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Your data matches 98 different statistics following compositions of up to 3 maps.
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Matching statistic: St001486
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(load all 4 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] => 2 = 1 + 1
[2] => 2 = 1 + 1
[1,1,1] => 2 = 1 + 1
[1,2] => 3 = 2 + 1
[2,1] => 3 = 2 + 1
[3] => 2 = 1 + 1
[1,1,1,1] => 2 = 1 + 1
[1,1,2] => 3 = 2 + 1
[1,2,1] => 4 = 3 + 1
[1,3] => 3 = 2 + 1
[2,1,1] => 3 = 2 + 1
[2,2] => 4 = 3 + 1
[3,1] => 3 = 2 + 1
[4] => 2 = 1 + 1
[1,1,1,1,1] => 2 = 1 + 1
[1,1,1,2] => 3 = 2 + 1
[1,1,2,1] => 4 = 3 + 1
[1,1,3] => 3 = 2 + 1
[1,2,1,1] => 4 = 3 + 1
[1,2,2] => 5 = 4 + 1
[1,3,1] => 4 = 3 + 1
[1,4] => 3 = 2 + 1
[2,1,1,1] => 3 = 2 + 1
[2,1,2] => 4 = 3 + 1
[2,2,1] => 5 = 4 + 1
[2,3] => 4 = 3 + 1
[3,1,1] => 3 = 2 + 1
[3,2] => 4 = 3 + 1
[4,1] => 3 = 2 + 1
[5] => 2 = 1 + 1
[1,1,1,1,1,1] => 2 = 1 + 1
[1,1,1,1,2] => 3 = 2 + 1
[1,1,1,2,1] => 4 = 3 + 1
[1,1,1,3] => 3 = 2 + 1
[1,1,2,1,1] => 4 = 3 + 1
[1,1,2,2] => 5 = 4 + 1
[1,1,3,1] => 4 = 3 + 1
[1,1,4] => 3 = 2 + 1
[1,2,1,1,1] => 4 = 3 + 1
[1,2,1,2] => 5 = 4 + 1
[1,2,2,1] => 6 = 5 + 1
[1,2,3] => 5 = 4 + 1
[1,3,1,1] => 4 = 3 + 1
[1,3,2] => 5 = 4 + 1
[1,4,1] => 4 = 3 + 1
[1,5] => 3 = 2 + 1
[2,1,1,1,1] => 3 = 2 + 1
[2,1,1,2] => 4 = 3 + 1
[2,1,2,1] => 5 = 4 + 1
[2,1,3] => 4 = 3 + 1
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: St001036
(load all 32 compositions to match this statistic)
(load all 32 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] => [1,0,1,0]
=> 0 = 1 - 1
[2] => [1,1,0,0]
=> 0 = 1 - 1
[1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[3] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
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)],2)
=> 1
[2] => [[2],[]]
=> ([(0,1)],2)
=> 1
[1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 2
[2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> 2
[3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 3
[1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 3
[3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,1,2] => [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 3
[1,1,3] => [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 3
[1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 4
[1,3,1] => [[3,3,1],[2]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 3
[1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[2,1,2] => [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 3
[2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 3
[3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 3
[4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 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
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 2
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> 3
[1,1,1,3] => [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 2
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> 3
[1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> 4
[1,1,3,1] => [[3,3,1,1],[2]]
=> ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
=> 3
[1,1,4] => [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 2
[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
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> 4
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> 5
[1,2,3] => [[4,2,1],[1]]
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> 4
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
=> 3
[1,3,2] => [[4,3,1],[2]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> 4
[1,4,1] => [[4,4,1],[3]]
=> ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> 3
[1,5] => [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 2
[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
[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] => [[3,3,2,2],[2,1,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 4
[2,1,3] => [[4,2,2],[1,1]]
=> ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
=> 3
Description
The number of maximal chains in a poset.
Matching statistic: St000482
Values
[1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2] => ([],2)
=> ([],1)
=> 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[3] => ([],3)
=> ([],1)
=> 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
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[4] => ([],4)
=> ([],1)
=> 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
[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
[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
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[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
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 2
[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] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[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
[2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[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
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[5] => ([],5)
=> ([],1)
=> 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
[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
[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
[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
[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
[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
[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
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[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
[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
[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
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[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
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[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
[1,5] => ([(4,5)],6)
=> ([(1,2)],3)
=> 2
[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
[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
[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
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
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: St001499
(load all 29 compositions to match this statistic)
(load all 29 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001499: Dyck paths ⟶ 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
St001499: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 1
[2] => [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 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]
=> 4
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 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
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[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]
=> 5
[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]
=> 5
[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]
=> 4
[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]
=> 4
[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]
=> 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]
=> 4
[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
[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]
=> 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]
=> 4
[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
[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
[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]
=> 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
[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]
=> 3
[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]
=> 2
[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]
=> 2
[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]
=> 4
[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]
=> 4
[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
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 3
Description
The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra.
