Your data matches 53 different statistics following compositions of up to 3 maps.
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Mp00071: Permutations descent compositionInteger compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[2,1] => [1,1] => 2
[1,3,2] => [2,1] => 3
[2,3,1] => [2,1] => 3
[3,2,1] => [1,1,1] => 2
[1,2,4,3] => [3,1] => 3
[1,3,4,2] => [3,1] => 3
[1,4,3,2] => [2,1,1] => 3
[2,1,4,3] => [1,2,1] => 4
[2,3,4,1] => [3,1] => 3
[2,4,3,1] => [2,1,1] => 3
[3,1,4,2] => [1,2,1] => 4
[3,2,4,1] => [1,2,1] => 4
[3,4,2,1] => [2,1,1] => 3
[4,1,3,2] => [1,2,1] => 4
[4,2,3,1] => [1,2,1] => 4
[4,3,2,1] => [1,1,1,1] => 2
[1,2,3,5,4] => [4,1] => 3
[1,2,4,5,3] => [4,1] => 3
[1,2,5,4,3] => [3,1,1] => 3
[1,3,2,5,4] => [2,2,1] => 5
[1,3,4,5,2] => [4,1] => 3
[1,3,5,4,2] => [3,1,1] => 3
[1,4,2,5,3] => [2,2,1] => 5
[1,4,3,5,2] => [2,2,1] => 5
[1,4,5,3,2] => [3,1,1] => 3
[1,5,2,4,3] => [2,2,1] => 5
[1,5,3,4,2] => [2,2,1] => 5
[1,5,4,3,2] => [2,1,1,1] => 3
[2,1,3,5,4] => [1,3,1] => 4
[2,1,4,5,3] => [1,3,1] => 4
[2,1,5,4,3] => [1,2,1,1] => 4
[2,3,1,5,4] => [2,2,1] => 5
[2,3,4,5,1] => [4,1] => 3
[2,3,5,4,1] => [3,1,1] => 3
[2,4,1,5,3] => [2,2,1] => 5
[2,4,3,5,1] => [2,2,1] => 5
[2,4,5,3,1] => [3,1,1] => 3
[2,5,1,4,3] => [2,2,1] => 5
[2,5,3,4,1] => [2,2,1] => 5
[2,5,4,3,1] => [2,1,1,1] => 3
[3,1,2,5,4] => [1,3,1] => 4
[3,1,4,5,2] => [1,3,1] => 4
[3,1,5,4,2] => [1,2,1,1] => 4
[3,2,1,5,4] => [1,1,2,1] => 4
[3,2,4,5,1] => [1,3,1] => 4
[3,2,5,4,1] => [1,2,1,1] => 4
[3,4,1,5,2] => [2,2,1] => 5
[3,4,2,5,1] => [2,2,1] => 5
[3,4,5,2,1] => [3,1,1] => 3
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.
