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Your data matches 88 different statistics following compositions of up to 3 maps.
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Matching statistic: St001036
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001036: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 0
[1,1] => [1,0,1,0]
=> 0
[2] => [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> 0
[1,2] => [1,0,1,1,0,0]
=> 1
[2,1] => [1,1,0,0,1,0]
=> 1
[3] => [1,1,1,0,0,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 3
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 4
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 3
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 3
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000691
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00280: Binary words —path rowmotion⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 0 => 0
[1,1] => 11 => 00 => 0
[2] => 10 => 11 => 0
[1,1,1] => 111 => 000 => 0
[1,2] => 110 => 111 => 0
[2,1] => 101 => 110 => 1
[3] => 100 => 011 => 1
[1,1,1,1] => 1111 => 0000 => 0
[1,1,2] => 1110 => 1111 => 0
[1,2,1] => 1101 => 1110 => 1
[1,3] => 1100 => 0111 => 1
[2,1,1] => 1011 => 1100 => 1
[2,2] => 1010 => 1101 => 2
[3,1] => 1001 => 0110 => 2
[4] => 1000 => 0011 => 1
[1,1,1,1,1] => 11111 => 00000 => 0
[1,1,1,2] => 11110 => 11111 => 0
[1,1,2,1] => 11101 => 11110 => 1
[1,1,3] => 11100 => 01111 => 1
[1,2,1,1] => 11011 => 11100 => 1
[1,2,2] => 11010 => 11101 => 2
[1,3,1] => 11001 => 01110 => 2
[1,4] => 11000 => 00111 => 1
[2,1,1,1] => 10111 => 11000 => 1
[2,1,2] => 10110 => 11011 => 2
[2,2,1] => 10101 => 11010 => 3
[2,3] => 10100 => 11001 => 2
[3,1,1] => 10011 => 01100 => 2
[3,2] => 10010 => 01101 => 3
[4,1] => 10001 => 00110 => 2
[5] => 10000 => 00011 => 1
[1,1,1,1,1,1] => 111111 => 000000 => 0
[1,1,1,1,2] => 111110 => 111111 => 0
[1,1,1,2,1] => 111101 => 111110 => 1
[1,1,1,3] => 111100 => 011111 => 1
[1,1,2,1,1] => 111011 => 111100 => 1
[1,1,2,2] => 111010 => 111101 => 2
[1,1,3,1] => 111001 => 011110 => 2
[1,1,4] => 111000 => 001111 => 1
[1,2,1,1,1] => 110111 => 111000 => 1
[1,2,1,2] => 110110 => 111011 => 2
[1,2,2,1] => 110101 => 111010 => 3
[1,2,3] => 110100 => 111001 => 2
[1,3,1,1] => 110011 => 011100 => 2
[1,3,2] => 110010 => 011101 => 3
[1,4,1] => 110001 => 001110 => 2
[1,5] => 110000 => 000111 => 1
[2,1,1,1,1] => 101111 => 110000 => 1
[2,1,1,2] => 101110 => 110111 => 2
[2,1,2,1] => 101101 => 110110 => 3
Description
The number of changes of a binary word.
This is the number of indices $i$ such that $w_i \neq w_{i+1}$.
