Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000996
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1] => 0
[1,0,1,0]
 => [1,2] => [1,2] => [1,2] => 0
[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] => 0
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => [2,3,1] => 2
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
 => [2,3,1] => [3,2,1] => [3,2,1] => 1
[1,1,1,0,0,0]
 => [3,1,2] => [3,1,2] => [1,3,2] => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => [2,3,4,1] => 3
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => [2,3,1,4] => 2
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,3,2] => [3,4,2,1] => 2
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [1,4,2,3] => [2,1,4,3] => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => [3,2,4,1] => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,3,2,1] => [4,3,2,1] => 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [4,2,1,3] => [3,1,4,2] => 2
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [3,1,2,4] => [1,3,2,4] => 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [4,3,1,2] => [1,4,3,2] => 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [4,1,3,2] => [2,4,3,1] => 2
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [4,1,2,3] => [1,2,4,3] => 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] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [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] => [1,2,4,3,5] => [2,3,4,1,5] => 3
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,5,4,3] => [3,4,5,2,1] => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [1,2,5,3,4] => [2,3,1,5,4] => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => [2,3,1,4,5] => 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [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] => [1,4,3,2,5] => [3,4,2,1,5] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,4,3,2] => [4,5,3,2,1] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [1,5,3,2,4] => [3,4,1,5,2] => 3
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [1,4,2,3,5] => [2,1,4,3,5] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [1,5,4,2,3] => [3,1,5,4,2] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [1,5,2,4,3] => [3,2,5,4,1] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [1,5,2,3,4] => [2,1,3,5,4] => 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] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [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] => [2,1,4,3,5] => [3,2,4,1,5] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,4,3] => [4,3,5,2,1] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [2,1,5,3,4] => [3,2,1,5,4] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,2,1,5,4] => [4,3,2,5,1] => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [5,3,2,1,4] => [4,3,1,5,2] => 2
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [4,2,1,3,5] => [3,1,4,2,5] => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [5,4,2,1,3] => [4,1,5,3,2] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [5,2,1,4,3] => [4,2,5,3,1] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [5,2,1,3,4] => [3,1,2,5,4] => 2
Description
The number of exclusive left-to-right maxima of a permutation. This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St001330
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 27%
Values
[1,0]
 => [1] => [1] => ([],1)
 => 1 = 0 + 1
[1,0,1,0]
 => [2,1] => [1,2] => ([],2)
 => 1 = 0 + 1
[1,1,0,0]
 => [1,2] => [2,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [2,3,1] => [1,2,3] => ([],3)
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [2,1,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,1,0,0,1,0]
 => [1,3,2] => [2,1,3] => ([(1,2)],3)
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [1,2,3,4] => ([],4)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 + 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,3,4] => ([(2,3)],4)
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => ([],5)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,3,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,4,5,2,3] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,4,2,3,5] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 3 = 2 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [4,1,5,2,3] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [4,5,1,2,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [4,1,2,3,5] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,2,5,3,4] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,5,2,3,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [5,1,2,3,4] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 2 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([],6)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,5,1,6] => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [2,3,4,1,6,5] => [5,2,3,4,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,3,4,6,1,5] => [6,2,3,4,1,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,3,4,1,5,6] => [5,2,3,4,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [2,3,1,5,6,4] => [4,2,3,1,5,6] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [2,3,1,5,4,6] => [4,2,3,6,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [2,3,5,1,6,4] => [5,2,3,1,4,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,3,5,6,1,4] => [6,2,3,1,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4,6] => [5,2,3,6,4,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [2,3,1,4,6,5] => [4,2,3,5,1,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,6,4,5] => [4,2,3,6,1,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,3,6,1,4,5] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,3,1,4,5,6] => [4,2,3,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,4,5,6,3] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [2,1,4,5,3,6] => [3,2,6,4,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,6,5] => [3,2,5,4,1,6] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,6,3,5] => [3,2,6,4,1,5] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,3,4,5,6,2] => [2,1,3,4,5,6] => ([(4,5)],6)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [3,1,4,5,6,2] => [3,1,2,4,5,6] => ([(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [3,4,1,5,6,2] => [4,1,2,3,5,6] => ([(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [3,4,5,1,6,2] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [3,4,5,6,1,2] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,2,4,5,6,3] => [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,4,2,5,6,3] => [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,4,5,2,6,3] => [2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,4,5,6,2,3] => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0,1,0]
 => [4,1,2,5,6,3] => [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => 3 = 2 + 1
[1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,2,3,5,6,4] => [2,3,4,1,5,6] => ([(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,2,5,3,6,4] => [2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,2,5,6,3,4] => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,2,3,4,6,5] => [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,2,3,6,4,5] => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [3,4,5,6,7,1,2] => [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,4,5,6,7,2,3] => [2,7,1,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 2 = 1 + 1
Description
The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St001232
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 64%
Values
[1,0]
 => [1] => [.,.]
 => [1,0]
 => 0
[1,0,1,0]
 => [2,1] => [[.,.],.]
 => [1,1,0,0]
 => 0
[1,1,0,0]
 => [1,2] => [.,[.,.]]
 => [1,0,1,0]
 => 1
[1,0,1,0,1,0]
 => [3,2,1] => [[[.,.],.],.]
