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Your data matches 57 different statistics following compositions of up to 3 maps.
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Matching statistic: St000691
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Mp00267: Signed permutations —signs⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => 1 => 0 = 2 - 2
[1,-2] => 01 => 1 = 3 - 2
[-1,-2] => 11 => 0 = 2 - 2
[2,-1] => 01 => 1 = 3 - 2
[-2,-1] => 11 => 0 = 2 - 2
[1,2,-3] => 001 => 1 = 3 - 2
[1,-2,-3] => 011 => 1 = 3 - 2
[-1,2,-3] => 101 => 2 = 4 - 2
[-1,-2,-3] => 111 => 0 = 2 - 2
[1,3,-2] => 001 => 1 = 3 - 2
[1,-3,-2] => 011 => 1 = 3 - 2
[-1,3,-2] => 101 => 2 = 4 - 2
[-1,-3,-2] => 111 => 0 = 2 - 2
[2,1,-3] => 001 => 1 = 3 - 2
[2,-1,-3] => 011 => 1 = 3 - 2
[-2,1,-3] => 101 => 2 = 4 - 2
[-2,-1,-3] => 111 => 0 = 2 - 2
[2,3,-1] => 001 => 1 = 3 - 2
[2,-3,-1] => 011 => 1 = 3 - 2
[-2,3,-1] => 101 => 2 = 4 - 2
[-2,-3,-1] => 111 => 0 = 2 - 2
[3,1,-2] => 001 => 1 = 3 - 2
[3,-1,-2] => 011 => 1 = 3 - 2
[-3,1,-2] => 101 => 2 = 4 - 2
[-3,-1,-2] => 111 => 0 = 2 - 2
[3,2,-1] => 001 => 1 = 3 - 2
[3,-2,-1] => 011 => 1 = 3 - 2
[-3,2,-1] => 101 => 2 = 4 - 2
[-3,-2,-1] => 111 => 0 = 2 - 2
[1,2,3,-4] => 0001 => 1 = 3 - 2
[1,2,-3,-4] => 0011 => 1 = 3 - 2
[1,-2,3,-4] => 0101 => 3 = 5 - 2
[1,-2,-3,-4] => 0111 => 1 = 3 - 2
[-1,2,3,-4] => 1001 => 2 = 4 - 2
[-1,2,-3,-4] => 1011 => 2 = 4 - 2
[-1,-2,3,-4] => 1101 => 2 = 4 - 2
[-1,-2,-3,-4] => 1111 => 0 = 2 - 2
[1,2,4,-3] => 0001 => 1 = 3 - 2
[1,2,-4,-3] => 0011 => 1 = 3 - 2
[1,-2,4,-3] => 0101 => 3 = 5 - 2
[1,-2,-4,-3] => 0111 => 1 = 3 - 2
[-1,2,4,-3] => 1001 => 2 = 4 - 2
[-1,2,-4,-3] => 1011 => 2 = 4 - 2
[-1,-2,4,-3] => 1101 => 2 = 4 - 2
[-1,-2,-4,-3] => 1111 => 0 = 2 - 2
[1,3,2,-4] => 0001 => 1 = 3 - 2
[1,3,-2,-4] => 0011 => 1 = 3 - 2
[1,-3,2,-4] => 0101 => 3 = 5 - 2
[1,-3,-2,-4] => 0111 => 1 = 3 - 2
[-1,3,2,-4] => 1001 => 2 = 4 - 2
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: St001486
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(load all 4 compositions to match this statistic)
Mp00267: Signed permutations —signs⟶ 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%
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 => [1,1] => 2
[1,-2] => 01 => [2,1] => 3
[-1,-2] => 11 => [1,1,1] => 2
[2,-1] => 01 => [2,1] => 3
[-2,-1] => 11 => [1,1,1] => 2
[1,2,-3] => 001 => [3,1] => 3
[1,-2,-3] => 011 => [2,1,1] => 3
[-1,2,-3] => 101 => [1,2,1] => 4
[-1,-2,-3] => 111 => [1,1,1,1] => 2
[1,3,-2] => 001 => [3,1] => 3
[1,-3,-2] => 011 => [2,1,1] => 3
[-1,3,-2] => 101 => [1,2,1] => 4
[-1,-3,-2] => 111 => [1,1,1,1] => 2
[2,1,-3] => 001 => [3,1] => 3
[2,-1,-3] => 011 => [2,1,1] => 3
[-2,1,-3] => 101 => [1,2,1] => 4
[-2,-1,-3] => 111 => [1,1,1,1] => 2
[2,3,-1] => 001 => [3,1] => 3
[2,-3,-1] => 011 => [2,1,1] => 3
[-2,3,-1] => 101 => [1,2,1] => 4
[-2,-3,-1] => 111 => [1,1,1,1] => 2
[3,1,-2] => 001 => [3,1] => 3
[3,-1,-2] => 011 => [2,1,1] => 3
[-3,1,-2] => 101 => [1,2,1] => 4
[-3,-1,-2] => 111 => [1,1,1,1] => 2
[3,2,-1] => 001 => [3,1] => 3
[3,-2,-1] => 011 => [2,1,1] => 3
[-3,2,-1] => 101 => [1,2,1] => 4
[-3,-2,-1] => 111 => [1,1,1,1] => 2
[1,2,3,-4] => 0001 => [4,1] => 3
[1,2,-3,-4] => 0011 => [3,1,1] => 3
[1,-2,3,-4] => 0101 => [2,2,1] => 5
[1,-2,-3,-4] => 0111 => [2,1,1,1] => 3
[-1,2,3,-4] => 1001 => [1,3,1] => 4
[-1,2,-3,-4] => 1011 => [1,2,1,1] => 4
[-1,-2,3,-4] => 1101 => [1,1,2,1] => 4
[-1,-2,-3,-4] => 1111 => [1,1,1,1,1] => 2
[1,2,4,-3] => 0001 => [4,1] => 3
[1,2,-4,-3] => 0011 => [3,1,1] => 3
