searching the database
Your data matches 105 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000691
(load all 2 compositions to match this statistic)
(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 = 2 - 2
[-1] => 1 => 0 = 2 - 2
[1,2] => 00 => 0 = 2 - 2
[1,-2] => 01 => 1 = 3 - 2
[-1,2] => 10 => 1 = 3 - 2
[-1,-2] => 11 => 0 = 2 - 2
[2,1] => 00 => 0 = 2 - 2
[2,-1] => 01 => 1 = 3 - 2
[-2,1] => 10 => 1 = 3 - 2
[-2,-1] => 11 => 0 = 2 - 2
[1,2,3] => 000 => 0 = 2 - 2
[1,2,-3] => 001 => 1 = 3 - 2
[1,-2,3] => 010 => 2 = 4 - 2
[1,-2,-3] => 011 => 1 = 3 - 2
[-1,2,3] => 100 => 1 = 3 - 2
[-1,2,-3] => 101 => 2 = 4 - 2
[-1,-2,3] => 110 => 1 = 3 - 2
[-1,-2,-3] => 111 => 0 = 2 - 2
[1,3,2] => 000 => 0 = 2 - 2
[1,3,-2] => 001 => 1 = 3 - 2
[1,-3,2] => 010 => 2 = 4 - 2
[1,-3,-2] => 011 => 1 = 3 - 2
[-1,3,2] => 100 => 1 = 3 - 2
[-1,3,-2] => 101 => 2 = 4 - 2
[-1,-3,2] => 110 => 1 = 3 - 2
[-1,-3,-2] => 111 => 0 = 2 - 2
[2,1,3] => 000 => 0 = 2 - 2
[2,1,-3] => 001 => 1 = 3 - 2
[2,-1,3] => 010 => 2 = 4 - 2
[2,-1,-3] => 011 => 1 = 3 - 2
[-2,1,3] => 100 => 1 = 3 - 2
[-2,1,-3] => 101 => 2 = 4 - 2
[-2,-1,3] => 110 => 1 = 3 - 2
[-2,-1,-3] => 111 => 0 = 2 - 2
[2,3,1] => 000 => 0 = 2 - 2
[2,3,-1] => 001 => 1 = 3 - 2
[2,-3,1] => 010 => 2 = 4 - 2
[2,-3,-1] => 011 => 1 = 3 - 2
[-2,3,1] => 100 => 1 = 3 - 2
[-2,3,-1] => 101 => 2 = 4 - 2
[-2,-3,1] => 110 => 1 = 3 - 2
[-2,-3,-1] => 111 => 0 = 2 - 2
[3,1,2] => 000 => 0 = 2 - 2
[3,1,-2] => 001 => 1 = 3 - 2
[3,-1,2] => 010 => 2 = 4 - 2
[3,-1,-2] => 011 => 1 = 3 - 2
[-3,1,2] => 100 => 1 = 3 - 2
[-3,1,-2] => 101 => 2 = 4 - 2
[-3,-1,2] => 110 => 1 = 3 - 2
[-3,-1,-2] => 111 => 0 = 2 - 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
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
[-1] => 1 => [1,1] => 2
[1,2] => 00 => [3] => 2
[1,-2] => 01 => [2,1] => 3
[-1,2] => 10 => [1,2] => 3
[-1,-2] => 11 => [1,1,1] => 2
[2,1] => 00 => [3] => 2
[2,-1] => 01 => [2,1] => 3
[-2,1] => 10 => [1,2] => 3
[-2,-1] => 11 => [1,1,1] => 2
[1,2,3] => 000 => [4] => 2
[1,2,-3] => 001 => [3,1] => 3
[1,-2,3] => 010 => [2,2] => 4
[1,-2,-3] => 011 => [2,1,1] => 3
[-1,2,3] => 100 => [1,3] => 3
[-1,2,-3] => 101 => [1,2,1] => 4
[-1,-2,3] => 110 => [1,1,2] => 3
[-1,-2,-3] => 111 => [1,1,1,1] => 2
[1,3,2] => 000 => [4] => 2
[1,3,-2] => 001 => [3,1] => 3
[1,-3,2] => 010 => [2,2] => 4
[1,-3,-2] => 011 => [2,1,1] => 3
[-1,3,2] => 100 => [1,3] => 3
[-1,3,-2] => 101 => [1,2,1] => 4
[-1,-3,2] => 110 => [1,1,2] => 3
[-1,-3,-2] => 111 => [1,1,1,1] => 2
[2,1,3] => 000 => [4] => 2
[2,1,-3] => 001 => [3,1] => 3
[2,-1,3] => 010 => [2,2] => 4
[2,-1,-3] => 011 => [2,1,1] => 3
[-2,1,3] => 100 => [1,3] => 3
[-2,1,-3] => 101 => [1,2,1] => 4
[-2,-1,3] => 110 => [1,1,2] => 3
[-2,-1,-3] => 111 => [1,1,1,1] => 2
[2,3,1] => 000 => [4] => 2
[2,3,-1] => 001 => [3,1] => 3
[2,-3,1] => 010 => [2,2] => 4
[2,-3,-1] => 011 => [2,1,1] => 3
[-2,3,1] => 100 => [1,3] => 3
[-2,3,-1] => 101 => [1,2,1] => 4
[-2,-3,1] => 110 => [1,1,2] => 3
[-2,-3,-1] => 111 => [1,1,1,1] => 2
[3,1,2] => 000 => [4] => 2
[3,1,-2] => 001 => [3,1] => 3
[3,-1,2] => 010 => [2,2] => 4
[3,-1,-2] => 011 => [2,1,1] => 3
[-3,1,2] => 100 => [1,3] => 3
[-3,1,-2] => 101 => [1,2,1] => 4
[-3,-1,2] => 110 => [1,1,2] => 3
