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Your data matches 334 different statistics following compositions of up to 3 maps.
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Matching statistic: St000691
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(load all 2 compositions to match this statistic)
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] => 0 => 0
[-1] => 1 => 0
[1,2] => 00 => 0
[1,-2] => 01 => 1
[-1,2] => 10 => 1
[-1,-2] => 11 => 0
[2,1] => 00 => 0
[2,-1] => 01 => 1
[-2,1] => 10 => 1
[-2,-1] => 11 => 0
[1,2,3] => 000 => 0
[1,2,-3] => 001 => 1
[1,-2,3] => 010 => 2
[1,-2,-3] => 011 => 1
[-1,2,3] => 100 => 1
[-1,2,-3] => 101 => 2
[-1,-2,3] => 110 => 1
[-1,-2,-3] => 111 => 0
[1,3,2] => 000 => 0
[1,3,-2] => 001 => 1
[1,-3,2] => 010 => 2
[1,-3,-2] => 011 => 1
[-1,3,2] => 100 => 1
[-1,3,-2] => 101 => 2
[-1,-3,2] => 110 => 1
[-1,-3,-2] => 111 => 0
[2,1,3] => 000 => 0
[2,1,-3] => 001 => 1
[2,-1,3] => 010 => 2
[2,-1,-3] => 011 => 1
[-2,1,3] => 100 => 1
[-2,1,-3] => 101 => 2
[-2,-1,3] => 110 => 1
[-2,-1,-3] => 111 => 0
[2,3,1] => 000 => 0
[2,3,-1] => 001 => 1
[2,-3,1] => 010 => 2
[2,-3,-1] => 011 => 1
[-2,3,1] => 100 => 1
[-2,3,-1] => 101 => 2
[-2,-3,1] => 110 => 1
[-2,-3,-1] => 111 => 0
[3,1,2] => 000 => 0
[3,1,-2] => 001 => 1
[3,-1,2] => 010 => 2
[3,-1,-2] => 011 => 1
[-3,1,2] => 100 => 1
[-3,1,-2] => 101 => 2
[-3,-1,2] => 110 => 1
[-3,-1,-2] => 111 => 0
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: St000288
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00267: Signed permutations —signs⟶ 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] => 0 => 1 => 1 = 0 + 1
[-1] => 1 => 1 => 1 = 0 + 1
[1,2] => 00 => 01 => 1 = 0 + 1
[1,-2] => 01 => 10 => 1 = 0 + 1
[-1,2] => 10 => 11 => 2 = 1 + 1
[-1,-2] => 11 => 11 => 2 = 1 + 1
[2,1] => 00 => 01 => 1 = 0 + 1
[2,-1] => 01 => 10 => 1 = 0 + 1
[-2,1] => 10 => 11 => 2 = 1 + 1
[-2,-1] => 11 => 11 => 2 = 1 + 1
[1,2,3] => 000 => 001 => 1 = 0 + 1
[1,2,-3] => 001 => 010 => 1 = 0 + 1
[1,-2,3] => 010 => 101 => 2 = 1 + 1
[1,-2,-3] => 011 => 101 => 2 = 1 + 1
[-1,2,3] => 100 => 101 => 2 = 1 + 1
[-1,2,-3] => 101 => 110 => 2 = 1 + 1
[-1,-2,3] => 110 => 111 => 3 = 2 + 1
[-1,-2,-3] => 111 => 111 => 3 = 2 + 1
[1,3,2] => 000 => 001 => 1 = 0 + 1
[1,3,-2] => 001 => 010 => 1 = 0 + 1
[1,-3,2] => 010 => 101 => 2 = 1 + 1
[1,-3,-2] => 011 => 101 => 2 = 1 + 1
[-1,3,2] => 100 => 101 => 2 = 1 + 1
[-1,3,-2] => 101 => 110 => 2 = 1 + 1
[-1,-3,2] => 110 => 111 => 3 = 2 + 1
[-1,-3,-2] => 111 => 111 => 3 = 2 + 1
[2,1,3] => 000 => 001 => 1 = 0 + 1
[2,1,-3] => 001 => 010 => 1 = 0 + 1
[2,-1,3] => 010 => 101 => 2 = 1 + 1
[2,-1,-3] => 011 => 101 => 2 = 1 + 1
[-2,1,3] => 100 => 101 => 2 = 1 + 1
[-2,1,-3] => 101 => 110 => 2 = 1 + 1
[-2,-1,3] => 110 => 111 => 3 = 2 + 1