Matching statistic: St000071
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000071: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00185: Skew partitions —cell poset⟶ Posets
St000071: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
=> ([],1)
=> 1 = 0 + 1
[1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[2] => [[2],[]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 2 = 1 + 1
[2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
[1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
[3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,1,2] => [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2 = 1 + 1
[1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 3 = 2 + 1
[1,1,3] => [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2 = 1 + 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 4 = 3 + 1
[1,3,1] => [[3,3,1],[2]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 3 = 2 + 1
[1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2 = 1 + 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[2,1,2] => [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 3 = 2 + 1
[2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4 = 3 + 1
[2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 3 = 2 + 1
[3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 3 = 2 + 1
[4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 2 = 1 + 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> 3 = 2 + 1
[1,1,1,3] => [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 2 = 1 + 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> 3 = 2 + 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> 4 = 3 + 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
=> 3 = 2 + 1
[1,1,4] => [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 2 = 1 + 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
=> 3 = 2 + 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> 4 = 3 + 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> 5 = 4 + 1
[1,2,3] => [[4,2,1],[1]]
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> 4 = 3 + 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,2] => [[4,3,1],[2]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> 4 = 3 + 1
[1,4,1] => [[4,4,1],[3]]
=> ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> 3 = 2 + 1
[1,5] => [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 2 = 1 + 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2 = 1 + 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> 3 = 2 + 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 4 = 3 + 1
Description
The number of maximal chains in a poset.
Matching statistic: St000482
Values
[1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
[2] => ([],2)
=> ([],1)
=> 1 = 0 + 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1 = 0 + 1
[3] => ([],3)
=> ([],1)
=> 1 = 0 + 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2 = 1 + 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2 = 1 + 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 0 + 1
[4] => ([],4)
=> ([],1)
=> 1 = 0 + 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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 2 = 1 + 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 1 + 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 1 + 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 0 + 1
[5] => ([],5)
=> ([],1)
=> 1 = 0 + 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)
=> ([(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 = 4 + 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)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 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)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 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)
=> ([(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 = 3 + 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 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)
=> ([(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 = 3 + 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 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)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 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)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,5] => ([(4,5)],6)
=> ([(1,2)],3)
=> 2 = 1 + 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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 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)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
Description
The (zero)-forcing number of a graph.
This is the minimal number of vertices initially coloured black, such that eventually all vertices of the graph are coloured black when using the following rule:
when $u$ is a black vertex of $G$, and exactly one neighbour $v$ of $u$ is white, then colour $v$ black.
Matching statistic: St000204
(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
St000204: 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
St000204: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [.,.]
=> 0
[1,1] => [1,0,1,0]
=> [1,2] => [.,[.,.]]
=> 0
[2] => [1,1,0,0]
=> [2,1] => [[.,.],.]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,2] => [1,0,1,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> 1
[2,1] => [1,1,0,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> 0
[3] => [1,1,1,0,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[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] => [[.,.],[.,[.,.]]]
=> 0
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> 1
[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] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> 3
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
=> 2
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> 3
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
=> 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]]
=> 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => [.,[.,[.,[[.,[.,.]],.]]]]
=> 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => [.,[.,[[.,.],[[.,.],.]]]]
=> 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,3,4,6] => [.,[.,[[.,[.,.]],[.,.]]]]
=> 2
[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] => [.,[[.,.],[.,[.,[.,.]]]]]
=> 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => [.,[[.,.],[.,[[.,.],.]]]]
=> 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> 2
[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] => [.,[[.,[.,.]],[.,[.,.]]]]
=> 2
[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] => [.,[[.,[.,[.,[.,.]]]],.]]
=> 4
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => [[.,.],[.,[.,[[.,.],.]]]]
=> 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => [[.,.],[.,[[.,.],[.,.]]]]
=> 1
Description
The number of internal nodes of a binary tree.
That is, the total number of nodes of the tree minus [[St000203]]. A counting formula for the total number of internal nodes across all binary trees of size $n$ is given in [1]. This is equivalent to the number of internal triangles in all triangulations of an $(n+1)$-gon.