[-3,-1,-2] => 111 => [1,1,1,1] => 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: St000288
(load all 8 compositions to match this statistic)
(load all 8 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 = 2 - 1
[-1] => 1 => 1 => 1 = 2 - 1
[1,2] => 00 => 01 => 1 = 2 - 1
[1,-2] => 01 => 10 => 1 = 2 - 1
[-1,2] => 10 => 11 => 2 = 3 - 1
[-1,-2] => 11 => 11 => 2 = 3 - 1
[2,1] => 00 => 01 => 1 = 2 - 1
[2,-1] => 01 => 10 => 1 = 2 - 1
[-2,1] => 10 => 11 => 2 = 3 - 1
[-2,-1] => 11 => 11 => 2 = 3 - 1
[1,2,3] => 000 => 001 => 1 = 2 - 1
[1,2,-3] => 001 => 010 => 1 = 2 - 1
[1,-2,3] => 010 => 101 => 2 = 3 - 1
[1,-2,-3] => 011 => 101 => 2 = 3 - 1
[-1,2,3] => 100 => 101 => 2 = 3 - 1
[-1,2,-3] => 101 => 110 => 2 = 3 - 1
[-1,-2,3] => 110 => 111 => 3 = 4 - 1
[-1,-2,-3] => 111 => 111 => 3 = 4 - 1
[1,3,2] => 000 => 001 => 1 = 2 - 1
[1,3,-2] => 001 => 010 => 1 = 2 - 1
[1,-3,2] => 010 => 101 => 2 = 3 - 1
[1,-3,-2] => 011 => 101 => 2 = 3 - 1
[-1,3,2] => 100 => 101 => 2 = 3 - 1
[-1,3,-2] => 101 => 110 => 2 = 3 - 1
[-1,-3,2] => 110 => 111 => 3 = 4 - 1
[-1,-3,-2] => 111 => 111 => 3 = 4 - 1
[2,1,3] => 000 => 001 => 1 = 2 - 1
[2,1,-3] => 001 => 010 => 1 = 2 - 1
[2,-1,3] => 010 => 101 => 2 = 3 - 1
[2,-1,-3] => 011 => 101 => 2 = 3 - 1
[-2,1,3] => 100 => 101 => 2 = 3 - 1
[-2,1,-3] => 101 => 110 => 2 = 3 - 1
[-2,-1,3] => 110 => 111 => 3 = 4 - 1
[-2,-1,-3] => 111 => 111 => 3 = 4 - 1
[2,3,1] => 000 => 001 => 1 = 2 - 1
[2,3,-1] => 001 => 010 => 1 = 2 - 1
[2,-3,1] => 010 => 101 => 2 = 3 - 1
[2,-3,-1] => 011 => 101 => 2 = 3 - 1
[-2,3,1] => 100 => 101 => 2 = 3 - 1
[-2,3,-1] => 101 => 110 => 2 = 3 - 1
[-2,-3,1] => 110 => 111 => 3 = 4 - 1
[-2,-3,-1] => 111 => 111 => 3 = 4 - 1
[3,1,2] => 000 => 001 => 1 = 2 - 1
[3,1,-2] => 001 => 010 => 1 = 2 - 1
[3,-1,2] => 010 => 101 => 2 = 3 - 1
[3,-1,-2] => 011 => 101 => 2 = 3 - 1
[-3,1,2] => 100 => 101 => 2 = 3 - 1
[-3,1,-2] => 101 => 110 => 2 = 3 - 1
[-3,-1,2] => 110 => 111 => 3 = 4 - 1
[-3,-1,-2] => 111 => 111 => 3 = 4 - 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
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] => 0 => [1] => [1,0]
=> 2
[-1] => 1 => [1] => [1,0]
=> 2
[1,2] => 00 => [2] => [1,1,0,0]
=> 2
[1,-2] => 01 => [1,1] => [1,0,1,0]
=> 3
[-1,2] => 10 => [1,1] => [1,0,1,0]
=> 3
[-1,-2] => 11 => [2] => [1,1,0,0]
=> 2
[2,1] => 00 => [2] => [1,1,0,0]
=> 2
[2,-1] => 01 => [1,1] => [1,0,1,0]
=> 3
[-2,1] => 10 => [1,1] => [1,0,1,0]
=> 3
[-2,-1] => 11 => [2] => [1,1,0,0]
=> 2
[1,2,3] => 000 => [3] => [1,1,1,0,0,0]
=> 2
[1,2,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[1,-2,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,-2,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-1,2,3] => 100 => [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] => 110 => [2,1] => [1,1,0,0,1,0]
=> 3
[-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]
=> 3
[1,-3,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,-3,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-1,3,2] => 100 => [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] => 110 => [2,1] => [1,1,0,0,1,0]
=> 3
[-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]
=> 3
