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Your data matches 77 different statistics following compositions of up to 3 maps.
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Matching statistic: St001486
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(load all 3 compositions to match this statistic)
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,1] => 2
[2] => 2
[1,1,1] => 2
[1,2] => 3
[2,1] => 3
[3] => 2
[1,1,1,1] => 2
[1,1,2] => 3
[1,2,1] => 4
[1,3] => 3
[2,1,1] => 3
[2,2] => 4
[3,1] => 3
[4] => 2
[1,1,1,1,1] => 2
[1,1,1,2] => 3
[1,1,2,1] => 4
[1,1,3] => 3
[1,2,1,1] => 4
[1,2,2] => 5
[1,3,1] => 4
[1,4] => 3
[2,1,1,1] => 3
[2,1,2] => 4
[2,2,1] => 5
[2,3] => 4
[3,1,1] => 3
[3,2] => 4
[4,1] => 3
[5] => 2
[1,1,1,1,1,1] => 2
[1,1,1,1,2] => 3
[1,1,1,2,1] => 4
[1,1,1,3] => 3
[1,1,2,1,1] => 4
[1,1,2,2] => 5
[1,1,3,1] => 4
[1,1,4] => 3
[1,2,1,1,1] => 4
[1,2,1,2] => 5
[1,2,2,1] => 6
[1,2,3] => 5
[1,3,1,1] => 4
[1,3,2] => 5
[1,4,1] => 4
[1,5] => 3
[2,1,1,1,1] => 3
[2,1,1,2] => 4
[2,1,2,1] => 5
Description
The number of corners of the ribbon associated with an integer composition.
We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell.
This statistic records the total number of corners of the ribbon shape.
Matching statistic: St001315
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001315: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001315: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 1
[1,1] => ([(0,1)],2)
=> 2
[2] => ([],2)
=> 2
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2] => ([(1,2)],3)
=> 3
[2,1] => ([(0,2),(1,2)],3)
=> 2
[3] => ([],3)
=> 3
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,3] => ([(2,3)],4)
=> 4
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,2] => ([(1,3),(2,3)],4)
=> 3
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4] => ([],4)
=> 4
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 4
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4] => ([(3,4)],5)
=> 5
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,3] => ([(2,4),(3,4)],5)
=> 4
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[5] => ([],5)
=> 5
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 5
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,5] => ([(4,5)],6)
=> 6
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
Description
The dissociation number of a graph.
Matching statistic: St001949
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001949: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001949: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 0 = 1 - 1
[1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[2] => ([],2)
=> 1 = 2 - 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,2] => ([(1,2)],3)
=> 1 = 2 - 1
[2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3] => ([],3)
=> 2 = 3 - 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,3] => ([(2,3)],4)
=> 2 = 3 - 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,2] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[4] => ([],4)
=> 3 = 4 - 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,4] => ([(3,4)],5)
=> 3 = 4 - 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[2,3] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[5] => ([],5)
=> 4 = 5 - 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,5] => ([(4,5)],6)
=> 4 = 5 - 1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
Description
The rigidity index of a graph.
A base of a permutation group is a set $B$ such that the pointwise stabilizer of $B$ is trivial. For example, a base of the symmetric group on $n$ letters must contain all but one letter.
This statistic yields the minimal size of a base for the automorphism group of a graph.
Matching statistic: St000288
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 1 => 1
[1,1] => 11 => 11 => 2
[2] => 10 => 11 => 2
[1,1,1] => 111 => 111 => 3
[1,2] => 110 => 111 => 3
[2,1] => 101 => 110 => 2
[3] => 100 => 101 => 2
[1,1,1,1] => 1111 => 1111 => 4
[1,1,2] => 1110 => 1111 => 4
[1,2,1] => 1101 => 1110 => 3
[1,3] => 1100 => 1101 => 3
[2,1,1] => 1011 => 1101 => 3
[2,2] => 1010 => 1101 => 3
[3,1] => 1001 => 1010 => 2
[4] => 1000 => 1001 => 2
[1,1,1,1,1] => 11111 => 11111 => 5
[1,1,1,2] => 11110 => 11111 => 5
[1,1,2,1] => 11101 => 11110 => 4
[1,1,3] => 11100 => 11101 => 4
[1,2,1,1] => 11011 => 11101 => 4
[1,2,2] => 11010 => 11101 => 4
[1,3,1] => 11001 => 11010 => 3
[1,4] => 11000 => 11001 => 3
[2,1,1,1] => 10111 => 11011 => 4
[2,1,2] => 10110 => 11011 => 4
[2,2,1] => 10101 => 11010 => 3
[2,3] => 10100 => 11001 => 3
[3,1,1] => 10011 => 10101 => 3
[3,2] => 10010 => 10101 => 3
[4,1] => 10001 => 10010 => 2
[5] => 10000 => 10001 => 2
[1,1,1,1,1,1] => 111111 => 111111 => 6
[1,1,1,1,2] => 111110 => 111111 => 6
[1,1,1,2,1] => 111101 => 111110 => 5
[1,1,1,3] => 111100 => 111101 => 5
[1,1,2,1,1] => 111011 => 111101 => 5
[1,1,2,2] => 111010 => 111101 => 5
[1,1,3,1] => 111001 => 111010 => 4
[1,1,4] => 111000 => 111001 => 4
[1,2,1,1,1] => 110111 => 111011 => 5
[1,2,1,2] => 110110 => 111011 => 5
[1,2,2,1] => 110101 => 111010 => 4
[1,2,3] => 110100 => 111001 => 4
[1,3,1,1] => 110011 => 110101 => 4
[1,3,2] => 110010 => 110101 => 4
[1,4,1] => 110001 => 110010 => 3
[1,5] => 110000 => 110001 => 3
[2,1,1,1,1] => 101111 => 110111 => 5
[2,1,1,2] => 101110 => 110111 => 5
[2,1,2,1] => 101101 => 110110 => 4
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000203
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000203: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000203: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [.,.]
