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Your data matches 153 different statistics following compositions of up to 3 maps.
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Matching statistic: St000142
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 0
[1,1,0,0]
=> [1,2] => [1,2] => [2]
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,3,2] => [2,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,4,2] => [2,2]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,4,3,2] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,4,3,2] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,4,3,2] => [2,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,1,5,4,2] => [2,2,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,5,1,4,2] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,5,4,2] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,5,4,2] => [2,2,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [4,1,5,3,2] => [2,2,1]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1,5,3,2] => [2,2,1]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,5,1,2,6,4] => [3,6,1,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,6,2,4] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,5,1,6,2,4] => [3,6,1,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,5,6,1,2,4] => [3,6,5,1,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4,6] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4,6] => [3,6,1,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4,6] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,6] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,6,4] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,6,4] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,5,2,6,4] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,5,6,2,4] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [3,6,1,2,4,5] => [3,6,1,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [3,1,2,6,4,5] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,5] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,6,2,4,5] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [3,1,2,4,6,5] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
Description
The number of even parts of a partition.
Matching statistic: St000259
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 0
[1,1,0,0]
=> [2] => [1,1] => ([(0,1)],2)
=> 1
[1,1,0,1,0,0]
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [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)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [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
[1,1,0,1,0,1,1,0,0,0]
=> [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
[1,1,0,1,1,0,0,0,1,0]
=> [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)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [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
[1,1,0,1,1,0,1,0,0,0]
=> [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
[1,1,0,1,1,1,0,0,0,0]
=> [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
[1,1,1,0,0,1,0,0,1,0]
=> [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)
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [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
[1,1,1,0,0,1,1,0,0,0]
=> [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
[1,1,1,0,1,0,0,0,1,0]
=> [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)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [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
[1,1,1,0,1,0,1,0,0,0]
=> [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
[1,1,1,0,1,1,0,0,0,0]
=> [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
[1,1,1,1,0,0,0,1,0,0]
=> [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
[1,1,1,1,0,0,1,0,0,0]
=> [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
[1,1,1,1,0,1,0,0,0,0]
=> [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
[1,1,1,1,1,0,0,0,0,0]
=> [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
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,1,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [4,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,2] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,2] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000260
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 0
[1,1,0,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6] => [2,6,1,5,4,3] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,3,1,4,6,5] => [2,6,1,5,4,3] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,6] => [2,6,1,5,4,3] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,3,1,5,6,4] => [2,6,1,5,4,3] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,1,6,5,4] => [2,6,1,5,4,3] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,1,5,6] => [2,6,5,1,4,3] => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5] => [2,6,5,1,4,3] => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,1,6] => [2,6,5,4,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,5,4,1,6] => [2,6,5,4,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,4,6,1] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,6,4,5,1] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,5,4,1] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,3,1,5,6] => [2,6,5,1,4,3] => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,3,1,6,5] => [2,6,5,1,4,3] => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,4,3,5,1,6] => [2,6,5,4,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,3,5,6,1] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,3,6,5,1] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,5,3,4,1,6] => [2,6,5,4,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St001176
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 0
[1,1,0,0]
=> [1,2] => [1,2] => [1,1]
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,3,2] => [2,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,4,2] => [2,2]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,4,3,2] => [3,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => [3,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => [3,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,4,3,2] => [3,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,4,3,2] => [3,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,1,5,4,2] => [3,2]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,5,1,4,2] => [3,1,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,5,4,2] => [3,2]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => [4,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,4,3,2] => [4,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,5,4,2] => [3,2]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => [4,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => [4,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [4,1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [4,1,5,3,2] => [3,2]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [1,5,4,3,2] => [4,1]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,5,4,3,2] => [4,1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1,5,3,2] => [3,2]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [1,5,4,3,2] => [4,1]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => [4,1]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => [4,1]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [1,5,4,3,2] => [4,1]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [1,5,4,3,2] => [4,1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => [4,1]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => [4,1]