[2,-1,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[2,-1,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-2,1,3] => 100 => [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] => 110 => [2,1] => [1,1,0,0,1,0]
=> 3
[-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]
=> 3
[2,-3,1] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[2,-3,-1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-2,3,1] => 100 => [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] => 110 => [2,1] => [1,1,0,0,1,0]
=> 3
[-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]
=> 3
[3,-1,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[3,-1,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-3,1,2] => 100 => [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] => 110 => [2,1] => [1,1,0,0,1,0]
=> 3
[-3,-1,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 2
Description
Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001226
Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001226: 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
St001226: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [1] => [1,0]
=> 2
[-1] => 1 => [1] => [1,0]
=> 2
[1,2] => 00 => [2] => [1,1,0,0]
=> 3
[1,-2] => 01 => [1,1] => [1,0,1,0]
=> 2
[-1,2] => 10 => [1,1] => [1,0,1,0]
=> 2
[-1,-2] => 11 => [2] => [1,1,0,0]
=> 3
[2,1] => 00 => [2] => [1,1,0,0]
=> 3
[2,-1] => 01 => [1,1] => [1,0,1,0]
=> 2
[-2,1] => 10 => [1,1] => [1,0,1,0]
=> 2
[-2,-1] => 11 => [2] => [1,1,0,0]
=> 3
[1,2,3] => 000 => [3] => [1,1,1,0,0,0]
=> 4
[1,2,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[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]
=> 3
[-1,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 3
[-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]
=> 3
[-1,-2,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 4
[1,3,2] => 000 => [3] => [1,1,1,0,0,0]
=> 4
[1,3,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[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]
=> 3
[-1,3,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 3
[-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]
=> 3
[-1,-3,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 4
[2,1,3] => 000 => [3] => [1,1,1,0,0,0]
=> 4
[2,1,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[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]
=> 3
[-2,1,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 3
[-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]
=> 3
[-2,-1,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 4
[2,3,1] => 000 => [3] => [1,1,1,0,0,0]
=> 4
[2,3,-1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[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]
=> 3
[-2,3,1] => 100 => [1,2] => [1,0,1,1,0,0]
=> 3
[-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]
=> 3
[-2,-3,-1] => 111 => [3] => [1,1,1,0,0,0]
=> 4
[3,1,2] => 000 => [3] => [1,1,1,0,0,0]
=> 4
[3,1,-2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[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]
=> 3
[-3,1,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 3
[-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]
=> 3
[-3,-1,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 4
Description
The 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.