[1,-2,4,-3] => 0101 => [2,2,1] => 5
[1,-2,-4,-3] => 0111 => [2,1,1,1] => 3
[-1,2,4,-3] => 1001 => [1,3,1] => 4
[-1,2,-4,-3] => 1011 => [1,2,1,1] => 4
[-1,-2,4,-3] => 1101 => [1,1,2,1] => 4
[-1,-2,-4,-3] => 1111 => [1,1,1,1,1] => 2
[1,3,2,-4] => 0001 => [4,1] => 3
[1,3,-2,-4] => 0011 => [3,1,1] => 3
[1,-3,2,-4] => 0101 => [2,2,1] => 5
[1,-3,-2,-4] => 0111 => [2,1,1,1] => 3
[-1,3,2,-4] => 1001 => [1,3,1] => 4
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: St000453
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000453: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000453: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => 1 => [1,1] => ([(0,1)],2)
=> 2
[1,-2] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 3
[-1,-2] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[2,-1] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 3
[-2,-1] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,-2,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,2,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[-1,-2,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,-3,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,3,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[-1,-3,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,-1,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-2,1,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[-2,-1,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,-1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,-3,-1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-2,3,-1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[-2,-3,-1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,-1,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-3,1,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[-3,-1,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,-1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,-2,-1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-3,2,-1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[-3,-2,-1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,3,-4] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,2,-3,-4] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,-2,3,-4] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,-2,-3,-4] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[-1,2,3,-4] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[-1,2,-3,-4] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[-1,-2,3,-4] => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[-1,-2,-3,-4] => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,4,-3] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,2,-4,-3] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,-2,4,-3] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,-2,-4,-3] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[-1,2,4,-3] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[-1,2,-4,-3] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[-1,-2,4,-3] => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[-1,-2,-4,-3] => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,-4] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,3,-2,-4] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,-3,2,-4] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,-3,-2,-4] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[-1,3,2,-4] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
Description
The number of distinct Laplacian eigenvalues of a graph.