Matching statistic: St000288
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00105: Binary words —complement⟶ 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
Mp00105: Binary words —complement⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 1 => 0 => 0
[1,1] => 11 => 11 => 00 => 0
[2] => 10 => 11 => 00 => 0
[1,1,1] => 111 => 111 => 000 => 0
[1,2] => 110 => 111 => 000 => 0
[2,1] => 101 => 110 => 001 => 1
[3] => 100 => 101 => 010 => 1
[1,1,1,1] => 1111 => 1111 => 0000 => 0
[1,1,2] => 1110 => 1111 => 0000 => 0
[1,2,1] => 1101 => 1110 => 0001 => 1
[1,3] => 1100 => 1101 => 0010 => 1
[2,1,1] => 1011 => 1101 => 0010 => 1
[2,2] => 1010 => 1101 => 0010 => 1
[3,1] => 1001 => 1010 => 0101 => 2
[4] => 1000 => 1001 => 0110 => 2
[1,1,1,1,1] => 11111 => 11111 => 00000 => 0
[1,1,1,2] => 11110 => 11111 => 00000 => 0
[1,1,2,1] => 11101 => 11110 => 00001 => 1
[1,1,3] => 11100 => 11101 => 00010 => 1
[1,2,1,1] => 11011 => 11101 => 00010 => 1
[1,2,2] => 11010 => 11101 => 00010 => 1
[1,3,1] => 11001 => 11010 => 00101 => 2
[1,4] => 11000 => 11001 => 00110 => 2
[2,1,1,1] => 10111 => 11011 => 00100 => 1
[2,1,2] => 10110 => 11011 => 00100 => 1
[2,2,1] => 10101 => 11010 => 00101 => 2
[2,3] => 10100 => 11001 => 00110 => 2
[3,1,1] => 10011 => 10101 => 01010 => 2
[3,2] => 10010 => 10101 => 01010 => 2
[4,1] => 10001 => 10010 => 01101 => 3
[5] => 10000 => 10001 => 01110 => 3
[1,1,1,1,1,1] => 111111 => 111111 => 000000 => 0
[1,1,1,1,2] => 111110 => 111111 => 000000 => 0
[1,1,1,2,1] => 111101 => 111110 => 000001 => 1
[1,1,1,3] => 111100 => 111101 => 000010 => 1
[1,1,2,1,1] => 111011 => 111101 => 000010 => 1
[1,1,2,2] => 111010 => 111101 => 000010 => 1
[1,1,3,1] => 111001 => 111010 => 000101 => 2
[1,1,4] => 111000 => 111001 => 000110 => 2
[1,2,1,1,1] => 110111 => 111011 => 000100 => 1
[1,2,1,2] => 110110 => 111011 => 000100 => 1
[1,2,2,1] => 110101 => 111010 => 000101 => 2
[1,2,3] => 110100 => 111001 => 000110 => 2
[1,3,1,1] => 110011 => 110101 => 001010 => 2
[1,3,2] => 110010 => 110101 => 001010 => 2
[1,4,1] => 110001 => 110010 => 001101 => 3
[1,5] => 110000 => 110001 => 001110 => 3
[2,1,1,1,1] => 101111 => 110111 => 001000 => 1
[2,1,1,2] => 101110 => 110111 => 001000 => 1
[2,1,2,1] => 101101 => 110110 => 001001 => 2
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St001486
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00280: Binary words —path rowmotion⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 0 => [2] => 2 = 0 + 2
[1,1] => 11 => 00 => [3] => 2 = 0 + 2
[2] => 10 => 11 => [1,1,1] => 2 = 0 + 2
[1,1,1] => 111 => 000 => [4] => 2 = 0 + 2
[1,2] => 110 => 111 => [1,1,1,1] => 2 = 0 + 2
[2,1] => 101 => 110 => [1,1,2] => 3 = 1 + 2
[3] => 100 => 011 => [2,1,1] => 3 = 1 + 2
[1,1,1,1] => 1111 => 0000 => [5] => 2 = 0 + 2
[1,1,2] => 1110 => 1111 => [1,1,1,1,1] => 2 = 0 + 2
[1,2,1] => 1101 => 1110 => [1,1,1,2] => 3 = 1 + 2
[1,3] => 1100 => 0111 => [2,1,1,1] => 3 = 1 + 2
[2,1,1] => 1011 => 1100 => [1,1,3] => 3 = 1 + 2
[2,2] => 1010 => 1101 => [1,1,2,1] => 4 = 2 + 2
[3,1] => 1001 => 0110 => [2,1,2] => 4 = 2 + 2
[4] => 1000 => 0011 => [3,1,1] => 3 = 1 + 2
[1,1,1,1,1] => 11111 => 00000 => [6] => 2 = 0 + 2
[1,1,1,2] => 11110 => 11111 => [1,1,1,1,1,1] => 2 = 0 + 2
[1,1,2,1] => 11101 => 11110 => [1,1,1,1,2] => 3 = 1 + 2