[-2,-1,-3] => 111 => 111 => 3 = 2 + 1
[2,3,1] => 000 => 001 => 1 = 0 + 1
[2,3,-1] => 001 => 010 => 1 = 0 + 1
[2,-3,1] => 010 => 101 => 2 = 1 + 1
[2,-3,-1] => 011 => 101 => 2 = 1 + 1
[-2,3,1] => 100 => 101 => 2 = 1 + 1
[-2,3,-1] => 101 => 110 => 2 = 1 + 1
[-2,-3,1] => 110 => 111 => 3 = 2 + 1
[-2,-3,-1] => 111 => 111 => 3 = 2 + 1
[3,1,2] => 000 => 001 => 1 = 0 + 1
[3,1,-2] => 001 => 010 => 1 = 0 + 1
[3,-1,2] => 010 => 101 => 2 = 1 + 1
[3,-1,-2] => 011 => 101 => 2 = 1 + 1
[-3,1,2] => 100 => 101 => 2 = 1 + 1
[-3,1,-2] => 101 => 110 => 2 = 1 + 1
[-3,-1,2] => 110 => 111 => 3 = 2 + 1
[-3,-1,-2] => 111 => 111 => 3 = 2 + 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St001486
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] => 0 => [2] => 2 = 0 + 2
[-1] => 1 => [1,1] => 2 = 0 + 2
[1,2] => 00 => [3] => 2 = 0 + 2
[1,-2] => 01 => [2,1] => 3 = 1 + 2
[-1,2] => 10 => [1,2] => 3 = 1 + 2
[-1,-2] => 11 => [1,1,1] => 2 = 0 + 2
[2,1] => 00 => [3] => 2 = 0 + 2
[2,-1] => 01 => [2,1] => 3 = 1 + 2
[-2,1] => 10 => [1,2] => 3 = 1 + 2
[-2,-1] => 11 => [1,1,1] => 2 = 0 + 2
[1,2,3] => 000 => [4] => 2 = 0 + 2
[1,2,-3] => 001 => [3,1] => 3 = 1 + 2
[1,-2,3] => 010 => [2,2] => 4 = 2 + 2
[1,-2,-3] => 011 => [2,1,1] => 3 = 1 + 2
[-1,2,3] => 100 => [1,3] => 3 = 1 + 2
[-1,2,-3] => 101 => [1,2,1] => 4 = 2 + 2
[-1,-2,3] => 110 => [1,1,2] => 3 = 1 + 2
[-1,-2,-3] => 111 => [1,1,1,1] => 2 = 0 + 2
[1,3,2] => 000 => [4] => 2 = 0 + 2
[1,3,-2] => 001 => [3,1] => 3 = 1 + 2
[1,-3,2] => 010 => [2,2] => 4 = 2 + 2
[1,-3,-2] => 011 => [2,1,1] => 3 = 1 + 2
[-1,3,2] => 100 => [1,3] => 3 = 1 + 2
[-1,3,-2] => 101 => [1,2,1] => 4 = 2 + 2
[-1,-3,2] => 110 => [1,1,2] => 3 = 1 + 2
[-1,-3,-2] => 111 => [1,1,1,1] => 2 = 0 + 2
[2,1,3] => 000 => [4] => 2 = 0 + 2
[2,1,-3] => 001 => [3,1] => 3 = 1 + 2
[2,-1,3] => 010 => [2,2] => 4 = 2 + 2
[2,-1,-3] => 011 => [2,1,1] => 3 = 1 + 2
[-2,1,3] => 100 => [1,3] => 3 = 1 + 2
[-2,1,-3] => 101 => [1,2,1] => 4 = 2 + 2
[-2,-1,3] => 110 => [1,1,2] => 3 = 1 + 2
[-2,-1,-3] => 111 => [1,1,1,1] => 2 = 0 + 2
[2,3,1] => 000 => [4] => 2 = 0 + 2
[2,3,-1] => 001 => [3,1] => 3 = 1 + 2
[2,-3,1] => 010 => [2,2] => 4 = 2 + 2
[2,-3,-1] => 011 => [2,1,1] => 3 = 1 + 2
[-2,3,1] => 100 => [1,3] => 3 = 1 + 2
[-2,3,-1] => 101 => [1,2,1] => 4 = 2 + 2
[-2,-3,1] => 110 => [1,1,2] => 3 = 1 + 2
[-2,-3,-1] => 111 => [1,1,1,1] => 2 = 0 + 2
[3,1,2] => 000 => [4] => 2 = 0 + 2
[3,1,-2] => 001 => [3,1] => 3 = 1 + 2
[3,-1,2] => 010 => [2,2] => 4 = 2 + 2
[3,-1,-2] => 011 => [2,1,1] => 3 = 1 + 2
[-3,1,2] => 100 => [1,3] => 3 = 1 + 2
[-3,1,-2] => 101 => [1,2,1] => 4 = 2 + 2
[-3,-1,2] => 110 => [1,1,2] => 3 = 1 + 2
[-3,-1,-2] => 111 => [1,1,1,1] => 2 = 0 + 2
Description
The number of corners of the ribbon associated with an integer composition.