=> 1
[1,1] => [1,0,1,0]
=> [1,2] => [.,[.,.]]
=> 2
[2] => [1,1,0,0]
=> [2,1] => [[.,.],.]
=> 2
[1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> 3
[1,2] => [1,0,1,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> 2
[2,1] => [1,1,0,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> 3
[3] => [1,1,1,0,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> 2
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 3
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> 2
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 4
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> 3
[3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> 3
[4] => [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> 2
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 5
[1,1,1,2] => [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
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 4
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 3
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> 3
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 5
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> 4
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> 4
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> 4
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> 3
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
=> 3
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> 2
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
=> 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]]
=> 5
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => [.,[.,[.,[[.,[.,.]],.]]]]
=> 4
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> 5
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => [.,[.,[[.,.],[[.,.],.]]]]
=> 4
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,3,4,6] => [.,[.,[[.,[.,.]],[.,.]]]]
=> 4
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,3,4,5] => [.,[.,[[.,[.,[.,.]]],.]]]
=> 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => [.,[[.,.],[.,[.,[.,.]]]]]
=> 5
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => [.,[[.,.],[.,[[.,.],.]]]]
=> 4
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> 4
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => [.,[[.,.],[[.,[.,.]],.]]]
=> 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,2,3,5,6] => [.,[[.,[.,.]],[.,[.,.]]]]
=> 4
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,2,3,6,5] => [.,[[.,[.,.]],[[.,.],.]]]
=> 3
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => [.,[[.,[.,[.,.]]],[.,.]]]
=> 3
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => [.,[[.,[.,[.,[.,.]]]],.]]
=> 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> 6
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => [[.,.],[.,[.,[[.,.],.]]]]
=> 5
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => [[.,.],[.,[[.,.],[.,.]]]]
=> 5
Description
The number of external nodes of a binary tree.
That is, the number of nodes that can be reached from the root by only left steps or only right steps, plus $1$ for the root node itself. A counting formula for the number of external node in all binary trees of size $n$ can be found in [1].
Matching statistic: St000236
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000236: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000236: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1] => 1
[1,1] => [1,0,1,0]
=> [1,0,1,0]
=> [1,2] => 2
[2] => [1,1,0,0]
=> [1,1,0,0]
=> [2,1] => 2
[1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 3
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 2
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 3
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 3
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 3
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 2
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 5
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 4
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => 4
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 4
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => 4
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => 3
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => 3
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 4
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 4
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 3
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 3
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 2
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 2
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 6
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,1,6] => 5
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => 5
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,1,5,6] => 5
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,2,3,4,6,1] => 5
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,3,4,1,6] => 4
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => 4
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6] => 5
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,2,3,5,6,1] => 5
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [4,2,3,5,1,6] => 4
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [6,3,4,2,5,1] => 4
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [4,2,3,1,5,6] => 4
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,3,4,2,6,1] => 4
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => 3
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => 3
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 5
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => 5
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [3,2,4,5,1,6] => 4
Description
The number of cyclical small weak excedances.
A cyclical small weak excedance is an index $i$ such that $\pi_i \in \{ i,i+1 \}$ considered cyclically.