[1,1,3] => 11100 => 01111 => [2,1,1,1,1] => 3 = 1 + 2
[1,2,1,1] => 11011 => 11100 => [1,1,1,3] => 3 = 1 + 2
[1,2,2] => 11010 => 11101 => [1,1,1,2,1] => 4 = 2 + 2
[1,3,1] => 11001 => 01110 => [2,1,1,2] => 4 = 2 + 2
[1,4] => 11000 => 00111 => [3,1,1,1] => 3 = 1 + 2
[2,1,1,1] => 10111 => 11000 => [1,1,4] => 3 = 1 + 2
[2,1,2] => 10110 => 11011 => [1,1,2,1,1] => 4 = 2 + 2
[2,2,1] => 10101 => 11010 => [1,1,2,2] => 5 = 3 + 2
[2,3] => 10100 => 11001 => [1,1,3,1] => 4 = 2 + 2
[3,1,1] => 10011 => 01100 => [2,1,3] => 4 = 2 + 2
[3,2] => 10010 => 01101 => [2,1,2,1] => 5 = 3 + 2
[4,1] => 10001 => 00110 => [3,1,2] => 4 = 2 + 2
[5] => 10000 => 00011 => [4,1,1] => 3 = 1 + 2
[1,1,1,1,1,1] => 111111 => 000000 => [7] => 2 = 0 + 2
[1,1,1,1,2] => 111110 => 111111 => [1,1,1,1,1,1,1] => 2 = 0 + 2
[1,1,1,2,1] => 111101 => 111110 => [1,1,1,1,1,2] => 3 = 1 + 2
[1,1,1,3] => 111100 => 011111 => [2,1,1,1,1,1] => 3 = 1 + 2
[1,1,2,1,1] => 111011 => 111100 => [1,1,1,1,3] => 3 = 1 + 2
[1,1,2,2] => 111010 => 111101 => [1,1,1,1,2,1] => 4 = 2 + 2
[1,1,3,1] => 111001 => 011110 => [2,1,1,1,2] => 4 = 2 + 2
[1,1,4] => 111000 => 001111 => [3,1,1,1,1] => 3 = 1 + 2
[1,2,1,1,1] => 110111 => 111000 => [1,1,1,4] => 3 = 1 + 2
[1,2,1,2] => 110110 => 111011 => [1,1,1,2,1,1] => 4 = 2 + 2
[1,2,2,1] => 110101 => 111010 => [1,1,1,2,2] => 5 = 3 + 2
[1,2,3] => 110100 => 111001 => [1,1,1,3,1] => 4 = 2 + 2
[1,3,1,1] => 110011 => 011100 => [2,1,1,3] => 4 = 2 + 2
[1,3,2] => 110010 => 011101 => [2,1,1,2,1] => 5 = 3 + 2
[1,4,1] => 110001 => 001110 => [3,1,1,2] => 4 = 2 + 2
[1,5] => 110000 => 000111 => [4,1,1,1] => 3 = 1 + 2
[2,1,1,1,1] => 101111 => 110000 => [1,1,5] => 3 = 1 + 2
[2,1,1,2] => 101110 => 110111 => [1,1,2,1,1,1] => 4 = 2 + 2
[2,1,2,1] => 101101 => 110110 => [1,1,2,1,2] => 5 = 3 + 2
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: St000388
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000388: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
St000388: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 1 = 0 + 1
[1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2] => ([],2)
=> 1 = 0 + 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3] => ([],3)
=> 1 = 0 + 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4] => ([],4)
=> 1 = 0 + 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)
=> 1 = 0 + 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[5] => ([],5)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 2 = 1 + 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 = 2 + 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 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 = 2 + 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 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 = 2 + 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 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)
=> 5 = 4 + 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 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 = 2 + 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5] => ([(4,5)],6)
=> 2 = 1 + 1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[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
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
Description
The number of orbits of vertices of a graph under automorphisms.
