,4,2,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[4,1,2,3] => [1,2,3] => [3] => ([],3)
=> 1
[4,1,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[4,2,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 2
[4,2,3,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[4,3,1,2] => [3,1,2] => [1,2] => ([(1,2)],3)
=> 2
[4,3,2,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,3,4,5] => [1,2,3,4] => [4] => ([],4)
=> 1
[1,2,3,5,4] => [1,2,3,4] => [4] => ([],4)
=> 1
[1,2,4,3,5] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,2,4,5,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,2,5,3,4] => [1,2,3,4] => [4] => ([],4)
=> 1
[1,2,5,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,3,2,4,5] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,3,2,5,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,3,4,2,5] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,3,4,5,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,3,5,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,3,5,4,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,4,2,3,5] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,4,2,5,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,4,3,2,5] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,4,3,5,2] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,4,5,2,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,4,5,3,2] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
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
Mp00252: Permutations —restriction⟶ Permutations
Mp00071: Permutations —descent 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%
Mp00071: Permutations —descent 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
[1,2] => [1] => [1] => ([],1)
=> 1
[2,1] => [1] => [1] => ([],1)
=> 1
[1,2,3] => [1,2] => [2] => ([],2)
=> 1
[1,3,2] => [1,2] => [2] => ([],2)
=> 1
[2,1,3] => [2,1] => [1,1] => ([(0,1)],2)
=> 2
[2,3,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 2
[3,1,2] => [1,2] => [2] => ([],2)
=> 1
[3,2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 2
[1,2,3,4] => [1,2,3] => [3] => ([],3)
=> 1
[1,2,4,3] => [1,2,3] => [3] => ([],3)
=> 1
[1,3,2,4] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,3,4,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,4,2,3] => [1,2,3] => [3] => ([],3)
=> 1
[1,4,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,3,4] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 2
[2,1,4,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 2
[2,3,1,4] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,3,4,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,4,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 2
[2,4,3,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,1,2,4] => [3,1,2] => [1,2] => ([(1,2)],3)
=> 2
[3,1,4,2] => [3,1,2] => [1,2] => ([(1,2)],3)
=> 2
[3,2,1,4] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[3,2,4,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[3,4,1,2] => [3,1,2] => [1,2] => ([(1,2)],3)
=> 2
[3,4,2,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[4,1,2,3] => [1,2,3] => [3] => ([],3)
=> 1
[4,1,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[4,2,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 2
[4,2,3,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[4,3,1,2] => [3,1,2] => [1,2] => ([(1,2)],3)
=> 2
[4,3,2,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,3,4,5] => [1,2,3,4] => [4] => ([],4)
=> 1
[1,2,3,5,4] => [1,2,3,4] => [4] => ([],4)
=> 1
[1,2,4,3,5] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,2,4,5,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,2,5,3,4] => [1,2,3,4] => [4] => ([],4)
=> 1
[1,2,5,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,3,2,4,5] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,3,2,5,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,3,4,2,5] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,3,4,5,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,3,5,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,3,5,4,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,4,2,3,5] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,4,2,5,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,4,3,2,5] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,4,3,5,2] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,4,5,2,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,4,5,3,2] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
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: St000167
Mp00252: Permutations —restriction⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000167: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000167: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [.,.]
=> [[]]
=> 1
[2,1] => [1] => [.,.]
=> [[]]
=> 1
[1,2,3] => [1,2] => [.,[.,.]]
=> [[[]]]
=> 1
[1,3,2] => [1,2] => [.,[.,.]]
=> [[[]]]
=> 1
[2,1,3] => [2,1] => [[.,.],.]
=> [[],[]]
=> 2
[2,3,1] => [2,1] => [[.,.],.]
=> [[],[]]
=> 2
[3,1,2] => [1,2] => [.,[.,.]]
=> [[[]]]
=> 1
[3,2,1] => [2,1] => [[.,.],.]
=> [[],[]]
=> 2
[1,2,3,4] => [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 1
[1,2,4,3] => [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 1
[1,3,2,4] => [1,3,2] => [.,[[.,.],.]]
=> [[[],[]]]
=> 2
[1,3,4,2] => [1,3,2] => [.,[[.,.],.]]
=> [[[],[]]]
=> 2
[1,4,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 1
[1,4,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [[[],[]]]
=> 2
[2,1,3,4] => [2,1,3] => [[.,.],[.,.]]
=> [[],[[]]]
=> 2
[2,1,4,3] => [2,1,3] => [[.,.],[.,.]]
=> [[],[[]]]
=> 2
[2,3,1,4] => [2,3,1] => [[.,[.,.]],.]
=> [[[]],[]]
=> 2
[2,3,4,1] => [2,3,1] => [[.,[.,.]],.]