We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell.
This statistic records the total number of corners of the ribbon shape.
Matching statistic: St000024
Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000024: 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
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [1] => [1,0]
=> 0
[-1] => 1 => [1] => [1,0]
=> 0
[1,2] => 00 => [2] => [1,1,0,0]
=> 1
[1,-2] => 01 => [1,1] => [1,0,1,0]
=> 0
[-1,2] => 10 => [1,1] => [1,0,1,0]
=> 0
[-1,-2] => 11 => [2] => [1,1,0,0]
=> 1
[2,1] => 00 => [2] => [1,1,0,0]
=> 1
[2,-1] => 01 => [1,1] => [1,0,1,0]
=> 0
[-2,1] => 10 => [1,1] => [1,0,1,0]
=> 0
[-2,-1] => 11 => [2] => [1,1,0,0]
=> 1
[1,2,3] => 000 => [3] => [1,1,1,0,0,0]
=> 2
[1,2,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[1,-2,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,-2,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[-1,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[-1,2,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[-1,-2,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-1,-2,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[1,3,2] => 000 => [3] => [1,1,1,0,0,0]
=> 2
[1,3,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[1,-3,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,-3,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[-1,3,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[-1,3,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[-1,-3,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-1,-3,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[2,1,3] => 000 => [3] => [1,1,1,0,0,0]
=> 2
[2,1,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[2,-1,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[2,-1,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[-2,1,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[-2,1,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[-2,-1,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-2,-1,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[2,3,1] => 000 => [3] => [1,1,1,0,0,0]
=> 2
[2,3,-1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[2,-3,1] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[2,-3,-1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[-2,3,1] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[-2,3,-1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[-2,-3,1] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-2,-3,-1] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[3,1,2] => 000 => [3] => [1,1,1,0,0,0]
=> 2
[3,1,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[3,-1,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[3,-1,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[-3,1,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[-3,1,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[-3,-1,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-3,-1,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 2
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000053
Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000053: 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
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [1] => [1,0]
=> 0
[-1] => 1 => [1] => [1,0]
=> 0
[1,2] => 00 => [2] => [1,1,0,0]
=> 0
[1,-2] => 01 => [1,1] => [1,0,1,0]
=> 1
[-1,2] => 10 => [1,1] => [1,0,1,0]
=> 1
[-1,-2] => 11 => [2] => [1,1,0,0]
=> 0
[2,1] => 00 => [2] => [1,1,0,0]
=> 0
[2,-1] => 01 => [1,1] => [1,0,1,0]
=> 1
[-2,1] => 10 => [1,1] => [1,0,1,0]
=> 1
[-2,-1] => 11 => [2] => [1,1,0,0]
=> 0
[1,2,3] => 000 => [3] => [1,1,1,0,0,0]
=> 0
[1,2,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[1,-2,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,-2,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[-1,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[-1,2,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[-1,-2,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-1,-2,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => 000 => [3] => [1,1,1,0,0,0]
=> 0
[1,3,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[1,-3,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,-3,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[-1,3,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[-1,3,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[-1,-3,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-1,-3,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 0
[2,1,3] => 000 => [3] => [1,1,1,0,0,0]
=> 0
[2,1,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[2,-1,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[2,-1,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[-2,1,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[-2,1,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[-2,-1,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-2,-1,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 0
[2,3,1] => 000 => [3] => [1,1,1,0,0,0]
=> 0
[2,3,-1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[2,-3,1] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[2,-3,-1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[-2,3,1] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[-2,3,-1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[-2,-3,1] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-2,-3,-1] => 111 => [3] => [1,1,1,0,0,0]
=> 0
[3,1,2] => 000 => [3] => [1,1,1,0,0,0]
=> 0
[3,1,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[3,-1,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[3,-1,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[-3,1,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[-3,1,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[-3,-1,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-3,-1,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 0
Description
The number of valleys of the Dyck path.