Matching statistic: St000777
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => 1 => [1,1] => ([(0,1)],2)
=> 2
[1,-2] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 3
[-1,-2] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[2,-1] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 3
[-2,-1] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,-2,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,2,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[-1,-2,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,-3,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,3,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[-1,-3,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,-1,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-2,1,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[-2,-1,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,-1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,-3,-1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-2,3,-1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[-2,-3,-1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,-1,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-3,1,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[-3,-1,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,-1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,-2,-1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-3,2,-1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[-3,-2,-1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,3,-4] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,2,-3,-4] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,-2,3,-4] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,-2,-3,-4] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[-1,2,3,-4] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[-1,2,-3,-4] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[-1,-2,3,-4] => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[-1,-2,-3,-4] => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,4,-3] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,2,-4,-3] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,-2,4,-3] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,-2,-4,-3] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[-1,2,4,-3] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[-1,2,-4,-3] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[-1,-2,4,-3] => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[-1,-2,-4,-3] => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,-4] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,3,-2,-4] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,-3,2,-4] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,-3,-2,-4] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[-1,3,2,-4] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St001028
Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001028: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001028: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => 1 => [1] => [1,0]
=> 2
[1,-2] => 01 => [1,1] => [1,0,1,0]
=> 3
[-1,-2] => 11 => [2] => [1,1,0,0]
=> 2
[2,-1] => 01 => [1,1] => [1,0,1,0]
=> 3
[-2,-1] => 11 => [2] => [1,1,0,0]
=> 2
[1,2,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[1,-2,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-1,2,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[-1,-2,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[1,3,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[1,-3,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-1,3,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[-1,-3,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[2,1,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[2,-1,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-2,1,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[-2,-1,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[2,3,-1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[2,-3,-1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-2,3,-1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[-2,-3,-1] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[3,1,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[3,-1,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-3,1,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[-3,-1,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[3,2,-1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[3,-2,-1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-3,2,-1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[-3,-2,-1] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[1,2,3,-4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,-3,-4] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[1,-2,3,-4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,-2,-3,-4] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[-1,2,3,-4] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[-1,2,-3,-4] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 4
[-1,-2,3,-4] => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[-1,-2,-3,-4] => 1111 => [4] => [1,1,1,1,0,0,0,0]
=> 2
[1,2,4,-3] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,-4,-3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[1,-2,4,-3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,-2,-4,-3] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[-1,2,4,-3] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[-1,2,-4,-3] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 4
[-1,-2,4,-3] => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[-1,-2,-4,-3] => 1111 => [4] => [1,1,1,1,0,0,0,0]
=> 2
[1,3,2,-4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,3,-2,-4] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[1,-3,2,-4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,-3,-2,-4] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[-1,3,2,-4] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
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
Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001290: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001290: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => 1 => [1] => [1,0]
=> 2
[1,-2] => 01 => [1,1] => [1,0,1,0]
=> 3
[-1,-2] => 11 => [2] => [1,1,0,0]
=> 2
[2,-1] => 01 => [1,1] => [1,0,1,0]
=> 3
[-2,-1] => 11 => [2] => [1,1,0,0]
=> 2
[1,2,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[1,-2,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-1,2,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[-1,-2,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[1,3,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[1,-3,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-1,3,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[-1,-3,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[2,1,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[2,-1,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-2,1,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[-2,-1,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[2,3,-1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[2,-3,-1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-2,3,-1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[-2,-3,-1] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[3,1,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[3,-1,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-3,1,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[-3,-1,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[3,2,-1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[3,-2,-1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-3,2,-1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[-3,-2,-1] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[1,2,3,-4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,-3,-4] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[1,-2,3,-4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,-2,-3,-4] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[-1,2,3,-4] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[-1,2,-3,-4] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 4
[-1,-2,3,-4] => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[-1,-2,-3,-4] => 1111 => [4] => [1,1,1,1,0,0,0,0]
=> 2
[1,2,4,-3] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,-4,-3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[1,-2,4,-3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,-2,-4,-3] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[-1,2,4,-3] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[-1,2,-4,-3] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 4
[-1,-2,4,-3] => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[-1,-2,-4,-3] => 1111 => [4] => [1,1,1,1,0,0,0,0]
=> 2
[1,3,2,-4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,3,-2,-4] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[1,-3,2,-4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,-3,-2,-4] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[-1,3,2,-4] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
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: St001505
Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001505: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001505: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => 1 => [1] => [1,0]
=> 2
[1,-2] => 01 => [1,1] => [1,0,1,0]
=> 3
[-1,-2] => 11 => [2] => [1,1,0,0]
=> 2
[2,-1] => 01 => [1,1] => [1,0,1,0]
=> 3
[-2,-1] => 11 => [2] => [1,1,0,0]
=> 2
[1,2,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[1,-2,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-1,2,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[-1,-2,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[1,3,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[1,-3,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-1,3,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[-1,-3,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[2,1,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[2,-1,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-2,1,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[-2,-1,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[2,3,-1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[2,-3,-1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-2,3,-1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[-2,-3,-1] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[3,1,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[3,-1,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-3,1,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[-3,-1,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[3,2,-1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[3,-2,-1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-3,2,-1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[-3,-2,-1] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[1,2,3,-4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,-3,-4] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[1,-2,3,-4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,-2,-3,-4] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[-1,2,3,-4] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[-1,2,-3,-4] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 4
[-1,-2,3,-4] => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[-1,-2,-3,-4] => 1111 => [4] => [1,1,1,1,0,0,0,0]
=> 2
[1,2,4,-3] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,-4,-3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[1,-2,4,-3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,-2,-4,-3] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[-1,2,4,-3] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[-1,2,-4,-3] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 4
[-1,-2,4,-3] => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[-1,-2,-4,-3] => 1111 => [4] => [1,1,1,1,0,0,0,0]
=> 2
[1,3,2,-4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,3,-2,-4] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[1,-3,2,-4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,-3,-2,-4] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[-1,3,2,-4] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
Description
The number of elements generated by the Dyck path as a map in the full transformation monoid.