Mp00071: Permutations descent compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000777: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 1
[2,1] => [1,1] => ([(0,1)],2)
=> 2
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 3
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,5,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[2,5,3,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[2,5,4,3,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,2,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[3,1,5,4,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[3,2,1,5,4] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[3,2,4,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[3,2,5,4,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[3,4,1,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[3,4,2,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000340
Mp00064: Permutations reversePermutations
Mp00071: Permutations descent compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> 0 = 1 - 1
[2,1] => [1,2] => [2] => [1,1,0,0]
=> 1 = 2 - 1
[1,3,2] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,3,1] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[3,2,1] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,4,3] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,3,4,2] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,4,3,2] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[2,1,4,3] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[2,3,4,1] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[2,4,3,1] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[3,1,4,2] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[3,2,4,1] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[3,4,2,1] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[4,1,3,2] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[4,2,3,1] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[4,3,2,1] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,3,2,5,4] => [4,5,2,3,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,3,4,5,2] => [2,5,4,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,3,5,4,2] => [2,4,5,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,4,3,5,2] => [2,5,3,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,4,5,3,2] => [2,3,5,4,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,5,2,4,3] => [3,4,2,5,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,5,3,4,2] => [2,4,3,5,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,5,4,3,2] => [2,3,4,5,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[2,1,3,5,4] => [4,5,3,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[2,1,4,5,3] => [3,5,4,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[2,1,5,4,3] => [3,4,5,1,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[2,3,1,5,4] => [4,5,1,3,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[2,3,4,5,1] => [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[2,3,5,4,1] => [1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[2,4,1,5,3] => [3,5,1,4,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[2,4,3,5,1] => [1,5,3,4,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[2,4,5,3,1] => [1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[2,5,1,4,3] => [3,4,1,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[2,5,3,4,1] => [1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[2,5,4,3,1] => [1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[3,1,2,5,4] => [4,5,2,1,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[3,1,4,5,2] => [2,5,4,1,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[3,1,5,4,2] => [2,4,5,1,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[3,2,1,5,4] => [4,5,1,2,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,2,4,5,1] => [1,5,4,2,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[3,2,5,4,1] => [1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[3,4,1,5,2] => [2,5,1,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[3,4,2,5,1] => [1,5,2,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[3,4,5,2,1] => [1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
Description
The number of non-final maximal constant sub-paths of length greater than one. This is the total number of occurrences of the patterns $110$ and $001$.
Mp00064: Permutations reversePermutations
Mp00071: Permutations descent compositionInteger compositions
Mp00094: Integer compositions to binary wordBinary words
St000691: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1 => 0 = 1 - 1
[2,1] => [1,2] => [2] => 10 => 1 = 2 - 1
[1,3,2] => [2,3,1] => [2,1] => 101 => 2 = 3 - 1
[2,3,1] => [1,3,2] => [2,1] => 101 => 2 = 3 - 1
[3,2,1] => [1,2,3] => [3] => 100 => 1 = 2 - 1
[1,2,4,3] => [3,4,2,1] => [2,1,1] => 1011 => 2 = 3 - 1
[1,3,4,2] => [2,4,3,1] => [2,1,1] => 1011 => 2 = 3 - 1
[1,4,3,2] => [2,3,4,1] => [3,1] => 1001 => 2 = 3 - 1
[2,1,4,3] => [3,4,1,2] => [2,2] => 1010 => 3 = 4 - 1
[2,3,4,1] => [1,4,3,2] => [2,1,1] => 1011 => 2 = 3 - 1
[2,4,3,1] => [1,3,4,2] => [3,1] => 1001 => 2 = 3 - 1
[3,1,4,2] => [2,4,1,3] => [2,2] => 1010 => 3 = 4 - 1
[3,2,4,1] => [1,4,2,3] => [2,2] => 1010 => 3 = 4 - 1
[3,4,2,1] => [1,2,4,3] => [3,1] => 1001 => 2 = 3 - 1
[4,1,3,2] => [2,3,1,4] => [2,2] => 1010 => 3 = 4 - 1
[4,2,3,1] => [1,3,2,4] => [2,2] => 1010 => 3 = 4 - 1
[4,3,2,1] => [1,2,3,4] => [4] => 1000 => 1 = 2 - 1
[1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => 10111 => 2 = 3 - 1
[1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => 10111 => 2 = 3 - 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,1,1] => 10011 => 2 = 3 - 1
[1,3,2,5,4] => [4,5,2,3,1] => [2,2,1] => 10101 => 4 = 5 - 1
[1,3,4,5,2] => [2,5,4,3,1] => [2,1,1,1] => 10111 => 2 = 3 - 1