That is the number of i such that $Ext_A^1(J,e_i J)=0$.
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] => 0 => [1] => [1,0]
=> 2
[-1] => 1 => [1] => [1,0]
=> 2
[1,2] => 00 => [2] => [1,1,0,0]
=> 2
[1,-2] => 01 => [1,1] => [1,0,1,0]
=> 3
[-1,2] => 10 => [1,1] => [1,0,1,0]
=> 3
[-1,-2] => 11 => [2] => [1,1,0,0]
=> 2
[2,1] => 00 => [2] => [1,1,0,0]
=> 2
[2,-1] => 01 => [1,1] => [1,0,1,0]
=> 3
[-2,1] => 10 => [1,1] => [1,0,1,0]
=> 3
[-2,-1] => 11 => [2] => [1,1,0,0]
=> 2
[1,2,3] => 000 => [3] => [1,1,1,0,0,0]
=> 2
[1,2,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[1,-2,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,-2,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-1,2,3] => 100 => [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] => 110 => [2,1] => [1,1,0,0,1,0]
=> 3
[-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]
=> 3
[1,-3,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,-3,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-1,3,2] => 100 => [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] => 110 => [2,1] => [1,1,0,0,1,0]
=> 3
[-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]
=> 3
[2,-1,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[2,-1,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-2,1,3] => 100 => [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] => 110 => [2,1] => [1,1,0,0,1,0]
=> 3
[-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]
=> 3
[2,-3,1] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[2,-3,-1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-2,3,1] => 100 => [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] => 110 => [2,1] => [1,1,0,0,1,0]
=> 3
[-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]
=> 3
[3,-1,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[3,-1,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-3,1,2] => 100 => [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] => 110 => [2,1] => [1,1,0,0,1,0]
=> 3
[-3,-1,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 2
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: St001315
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001315: 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
St001315: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [2] => ([],2)
=> 2
[-1] => 1 => [1,1] => ([(0,1)],2)
=> 2
[1,2] => 00 => [3] => ([],3)
=> 3
[1,-2] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[-1,2] => 10 => [1,2] => ([(1,2)],3)
=> 3
[-1,-2] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[2,1] => 00 => [3] => ([],3)
=> 3
[2,-1] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[-2,1] => 10 => [1,2] => ([(1,2)],3)
=> 3
[-2,-1] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3] => 000 => [4] => ([],4)
=> 4
[1,2,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,-2,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 3
[1,-2,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[-1,2,3] => 100 => [1,3] => ([(2,3)],4)
=> 4
[-1,2,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,-2,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[-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] => 000 => [4] => ([],4)
=> 4
[1,3,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,-3,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 3
[1,-3,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[-1,3,2] => 100 => [1,3] => ([(2,3)],4)
=> 4
[-1,3,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,-3,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[-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] => 000 => [4] => ([],4)
=> 4
[2,1,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,-1,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 3
[2,-1,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[-2,1,3] => 100 => [1,3] => ([(2,3)],4)
=> 4
[-2,1,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-2,-1,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[-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] => 000 => [4] => ([],4)
=> 4
[2,3,-1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,-3,1] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 3
[2,-3,-1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[-2,3,1] => 100 => [1,3] => ([(2,3)],4)
=> 4
[-2,3,-1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-2,-3,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[-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] => 000 => [4] => ([],4)
=> 4
[3,1,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,-1,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 3
[3,-1,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[-3,1,2] => 100 => [1,3] => ([(2,3)],4)
=> 4
[-3,1,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-3,-1,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[-3,-1,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
Description
The dissociation number of a graph.