=> [[[]],[]]
=> 2
[2,4,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [[],[[]]]
=> 2
[2,4,3,1] => [2,3,1] => [[.,[.,.]],.]
=> [[[]],[]]
=> 2
[3,1,2,4] => [3,1,2] => [[.,.],[.,.]]
=> [[],[[]]]
=> 2
[3,1,4,2] => [3,1,2] => [[.,.],[.,.]]
=> [[],[[]]]
=> 2
[3,2,1,4] => [3,2,1] => [[[.,.],.],.]
=> [[],[],[]]
=> 3
[3,2,4,1] => [3,2,1] => [[[.,.],.],.]
=> [[],[],[]]
=> 3
[3,4,1,2] => [3,1,2] => [[.,.],[.,.]]
=> [[],[[]]]
=> 2
[3,4,2,1] => [3,2,1] => [[[.,.],.],.]
=> [[],[],[]]
=> 3
[4,1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 1
[4,1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [[[],[]]]
=> 2
[4,2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [[],[[]]]
=> 2
[4,2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> [[[]],[]]
=> 2
[4,3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> [[],[[]]]
=> 2
[4,3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [[],[],[]]
=> 3
[1,2,3,4,5] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1
[1,2,3,5,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1
[1,2,4,3,5] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> 2
[1,2,4,5,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> 2
[1,2,5,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1
[1,2,5,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> 2
[1,3,2,4,5] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2
[1,3,2,5,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2
[1,3,4,2,5] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> 2
[1,3,4,5,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> 2
[1,3,5,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2
[1,3,5,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> 2
[1,4,2,3,5] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2
[1,4,2,5,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2
[1,4,3,2,5] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [[[],[],[]]]
=> 3
[1,4,3,5,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [[[],[],[]]]
=> 3
[1,4,5,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2
[1,4,5,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [[[],[],[]]]
=> 3
Description
The number of leaves of an ordered tree.
This is the number of nodes which do not have any children.
Matching statistic: St000172
Mp00252: Permutations —restriction⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000172: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000172: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => ([],1)
=> 1
[2,1] => [1] => [1] => ([],1)
=> 1
[1,2,3] => [1,2] => [2] => ([],2)
=> 1
[1,3,2] => [1,2] => [2] => ([],2)
=> 1
[2,1,3] => [2,1] => [1,1] => ([(0,1)],2)
=> 2
[2,3,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 2
[3,1,2] => [1,2] => [2] => ([],2)
=> 1
[3,2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 2
[1,2,3,4] => [1,2,3] => [3] => ([],3)
=> 1
[1,2,4,3] => [1,2,3] => [3] => ([],3)
=> 1
[1,3,2,4] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,3,4,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,4,2,3] => [1,2,3] => [3] => ([],3)
=> 1
[1,4,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,3,4] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 2
[2,1,4,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 2
[2,3,1,4] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,3,4,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,4,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 2
[2,4,3,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,1,2,4] => [3,1,2] => [1,2] => ([(1,2)],3)
=> 2
[3,1,4,2] => [3,1,2] => [1,2] => ([(1,2)],3)
=> 2
[3,2,1,4] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[3,2,4,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[3,4,1,2] => [3,1,2] => [1,2] => ([(1,2)],3)
=> 2
[3,4,2,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[4,1,2,3] => [1,2,3] => [3] => ([],3)
=> 1
[4,1,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[4,2,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 2
[4,2,3,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[4,3,1,2] => [3,1,2] => [1,2] => ([(1,2)],3)
=> 2
[4,3,2,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,3,4,5] => [1,2,3,4] => [4] => ([],4)
=> 1
[1,2,3,5,4] => [1,2,3,4] => [4] => ([],4)
=> 1
[1,2,4,3,5] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,2,4,5,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,2,5,3,4] => [1,2,3,4] => [4] => ([],4)
=> 1
[1,2,5,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,3,2,4,5] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,3,2,5,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,3,4,2,5] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,3,4,5,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,3,5,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,3,5,4,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,4,2,3,5] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,4,2,5,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,4,3,2,5] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,4,3,5,2] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,4,5,2,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,4,5,3,2] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
Description
The Grundy number of a graph.
The Grundy number $\Gamma(G)$ is defined to be the largest $k$ such that $G$ admits a greedy $k$-coloring. Any order of the vertices of $G$ induces a greedy coloring by assigning to the $i$-th vertex in this order the smallest positive integer such that the partial coloring remains a proper coloring.
In particular, we have that $\chi(G) \leq \Gamma(G) \leq \Delta(G) + 1$, where $\chi(G)$ is the chromatic number of $G$ ([[St000098]]), and where $\Delta(G)$ is the maximal degree of a vertex of $G$ ([[St000171]]).