[1,3,5,4,2] => [2,4,5,3,1] => [3,1,1] => 10011 => 2 = 3 - 1
[1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => 10101 => 4 = 5 - 1
[1,4,3,5,2] => [2,5,3,4,1] => [2,2,1] => 10101 => 4 = 5 - 1
[1,4,5,3,2] => [2,3,5,4,1] => [3,1,1] => 10011 => 2 = 3 - 1
[1,5,2,4,3] => [3,4,2,5,1] => [2,2,1] => 10101 => 4 = 5 - 1
[1,5,3,4,2] => [2,4,3,5,1] => [2,2,1] => 10101 => 4 = 5 - 1
[1,5,4,3,2] => [2,3,4,5,1] => [4,1] => 10001 => 2 = 3 - 1
[2,1,3,5,4] => [4,5,3,1,2] => [2,1,2] => 10110 => 3 = 4 - 1
[2,1,4,5,3] => [3,5,4,1,2] => [2,1,2] => 10110 => 3 = 4 - 1
[2,1,5,4,3] => [3,4,5,1,2] => [3,2] => 10010 => 3 = 4 - 1
[2,3,1,5,4] => [4,5,1,3,2] => [2,2,1] => 10101 => 4 = 5 - 1
[2,3,4,5,1] => [1,5,4,3,2] => [2,1,1,1] => 10111 => 2 = 3 - 1
[2,3,5,4,1] => [1,4,5,3,2] => [3,1,1] => 10011 => 2 = 3 - 1
[2,4,1,5,3] => [3,5,1,4,2] => [2,2,1] => 10101 => 4 = 5 - 1
[2,4,3,5,1] => [1,5,3,4,2] => [2,2,1] => 10101 => 4 = 5 - 1
[2,4,5,3,1] => [1,3,5,4,2] => [3,1,1] => 10011 => 2 = 3 - 1
[2,5,1,4,3] => [3,4,1,5,2] => [2,2,1] => 10101 => 4 = 5 - 1
[2,5,3,4,1] => [1,4,3,5,2] => [2,2,1] => 10101 => 4 = 5 - 1
[2,5,4,3,1] => [1,3,4,5,2] => [4,1] => 10001 => 2 = 3 - 1
[3,1,2,5,4] => [4,5,2,1,3] => [2,1,2] => 10110 => 3 = 4 - 1
[3,1,4,5,2] => [2,5,4,1,3] => [2,1,2] => 10110 => 3 = 4 - 1
[3,1,5,4,2] => [2,4,5,1,3] => [3,2] => 10010 => 3 = 4 - 1
[3,2,1,5,4] => [4,5,1,2,3] => [2,3] => 10100 => 3 = 4 - 1
[3,2,4,5,1] => [1,5,4,2,3] => [2,1,2] => 10110 => 3 = 4 - 1
[3,2,5,4,1] => [1,4,5,2,3] => [3,2] => 10010 => 3 = 4 - 1
[3,4,1,5,2] => [2,5,1,4,3] => [2,2,1] => 10101 => 4 = 5 - 1
[3,4,2,5,1] => [1,5,2,4,3] => [2,2,1] => 10101 => 4 = 5 - 1
[3,4,5,2,1] => [1,2,5,4,3] => [3,1,1] => 10011 => 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: St001504
Mp00071: Permutations descent compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00229: Dyck paths Delest-ViennotDyck paths
St001504: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 2 = 1 + 1
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 4 = 3 + 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[1,5,2,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[1,5,3,4,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[2,1,3,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[2,1,4,5,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[2,1,5,4,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[2,3,1,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[2,3,4,5,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[2,3,5,4,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[2,4,1,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[2,4,3,5,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[2,4,5,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[2,5,1,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[2,5,3,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[2,5,4,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[3,1,2,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[3,1,4,5,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[3,1,5,4,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[3,2,1,5,4] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[3,2,4,5,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[3,2,5,4,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[3,4,1,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[3,4,2,5,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[3,4,5,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
Description
The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path.
Mp00071: Permutations descent compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
St001035: Dyck paths ⟶ ℤResult quality: 86% values known / values provided: 100%distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1,0]
=> ? = 1 - 2
[2,1] => [1,1] => [1,0,1,0]
=> 0 = 2 - 2
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1 = 3 - 2
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1 = 3 - 2
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 2 - 2
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 3 - 2
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 3 - 2
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 3 - 2
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,5,2,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[1,5,3,4,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[2,1,3,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[2,1,4,5,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[2,1,5,4,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[2,3,1,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[2,3,4,5,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 3 - 2
[2,3,5,4,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 3 - 2
[2,4,1,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[2,4,3,5,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[2,4,5,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 3 - 2
[2,5,1,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[2,5,3,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[2,5,4,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[3,1,2,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[3,1,4,5,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[3,1,5,4,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[3,2,1,5,4] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[3,2,4,5,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[3,2,5,4,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[3,4,1,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[3,4,2,5,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[3,4,5,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 3 - 2
[3,5,1,4,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 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].