Matching statistic: St000272
Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000272: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000272: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [1] => ([],1)
=> 0
[-1] => 1 => [1] => ([],1)
=> 0
[1,2] => 00 => [2] => ([],2)
=> 0
[1,-2] => 01 => [1,1] => ([(0,1)],2)
=> 1
[-1,2] => 10 => [1,1] => ([(0,1)],2)
=> 1
[-1,-2] => 11 => [2] => ([],2)
=> 0
[2,1] => 00 => [2] => ([],2)
=> 0
[2,-1] => 01 => [1,1] => ([(0,1)],2)
=> 1
[-2,1] => 10 => [1,1] => ([(0,1)],2)
=> 1
[-2,-1] => 11 => [2] => ([],2)
=> 0
[1,2,3] => 000 => [3] => ([],3)
=> 0
[1,2,-3] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,-2,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,-2,-3] => 011 => [1,2] => ([(1,2)],3)
=> 1
[-1,2,3] => 100 => [1,2] => ([(1,2)],3)
=> 1
[-1,2,-3] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[-1,-2,3] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[-1,-2,-3] => 111 => [3] => ([],3)
=> 0
[1,3,2] => 000 => [3] => ([],3)
=> 0
[1,3,-2] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,-3,2] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,-3,-2] => 011 => [1,2] => ([(1,2)],3)
=> 1
[-1,3,2] => 100 => [1,2] => ([(1,2)],3)
=> 1
[-1,3,-2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[-1,-3,2] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[-1,-3,-2] => 111 => [3] => ([],3)
=> 0
[2,1,3] => 000 => [3] => ([],3)
=> 0
[2,1,-3] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,-1,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[2,-1,-3] => 011 => [1,2] => ([(1,2)],3)
=> 1
[-2,1,3] => 100 => [1,2] => ([(1,2)],3)
=> 1
[-2,1,-3] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[-2,-1,3] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[-2,-1,-3] => 111 => [3] => ([],3)
=> 0
[2,3,1] => 000 => [3] => ([],3)
=> 0
[2,3,-1] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,-3,1] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[2,-3,-1] => 011 => [1,2] => ([(1,2)],3)
=> 1
[-2,3,1] => 100 => [1,2] => ([(1,2)],3)
=> 1
[-2,3,-1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[-2,-3,1] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[-2,-3,-1] => 111 => [3] => ([],3)
=> 0
[3,1,2] => 000 => [3] => ([],3)
=> 0
[3,1,-2] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[3,-1,2] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[3,-1,-2] => 011 => [1,2] => ([(1,2)],3)
=> 1
[-3,1,2] => 100 => [1,2] => ([(1,2)],3)
=> 1
[-3,1,-2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[-3,-1,2] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[-3,-1,-2] => 111 => [3] => ([],3)
=> 0
Description
The treewidth of a graph.
A graph has treewidth zero if and only if it has no edges. A connected graph has treewidth at most one if and only if it is a tree. A connected graph has treewidth at most two if and only if it is a series-parallel graph.