=> 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => [4,2]
=> 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,5,1,2,6,4] => [3,6,1,5,4,2] => [4,1,1]
=> 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,6,2,4] => [3,1,6,5,4,2] => [4,2]
=> 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,5,1,6,2,4] => [3,6,1,5,4,2] => [4,1,1]
=> 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,5,6,1,2,4] => [3,6,5,1,4,2] => [4,1,1]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4,6] => [3,1,6,5,4,2] => [4,2]
=> 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4,6] => [3,6,1,5,4,2] => [4,1,1]
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4,6] => [3,1,6,5,4,2] => [4,2]
=> 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,6] => [1,6,5,4,3,2] => [5,1]
=> 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => [1,6,5,4,3,2] => [5,1]
=> 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,6,4] => [3,1,6,5,4,2] => [4,2]
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,6,4] => [1,6,5,4,3,2] => [5,1]
=> 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,5,2,6,4] => [1,6,5,4,3,2] => [5,1]
=> 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,5,6,2,4] => [1,6,5,4,3,2] => [5,1]
=> 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => [4,2]
=> 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [3,6,1,2,4,5] => [3,6,1,5,4,2] => [4,1,1]
=> 2
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [3,1,2,6,4,5] => [3,1,6,5,4,2] => [4,2]
=> 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,5] => [1,6,5,4,3,2] => [5,1]
=> 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,6,2,4,5] => [1,6,5,4,3,2] => [5,1]
=> 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [3,1,2,4,6,5] => [3,1,6,5,4,2] => [4,2]
=> 2
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St001251
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 0
[1,1,0,0]
=> [1,2] => [1,2] => [2]
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,3,2] => [2,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,4,2] => [2,2]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,4,3,2] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,4,3,2] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,4,3,2] => [2,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,1,5,4,2] => [2,2,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,5,1,4,2] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,5,4,2] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,5,4,2] => [2,2,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [4,1,5,3,2] => [2,2,1]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1,5,3,2] => [2,2,1]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,5,1,2,6,4] => [3,6,1,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,6,2,4] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,5,1,6,2,4] => [3,6,1,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,5,6,1,2,4] => [3,6,5,1,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4,6] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4,6] => [3,6,1,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4,6] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,6] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,6,4] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,6,4] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,5,2,6,4] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,5,6,2,4] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [3,6,1,2,4,5] => [3,6,1,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [3,1,2,6,4,5] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,5] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,6,2,4,5] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [3,1,2,4,6,5] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
Description
The number of parts of a partition that are not congruent 1 modulo 3.
Matching statistic: St001252
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001252: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001252: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 0
[1,1,0,0]
=> [1,2] => [1,2] => [2]
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,3,2] => [2,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,4,2] => [2,2]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,4,3,2] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,4,3,2] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,4,3,2] => [2,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,1,5,4,2] => [2,2,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,5,1,4,2] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,5,4,2] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,5,4,2] => [2,2,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [4,1,5,3,2] => [2,2,1]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1,5,3,2] => [2,2,1]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,5,1,2,6,4] => [3,6,1,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,6,2,4] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,5,1,6,2,4] => [3,6,1,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,5,6,1,2,4] => [3,6,5,1,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4,6] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4,6] => [3,6,1,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4,6] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,6] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,6,4] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,6,4] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,5,2,6,4] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,5,6,2,4] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [3,6,1,2,4,5] => [3,6,1,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [3,1,2,6,4,5] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,5] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,6,2,4,5] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [3,1,2,4,6,5] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
Description
Half the sum of the even parts of a partition.