Matching statistic: St001951
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001951: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
St001951: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 1 = 0 + 1
[1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2] => ([],2)
=> 1 = 0 + 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3] => ([],3)
=> 1 = 0 + 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4] => ([],4)
=> 1 = 0 + 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)
=> 1 = 0 + 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[5] => ([],5)
=> 1 = 0 + 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)
=> 1 = 0 + 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)
=> 2 = 1 + 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 = 2 + 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 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 = 2 + 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 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 = 2 + 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 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)
=> 5 = 4 + 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 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 = 2 + 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5] => ([(4,5)],6)
=> 2 = 1 + 1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[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
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
Description
The number of factors in the disjoint direct product decomposition of the automorphism group of a graph.
The disjoint direct product decomposition of a permutation group factors the group corresponding to the product $(G, X) \ast (H, Y) = (G\times H, Z)$, where $Z$ is the disjoint union of $X$ and $Y$.
In particular, for an asymmetric graph, i.e., with trivial automorphism group, this statistic equals the number of vertices, because the trivial action factors completely.
Matching statistic: St001035
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> ? = 0
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> 3
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 3
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 3
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 2
Description
The convexity degree of the parallelogram polyomino associated with the Dyck path.
A parallelogram polyomino is $k$-convex if $k$ is the maximal number of turns an axis-parallel path must take to connect two cells of the polyomino.
For example, any rotation of a Ferrers shape has convexity degree at most one.
The (bivariate) generating function is given in Theorem 2 of [1].
The following 78 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St001304The number of maximally independent sets of vertices of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001280The number of parts of an integer partition that are at least two. St000473The number of parts of a partition that are strictly bigger than the number of ones. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000159The number of distinct parts of the integer partition. St000142The number of even parts of a partition. St000646The number of big ascents of a permutation. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St000619The number of cyclic descents of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000648The number of 2-excedences of a permutation. St000444The length of the maximal rise of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000312The number of leaves in a graph. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000711The number of big exceedences of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000993The multiplicity of the largest part of an integer partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. 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 $eA$. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000242The number of indices that are not cyclical small weak excedances. St000871The number of very big ascents of a permutation. St000636The hull number of a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001883The mutual visibility number of a graph. St001388The number of non-attacking neighbors of a permutation. St000454The largest eigenvalue of a graph if it is integral. St001315The dissociation number of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001638The book thickness of a graph. St001083The number of boxed occurrences of 132 in a permutation. St000460The hook length of the last cell along the main diagonal of an integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000120The number of left tunnels of a Dyck path. St000366The number of double descents of a permutation. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000672The number of minimal elements in Bruhat order not less than the permutation. St000365The number of double ascents of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001902The number of potential covers of a poset. St000741The Colin de Verdière graph invariant. St001537The number of cyclic crossings of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000681The Grundy value of Chomp on Ferrers diagrams. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001516The number of cyclic bonds of a permutation. 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. St001118The acyclic chromatic index of a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000668The least common multiple of the parts of the partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000937The number of positive values of the symmetric group character corresponding to the partition. St001568The smallest positive integer that does not appear twice in the partition. St001624The breadth of a lattice. St001060The distinguishing index of a graph. St000670The reversal length of a permutation. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001637The number of (upper) dissectors of a poset.
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