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] => 0 => [1] => [1,0]
=> 2
[-1] => 1 => [1] => [1,0]
=> 2
[1,2] => 00 => [2] => [1,1,0,0]
=> 2
[1,-2] => 01 => [1,1] => [1,0,1,0]
=> 3
[-1,2] => 10 => [1,1] => [1,0,1,0]
=> 3
[-1,-2] => 11 => [2] => [1,1,0,0]
=> 2
[2,1] => 00 => [2] => [1,1,0,0]
=> 2
[2,-1] => 01 => [1,1] => [1,0,1,0]
=> 3
[-2,1] => 10 => [1,1] => [1,0,1,0]
=> 3
[-2,-1] => 11 => [2] => [1,1,0,0]
=> 2
[1,2,3] => 000 => [3] => [1,1,1,0,0,0]
=> 2
[1,2,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[1,-2,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,-2,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-1,2,3] => 100 => [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] => 110 => [2,1] => [1,1,0,0,1,0]
=> 3
[-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]
=> 3
[1,-3,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,-3,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-1,3,2] => 100 => [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] => 110 => [2,1] => [1,1,0,0,1,0]
=> 3
[-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]
=> 3
[2,-1,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[2,-1,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-2,1,3] => 100 => [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] => 110 => [2,1] => [1,1,0,0,1,0]
=> 3
[-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]
=> 3
[2,-3,1] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[2,-3,-1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-2,3,1] => 100 => [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] => 110 => [2,1] => [1,1,0,0,1,0]
=> 3
[-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]
=> 3
[3,-1,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[3,-1,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[-3,1,2] => 100 => [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] => 110 => [2,1] => [1,1,0,0,1,0]
=> 3
[-3,-1,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 2
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] => 0 => [1] => [1]
=> 1 = 2 - 1
[-1] => 1 => [1] => [1]
=> 1 = 2 - 1
[1,2] => 00 => [2] => [2]
=> 1 = 2 - 1
[1,-2] => 01 => [1,1] => [1,1]
=> 2 = 3 - 1
[-1,2] => 10 => [1,1] => [1,1]
=> 2 = 3 - 1
[-1,-2] => 11 => [2] => [2]
=> 1 = 2 - 1
[2,1] => 00 => [2] => [2]
=> 1 = 2 - 1
[2,-1] => 01 => [1,1] => [1,1]
=> 2 = 3 - 1
[-2,1] => 10 => [1,1] => [1,1]
=> 2 = 3 - 1
[-2,-1] => 11 => [2] => [2]
=> 1 = 2 - 1
[1,2,3] => 000 => [3] => [3]
=> 1 = 2 - 1
[1,2,-3] => 001 => [2,1] => [2,1]
=> 2 = 3 - 1
[1,-2,3] => 010 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[1,-2,-3] => 011 => [1,2] => [2,1]
=> 2 = 3 - 1
[-1,2,3] => 100 => [1,2] => [2,1]
=> 2 = 3 - 1
[-1,2,-3] => 101 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[-1,-2,3] => 110 => [2,1] => [2,1]
=> 2 = 3 - 1
[-1,-2,-3] => 111 => [3] => [3]
=> 1 = 2 - 1
[1,3,2] => 000 => [3] => [3]
=> 1 = 2 - 1
[1,3,-2] => 001 => [2,1] => [2,1]
=> 2 = 3 - 1
[1,-3,2] => 010 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[1,-3,-2] => 011 => [1,2] => [2,1]
=> 2 = 3 - 1
[-1,3,2] => 100 => [1,2] => [2,1]
=> 2 = 3 - 1
[-1,3,-2] => 101 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[-1,-3,2] => 110 => [2,1] => [2,1]
=> 2 = 3 - 1
[-1,-3,-2] => 111 => [3] => [3]
=> 1 = 2 - 1
[2,1,3] => 000 => [3] => [3]
=> 1 = 2 - 1
[2,1,-3] => 001 => [2,1] => [2,1]
=> 2 = 3 - 1
[2,-1,3] => 010 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[2,-1,-3] => 011 => [1,2] => [2,1]
=> 2 = 3 - 1
[-2,1,3] => 100 => [1,2] => [2,1]
=> 2 = 3 - 1
[-2,1,-3] => 101 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[-2,-1,3] => 110 => [2,1] => [2,1]
=> 2 = 3 - 1
[-2,-1,-3] => 111 => [3] => [3]
=> 1 = 2 - 1
[2,3,1] => 000 => [3] => [3]
=> 1 = 2 - 1
[2,3,-1] => 001 => [2,1] => [2,1]
=> 2 = 3 - 1
[2,-3,1] => 010 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[2,-3,-1] => 011 => [1,2] => [2,1]
=> 2 = 3 - 1
[-2,3,1] => 100 => [1,2] => [2,1]
=> 2 = 3 - 1
[-2,3,-1] => 101 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[-2,-3,1] => 110 => [2,1] => [2,1]
=> 2 = 3 - 1
[-2,-3,-1] => 111 => [3] => [3]
=> 1 = 2 - 1
[3,1,2] => 000 => [3] => [3]
=> 1 = 2 - 1
[3,1,-2] => 001 => [2,1] => [2,1]
=> 2 = 3 - 1
[3,-1,2] => 010 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[3,-1,-2] => 011 => [1,2] => [2,1]
=> 2 = 3 - 1
[-3,1,2] => 100 => [1,2] => [2,1]
=> 2 = 3 - 1
[-3,1,-2] => 101 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[-3,-1,2] => 110 => [2,1] => [2,1]