Matching statistic: St001028
Mp00109: Permutations descent wordBinary words
Mp00097: Binary words delta morphismInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
St001028: Dyck paths ⟶ ℤResult quality: 86% values known / values provided: 100%distinct values known / distinct values provided: 86%
Values
[1] => => [] => ?
=> ? = 1
[2,1] => 1 => [1] => [1,0]
=> 2
[1,3,2] => 01 => [1,1] => [1,0,1,0]
=> 3
[2,3,1] => 01 => [1,1] => [1,0,1,0]
=> 3
[3,2,1] => 11 => [2] => [1,1,0,0]
=> 2
[1,2,4,3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[1,3,4,2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[1,4,3,2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[2,1,4,3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[2,3,4,1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[2,4,3,1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[3,1,4,2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[3,2,4,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[3,4,2,1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[4,1,3,2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[4,2,3,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[4,3,2,1] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[1,2,3,5,4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,4,5,3] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,5,4,3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[1,3,2,5,4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,3,4,5,2] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,3,5,4,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[1,4,2,5,3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,4,3,5,2] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,4,5,3,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[1,5,2,4,3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,5,3,4,2] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,5,4,3,2] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[2,1,3,5,4] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[2,1,4,5,3] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[2,1,5,4,3] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 4
[2,3,1,5,4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[2,3,4,5,1] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[2,3,5,4,1] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[2,4,1,5,3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[2,4,3,5,1] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[2,4,5,3,1] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[2,5,1,4,3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[2,5,3,4,1] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[2,5,4,3,1] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[3,1,2,5,4] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[3,1,4,5,2] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[3,1,5,4,2] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 4
[3,2,1,5,4] => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[3,2,4,5,1] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[3,2,5,4,1] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 4
[3,4,1,5,2] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[3,4,2,5,1] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[3,4,5,2,1] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[3,5,1,4,2] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
Description
Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001290
Mp00109: Permutations descent wordBinary words
Mp00097: Binary words delta morphismInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
St001290: Dyck paths ⟶ ℤResult quality: 86% values known / values provided: 100%distinct values known / distinct values provided: 86%
Values
[1] => => [] => ?
=> ? = 1
[2,1] => 1 => [1] => [1,0]
=> 2
[1,3,2] => 01 => [1,1] => [1,0,1,0]
=> 3
[2,3,1] => 01 => [1,1] => [1,0,1,0]
=> 3
[3,2,1] => 11 => [2] => [1,1,0,0]
=> 2
[1,2,4,3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[1,3,4,2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[1,4,3,2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[2,1,4,3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[2,3,4,1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[2,4,3,1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[3,1,4,2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[3,2,4,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[3,4,2,1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[4,1,3,2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[4,2,3,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[4,3,2,1] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[1,2,3,5,4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,4,5,3] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,5,4,3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[1,3,2,5,4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,3,4,5,2] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,3,5,4,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[1,4,2,5,3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,4,3,5,2] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,4,5,3,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[1,5,2,4,3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,5,3,4,2] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,5,4,3,2] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[2,1,3,5,4] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[2,1,4,5,3] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[2,1,5,4,3] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 4
[2,3,1,5,4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[2,3,4,5,1] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[2,3,5,4,1] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[2,4,1,5,3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[2,4,3,5,1] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[2,4,5,3,1] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[2,5,1,4,3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[2,5,3,4,1] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[2,5,4,3,1] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[3,1,2,5,4] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[3,1,4,5,2] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[3,1,5,4,2] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 4
[3,2,1,5,4] => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[3,2,4,5,1] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[3,2,5,4,1] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 4
[3,4,1,5,2] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[3,4,2,5,1] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[3,4,5,2,1] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[3,5,1,4,2] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
Description
The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A.
