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Matching statistic: St001486
Mp00071: Permutations —descent composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [2] => 2
[2,1] => [1,1] => 2
[1,2,3] => [3] => 2
[1,3,2] => [2,1] => 3
[2,1,3] => [1,2] => 3
[2,3,1] => [2,1] => 3
[3,1,2] => [1,2] => 3
[3,2,1] => [1,1,1] => 2
[1,2,3,4] => [4] => 2
[1,2,4,3] => [3,1] => 3
[1,3,2,4] => [2,2] => 4
[1,3,4,2] => [3,1] => 3
[1,4,2,3] => [2,2] => 4
[1,4,3,2] => [2,1,1] => 3
[2,1,3,4] => [1,3] => 3
[2,1,4,3] => [1,2,1] => 4
[2,3,1,4] => [2,2] => 4
[2,3,4,1] => [3,1] => 3
[2,4,1,3] => [2,2] => 4
[2,4,3,1] => [2,1,1] => 3
[3,1,2,4] => [1,3] => 3
[3,1,4,2] => [1,2,1] => 4
[3,2,1,4] => [1,1,2] => 3
[3,2,4,1] => [1,2,1] => 4
[3,4,1,2] => [2,2] => 4
[3,4,2,1] => [2,1,1] => 3
[4,1,2,3] => [1,3] => 3
[4,1,3,2] => [1,2,1] => 4
[4,2,1,3] => [1,1,2] => 3
[4,2,3,1] => [1,2,1] => 4
[4,3,1,2] => [1,1,2] => 3
[4,3,2,1] => [1,1,1,1] => 2
[1,2,3,4,5] => [5] => 2
[1,2,3,5,4] => [4,1] => 3
[1,2,4,3,5] => [3,2] => 4
[1,2,4,5,3] => [4,1] => 3
[1,2,5,3,4] => [3,2] => 4
[1,2,5,4,3] => [3,1,1] => 3
[1,3,2,4,5] => [2,3] => 4
[1,3,2,5,4] => [2,2,1] => 5
[1,3,4,2,5] => [3,2] => 4
[1,3,4,5,2] => [4,1] => 3
[1,3,5,2,4] => [3,2] => 4
[1,3,5,4,2] => [3,1,1] => 3
[1,4,2,3,5] => [2,3] => 4
[1,4,2,5,3] => [2,2,1] => 5
[1,4,3,2,5] => [2,1,2] => 4
[1,4,3,5,2] => [2,2,1] => 5
[1,4,5,2,3] => [3,2] => 4
Description
The number of corners of the ribbon associated with an integer composition.
We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell.
This statistic records the total number of corners of the ribbon shape.
Matching statistic: St000691
Mp00109: Permutations —descent word⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
St000691: Binary words ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Values
[1] => => ? = 1 - 2
[1,2] => 0 => 0 = 2 - 2
[2,1] => 1 => 0 = 2 - 2
[1,2,3] => 00 => 0 = 2 - 2
[1,3,2] => 01 => 1 = 3 - 2
[2,1,3] => 10 => 1 = 3 - 2
[2,3,1] => 01 => 1 = 3 - 2
[3,1,2] => 10 => 1 = 3 - 2
[3,2,1] => 11 => 0 = 2 - 2
[1,2,3,4] => 000 => 0 = 2 - 2
[1,2,4,3] => 001 => 1 = 3 - 2
[1,3,2,4] => 010 => 2 = 4 - 2
[1,3,4,2] => 001 => 1 = 3 - 2
[1,4,2,3] => 010 => 2 = 4 - 2
[1,4,3,2] => 011 => 1 = 3 - 2
[2,1,3,4] => 100 => 1 = 3 - 2
[2,1,4,3] => 101 => 2 = 4 - 2
[2,3,1,4] => 010 => 2 = 4 - 2
[2,3,4,1] => 001 => 1 = 3 - 2
[2,4,1,3] => 010 => 2 = 4 - 2
[2,4,3,1] => 011 => 1 = 3 - 2
[3,1,2,4] => 100 => 1 = 3 - 2
[3,1,4,2] => 101 => 2 = 4 - 2
[3,2,1,4] => 110 => 1 = 3 - 2
[3,2,4,1] => 101 => 2 = 4 - 2
[3,4,1,2] => 010 => 2 = 4 - 2
[3,4,2,1] => 011 => 1 = 3 - 2
[4,1,2,3] => 100 => 1 = 3 - 2
[4,1,3,2] => 101 => 2 = 4 - 2
[4,2,1,3] => 110 => 1 = 3 - 2
[4,2,3,1] => 101 => 2 = 4 - 2
[4,3,1,2] => 110 => 1 = 3 - 2
[4,3,2,1] => 111 => 0 = 2 - 2
[1,2,3,4,5] => 0000 => 0 = 2 - 2
[1,2,3,5,4] => 0001 => 1 = 3 - 2
[1,2,4,3,5] => 0010 => 2 = 4 - 2
[1,2,4,5,3] => 0001 => 1 = 3 - 2
[1,2,5,3,4] => 0010 => 2 = 4 - 2
[1,2,5,4,3] => 0011 => 1 = 3 - 2
[1,3,2,4,5] => 0100 => 2 = 4 - 2
[1,3,2,5,4] => 0101 => 3 = 5 - 2
[1,3,4,2,5] => 0010 => 2 = 4 - 2
[1,3,4,5,2] => 0001 => 1 = 3 - 2
[1,3,5,2,4] => 0010 => 2 = 4 - 2
[1,3,5,4,2] => 0011 => 1 = 3 - 2
[1,4,2,3,5] => 0100 => 2 = 4 - 2
[1,4,2,5,3] => 0101 => 3 = 5 - 2
[1,4,3,2,5] => 0110 => 2 = 4 - 2
[1,4,3,5,2] => 0101 => 3 = 5 - 2
[1,4,5,2,3] => 0010 => 2 = 4 - 2
[1,4,5,3,2] => 0011 => 1 = 3 - 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: St001028
Mp00109: Permutations —descent word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001028: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001028: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Values
[1] => => [] => ?