Matching statistic: St000288
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00071: Permutations —descent 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%
Mp00071: Permutations —descent 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
[1,2] => [1] => [1] => 1 => 1
[2,1] => [1] => [1] => 1 => 1
[1,2,3] => [1,2] => [2] => 10 => 1
[1,3,2] => [1,2] => [2] => 10 => 1
[2,1,3] => [2,1] => [1,1] => 11 => 2
[2,3,1] => [2,1] => [1,1] => 11 => 2
[3,1,2] => [1,2] => [2] => 10 => 1
[3,2,1] => [2,1] => [1,1] => 11 => 2
[1,2,3,4] => [1,2,3] => [3] => 100 => 1
[1,2,4,3] => [1,2,3] => [3] => 100 => 1
[1,3,2,4] => [1,3,2] => [2,1] => 101 => 2
[1,3,4,2] => [1,3,2] => [2,1] => 101 => 2
[1,4,2,3] => [1,2,3] => [3] => 100 => 1
[1,4,3,2] => [1,3,2] => [2,1] => 101 => 2
[2,1,3,4] => [2,1,3] => [1,2] => 110 => 2
[2,1,4,3] => [2,1,3] => [1,2] => 110 => 2
[2,3,1,4] => [2,3,1] => [2,1] => 101 => 2
[2,3,4,1] => [2,3,1] => [2,1] => 101 => 2
[2,4,1,3] => [2,1,3] => [1,2] => 110 => 2
[2,4,3,1] => [2,3,1] => [2,1] => 101 => 2
[3,1,2,4] => [3,1,2] => [1,2] => 110 => 2
[3,1,4,2] => [3,1,2] => [1,2] => 110 => 2
[3,2,1,4] => [3,2,1] => [1,1,1] => 111 => 3
[3,2,4,1] => [3,2,1] => [1,1,1] => 111 => 3
[3,4,1,2] => [3,1,2] => [1,2] => 110 => 2
[3,4,2,1] => [3,2,1] => [1,1,1] => 111 => 3
[4,1,2,3] => [1,2,3] => [3] => 100 => 1
[4,1,3,2] => [1,3,2] => [2,1] => 101 => 2
[4,2,1,3] => [2,1,3] => [1,2] => 110 => 2
[4,2,3,1] => [2,3,1] => [2,1] => 101 => 2
[4,3,1,2] => [3,1,2] => [1,2] => 110 => 2
[4,3,2,1] => [3,2,1] => [1,1,1] => 111 => 3
[1,2,3,4,5] => [1,2,3,4] => [4] => 1000 => 1
[1,2,3,5,4] => [1,2,3,4] => [4] => 1000 => 1
[1,2,4,3,5] => [1,2,4,3] => [3,1] => 1001 => 2
[1,2,4,5,3] => [1,2,4,3] => [3,1] => 1001 => 2
[1,2,5,3,4] => [1,2,3,4] => [4] => 1000 => 1
[1,2,5,4,3] => [1,2,4,3] => [3,1] => 1001 => 2
[1,3,2,4,5] => [1,3,2,4] => [2,2] => 1010 => 2
[1,3,2,5,4] => [1,3,2,4] => [2,2] => 1010 => 2
[1,3,4,2,5] => [1,3,4,2] => [3,1] => 1001 => 2
[1,3,4,5,2] => [1,3,4,2] => [3,1] => 1001 => 2
[1,3,5,2,4] => [1,3,2,4] => [2,2] => 1010 => 2
[1,3,5,4,2] => [1,3,4,2] => [3,1] => 1001 => 2
[1,4,2,3,5] => [1,4,2,3] => [2,2] => 1010 => 2
[1,4,2,5,3] => [1,4,2,3] => [2,2] => 1010 => 2
[1,4,3,2,5] => [1,4,3,2] => [2,1,1] => 1011 => 3
[1,4,3,5,2] => [1,4,3,2] => [2,1,1] => 1011 => 3
[1,4,5,2,3] => [1,4,2,3] => [2,2] => 1010 => 2
[1,4,5,3,2] => [1,4,3,2] => [2,1,1] => 1011 => 3
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000507
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1]
=> [[1]]
=> 1
[2,1] => [1] => [1]
=> [[1]]
=> 1
[1,2,3] => [1,2] => [1,1]
=> [[1],[2]]
=> 1
[1,3,2] => [1,2] => [1,1]
=> [[1],[2]]
=> 1
[2,1,3] => [2,1] => [2]
=> [[1,2]]
=> 2
[2,3,1] => [2,1] => [2]
=> [[1,2]]
=> 2
[3,1,2] => [1,2] => [1,1]
=> [[1],[2]]
=> 1
[3,2,1] => [2,1] => [2]
=> [[1,2]]
=> 2
[1,2,3,4] => [1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,2,4,3] => [1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,3,2,4] => [1,3,2] => [2,1]
=> [[1,2],[3]]
=> 2
[1,3,4,2] => [1,3,2] => [2,1]
=> [[1,2],[3]]
=> 2
[1,4,2,3] => [1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,4,3,2] => [1,3,2] => [2,1]
=> [[1,2],[3]]
=> 2
[2,1,3,4] => [2,1,3] => [2,1]
=> [[1,2],[3]]
=> 2
[2,1,4,3] => [2,1,3] => [2,1]
=> [[1,2],[3]]
=> 2
[2,3,1,4] => [2,3,1] => [2,1]
=> [[1,2],[3]]
=> 2
[2,3,4,1] => [2,3,1] => [2,1]
=> [[1,2],[3]]
=> 2
[2,4,1,3] => [2,1,3] => [2,1]
=> [[1,2],[3]]
=> 2
[2,4,3,1] => [2,3,1] => [2,1]
=> [[1,2],[3]]
=> 2
[3,1,2,4] => [3,1,2] => [2,1]
=> [[1,2],[3]]
=> 2
[3,1,4,2] => [3,1,2] => [2,1]
=> [[1,2],[3]]
=> 2
[3,2,1,4] => [3,2,1] => [3]
=> [[1,2,3]]
=> 3
[3,2,4,1] => [3,2,1] => [3]
=> [[1,2,3]]
=> 3
[3,4,1,2] => [3,1,2] => [2,1]
=> [[1,2],[3]]
=> 2
[3,4,2,1] => [3,2,1] => [3]
=> [[1,2,3]]
=> 3
[4,1,2,3] => [1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 1
[4,1,3,2] => [1,3,2] => [2,1]
=> [[1,2],[3]]
=> 2
[4,2,1,3] => [2,1,3] => [2,1]
=> [[1,2],[3]]
=> 2
[4,2,3,1] => [2,3,1] => [2,1]
=> [[1,2],[3]]
=> 2
[4,3,1,2] => [3,1,2] => [2,1]
=> [[1,2],[3]]
=> 2
[4,3,2,1] => [3,2,1] => [3]
=> [[1,2,3]]
=> 3
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,3,4,2,5] => [1,3,4,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,3,4,5,2] => [1,3,4,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,3,5,4,2] => [1,3,4,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,4,2,3,5] => [1,4,2,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,4,2,5,3] => [1,4,2,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,4,3,2,5] => [1,4,3,2] => [3,1]
=> [[1,2,3],[4]]
=> 3
[1,4,3,5,2] => [1,4,3,2] => [3,1]
=> [[1,2,3],[4]]
=> 3