Matching statistic: St000259
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [2,1] => ([(0,1)],2)
=> 1
[1,1] => [1,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[2] => [1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[3] => [1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 3
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 4
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 3
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 3
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 4
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 3
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 3
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 3
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 4
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 5
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 4
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 4
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 5
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 6
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 5
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 4
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 5
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 4
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 3
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 3
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 4
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 5
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000390
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 10 => 01 => 1
[1,1] => [1,0,1,0]
=> 1010 => 0101 => 2
[2] => [1,1,0,0]
=> 1100 => 1001 => 2
[1,1,1] => [1,0,1,0,1,0]
=> 101010 => 010101 => 3
[1,2] => [1,0,1,1,0,0]
=> 101100 => 011001 => 2
[2,1] => [1,1,0,0,1,0]
=> 110010 => 100101 => 3
[3] => [1,1,1,0,0,0]
=> 111000 => 110001 => 2
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 10101010 => 01010101 => 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 10101100 => 01011001 => 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 10110010 => 01100101 => 3
[1,3] => [1,0,1,1,1,0,0,0]
=> 10111000 => 01110001 => 2
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 11001010 => 10010101 => 4
[2,2] => [1,1,0,0,1,1,0,0]
=> 11001100 => 10011001 => 3
[3,1] => [1,1,1,0,0,0,1,0]
=> 11100010 => 11000101 => 3
[4] => [1,1,1,1,0,0,0,0]
=> 11110000 => 11100001 => 2
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 0101010101 => 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 0101011001 => 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 0101100101 => 4
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 0101110001 => 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 0110010101 => 4
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 0110011001 => 3
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 0111000101 => 3
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 0111100001 => 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 1001010101 => 5
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 1001011001 => 4
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 1001100101 => 4
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 1001110001 => 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 1100010101 => 4
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 1100011001 => 3
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => 1110000101 => 3
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => 1111000001 => 2
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => 010101010101 => 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => 010101011001 => 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 101010110010 => 010101100101 => 5
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 101010111000 => 010101110001 => 4
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 101011001010 => 010110010101 => 5
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 101011001100 => 010110011001 => 4
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 101011100010 => 010111000101 => 4
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 101011110000 => 010111100001 => 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 101100101010 => 011001010101 => 5
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 101100101100 => 011001011001 => 4
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 101100110010 => 011001100101 => 4
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 101100111000 => 011001110001 => 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 101110001010 => 011100010101 => 4
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 101110001100 => 011100011001 => 3
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 101111000010 => 011110000101 => 3
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => 011111000001 => 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 110010101010 => 100101010101 => 6
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 110010101100 => 100101011001 => 5
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 110010110010 => 100101100101 => 5
Description
The number of runs of ones in a binary word.
Matching statistic: St000725
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000725: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000725: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1] => 1
[1,1] => [1,0,1,0]
=> [1,0,1,0]
=> [1,2] => 2
[2] => [1,1,0,0]
=> [1,1,0,0]
=> [2,1] => 2
[1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 3
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 2
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 3
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 2
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 3
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 3
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 5
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 4
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 3
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 4
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => 4
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => 3
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 3
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 3
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => 4
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 4
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 4
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 6
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,1,6] => 5
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,1,2] => 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,1,5,6] => 4
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,1,6,2] => 5
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,1,2,6] => 5
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,5,6,1,2,3] => 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6] => 3
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,4,1,5,6,2] => 4
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [3,4,1,5,2,6] => 4
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,5,1,6,2,3] => 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [3,4,1,2,5,6] => 4
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,5,1,2,6,3] => 5
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,1,2,3,6] => 5
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [5,6,1,2,3,4] => 4
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 2
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,2] => 3
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [3,1,4,5,2,6] => 3
Description
The smallest label of a leaf of the increasing binary tree associated to a permutation.
Matching statistic: St001120
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001120: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001120: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [2,1] => ([(0,1)],2)
=> 1
[1,1] => [1,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[2] => [1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[3] => [1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 3
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 4
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 3
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 3
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 4
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 3
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 3
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 3
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 4
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 5
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 4
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 4
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 5
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 6
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 5
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 4
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 5
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 4
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 3
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 3
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 4
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 5
Description
The length of a longest path in a graph.
The following 67 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000552The number of cut vertices of a graph. St000691The number of changes of a binary word. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001093The detour number of a graph. St001405The number of bonds in a permutation. St001692The number of vertices with higher degree than the average degree in a graph. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000619The number of cyclic descents of a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000646The number of big ascents of a permutation. St000711The number of big exceedences of a permutation. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St001388The number of non-attacking neighbors of a permutation. St000741The Colin de Verdière graph invariant. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St001875The number of simple modules with projective dimension at most 1. St000670The reversal length of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001488The number of corners of a skew partition. St000831The number of indices that are either descents or recoils. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000956The maximal displacement of a permutation. St001649The length of a longest trail in a graph. St001180Number of indecomposable injective modules with projective dimension at most 1. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001960The number of descents of a permutation minus one if its first entry is not one. St001557The number of inversions of the second entry of a permutation. St001720The minimal length of a chain of small intervals in a lattice. St001090The number of pop-stack-sorts needed to sort a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000392The length of the longest run of ones in a binary word. St000628The balance of a binary word. St001626The number of maximal proper sublattices of a lattice. St000264The girth of a graph, which is not a tree. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001060The distinguishing index of a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000834The number of right outer peaks of a permutation. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000717The number of ordinal summands of a poset. St000454The largest eigenvalue of a graph if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000075The orbit size of a standard tableau under promotion. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000099The number of valleys of a permutation, including the boundary. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000023The number of inner peaks of a permutation. St000779The tier of a permutation. St001469The holeyness of a permutation. St001520The number of strict 3-descents. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000735The last entry on the main diagonal of a standard tableau. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001645The pebbling number of a connected graph. St000260The radius of a connected graph.
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