Matching statistic: St000010
Mp00109: Permutations descent wordBinary words
Mp00097: Binary words delta morphismInteger compositions
Mp00040: Integer compositions to partitionInteger partitions
St000010: Integer partitions ⟶ ℤResult quality: 86% values known / values provided: 100%distinct values known / distinct values provided: 86%
Values
[1] => => [] => ?
=> ? = 1 - 1
[2,1] => 1 => [1] => [1]
=> 1 = 2 - 1
[1,3,2] => 01 => [1,1] => [1,1]
=> 2 = 3 - 1
[2,3,1] => 01 => [1,1] => [1,1]
=> 2 = 3 - 1
[3,2,1] => 11 => [2] => [2]
=> 1 = 2 - 1
[1,2,4,3] => 001 => [2,1] => [2,1]
=> 2 = 3 - 1
[1,3,4,2] => 001 => [2,1] => [2,1]
=> 2 = 3 - 1
[1,4,3,2] => 011 => [1,2] => [2,1]
=> 2 = 3 - 1
[2,1,4,3] => 101 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[2,3,4,1] => 001 => [2,1] => [2,1]
=> 2 = 3 - 1
[2,4,3,1] => 011 => [1,2] => [2,1]
=> 2 = 3 - 1
[3,1,4,2] => 101 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[3,2,4,1] => 101 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[3,4,2,1] => 011 => [1,2] => [2,1]
=> 2 = 3 - 1
[4,1,3,2] => 101 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[4,2,3,1] => 101 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[4,3,2,1] => 111 => [3] => [3]
=> 1 = 2 - 1
[1,2,3,5,4] => 0001 => [3,1] => [3,1]
=> 2 = 3 - 1
[1,2,4,5,3] => 0001 => [3,1] => [3,1]
=> 2 = 3 - 1
[1,2,5,4,3] => 0011 => [2,2] => [2,2]
=> 2 = 3 - 1
[1,3,2,5,4] => 0101 => [1,1,1,1] => [1,1,1,1]
=> 4 = 5 - 1
[1,3,4,5,2] => 0001 => [3,1] => [3,1]
=> 2 = 3 - 1
[1,3,5,4,2] => 0011 => [2,2] => [2,2]
=> 2 = 3 - 1
[1,4,2,5,3] => 0101 => [1,1,1,1] => [1,1,1,1]
=> 4 = 5 - 1
[1,4,3,5,2] => 0101 => [1,1,1,1] => [1,1,1,1]
=> 4 = 5 - 1
[1,4,5,3,2] => 0011 => [2,2] => [2,2]
=> 2 = 3 - 1
[1,5,2,4,3] => 0101 => [1,1,1,1] => [1,1,1,1]
=> 4 = 5 - 1
[1,5,3,4,2] => 0101 => [1,1,1,1] => [1,1,1,1]
=> 4 = 5 - 1
[1,5,4,3,2] => 0111 => [1,3] => [3,1]
=> 2 = 3 - 1
[2,1,3,5,4] => 1001 => [1,2,1] => [2,1,1]
=> 3 = 4 - 1
[2,1,4,5,3] => 1001 => [1,2,1] => [2,1,1]
=> 3 = 4 - 1
[2,1,5,4,3] => 1011 => [1,1,2] => [2,1,1]
=> 3 = 4 - 1
[2,3,1,5,4] => 0101 => [1,1,1,1] => [1,1,1,1]
=> 4 = 5 - 1
[2,3,4,5,1] => 0001 => [3,1] => [3,1]
=> 2 = 3 - 1
[2,3,5,4,1] => 0011 => [2,2] => [2,2]
=> 2 = 3 - 1
[2,4,1,5,3] => 0101 => [1,1,1,1] => [1,1,1,1]
=> 4 = 5 - 1
[2,4,3,5,1] => 0101 => [1,1,1,1] => [1,1,1,1]
=> 4 = 5 - 1
[2,4,5,3,1] => 0011 => [2,2] => [2,2]
=> 2 = 3 - 1
[2,5,1,4,3] => 0101 => [1,1,1,1] => [1,1,1,1]
=> 4 = 5 - 1
[2,5,3,4,1] => 0101 => [1,1,1,1] => [1,1,1,1]
=> 4 = 5 - 1
[2,5,4,3,1] => 0111 => [1,3] => [3,1]
=> 2 = 3 - 1
[3,1,2,5,4] => 1001 => [1,2,1] => [2,1,1]
=> 3 = 4 - 1
[3,1,4,5,2] => 1001 => [1,2,1] => [2,1,1]
=> 3 = 4 - 1
[3,1,5,4,2] => 1011 => [1,1,2] => [2,1,1]
=> 3 = 4 - 1
[3,2,1,5,4] => 1101 => [2,1,1] => [2,1,1]
=> 3 = 4 - 1
[3,2,4,5,1] => 1001 => [1,2,1] => [2,1,1]
=> 3 = 4 - 1
[3,2,5,4,1] => 1011 => [1,1,2] => [2,1,1]
=> 3 = 4 - 1
[3,4,1,5,2] => 0101 => [1,1,1,1] => [1,1,1,1]
=> 4 = 5 - 1
[3,4,2,5,1] => 0101 => [1,1,1,1] => [1,1,1,1]
=> 4 = 5 - 1
[3,4,5,2,1] => 0011 => [2,2] => [2,2]
=> 2 = 3 - 1
[3,5,1,4,2] => 0101 => [1,1,1,1] => [1,1,1,1]
=> 4 = 5 - 1
Description
The length of the partition.
Matching statistic: St000011
Mp00109: Permutations descent wordBinary words
Mp00097: Binary words delta morphismInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
St000011: Dyck paths ⟶ ℤResult quality: 86% values known / values provided: 100%distinct values known / distinct values provided: 86%
Values
[1] => => [] => ?
=> ? = 1 - 1
[2,1] => 1 => [1] => [1,0]
=> 1 = 2 - 1