=> ? = 1
[1,2] => 0 => [1] => [1,0]
=> 2
[2,1] => 1 => [1] => [1,0]
=> 2
[1,2,3] => 00 => [2] => [1,1,0,0]
=> 2
[1,3,2] => 01 => [1,1] => [1,0,1,0]
=> 3
[2,1,3] => 10 => [1,1] => [1,0,1,0]
=> 3
[2,3,1] => 01 => [1,1] => [1,0,1,0]
=> 3
[3,1,2] => 10 => [1,1] => [1,0,1,0]
=> 3
[3,2,1] => 11 => [2] => [1,1,0,0]
=> 2
[1,2,3,4] => 000 => [3] => [1,1,1,0,0,0]
=> 2
[1,2,4,3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[1,3,2,4] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,3,4,2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[1,4,2,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,4,3,2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[2,1,3,4] => 100 => [1,2] => [1,0,1,1,0,0]
=> 3
[2,1,4,3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[2,3,1,4] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[2,3,4,1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[2,4,1,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[2,4,3,1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[3,1,2,4] => 100 => [1,2] => [1,0,1,1,0,0]
=> 3
[3,1,4,2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[3,2,1,4] => 110 => [2,1] => [1,1,0,0,1,0]
=> 3
[3,2,4,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[3,4,1,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[3,4,2,1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[4,1,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 3
[4,1,3,2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[4,2,1,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 3
[4,2,3,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[4,3,1,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 3
[4,3,2,1] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[1,2,3,4,5] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> 2
[1,2,3,5,4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,4,3,5] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[1,2,4,5,3] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,5,3,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[1,2,5,4,3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[1,3,2,4,5] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 4
[1,3,2,5,4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,3,4,2,5] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[1,3,4,5,2] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,3,5,2,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[1,3,5,4,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[1,4,2,3,5] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 4
[1,4,2,5,3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,4,3,2,5] => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[1,4,3,5,2] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,4,5,2,3] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[1,4,5,3,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
Description
Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001290
Mp00109: Permutations —descent word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001290: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001290: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Values
[1] => => [] => ?
=> ? = 1
[1,2] => 0 => [1] => [1,0]
=> 2
[2,1] => 1 => [1] => [1,0]
=> 2
[1,2,3] => 00 => [2] => [1,1,0,0]
=> 2
[1,3,2] => 01 => [1,1] => [1,0,1,0]
=> 3
[2,1,3] => 10 => [1,1] => [1,0,1,0]
=> 3
[2,3,1] => 01 => [1,1] => [1,0,1,0]
=> 3
[3,1,2] => 10 => [1,1] => [1,0,1,0]
=> 3
[3,2,1] => 11 => [2] => [1,1,0,0]
=> 2
[1,2,3,4] => 000 => [3] => [1,1,1,0,0,0]
=> 2
[1,2,4,3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[1,3,2,4] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,3,4,2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[1,4,2,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,4,3,2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[2,1,3,4] => 100 => [1,2] => [1,0,1,1,0,0]
=> 3
[2,1,4,3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[2,3,1,4] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[2,3,4,1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 3
[2,4,1,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[2,4,3,1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[3,1,2,4] => 100 => [1,2] => [1,0,1,1,0,0]
=> 3
[3,1,4,2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[3,2,1,4] => 110 => [2,1] => [1,1,0,0,1,0]
=> 3
[3,2,4,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[3,4,1,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[3,4,2,1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 3
[4,1,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 3
[4,1,3,2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[4,2,1,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 3
[4,2,3,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[4,3,1,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 3
[4,3,2,1] => 111 => [3] => [1,1,1,0,0,0]
=> 2
[1,2,3,4,5] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> 2
[1,2,3,5,4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,4,3,5] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[1,2,4,5,3] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,5,3,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[1,2,5,4,3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[1,3,2,4,5] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 4
[1,3,2,5,4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,3,4,2,5] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[1,3,4,5,2] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,3,5,2,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[1,3,5,4,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[1,4,2,3,5] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 4
[1,4,2,5,3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,4,3,2,5] => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[1,4,3,5,2] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,4,5,2,3] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[1,4,5,3,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
Description
The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A.