We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.
Matching statistic: St000288
(load all 11 compositions to match this statistic)
(load all 11 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] => 11 => 11 => 2 = 1 + 1
[2] => 10 => 11 => 2 = 1 + 1
[1,1,1] => 111 => 111 => 3 = 2 + 1
[1,2] => 110 => 111 => 3 = 2 + 1
[2,1] => 101 => 110 => 2 = 1 + 1
[3] => 100 => 101 => 2 = 1 + 1
[1,1,1,1] => 1111 => 1111 => 4 = 3 + 1
[1,1,2] => 1110 => 1111 => 4 = 3 + 1
[1,2,1] => 1101 => 1110 => 3 = 2 + 1
[1,3] => 1100 => 1101 => 3 = 2 + 1
[2,1,1] => 1011 => 1101 => 3 = 2 + 1
[2,2] => 1010 => 1101 => 3 = 2 + 1
[3,1] => 1001 => 1010 => 2 = 1 + 1
[4] => 1000 => 1001 => 2 = 1 + 1
[1,1,1,1,1] => 11111 => 11111 => 5 = 4 + 1
[1,1,1,2] => 11110 => 11111 => 5 = 4 + 1
[1,1,2,1] => 11101 => 11110 => 4 = 3 + 1
[1,1,3] => 11100 => 11101 => 4 = 3 + 1
[1,2,1,1] => 11011 => 11101 => 4 = 3 + 1
[1,2,2] => 11010 => 11101 => 4 = 3 + 1
[1,3,1] => 11001 => 11010 => 3 = 2 + 1
[1,4] => 11000 => 11001 => 3 = 2 + 1
[2,1,1,1] => 10111 => 11011 => 4 = 3 + 1
[2,1,2] => 10110 => 11011 => 4 = 3 + 1
[2,2,1] => 10101 => 11010 => 3 = 2 + 1
[2,3] => 10100 => 11001 => 3 = 2 + 1
[3,1,1] => 10011 => 10101 => 3 = 2 + 1
[3,2] => 10010 => 10101 => 3 = 2 + 1
[4,1] => 10001 => 10010 => 2 = 1 + 1
[5] => 10000 => 10001 => 2 = 1 + 1
[1,1,1,1,1,1] => 111111 => 111111 => 6 = 5 + 1
[1,1,1,1,2] => 111110 => 111111 => 6 = 5 + 1
[1,1,1,2,1] => 111101 => 111110 => 5 = 4 + 1
[1,1,1,3] => 111100 => 111101 => 5 = 4 + 1
[1,1,2,1,1] => 111011 => 111101 => 5 = 4 + 1
[1,1,2,2] => 111010 => 111101 => 5 = 4 + 1
[1,1,3,1] => 111001 => 111010 => 4 = 3 + 1
[1,1,4] => 111000 => 111001 => 4 = 3 + 1
[1,2,1,1,1] => 110111 => 111011 => 5 = 4 + 1
[1,2,1,2] => 110110 => 111011 => 5 = 4 + 1
[1,2,2,1] => 110101 => 111010 => 4 = 3 + 1
[1,2,3] => 110100 => 111001 => 4 = 3 + 1
[1,3,1,1] => 110011 => 110101 => 4 = 3 + 1
[1,3,2] => 110010 => 110101 => 4 = 3 + 1
[1,4,1] => 110001 => 110010 => 3 = 2 + 1
[1,5] => 110000 => 110001 => 3 = 2 + 1
[2,1,1,1,1] => 101111 => 110111 => 5 = 4 + 1
[2,1,1,2] => 101110 => 110111 => 5 = 4 + 1
[2,1,2,1] => 101101 => 110110 => 4 = 3 + 1
[2,1,3] => 101100 => 110101 => 4 = 3 + 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000691
(load all 9 compositions to match this statistic)
(load all 9 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] => 11 => 00 => 0 = 1 - 1
[2] => 10 => 11 => 0 = 1 - 1
[1,1,1] => 111 => 000 => 0 = 1 - 1
[1,2] => 110 => 111 => 0 = 1 - 1
[2,1] => 101 => 110 => 1 = 2 - 1
[3] => 100 => 011 => 1 = 2 - 1
[1,1,1,1] => 1111 => 0000 => 0 = 1 - 1
[1,1,2] => 1110 => 1111 => 0 = 1 - 1
[1,2,1] => 1101 => 1110 => 1 = 2 - 1
[1,3] => 1100 => 0111 => 1 = 2 - 1
[2,1,1] => 1011 => 1100 => 1 = 2 - 1
[2,2] => 1010 => 1101 => 2 = 3 - 1
[3,1] => 1001 => 0110 => 2 = 3 - 1
[4] => 1000 => 0011 => 1 = 2 - 1
[1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
[1,1,1,2] => 11110 => 11111 => 0 = 1 - 1
[1,1,2,1] => 11101 => 11110 => 1 = 2 - 1
[1,1,3] => 11100 => 01111 => 1 = 2 - 1
[1,2,1,1] => 11011 => 11100 => 1 = 2 - 1
[1,2,2] => 11010 => 11101 => 2 = 3 - 1
[1,3,1] => 11001 => 01110 => 2 = 3 - 1
[1,4] => 11000 => 00111 => 1 = 2 - 1
[2,1,1,1] => 10111 => 11000 => 1 = 2 - 1
[2,1,2] => 10110 => 11011 => 2 = 3 - 1
[2,2,1] => 10101 => 11010 => 3 = 4 - 1
[2,3] => 10100 => 11001 => 2 = 3 - 1
[3,1,1] => 10011 => 01100 => 2 = 3 - 1
[3,2] => 10010 => 01101 => 3 = 4 - 1
[4,1] => 10001 => 00110 => 2 = 3 - 1
[5] => 10000 => 00011 => 1 = 2 - 1
[1,1,1,1,1,1] => 111111 => 000000 => 0 = 1 - 1
[1,1,1,1,2] => 111110 => 111111 => 0 = 1 - 1
[1,1,1,2,1] => 111101 => 111110 => 1 = 2 - 1
[1,1,1,3] => 111100 => 011111 => 1 = 2 - 1
[1,1,2,1,1] => 111011 => 111100 => 1 = 2 - 1
[1,1,2,2] => 111010 => 111101 => 2 = 3 - 1
[1,1,3,1] => 111001 => 011110 => 2 = 3 - 1
[1,1,4] => 111000 => 001111 => 1 = 2 - 1
[1,2,1,1,1] => 110111 => 111000 => 1 = 2 - 1
[1,2,1,2] => 110110 => 111011 => 2 = 3 - 1
[1,2,2,1] => 110101 => 111010 => 3 = 4 - 1
[1,2,3] => 110100 => 111001 => 2 = 3 - 1
[1,3,1,1] => 110011 => 011100 => 2 = 3 - 1
[1,3,2] => 110010 => 011101 => 3 = 4 - 1
[1,4,1] => 110001 => 001110 => 2 = 3 - 1
[1,5] => 110000 => 000111 => 1 = 2 - 1
[2,1,1,1,1] => 101111 => 110000 => 1 = 2 - 1
[2,1,1,2] => 101110 => 110111 => 2 = 3 - 1
[2,1,2,1] => 101101 => 110110 => 3 = 4 - 1
[2,1,3] => 101100 => 110011 => 2 = 3 - 1
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: St001035
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[2] => [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 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 = 3 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 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 = 3 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[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
[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
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[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
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 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
[1,1,1,1,1,1] => [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
[1,1,1,1,2] => [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 = 2 - 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 = 3 - 1
[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 = 2 - 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 = 3 - 1
[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 = 4 - 1
[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 = 3 - 1
[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 = 2 - 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 = 3 - 1
[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 = 4 - 1
[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 = 5 - 1
[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 = 4 - 1
[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 = 3 - 1
[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 = 4 - 1
[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 = 3 - 1
[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
[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
[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 = 3 - 1
[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 = 4 - 1
[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 = 3 - 1
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].
Matching statistic: St001027
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001027: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001027: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[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]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
[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]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 4
[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]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 4
[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]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 3
[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]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 4
[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]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 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]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 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]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 3
[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]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
[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]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 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]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 2
[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]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1
[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]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 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]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 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]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 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]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 3
[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]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[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]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 3
[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]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 3
[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]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 2
[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]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 3
Description
Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path.
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] => [1,0,1,0]
=> [1,2] => [.,[.,.]]
