Your data matches 40 different statistics following compositions of up to 3 maps.
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Matching statistic: St000062
(load all 11 compositions to match this statistic)
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => 1
[1,0,1,0]
 => [1,2] => [1,2] => 2
[1,1,0,0]
 => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 3
[1,0,1,1,0,0]
 => [1,3,2] => [2,3,1] => 2
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => 2
[1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => 2
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,3,4,1] => 3
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,3,1,4] => 3
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,4,1,3] => 2
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [3,4,2,1] => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => 3
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [3,2,4,1] => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1,2,4] => 3
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,1,2,3] => 3
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [4,2,3,1] => 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [4,2,1,3] => 2
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [4,3,1,2] => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,3,4,5,1] => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,3,4,1,5] => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,3,5,1,4] => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,4,5,2,1] => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,3,1,4,5] => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [3,4,2,5,1] => 3
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [2,4,1,3,5] => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,5,1,3,4] => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,5,2,4,1] => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [3,4,2,1,5] => 3
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,5,2,1,4] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [3,5,4,1,2] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [4,5,3,2,1] => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [3,2,4,5,1] => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [3,2,4,1,5] => 3
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,2,5,1,4] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [4,3,5,2,1] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,2,4,5] => 4
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [4,2,3,5,1] => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => 4
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [5,2,3,4,1] => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [4,2,3,1,5] => 3
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [5,2,3,1,4] => 3
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [5,2,4,1,3] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [5,3,4,2,1] => 2
Description
The length of the longest increasing subsequence of the permutation.
Matching statistic: St000093
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => ([],1)
 => 1
[1,0,1,0]
 => [1,2] => [1,2] => ([],2)
 => 2
[1,1,0,0]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => ([],3)
 => 3
[1,0,1,1,0,0]
 => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => 2
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => 2
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 4
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 3
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 3
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 3
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 2
[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)
 => 2
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[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
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
Description
The cardinality of a maximal independent set of vertices of a graph. An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000147
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1]
 => 1
[1,0,1,0]
 => [1,2] => [1,2] => [2]
 => 2
[1,1,0,0]
 => [2,1] => [2,1] => [1,1]
 => 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => [3]
 => 3
[1,0,1,1,0,0]
 => [1,3,2] => [3,1,2] => [2,1]
 => 2
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [2,1]
 => 2
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => [2,1]
 => 2
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => [1,1,1]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [4]
 => 4
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [4,1,2,3] => [3,1]
 => 3
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [3,1,2,4] => [3,1]
 => 3
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1,4,2] => [2,2]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [4,3,1,2] => [2,1,1]
 => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [3,1]
 => 3
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [4,2,1,3] => [2,1,1]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,3,1,4] => [3,1]
 => 3
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,3,4,1] => [3,1]
 => 3
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [4,2,3,1] => [2,1,1]
 => 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => [2,1,1]
 => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => [2,1,1]
 => 2
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [3,4,2,1] => [2,1,1]
 => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => [1,1,1,1]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [5]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [5,1,2,3,4] => [4,1]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [4,1,2,3,5] => [4,1]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,1,2,5,3] => [3,2]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [5,4,1,2,3] => [3,1,1]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [3,1,2,4,5] => [4,1]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [5,3,1,2,4] => [3,1,1]
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,1,4,2,5] => [3,2]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [3,1,4,5,2] => [3,2]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [5,3,1,4,2] => [2,2,1]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [4,3,1,2,5] => [3,1,1]
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [4,3,1,5,2] => [2,2,1]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [4,5,1,3,2] => [2,2,1]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [5,4,3,1,2] => [2,1,1,1]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [4,1]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [5,2,1,3,4] => [3,1,1]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [4,2,1,3,5] => [3,1,1]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [4,2,1,5,3] => [2,2,1]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [5,4,2,1,3] => [2,1,1,1]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3,1,4,5] => [4,1]
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [5,2,3,1,4] => [3,1,1]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => [4,1]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => [4,1]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [5,2,3,4,1] => [3,1,1]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [4,2,3,1,5] => [3,1,1]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [4,2,3,5,1] => [3,1,1]
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [4,2,5,3,1] => [2,2,1]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [5,4,2,3,1] => [2,1,1,1]
 => 2
Description
The largest part of an integer partition.