Matching statistic: St001280
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 0
[1,1,0,0]
=> [1,2] => [1,2] => [2]
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,3,2] => [2,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,4,2] => [2,2]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,4,3,2] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,4,3,2] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,4,3,2] => [2,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,1,5,4,2] => [2,2,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,5,1,4,2] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,5,4,2] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,5,4,2] => [2,2,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [4,1,5,3,2] => [2,2,1]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1,5,3,2] => [2,2,1]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,5,1,2,6,4] => [3,6,1,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,6,2,4] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,5,1,6,2,4] => [3,6,1,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,5,6,1,2,4] => [3,6,5,1,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4,6] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4,6] => [3,6,1,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4,6] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,6] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,6,4] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,6,4] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,5,2,6,4] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,5,6,2,4] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [3,6,1,2,4,5] => [3,6,1,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [3,1,2,6,4,5] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,5] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,6,2,4,5] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [3,1,2,4,6,5] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001657
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 0
[1,1,0,0]
=> [1,2] => [1,2] => [2]
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,3,2] => [2,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,4,2] => [2,2]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,4,3,2] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,4,3,2] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,4,3,2] => [2,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,1,5,4,2] => [2,2,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,5,1,4,2] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,5,4,2] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,5,4,2] => [2,2,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [4,1,5,3,2] => [2,2,1]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1,5,3,2] => [2,2,1]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,5,1,2,6,4] => [3,6,1,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,6,2,4] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,5,1,6,2,4] => [3,6,1,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,5,6,1,2,4] => [3,6,5,1,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4,6] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4,6] => [3,6,1,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4,6] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,6] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,6,4] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,6,4] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,5,2,6,4] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,5,6,2,4] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [3,6,1,2,4,5] => [3,6,1,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [3,1,2,6,4,5] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,5] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,6,2,4,5] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [3,1,2,4,6,5] => [3,1,6,5,4,2] => [2,2,1,1]
=> 2
Description
The number of twos in an integer partition.
The total number of twos in all partitions of $n$ is equal to the total number of singletons [[St001484]] in all partitions of $n-1$, see [1].
Matching statistic: St000203
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000203: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000203: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [.,.]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,2] => [1,2] => [.,[.,.]]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,3,2] => [.,[[.,.],.]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,1,5,4,2] => [[.,[.,.]],[[.,.],.]]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,5,4,2] => [[.,[.,.]],[[.,.],.]]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,5,4,2] => [[.,[.,.]],[[.,.],.]]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [4,1,5,3,2] => [[.,[[.,.],.]],[.,.]]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1,5,3,2] => [[.,[[.,.],.]],[.,.]]
=> 3 = 2 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,5,1,2,6,4] => [3,6,1,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,6,2,4] => [3,1,6,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,5,1,6,2,4] => [3,6,1,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,5,6,1,2,4] => [3,6,5,1,4,2] => [[.,[.,.]],[[[.,.],.],.]]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4,6] => [3,1,6,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4,6] => [3,6,1,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4,6] => [3,1,6,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,6] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,6,4] => [3,1,6,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,6,4] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,5,2,6,4] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,5,6,2,4] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [3,6,1,2,4,5] => [3,6,1,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [3,1,2,6,4,5] => [3,1,6,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,5] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,6,2,4,5] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [3,1,2,4,6,5] => [3,1,6,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
=> 3 = 2 + 1
Description
The number of external nodes of a binary tree.
That is, the number of nodes that can be reached from the root by only left steps or only right steps, plus $1$ for the root node itself. A counting formula for the number of external node in all binary trees of size $n$ can be found in [1].
Matching statistic: St000321
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000321: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000321: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,2] => [1,2] => [2]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,4,2] => [2,2]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,4,3,2] => [2,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,4,3,2] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,4,3,2] => [2,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,1,5,4,2] => [2,2,1]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,5,1,4,2] => [2,2,1]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,5,4,2] => [2,2,1]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,4,3,2] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,5,4,2] => [2,2,1]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [4,1,5,3,2] => [2,2,1]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [1,5,4,3,2] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,5,4,3,2] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1,5,3,2] => [2,2,1]
=> 3 = 2 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [1,5,4,3,2] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => [2,2,1,1]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,5,1,2,6,4] => [3,6,1,5,4,2] => [2,2,1,1]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,6,2,4] => [3,1,6,5,4,2] => [2,2,1,1]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,5,1,6,2,4] => [3,6,1,5,4,2] => [2,2,1,1]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,5,6,1,2,4] => [3,6,5,1,4,2] => [2,2,1,1]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4,6] => [3,1,6,5,4,2] => [2,2,1,1]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4,6] => [3,6,1,5,4,2] => [2,2,1,1]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4,6] => [3,1,6,5,4,2] => [2,2,1,1]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,6] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,6,4] => [3,1,6,5,4,2] => [2,2,1,1]
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,6,4] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,5,2,6,4] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,5,6,2,4] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => [2,2,1,1]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [3,6,1,2,4,5] => [3,6,1,5,4,2] => [2,2,1,1]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [3,1,2,6,4,5] => [3,1,6,5,4,2] => [2,2,1,1]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,5] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,6,2,4,5] => [1,6,5,4,3,2] => [2,1,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [3,1,2,4,6,5] => [3,1,6,5,4,2] => [2,2,1,1]
=> 3 = 2 + 1
Description
The number of integer partitions of n that are dominated by an integer partition.