We view the resolution quiver of a Dyck path (corresponding to an LNakayamaalgebra) as a transformation and associate to it the submonoid generated by this map in the full transformation monoid.
Matching statistic: St000010
Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => 1 => [1] => [1]
=> 1 = 2 - 1
[1,-2] => 01 => [1,1] => [1,1]
=> 2 = 3 - 1
[-1,-2] => 11 => [2] => [2]
=> 1 = 2 - 1
[2,-1] => 01 => [1,1] => [1,1]
=> 2 = 3 - 1
[-2,-1] => 11 => [2] => [2]
=> 1 = 2 - 1
[1,2,-3] => 001 => [2,1] => [2,1]
=> 2 = 3 - 1
[1,-2,-3] => 011 => [1,2] => [2,1]
=> 2 = 3 - 1
[-1,2,-3] => 101 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[-1,-2,-3] => 111 => [3] => [3]
=> 1 = 2 - 1
[1,3,-2] => 001 => [2,1] => [2,1]
=> 2 = 3 - 1
[1,-3,-2] => 011 => [1,2] => [2,1]
=> 2 = 3 - 1
[-1,3,-2] => 101 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[-1,-3,-2] => 111 => [3] => [3]
=> 1 = 2 - 1
[2,1,-3] => 001 => [2,1] => [2,1]
=> 2 = 3 - 1
[2,-1,-3] => 011 => [1,2] => [2,1]
=> 2 = 3 - 1
[-2,1,-3] => 101 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[-2,-1,-3] => 111 => [3] => [3]
=> 1 = 2 - 1
[2,3,-1] => 001 => [2,1] => [2,1]
=> 2 = 3 - 1
[2,-3,-1] => 011 => [1,2] => [2,1]
=> 2 = 3 - 1
[-2,3,-1] => 101 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[-2,-3,-1] => 111 => [3] => [3]
=> 1 = 2 - 1
[3,1,-2] => 001 => [2,1] => [2,1]
=> 2 = 3 - 1
[3,-1,-2] => 011 => [1,2] => [2,1]
=> 2 = 3 - 1
[-3,1,-2] => 101 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[-3,-1,-2] => 111 => [3] => [3]
=> 1 = 2 - 1
[3,2,-1] => 001 => [2,1] => [2,1]
=> 2 = 3 - 1
[3,-2,-1] => 011 => [1,2] => [2,1]
=> 2 = 3 - 1
[-3,2,-1] => 101 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[-3,-2,-1] => 111 => [3] => [3]
=> 1 = 2 - 1
[1,2,3,-4] => 0001 => [3,1] => [3,1]
=> 2 = 3 - 1
[1,2,-3,-4] => 0011 => [2,2] => [2,2]
=> 2 = 3 - 1
[1,-2,3,-4] => 0101 => [1,1,1,1] => [1,1,1,1]
=> 4 = 5 - 1
[1,-2,-3,-4] => 0111 => [1,3] => [3,1]
=> 2 = 3 - 1
[-1,2,3,-4] => 1001 => [1,2,1] => [2,1,1]
=> 3 = 4 - 1
[-1,2,-3,-4] => 1011 => [1,1,2] => [2,1,1]
=> 3 = 4 - 1
[-1,-2,3,-4] => 1101 => [2,1,1] => [2,1,1]
=> 3 = 4 - 1
[-1,-2,-3,-4] => 1111 => [4] => [4]
=> 1 = 2 - 1
[1,2,4,-3] => 0001 => [3,1] => [3,1]
=> 2 = 3 - 1
[1,2,-4,-3] => 0011 => [2,2] => [2,2]
=> 2 = 3 - 1
[1,-2,4,-3] => 0101 => [1,1,1,1] => [1,1,1,1]
=> 4 = 5 - 1
[1,-2,-4,-3] => 0111 => [1,3] => [3,1]
=> 2 = 3 - 1
[-1,2,4,-3] => 1001 => [1,2,1] => [2,1,1]
=> 3 = 4 - 1
[-1,2,-4,-3] => 1011 => [1,1,2] => [2,1,1]
=> 3 = 4 - 1
[-1,-2,4,-3] => 1101 => [2,1,1] => [2,1,1]
=> 3 = 4 - 1
[-1,-2,-4,-3] => 1111 => [4] => [4]
=> 1 = 2 - 1
[1,3,2,-4] => 0001 => [3,1] => [3,1]
=> 2 = 3 - 1
[1,3,-2,-4] => 0011 => [2,2] => [2,2]
=> 2 = 3 - 1
[1,-3,2,-4] => 0101 => [1,1,1,1] => [1,1,1,1]
=> 4 = 5 - 1
[1,-3,-2,-4] => 0111 => [1,3] => [3,1]
=> 2 = 3 - 1
[-1,3,2,-4] => 1001 => [1,2,1] => [2,1,1]
=> 3 = 4 - 1
Description
The length of the partition.
Matching statistic: St000011
Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => 1 => [1] => [1,0]
=> 1 = 2 - 1
[1,-2] => 01 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[-1,-2] => 11 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[2,-1] => 01 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[-2,-1] => 11 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[1,2,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,-2,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-1,2,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[-1,-2,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,3,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,-3,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-1,3,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[-1,-3,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,-1,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-2,1,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[-2,-1,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,3,-1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,-3,-1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-2,3,-1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[-2,-3,-1] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[3,1,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[3,-1,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-3,1,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[-3,-1,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[3,2,-1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[3,-2,-1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-3,2,-1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[-3,-2,-1] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,3,-4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,2,-3,-4] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,-2,3,-4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,-2,-3,-4] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[-1,2,3,-4] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[-1,2,-3,-4] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[-1,-2,3,-4] => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[-1,-2,-3,-4] => 1111 => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,4,-3] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,2,-4,-3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,-2,4,-3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,-2,-4,-3] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[-1,2,4,-3] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[-1,2,-4,-3] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[-1,-2,4,-3] => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[-1,-2,-4,-3] => 1111 => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,2,-4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,3,-2,-4] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,-3,2,-4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,-3,-2,-4] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[-1,3,2,-4] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 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.
Matching statistic: St000015
Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => 1 => [1] => [1,0]
=> 1 = 2 - 1
[1,-2] => 01 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[-1,-2] => 11 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[2,-1] => 01 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[-2,-1] => 11 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[1,2,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,-2,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-1,2,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[-1,-2,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,3,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,-3,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-1,3,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[-1,-3,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,-1,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-2,1,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[-2,-1,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,3,-1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,-3,-1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-2,3,-1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[-2,-3,-1] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[3,1,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[3,-1,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-3,1,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[-3,-1,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[3,2,-1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[3,-2,-1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-3,2,-1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[-3,-2,-1] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,3,-4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,2,-3,-4] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,-2,3,-4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,-2,-3,-4] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[-1,2,3,-4] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[-1,2,-3,-4] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[-1,-2,3,-4] => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[-1,-2,-3,-4] => 1111 => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,4,-3] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,2,-4,-3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,-2,4,-3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,-2,-4,-3] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[-1,2,4,-3] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[-1,2,-4,-3] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[-1,-2,4,-3] => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[-1,-2,-4,-3] => 1111 => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,2,-4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,3,-2,-4] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,-3,2,-4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,-3,-2,-4] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[-1,3,2,-4] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
Description
The number of peaks of a Dyck path.
The following 47 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St000388The number of orbits of vertices of a graph under automorphisms. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. 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. St001352The number of internal nodes in the modular decomposition 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. St001951The number of factors in the disjoint direct product decomposition of the automorphism group 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. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. 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. St001812The biclique partition number of a graph. St000806The semiperimeter of the associated bargraph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001330The hat guessing number of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001430The number of positive entries in a signed permutation. St001429The number of negative entries in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001488The number of corners of a skew partition. St000307The number of rowmotion orbits of a poset. St000632The jump number of the poset.
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