Matching statistic: St000306
Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000306: 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
St000306: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [1] => [1,0]
=> 0
[-1] => 1 => [1] => [1,0]
=> 0
[1,2] => 00 => [2] => [1,1,0,0]
=> 0
[1,-2] => 01 => [1,1] => [1,0,1,0]
=> 1
[-1,2] => 10 => [1,1] => [1,0,1,0]
=> 1
[-1,-2] => 11 => [2] => [1,1,0,0]
=> 0
[2,1] => 00 => [2] => [1,1,0,0]
=> 0
[2,-1] => 01 => [1,1] => [1,0,1,0]
=> 1
[-2,1] => 10 => [1,1] => [1,0,1,0]
=> 1
[-2,-1] => 11 => [2] => [1,1,0,0]
=> 0
[1,2,3] => 000 => [3] => [1,1,1,0,0,0]
=> 0
[1,2,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[1,-2,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,-2,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[-1,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[-1,2,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[-1,-2,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-1,-2,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => 000 => [3] => [1,1,1,0,0,0]
=> 0
[1,3,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[1,-3,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,-3,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[-1,3,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[-1,3,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[-1,-3,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-1,-3,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 0
[2,1,3] => 000 => [3] => [1,1,1,0,0,0]
=> 0
[2,1,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[2,-1,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[2,-1,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[-2,1,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[-2,1,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[-2,-1,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-2,-1,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 0
[2,3,1] => 000 => [3] => [1,1,1,0,0,0]
=> 0
[2,3,-1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[2,-3,1] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[2,-3,-1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[-2,3,1] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[-2,3,-1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[-2,-3,1] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-2,-3,-1] => 111 => [3] => [1,1,1,0,0,0]
=> 0
[3,1,2] => 000 => [3] => [1,1,1,0,0,0]
=> 0
[3,1,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[3,-1,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[3,-1,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[-3,1,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[-3,1,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[-3,-1,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-3,-1,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 0
Description
The bounce count of a Dyck path.
For a Dyck path $D$ of length $2n$, this is the number of points $(i,i)$ for $1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of $D$.
Matching statistic: St000340
Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000340: 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
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [1] => [1,0]
=> 0
[-1] => 1 => [1] => [1,0]
=> 0
[1,2] => 00 => [2] => [1,1,0,0]
=> 1
[1,-2] => 01 => [1,1] => [1,0,1,0]
=> 0
[-1,2] => 10 => [1,1] => [1,0,1,0]
=> 0
[-1,-2] => 11 => [2] => [1,1,0,0]
=> 1
[2,1] => 00 => [2] => [1,1,0,0]
=> 1
[2,-1] => 01 => [1,1] => [1,0,1,0]
=> 0
[-2,1] => 10 => [1,1] => [1,0,1,0]
=> 0
[-2,-1] => 11 => [2] => [1,1,0,0]
=> 1
[1,2,3] => 000 => [3] => [1,1,1,0,0,0]
=> 1
[1,2,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2
[1,-2,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,-2,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[-1,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[-1,2,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[-1,-2,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2
[-1,-2,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 1
[1,3,2] => 000 => [3] => [1,1,1,0,0,0]
=> 1
[1,3,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2
[1,-3,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,-3,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[-1,3,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[-1,3,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[-1,-3,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2
[-1,-3,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 1
[2,1,3] => 000 => [3] => [1,1,1,0,0,0]
=> 1
[2,1,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2
[2,-1,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[2,-1,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[-2,1,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[-2,1,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[-2,-1,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2
[-2,-1,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 1
[2,3,1] => 000 => [3] => [1,1,1,0,0,0]
=> 1
[2,3,-1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2
[2,-3,1] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[2,-3,-1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[-2,3,1] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[-2,3,-1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[-2,-3,1] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2
[-2,-3,-1] => 111 => [3] => [1,1,1,0,0,0]
=> 1
[3,1,2] => 000 => [3] => [1,1,1,0,0,0]
=> 1
[3,1,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2
[3,-1,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[3,-1,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[-3,1,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[-3,1,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[-3,-1,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2
[-3,-1,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 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 $001$.