 => [1,1,1,0,0,0]
 => 0
[1,0,1,1,0,0]
 => [2,3,1] => [[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => 2
[1,1,0,0,1,0]
 => [3,1,2] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => [2,1,3] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => 3
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => ? = 2
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => ? = 2
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => ? = 2
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => ? = 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => ? = 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => ? = 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [[[[.,.],[.,.]],.],.]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 3
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 3
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 3
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 2
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [[[.,.],[.,[.,.]]],.]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 2
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 2
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 3
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 2
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 2
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 2
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 2
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 2
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 2
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 2
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 2
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? = 3
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? = 2
[1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 2
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 2
[1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 2
[1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? = 3
[1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 3
[1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 2
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,2,1] => [[[[[.,[.,.]],.],.],.],.]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,2,1] => [[[[[.,.],[.,.]],.],.],.]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,6,3,2,1] => [[[[[.,.],[.,.]],.],.],.]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => [[[[.,[.,[.,.]]],.],.],.]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 4
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,2,1] => [[[[[.,.],.],[.,.]],.],.]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,2,1] => [[[[.,[.,.]],[.,.]],.],.]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => ? = 4
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [6,4,3,5,2,1] => [[[[[.,.],.],[.,.]],.],.]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [5,4,3,6,2,1] => [[[[[.,.],.],[.,.]],.],.]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,2,1] => [[[[.,[.,.]],[.,.]],.],.]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => ? = 4
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [6,3,4,5,2,1] => [[[[.,.],[.,[.,.]]],.],.]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,2,1] => [[[[.,.],[.,[.,.]]],.],.]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => ? = 3
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [4,3,5,6,2,1] => [[[[.,.],[.,[.,.]]],.],.]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => [[[.,[.,[.,[.,.]]]],.],.]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 3
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,3,1] => [[[[[.,.],.],.],[.,.]],.]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,6,4,2,3,1] => [[[[.,[.,.]],.],[.,.]],.]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 4
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,5,2,3,1] => [[[[.,.],[.,.]],[.,.]],.]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [5,4,6,2,3,1] => [[[[.,.],[.,.]],[.,.]],.]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 3
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,3,1] => [[[.,[.,[.,.]]],[.,.]],.]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 3
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,4,1] => [[[[[.,.],.],.],[.,.]],.]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,5,1] => [[[[[.,.],.],.],[.,.]],.]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,6,1] => [[[[[.,.],.],.],[.,.]],.]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,2] => [[[[[.,.],.],.],.],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,1,3] => [[[[[.,.],.],.],.],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,1,4] => [[[[[.,.],.],.],.],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,1,5] => [[[[[.,.],.],.],.],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,1,6] => [[[[[.,.],.],.],.],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],.]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,3,2,1] => [[[[[[.,[.,.]],.],.],.],.],.]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 6
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,3,2,1] => [[[[[[.,.],[.,.]],.],.],.],.]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => 5
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [[[[[[.,.],.],[.,.]],.],.],.]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => 4
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,2,1] => [[[[[[.,.],.],.],[.,.]],.],.]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => 3
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,6,2,1] => [[[[[[.,.],.],.],[.,.]],.],.]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => 3
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,3,1] => [[[[[[.,.],.],.],.],[.,.]],.]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,2,6,1] => [[[[[[.,.],.],.],.],[.,.]],.]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,1,2] => [[[[[[.,.],.],.],.],.],[.,.]]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001905
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001905: Parking functions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 36%
Values
[1,0]
 => [1] => [1] => [1] => 0
[1,0,1,0]
 => [1,2] => [1,2] => [1,2] => 0
[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] => 0
[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] => 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => [2,3,1] => 1
[1,1,1,0,0,0]
 => [3,1,2] => [1,3,2] => [1,3,2] => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[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] => 2
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,4,1,2] => [3,4,1,2] => 2
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,4,1,3] => [2,4,1,3] => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,3,4,1] => [2,3,4,1] => 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 2
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [1,3,4,2] => [1,3,4,2] => 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 2
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 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] => ? = 0
[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] => ? = 3
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 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] => ? = 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 3
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 3
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [1,4,5,2,3] => [1,4,5,2,3] => ? = 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [4,1,5,2,3] => [4,1,5,2,3] => ? = 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 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] => ? = 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [5,2,3,1,4] => [5,2,3,1,4] => ? = 2
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [4,2,5,1,3] => [4,2,5,1,3] => ? = 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [4,5,2,1,3] => [4,5,2,1,3] => ? = 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 2
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => [1,3,5,2,4] => [1,3,5,2,4] => ? = 3
[1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => ? = 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => [1,3,4,5,2] => [1,3,4,5,2] => ? = 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 2
[1,1,1,0,1,0,0,0,1,0]
 => [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2
[1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => [3,1,4,5,2] => [3,1,4,5,2] => ? = 2
[1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => [3,4,1,5,2] => [3,4,1,5,2] => ? = 2
[1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => ? = 3
[1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1
[1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 3
[1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 2
[1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? = 5
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [5,1,2,3,4,6] => ? = 4
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [5,6,1,2,3,4] => [5,6,1,2,3,4] => ? = 4
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,6,2,3,4,5] => ? = 4
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => [4,1,2,3,5,6] => [4,1,2,3,5,6] => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => [4,6,1,2,3,5] => [4,6,1,2,3,5] => ? = 4
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,4,5,3,6] => [4,5,1,2,3,6] => [4,5,1,2,3,6] => ? = 3
Description
The number of preferred parking spots in a parking function less than the index of the car. Let $(a_1,\dots,a_n)$ be a parking function. Then this statistic returns the number of indices $1\leq i\leq n$ such that $a_i < i$.