Matching statistic: St000308
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1] => 1
[1,0,1,0]
 => [1,2] => [1,2] => [1,2] => 2
[1,1,0,0]
 => [2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => [1,2,3] => 3
[1,0,1,1,0,0]
 => [1,3,2] => [3,1,2] => [3,1,2] => 2
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [2,1,3] => 2
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => [1,3,2] => 2
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => [3,2,1] => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 3
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [3,1,2,4] => [3,1,2,4] => 3
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1,4,2] => [2,1,4,3] => 2
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [4,3,1,2] => [4,3,1,2] => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 3
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [4,2,1,3] => [4,2,1,3] => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,3,1,4] => [1,3,2,4] => 3
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,3,4,1] => [1,2,4,3] => 3
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [4,2,3,1] => [4,1,3,2] => 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => [2,1,4,3] => 2
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [3,4,2,1] => [1,4,3,2] => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,1,2,5,3] => [3,1,2,5,4] => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [5,3,1,2,4] => [5,3,1,2,4] => 3
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,1,4,2,5] => [2,1,4,3,5] => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [3,1,4,5,2] => [2,1,3,5,4] => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [5,3,1,4,2] => [5,2,1,4,3] => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => 3
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [4,3,1,5,2] => [3,2,1,5,4] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [4,5,1,3,2] => [2,5,1,4,3] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [5,4,3,1,2] => [5,4,3,1,2] => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [5,2,1,3,4] => [5,2,1,3,4] => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => 3
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [4,2,1,5,3] => [3,2,1,5,4] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [5,4,2,1,3] => [5,4,2,1,3] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3,1,4,5] => [1,3,2,4,5] => 4
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [5,2,3,1,4] => [5,1,3,2,4] => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => [1,2,4,3,5] => 4
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,5,4] => 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [5,2,3,4,1] => [5,1,2,4,3] => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [4,2,3,1,5] => [4,1,3,2,5] => 3
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [4,2,3,5,1] => [3,1,2,5,4] => 3
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [4,2,5,3,1] => [2,1,5,4,3] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [5,4,2,3,1] => [5,4,1,3,2] => 2
Description
The height of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1]. The statistic is given by the height of this tree. See also [[St000325]] for the width of this tree.
Matching statistic: St000528
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => ([],1)
 => 1
[1,0,1,0]
 => [1,2] => [1,2] => ([(0,1)],2)
 => 2
[1,1,0,0]
 => [2,1] => [2,1] => ([],2)
 => 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0]
 => [1,3,2] => [3,1,2] => ([(1,2)],3)
 => 2
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
 => 2
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => ([(1,2)],3)
 => 2
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => ([],3)
 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 3
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => 3
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 2
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [4,3,1,2] => ([(2,3)],4)
 => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 3
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 3
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 3
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [4,2,3,1] => ([(2,3)],4)
 => 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 2
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
 => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => ([],4)
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [5,4,3,1,2] => ([(3,4)],5)
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [5,4,2,3,1] => ([(3,4)],5)
 => 2
Description
The height of a poset. This equals the rank of the poset [[St000080]] plus one.
Matching statistic: St000991
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1] => 1
[1,0,1,0]
 => [1,2] => [1,2] => [1,2] => 2
[1,1,0,0]
 => [2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => [1,2,3] => 3
[1,0,1,1,0,0]
 => [1,3,2] => [3,1,2] => [3,1,2] => 2
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [2,1,3] => 2
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => [1,3,2] => 2
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => [3,2,1] => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 3
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [3,1,2,4] => [3,1,2,4] => 3
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1,4,2] => [2,1,4,3] => 2
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [4,3,1,2] => [4,3,1,2] => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 3
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [4,2,1,3] => [4,2,1,3] => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,3,1,4] => [1,3,2,4] => 3
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,3,4,1] => [1,2,4,3] => 3
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [4,2,3,1] => [4,1,3,2] => 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => [2,1,4,3] => 2
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [3,4,2,1] => [1,4,3,2] => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,1,2,5,3] => [3,1,2,5,4] => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [5,3,1,2,4] => [5,3,1,2,4] => 3
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,1,4,2,5] => [2,1,4,3,5] => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [3,1,4,5,2] => [2,1,3,5,4] => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [5,3,1,4,2] => [5,2,1,4,3] => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => 3
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [4,3,1,5,2] => [3,2,1,5,4] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [4,5,1,3,2] => [2,5,1,4,3] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [5,4,3,1,2] => [5,4,3,1,2] => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [5,2,1,3,4] => [5,2,1,3,4] => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => 3
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [4,2,1,5,3] => [3,2,1,5,4] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [5,4,2,1,3] => [5,4,2,1,3] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3,1,4,5] => [1,3,2,4,5] => 4
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [5,2,3,1,4] => [5,1,3,2,4] => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => [1,2,4,3,5] => 4
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,5,4] => 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [5,2,3,4,1] => [5,1,2,4,3] => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [4,2,3,1,5] => [4,1,3,2,5] => 3
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [4,2,3,5,1] => [3,1,2,5,4] => 3
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [4,2,5,3,1] => [2,1,5,4,3] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [5,4,2,3,1] => [5,4,1,3,2] => 2
Description
The number of right-to-left minima of a permutation. For the number of left-to-right maxima, see [[St000314]].