=> 2 = 3 - 1
[-3,-1,-2] => 111 => [3] => [3]
=> 1 = 2 - 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] => 0 => [1] => [1,0]
=> 1 = 2 - 1
[-1] => 1 => [1] => [1,0]
=> 1 = 2 - 1
[1,2] => 00 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[1,-2] => 01 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[-1,2] => 10 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[-1,-2] => 11 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[2,1] => 00 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[2,-1] => 01 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[-2,1] => 10 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[-2,-1] => 11 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[1,2,3] => 000 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,-3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,-2,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,-2,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-1,2,3] => 100 => [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] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[-1,-2,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,3,2] => 000 => [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] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,-3,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-1,3,2] => 100 => [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] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[-1,-3,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,3] => 000 => [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] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[2,-1,-3] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-2,1,3] => 100 => [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] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[-2,-1,-3] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,3,1] => 000 => [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] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[2,-3,-1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-2,3,1] => 100 => [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] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[-2,-3,-1] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[3,1,2] => 000 => [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] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[3,-1,-2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[-3,1,2] => 100 => [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] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[-3,-1,-2] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
The following 95 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000015The number of peaks of a Dyck path. 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. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000272The treewidth of a graph. St000306The bounce count of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000362The size of a minimal vertex cover of a graph. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. 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. St001480The number of simple summands of the module J^2/J^3. St000806The semiperimeter of the associated bargraph. St000741The Colin de Verdière graph invariant. St000259The diameter of a connected graph. 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. St000454The largest eigenvalue of a graph if it is integral. St001875The number of simple modules with projective dimension at most 1. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000438The position of the last up step in a Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001838The number of nonempty primitive factors of a binary word. St000264The girth of a graph, which is not a tree. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000455The second largest eigenvalue of a graph if it is integral. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000260The radius of a connected graph. St000981The length of the longest zigzag subpath. St001488The number of corners of a skew partition. St000307The number of rowmotion orbits of a poset. St000632The jump number of the poset. St001645The pebbling number of a connected graph. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001568The smallest positive integer that does not appear twice in the partition.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!