=> 2 = 1 + 1
[2] => [1,1,0,0]
=> [2,1] => [[.,.],.]
=> 2 = 1 + 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> 3 = 2 + 1
[1,2] => [1,0,1,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> 2 = 1 + 1
[2,1] => [1,1,0,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> 3 = 2 + 1
[3] => [1,1,1,0,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> 2 = 1 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 4 = 3 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 3 = 2 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 3 = 2 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> 2 = 1 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 4 = 3 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> 3 = 2 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> 3 = 2 + 1
[4] => [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> 2 = 1 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 5 = 4 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 4 = 3 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 4 = 3 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 3 = 2 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 4 = 3 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 3 = 2 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> 3 = 2 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> 2 = 1 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 5 = 4 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> 4 = 3 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> 4 = 3 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> 3 = 2 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> 4 = 3 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> 3 = 2 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
=> 3 = 2 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> 2 = 1 + 1
[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 = 5 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
=> 5 = 4 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]]
=> 5 = 4 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => [.,[.,[.,[[.,[.,.]],.]]]]
=> 4 = 3 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> 5 = 4 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => [.,[.,[[.,.],[[.,.],.]]]]
=> 4 = 3 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,3,4,6] => [.,[.,[[.,[.,.]],[.,.]]]]
=> 4 = 3 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,3,4,5] => [.,[.,[[.,[.,[.,.]]],.]]]
=> 3 = 2 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => [.,[[.,.],[.,[.,[.,.]]]]]
=> 5 = 4 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => [.,[[.,.],[.,[[.,.],.]]]]
=> 4 = 3 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> 4 = 3 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => [.,[[.,.],[[.,[.,.]],.]]]
=> 3 = 2 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,2,3,5,6] => [.,[[.,[.,.]],[.,[.,.]]]]
=> 4 = 3 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,2,3,6,5] => [.,[[.,[.,.]],[[.,.],.]]]
=> 3 = 2 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => [.,[[.,[.,[.,.]]],[.,.]]]
=> 3 = 2 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => [.,[[.,[.,[.,[.,.]]]],.]]
=> 2 = 1 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> 6 = 5 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => [[.,.],[.,[.,[[.,.],.]]]]
=> 5 = 4 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => [[.,.],[.,[[.,.],[.,.]]]]
=> 5 = 4 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,4,5] => [[.,.],[.,[[.,[.,.]],.]]]
=> 4 = 3 + 1
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].
The following 88 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000204The number of internal nodes of a binary tree. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000388The number of orbits of vertices of a graph under automorphisms. St001949The rigidity index of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001304The number of maximally independent sets of vertices of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001352The number of internal nodes in the modular decomposition of a graph. St001315The dissociation number of a graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000142The number of even parts of a partition. St000646The number of big ascents of a permutation. St000619The number of cyclic descents of a permutation. St001405The number of bonds in a permutation. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 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. 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. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. 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. St000292The number of ascents of a binary word. St000552The number of cut vertices of a graph. St001692The number of vertices with higher degree than the average degree in a graph. St000236The number of cyclical small weak excedances. St000242The number of indices that are not cyclical small weak excedances. St000259The diameter of a connected graph. St000636The hull number of a graph. St000871The number of very big ascents of a permutation. St001120The length of a longest path in a graph. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001388The number of non-attacking neighbors of a permutation. 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. St001093The detour number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001716The 1-improper chromatic number of a graph. St000741The Colin de Verdière graph invariant. St001875The number of simple modules with projective dimension at most 1. St001083The number of boxed occurrences of 132 in a permutation. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St000672The number of minimal elements in Bruhat order not less than the permutation. St000366The number of double descents of a permutation. St000670The reversal length of a permutation. St000742The number of big ascents of a permutation after prepending zero. 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. St000365The number of double ascents of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St001902The number of potential covers of a poset. St000993The multiplicity of the largest part of an integer partition. St001488The number of corners of a skew partition. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. 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. 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. St001960The number of descents of a permutation minus one if its first entry is not one. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000956The maximal displacement of a permutation. St001516The number of cyclic bonds of a permutation. St001557The number of inversions of the second entry 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. St001720The minimal length of a chain of small intervals in a lattice. St001090The number of pop-stack-sorts needed to sort a permutation. St001624The breadth of a lattice. St001060The distinguishing index of a graph. St000455The second largest eigenvalue of a graph if it is integral. 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. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. 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. St001637The number of (upper) dissectors of a poset. St000735The last entry on the main diagonal of a standard tableau.
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