Matching statistic: St001337
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001337: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => ([],1)
 => 1
[1,0,1,0]
 => [1,2] => [1,2] => ([],2)
 => 2
[1,1,0,0]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => ([],3)
 => 3
[1,0,1,1,0,0]
 => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => 2
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => 2
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 4
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 3
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 3
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 3
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 2
[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)
 => 2
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[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
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
Description
The upper domination number of a graph. This is the maximum cardinality of a minimal dominating set of $G$. The smallest graph with different upper irredundance number and upper domination number has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [1].
Matching statistic: St001338
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001338: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => ([],1)
 => 1
[1,0,1,0]
 => [1,2] => [1,2] => ([],2)
 => 2
[1,1,0,0]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => ([],3)
 => 3
[1,0,1,1,0,0]
 => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => 2
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => 2
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 4
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 3
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 3
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 3
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 2
[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)
 => 2
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[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
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
Description
The upper irredundance number of a graph. A set $S$ of vertices is irredundant, if there is no vertex in $S$, whose closed neighbourhood is contained in the union of the closed neighbourhoods of the other vertices of $S$. The upper irredundance number is the largest size of a maximal irredundant set. The smallest graph with different upper irredundance number and upper domination number [[St001337]] has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [2].
Matching statistic: St000080
(load all 2 compositions to match this statistic)
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000080: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => ([],1)
 => 0 = 1 - 1
[1,0,1,0]
 => [1,2] => [1,2] => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,0]
 => [2,1] => [2,1] => ([],2)
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0]
 => [1,3,2] => [3,1,2] => ([(1,2)],3)
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => ([(1,2)],3)
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => ([],3)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [4,3,1,2] => ([(2,3)],4)
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [4,2,3,1] => ([(2,3)],4)
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => ([],4)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [5,4,3,1,2] => ([(3,4)],5)
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
 => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [5,4,2,3,1] => ([(3,4)],5)
 => 1 = 2 - 1
Description
The rank of the poset.
Matching statistic: St000319
(load all 2 compositions to match this statistic)
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000319: 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
[1,0,1,0]
 => [1,2] => [1,2] => [2]
 => 1 = 2 - 1
[1,1,0,0]
 => [2,1] => [2,1] => [1,1]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => [3]
 => 2 = 3 - 1
[1,0,1,1,0,0]
 => [1,3,2] => [3,1,2] => [2,1]
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [2,1]
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => [2,1]
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => [1,1,1]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [4]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [4,1,2,3] => [3,1]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [3,1,2,4] => [3,1]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1,4,2] => [2,2]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [4,3,1,2] => [2,1,1]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [3,1]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [4,2,1,3] => [2,1,1]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,3,1,4] => [3,1]
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,3,4,1] => [3,1]
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [4,2,3,1] => [2,1,1]
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => [2,1,1]
 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => [2,1,1]
 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [3,4,2,1] => [2,1,1]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => [1,1,1,1]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [5]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [5,1,2,3,4] => [4,1]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [4,1,2,3,5] => [4,1]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,1,2,5,3] => [3,2]
 => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [5,4,1,2,3] => [3,1,1]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [3,1,2,4,5] => [4,1]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [5,3,1,2,4] => [3,1,1]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,1,4,2,5] => [3,2]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [3,1,4,5,2] => [3,2]
 => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [5,3,1,4,2] => [2,2,1]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [4,3,1,2,5] => [3,1,1]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [4,3,1,5,2] => [2,2,1]
 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [4,5,1,3,2] => [2,2,1]
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [5,4,3,1,2] => [2,1,1,1]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [4,1]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [5,2,1,3,4] => [3,1,1]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [4,2,1,3,5] => [3,1,1]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [4,2,1,5,3] => [2,2,1]
 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [5,4,2,1,3] => [2,1,1,1]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3,1,4,5] => [4,1]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [5,2,3,1,4] => [3,1,1]
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => [4,1]
 => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => [4,1]
 => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [5,2,3,4,1] => [3,1,1]
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [4,2,3,1,5] => [3,1,1]
 => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [4,2,3,5,1] => [3,1,1]
 => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [4,2,5,3,1] => [2,2,1]
 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [5,4,2,3,1] => [2,1,1,1]
 => 1 = 2 - 1
Description
The spin of an integer partition. The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions $$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$ The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross. This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
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St000320The dinv adjustment of an integer partition. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001717The largest size of an interval in a poset. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000451The length of the longest pattern of the form k 1 2. St000035The number of left outer peaks of a permutation. St000153The number of adjacent cycles of a permutation. St000166The depth minus 1 of an ordered tree. St000374The number of exclusive right-to-left minima of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000996The number of exclusive left-to-right maxima of a permutation. St001589The nesting number of a perfect matching. St000094The depth of an ordered tree. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001864The number of excedances of a signed permutation. St000983The length of the longest alternating subword. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001935The number of ascents in a parking function. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001624The breadth of a lattice. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001877Number of indecomposable injective modules with projective dimension 2.