[1,3,2] => 01 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[2,3,1] => 01 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[3,2,1] => 11 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[1,2,4,3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,3,4,2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,3,2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,1,4,3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[2,3,4,1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,4,3,1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[3,1,4,2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[3,2,4,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[3,4,2,1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[4,1,3,2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[4,2,3,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[4,3,2,1] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,3,5,4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,2,4,5,3] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,2,5,4,3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,3,2,5,4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,3,4,5,2] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,3,5,4,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,4,2,5,3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,4,3,5,2] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,4,5,3,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,5,2,4,3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,5,3,4,2] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,5,4,3,2] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[2,1,3,5,4] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[2,1,4,5,3] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[2,1,5,4,3] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[2,3,1,5,4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[2,3,4,5,1] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[2,3,5,4,1] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[2,4,1,5,3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[2,4,3,5,1] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[2,4,5,3,1] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[2,5,1,4,3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[2,5,3,4,1] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[2,5,4,3,1] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[3,1,2,5,4] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[3,1,4,5,2] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[3,1,5,4,2] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[3,2,1,5,4] => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[3,2,4,5,1] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[3,2,5,4,1] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[3,4,1,5,2] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[3,4,2,5,1] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[3,4,5,2,1] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[3,5,1,4,2] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
Description
The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.
The following 43 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000015The number of peaks of a Dyck path. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000288The number of ones in a binary word. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000053The number of valleys of the Dyck path. St000272The treewidth of a graph. St000306The bounce count of a Dyck path. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001358The largest degree of a regular subgraph of a graph. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000453The number of distinct Laplacian eigenvalues of a graph. St001812The biclique partition number of a graph. St000796The stat' of a permutation. St001330The hat guessing number of a graph. St000638The number of up-down runs of a permutation. St000388The number of orbits of vertices of a graph under automorphisms. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000455The second largest eigenvalue of a graph if it is integral. St001488The number of corners of a skew partition. St000831The number of indices that are either descents or recoils.