[1,4,5,2,3] => [1,4,2,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,4,5,3,2] => [1,4,3,2] => [3,1]
=> [[1,2,3],[4]]
=> 3
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St001007
Mp00252: Permutations —restriction⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [.,.]
=> [1,0]
=> 1
[2,1] => [1] => [.,.]
=> [1,0]
=> 1
[1,2,3] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[1,3,2] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[2,1,3] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 2
[2,3,1] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 2
[3,1,2] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[3,2,1] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 2
[1,2,3,4] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1
[1,2,4,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1
[1,3,2,4] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2
[1,3,4,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2
[1,4,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1
[1,4,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2
[2,1,3,4] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[2,1,4,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[2,3,1,4] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 2
[2,3,4,1] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 2
[2,4,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[2,4,3,1] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 2
[3,1,2,4] => [3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[3,1,4,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[3,2,1,4] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 3
[3,2,4,1] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 3
[3,4,1,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[3,4,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 3
[4,1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1
[4,1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2
[4,2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[4,2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 2
[4,3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[4,3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 3
[1,2,3,4,5] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3,5] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,2,4,5,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,2,5,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,2,5,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,3,2,4,5] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,3,2,5,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,3,4,2,5] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,3,4,5,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,3,5,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,3,5,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,4,2,3,5] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,4,2,5,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,4,3,2,5] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,4,3,5,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,4,5,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,4,5,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 3
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
The following 48 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001176The size of a partition minus its first part. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000703The number of deficiencies of a permutation. St000354The number of recoils of a permutation. St000662The staircase size of the code of a permutation. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001489The maximum of the number of descents and the number of inverse descents. St000325The width of the tree associated to a permutation. St000021The number of descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000314The number of left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St000822The Hadwiger number of the graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000168The number of internal nodes of an ordered tree. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St000702The number of weak deficiencies of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St001812The biclique partition number of a graph. St001427The number of descents of a signed permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001330The hat guessing number of a graph. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001946The number of descents in a parking function. St001935The number of ascents in a parking function.
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