A partition $\lambda = (\lambda_1,\ldots,\lambda_n) \vdash n$ dominates a partition $\mu = (\mu_1,\ldots,\mu_n) \vdash n$ if $\sum_{i=1}^k (\lambda_i - \mu_i) \geq 0$ for all $k$.
The following 143 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000345The number of refinements of a partition. St000507The number of ascents of a standard tableau. St000935The number of ordered refinements of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001486The number of corners of the ribbon associated with an integer composition. St000542The number of left-to-right-minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000068The number of minimal elements in a poset. St000092The number of outer peaks of a permutation. St000314The number of left-to-right-maxima of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000504The cardinality of the first block of a set partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000659The number of rises of length at least 2 of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000701The protection number of a binary tree. St000991The number of right-to-left minima of a permutation. St000993The multiplicity of the largest part of an integer partition. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000291The number of descents of a binary word. St000297The number of leading ones in a binary word. St000674The number of hills of a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000877The depth of the binary word interpreted as a path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001500The global dimension of magnitude 1 Nakayama algebras. St001730The number of times the path corresponding to a binary word crosses the base line. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000061The number of nodes on the left branch of a binary tree. St000353The number of inner valleys of a permutation. St000779The tier of a permutation. St001093The detour 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 $K[x]/(x^n)$. St001498The normalised height of a Nakayama algebra with magnitude 1. St000846The maximal number of elements covering an element of a poset. St000455The second largest eigenvalue of a graph if it is integral. St000141The maximum drop size of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000451The length of the longest pattern of the form k 1 2. St000245The number of ascents of a permutation. St000834The number of right outer peaks of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000028The number of stack-sorts needed to sort a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000662The staircase size of the code of a permutation. St000906The length of the shortest maximal chain in a poset. St000035The number of left outer peaks of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000456The monochromatic index of a connected graph. St000990The first ascent of a permutation. St000781The number of proper colouring schemes of a Ferrers diagram. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001568The smallest positive integer that does not appear twice in the partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000929The constant term of the character polynomial of an integer partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001545The second Elser number of a connected graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000007The number of saliances of the permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000845The maximal number of elements covered by an element in a poset. St000100The number of linear extensions of a poset. St001330The hat guessing number of a graph. St000654The first descent of a permutation. St000021The number of descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001665The number of pure excedances of a permutation. St001726The number of visible inversions of a permutation. St001737The number of descents of type 2 in a permutation. St001874Lusztig's a-function for the symmetric group. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000873The aix statistic of a permutation. St000989The number of final rises of a permutation. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001642The Prague dimension of a graph. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St000702The number of weak deficiencies of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001220The width of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000624The normalized sum of the minimal distances to a greater element. St000646The number of big ascents of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St000872The number of very big descents of a permutation. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001481The minimal height of a peak of a Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000741The Colin de Verdière graph invariant. St001877Number of indecomposable injective modules with projective dimension 2. St001722The number of minimal chains with small intervals between a binary word and the top element. St000264The girth of a graph, which is not a tree. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001095The number of non-isomorphic posets with precisely one further covering relation. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001488The number of corners of a skew partition. St001569The maximal modular displacement of a permutation. St001668The number of points of the poset minus the width of the poset. St001520The number of strict 3-descents. St001557The number of inversions of the second entry of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1.
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