Matching statistic: St000362
Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000362: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000362: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [1] => ([],1)
=> 0
[-1] => 1 => [1] => ([],1)
=> 0
[1,2] => 00 => [2] => ([],2)
=> 0
[1,-2] => 01 => [1,1] => ([(0,1)],2)
=> 1
[-1,2] => 10 => [1,1] => ([(0,1)],2)
=> 1
[-1,-2] => 11 => [2] => ([],2)
=> 0
[2,1] => 00 => [2] => ([],2)
=> 0
[2,-1] => 01 => [1,1] => ([(0,1)],2)
=> 1
[-2,1] => 10 => [1,1] => ([(0,1)],2)
=> 1
[-2,-1] => 11 => [2] => ([],2)
=> 0
[1,2,3] => 000 => [3] => ([],3)
=> 0
[1,2,-3] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,-2,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,-2,-3] => 011 => [1,2] => ([(1,2)],3)
=> 1
[-1,2,3] => 100 => [1,2] => ([(1,2)],3)
=> 1
[-1,2,-3] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[-1,-2,3] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[-1,-2,-3] => 111 => [3] => ([],3)
=> 0
[1,3,2] => 000 => [3] => ([],3)
=> 0
[1,3,-2] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,-3,2] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,-3,-2] => 011 => [1,2] => ([(1,2)],3)
=> 1
[-1,3,2] => 100 => [1,2] => ([(1,2)],3)
=> 1
[-1,3,-2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[-1,-3,2] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[-1,-3,-2] => 111 => [3] => ([],3)
=> 0
[2,1,3] => 000 => [3] => ([],3)
=> 0
[2,1,-3] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,-1,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[2,-1,-3] => 011 => [1,2] => ([(1,2)],3)
=> 1
[-2,1,3] => 100 => [1,2] => ([(1,2)],3)
=> 1
[-2,1,-3] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[-2,-1,3] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[-2,-1,-3] => 111 => [3] => ([],3)
=> 0
[2,3,1] => 000 => [3] => ([],3)
=> 0
[2,3,-1] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,-3,1] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[2,-3,-1] => 011 => [1,2] => ([(1,2)],3)
=> 1
[-2,3,1] => 100 => [1,2] => ([(1,2)],3)
=> 1
[-2,3,-1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[-2,-3,1] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[-2,-3,-1] => 111 => [3] => ([],3)
=> 0
[3,1,2] => 000 => [3] => ([],3)
=> 0
[3,1,-2] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[3,-1,2] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[3,-1,-2] => 011 => [1,2] => ([(1,2)],3)
=> 1
[-3,1,2] => 100 => [1,2] => ([(1,2)],3)
=> 1
[-3,1,-2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[-3,-1,2] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[-3,-1,-2] => 111 => [3] => ([],3)
=> 0
Description
The size of a minimal vertex cover of a graph.
A '''vertex cover''' of a graph $G$ is a subset $S$ of the vertices of $G$ such that each edge of $G$ contains at least one vertex of $S$. Finding a minimal vertex cover is an NP-hard optimization problem.
Matching statistic: St000394
Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000394: 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
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [1] => [1,0]
=> 0
[-1] => 1 => [1] => [1,0]
=> 0
[1,2] => 00 => [2] => [1,1,0,0]
=> 1
[1,-2] => 01 => [1,1] => [1,0,1,0]
=> 0
[-1,2] => 10 => [1,1] => [1,0,1,0]
=> 0
[-1,-2] => 11 => [2] => [1,1,0,0]
=> 1
[2,1] => 00 => [2] => [1,1,0,0]
=> 1
[2,-1] => 01 => [1,1] => [1,0,1,0]
=> 0
[-2,1] => 10 => [1,1] => [1,0,1,0]
=> 0
[-2,-1] => 11 => [2] => [1,1,0,0]
=> 1
[1,2,3] => 000 => [3] => [1,1,1,0,0,0]
=> 2
[1,2,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[1,-2,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,-2,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[-1,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[-1,2,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[-1,-2,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-1,-2,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[1,3,2] => 000 => [3] => [1,1,1,0,0,0]
=> 2
[1,3,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[1,-3,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,-3,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[-1,3,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[-1,3,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[-1,-3,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-1,-3,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[2,1,3] => 000 => [3] => [1,1,1,0,0,0]
=> 2
[2,1,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[2,-1,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[2,-1,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[-2,1,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[-2,1,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[-2,-1,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-2,-1,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[2,3,1] => 000 => [3] => [1,1,1,0,0,0]
=> 2
[2,3,-1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[2,-3,1] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[2,-3,-1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[-2,3,1] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[-2,3,-1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[-2,-3,1] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-2,-3,-1] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[3,1,2] => 000 => [3] => [1,1,1,0,0,0]
=> 2
[3,1,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1
[3,-1,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[3,-1,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1
[-3,1,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1
[-3,1,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[-3,-1,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[-3,-1,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 2
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
The following 324 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000536The pathwidth of a graph. 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. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. 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. St001971The number of negative eigenvalues of the adjacency matrix of the graph. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000388The number of orbits of vertices of a graph under automorphisms. St000443The number of long tunnels of a Dyck path. St000453The number of distinct Laplacian eigenvalues of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000822The Hadwiger number of the graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. St001187The number of simple modules with grade at least one 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:
St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number 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. St001674The number of vertices of the largest induced star graph in the graph. St001949The rigidity index 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. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001315The dissociation number of a graph. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001983The number of indecomposable injective modules that are pure. St001480The number of simple summands of the module J^2/J^3. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000741The Colin de Verdière graph invariant. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000216The absolute length of a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000619The number of cyclic descents of a permutation. St000653The last descent of a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St000259The diameter of a connected graph. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001330The hat guessing number of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000681The Grundy value of Chomp on Ferrers diagrams. St000937The number of positive values of the symmetric group character corresponding to the partition. St001527The cyclic permutation representation number of an integer partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001389The number of partitions of the same length below the given integer partition. St000454The largest eigenvalue of a graph if it is integral. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001568The smallest positive integer that does not appear twice in the partition. St001249Sum of the odd parts of a partition. St001933The largest multiplicity of a part in an integer partition. St001060The distinguishing index of a graph. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000063The number of linear extensions of a certain poset defined for an integer partition. St000088The row sums of the character table of the symmetric group. St000108The number of partitions contained in the given partition. St000144The pyramid weight of the Dyck path. St000147The largest part of an integer partition. St000148The number of odd parts of a partition. St000160The multiplicity of the smallest part of a partition. St000228The size of a partition. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000296The length of the symmetric border of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000335The difference of lower and upper interactions. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000378The diagonal inversion number of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000392The length of the longest run of ones in a binary word. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000444The length of the maximal rise of a Dyck path. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000519The largest length of a factor maximising the subword complexity. St000531The leading coefficient of the rook polynomial of an integer partition. St000532The total number of rook placements on a Ferrers board. St000548The number of different non-empty partial sums of an integer partition. St000549The number of odd partial sums of an integer partition. St000644The number of graphs with given frequency partition. St000667The greatest common divisor of the parts of the partition. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000753The Grundy value for the game of Kayles on a binary word. St000759The smallest missing part in an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000922The minimal number such that all substrings of this length are unique. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000932The number of occurrences of the pattern UDU in a Dyck path. St000935The number of ordered refinements of an integer partition. St000947The major index east count of a Dyck path. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000982The length of the longest constant subword. St000992The alternating sum of the parts of an integer partition. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001161The major index north count of a Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001360The number of covering relations in Young's lattice below a partition. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001372The length of a longest cyclic run of ones of a binary word. St001400The total number of Littlewood-Richardson tableaux of given shape. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001462The number of factors of a standard tableaux under concatenation. St001471The magnitude of a Dyck path. St001485The modular major index of a binary word. St001488The number of corners of a skew partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001523The degree of symmetry of a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001530The depth of a Dyck path. St001571The Cartan determinant of the integer partition. St001614The cyclic permutation representation number of a skew partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001721The degree of a binary word. St001733The number of weak left to right maxima of a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St001814The number of partitions interlacing the given partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001955The number of natural descents for set-valued two row standard Young tableaux. St001959The product of the heights of the peaks of a Dyck path. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000284The Plancherel distribution on integer partitions. St000770The major index of an integer partition when read from bottom to top. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000264The girth of a graph, which is not a tree. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000929The constant term of the character polynomial of an integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000455The second largest eigenvalue of a graph if it is integral. St000782The indicator function of whether a given perfect matching is an L & P matching. St000260The radius of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000442The maximal area to the right of an up step of a Dyck path. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000674The number of hills of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000706The product of the factorials of the multiplicities of an integer partition. St000735The last entry on the main diagonal of a standard tableau. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000874The position of the last double rise in a Dyck path. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000946The sum of the skew hook positions in a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St000984The number of boxes below precisely one peak. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001204Call 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. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. 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. St000632The jump number of the poset. St000307The number of rowmotion orbits of a poset. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001176The size of a partition minus its first part. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001432The order dimension of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St001982The number of orbits of the action of a permutation of given cycle type on the set of edges of the complete graph. St001645The pebbling number of a connected graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
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