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Your data matches 177 different statistics following compositions of up to 3 maps.
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Matching statistic: St001761
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001761: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001761: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,2] => 0 = 1 - 1
[1,1,0,0]
=> [2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,1,2] => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 1 = 2 - 1
Description
The maximal multiplicity of a letter in a reduced word of a permutation.
For example, the permutation $3421$ has the reduced word $s_2 s_1 s_2 s_3 s_2$, where $s_2$ appears three times.
Matching statistic: St000451
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000451: 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] => 1
[1,1,0,0]
=> [2,1] => [2,1] => 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 2
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => 2
[1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,3,2,1] => 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,4,2,1] => 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,3,1] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,2,3,1] => 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,4,1] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,5,3,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,3,4,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,3,5,4,2] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,3,4,2] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,5,2] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,5,3,2,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,4,2,1,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,5,4,2,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [5,3,4,2,1] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,3,5,2,1] => 2
Description
The length of the longest pattern of the form k 1 2...(k-1).
Matching statistic: St001261
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001261: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St001261: Graphs ⟶ 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] => ([],2)
=> 1
[1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 2
[1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 3
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ([(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] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(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] => ([(3,4)],5)
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => ([(0,4),(1,3),(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] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
Description
The Castelnuovo-Mumford regularity of a graph.
Matching statistic: St001022
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001022: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001022: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
Description
Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001031
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001031: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001031: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
Description
The height of the bicoloured Motzkin path associated with the Dyck path.
Matching statistic: St001277
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001277: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St001277: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> 0 = 1 - 1
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0]
=> [1,2] => ([],2)
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
Description
The degeneracy of a graph.
The degeneracy of a graph $G$ is the maximum of the minimum degrees of the (vertex induced) subgraphs of $G$.
Matching statistic: St001280
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ 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%
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]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [2]
=> 1 = 2 - 1
[1,1,0,0]
=> [2,1] => [1,1]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [3]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,1,1]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,1]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,1]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,1,1]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [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] => [5]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,1]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [4,1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [4,1]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,2,1]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,1]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,1]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,1,1,1]
=> 1 = 2 - 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001358
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001358: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St001358: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> 0 = 1 - 1
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0]
=> [1,2] => ([],2)
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
Description
The largest degree of a regular subgraph of a graph.
For $k > 2$, it is an NP-complete problem to determine whether a graph has a $k$-regular subgraph, see [1].
Matching statistic: St001393
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001393: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St001393: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> 0 = 1 - 1
[1,0,1,0]
=> [1,2] => ([],2)
=> 0 = 1 - 1
[1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
Description
The induced matching number of a graph.
An induced matching of a graph is a set of independent edges which is an induced subgraph. This statistic records the maximal number of edges in an induced matching.
Matching statistic: St001665
(load all 31 compositions to match this statistic)
(load all 31 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St001665: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St001665: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [2,1] => 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [3,2,1] => [2,3,1] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [2,3,4,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [2,4,3,1] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,4,2,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,3,2,1] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1,4,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,1,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,1,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,1,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,1,2,4] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [2,3,4,5,1] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [2,3,5,4,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [2,4,5,3,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [2,5,4,3,1] => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [2,5,3,4,1] => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [3,4,2,5,1] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,5,2,4,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [4,3,5,2,1] => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,3,4,2,1] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,4,3,2,1] => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [4,5,2,3,1] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,4,2,3,1] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,3,2,4,1] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [3,1,4,5,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,1,5,4,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [4,1,5,3,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [5,1,4,3,2] => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,1,3,4,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [2,4,1,5,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [2,5,1,4,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [2,3,5,1,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [2,3,4,1,5] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [2,4,3,1,5] => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,5,2,1,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,4,2,1,5] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,3,2,1,5] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => 1 = 2 - 1
Description
The number of pure excedances of a permutation.
A pure excedance of a permutation $\pi$ is a position $i < \pi_i$ such that there is no $j < i$ with $i\leq \pi_j < \pi_i$.
The following 167 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000010The length of the partition. St000013The height of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000308The height of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000527The width of the poset. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001062The maximal size of a block of a set partition. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001471The magnitude of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000352The Elizalde-Pak rank of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000387The matching number of a graph. St000703The number of deficiencies of a permutation. St000834The number of right outer peaks of a permutation. St000845The maximal number of elements covered by an element in a poset. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000251The number of nonsingleton blocks of a set partition. St000390The number of runs of ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000919The number of maximal left branches of a binary tree. St001114The number of odd descents of a permutation. St001621The number of atoms of a lattice. 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 $eA$. St000808The number of up steps of the associated bargraph. St000884The number of isolated descents of a permutation. St000640The rank of the largest boolean interval in a poset. St001330The hat guessing number of a graph. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000454The largest eigenvalue of a graph if it is integral. St000015The number of peaks of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001203We associate to 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 Dyck path as follows:
St001530The depth of a Dyck path. St000260The radius of a connected graph. St000259The diameter of a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000470The number of runs in a permutation. St000619The number of cyclic descents of a permutation. St001115The number of even descents of a permutation. St000035The number of left outer peaks of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001728The number of invisible descents of a permutation. St000647The number of big descents of a permutation. St000542The number of left-to-right-minima of a permutation. St001060The distinguishing index of a graph. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001394The genus of a permutation. St001644The dimension of a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001427The number of descents of a signed permutation. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St000099The number of valleys of a permutation, including the boundary. St000325The width of the tree associated to a permutation. St000822The Hadwiger number of the graph. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001732The number of peaks visible from the left. St001734The lettericity of a graph. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000664The number of right ropes of a permutation. St000779The tier of a permutation. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001431Half of the Loewy length minus one of a modified stable Auslander algebra 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. St000862The number of parts of the shifted shape of a permutation. St000846The maximal number of elements covering an element of a poset. St001545The second Elser number of a connected graph. St000306The bounce count of a Dyck path. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001432The order dimension of the partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000474Dyson's crank of a partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000667The greatest common divisor of the parts of the partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001279The sum of the parts of an integer partition that are at least two. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000660The number of rises of length at least 3 of a Dyck path. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St000264The girth of a graph, which is not a tree. St001597The Frobenius rank of a skew partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000007The number of saliances of the permutation. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001624The breadth of a lattice. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000628The balance of a binary word. St000758The length of the longest staircase fitting into an integer composition. St000805The number of peaks of the associated bargraph. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000486The number of cycles of length at least 3 of a permutation. St000632The jump number of the poset. St000807The sum of the heights of the valleys of the associated bargraph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001335The cardinality of a minimal cycle-isolating set of a graph. St001413Half the length of the longest even length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001470The cyclic holeyness of a permutation. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001935The number of ascents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001487The number of inner corners of a skew partition.
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