Matching statistic: St000010
Mp00109: Permutations —descent word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Values
[1] => => [] => ?
=> ? = 1 - 1
[1,2] => 0 => [1] => [1]
=> 1 = 2 - 1
[2,1] => 1 => [1] => [1]
=> 1 = 2 - 1
[1,2,3] => 00 => [2] => [2]
=> 1 = 2 - 1
[1,3,2] => 01 => [1,1] => [1,1]
=> 2 = 3 - 1
[2,1,3] => 10 => [1,1] => [1,1]
=> 2 = 3 - 1
[2,3,1] => 01 => [1,1] => [1,1]
=> 2 = 3 - 1
[3,1,2] => 10 => [1,1] => [1,1]
=> 2 = 3 - 1
[3,2,1] => 11 => [2] => [2]
=> 1 = 2 - 1
[1,2,3,4] => 000 => [3] => [3]
=> 1 = 2 - 1
[1,2,4,3] => 001 => [2,1] => [2,1]
=> 2 = 3 - 1
[1,3,2,4] => 010 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[1,3,4,2] => 001 => [2,1] => [2,1]
=> 2 = 3 - 1
[1,4,2,3] => 010 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[1,4,3,2] => 011 => [1,2] => [2,1]
=> 2 = 3 - 1
[2,1,3,4] => 100 => [1,2] => [2,1]
=> 2 = 3 - 1
[2,1,4,3] => 101 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[2,3,1,4] => 010 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[2,3,4,1] => 001 => [2,1] => [2,1]
=> 2 = 3 - 1
[2,4,1,3] => 010 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[2,4,3,1] => 011 => [1,2] => [2,1]
=> 2 = 3 - 1
[3,1,2,4] => 100 => [1,2] => [2,1]
=> 2 = 3 - 1
[3,1,4,2] => 101 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[3,2,1,4] => 110 => [2,1] => [2,1]
=> 2 = 3 - 1
[3,2,4,1] => 101 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[3,4,1,2] => 010 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[3,4,2,1] => 011 => [1,2] => [2,1]
=> 2 = 3 - 1
[4,1,2,3] => 100 => [1,2] => [2,1]
=> 2 = 3 - 1
[4,1,3,2] => 101 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[4,2,1,3] => 110 => [2,1] => [2,1]
=> 2 = 3 - 1
[4,2,3,1] => 101 => [1,1,1] => [1,1,1]
=> 3 = 4 - 1
[4,3,1,2] => 110 => [2,1] => [2,1]
=> 2 = 3 - 1
[4,3,2,1] => 111 => [3] => [3]
=> 1 = 2 - 1
[1,2,3,4,5] => 0000 => [4] => [4]
=> 1 = 2 - 1
[1,2,3,5,4] => 0001 => [3,1] => [3,1]
=> 2 = 3 - 1
[1,2,4,3,5] => 0010 => [2,1,1] => [2,1,1]
=> 3 = 4 - 1
[1,2,4,5,3] => 0001 => [3,1] => [3,1]
=> 2 = 3 - 1
[1,2,5,3,4] => 0010 => [2,1,1] => [2,1,1]
=> 3 = 4 - 1
[1,2,5,4,3] => 0011 => [2,2] => [2,2]
=> 2 = 3 - 1
[1,3,2,4,5] => 0100 => [1,1,2] => [2,1,1]
=> 3 = 4 - 1
[1,3,2,5,4] => 0101 => [1,1,1,1] => [1,1,1,1]
=> 4 = 5 - 1
[1,3,4,2,5] => 0010 => [2,1,1] => [2,1,1]
=> 3 = 4 - 1
[1,3,4,5,2] => 0001 => [3,1] => [3,1]
=> 2 = 3 - 1
[1,3,5,2,4] => 0010 => [2,1,1] => [2,1,1]
=> 3 = 4 - 1
[1,3,5,4,2] => 0011 => [2,2] => [2,2]
=> 2 = 3 - 1
[1,4,2,3,5] => 0100 => [1,1,2] => [2,1,1]
=> 3 = 4 - 1
[1,4,2,5,3] => 0101 => [1,1,1,1] => [1,1,1,1]
=> 4 = 5 - 1
[1,4,3,2,5] => 0110 => [1,2,1] => [2,1,1]
=> 3 = 4 - 1
[1,4,3,5,2] => 0101 => [1,1,1,1] => [1,1,1,1]
=> 4 = 5 - 1
[1,4,5,2,3] => 0010 => [2,1,1] => [2,1,1]
=> 3 = 4 - 1
[1,4,5,3,2] => 0011 => [2,2] => [2,2]
=> 2 = 3 - 1
Description
The length of the partition.
Matching statistic: St000011
Mp00109: Permutations —descent word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Values
[1] => => [] => ?
=> ? = 1 - 1
[1,2] => 0 => [1] => [1,0]
=> 1 = 2 - 1
[2,1] => 1 => [1] => [1,0]
=> 1 = 2 - 1
[1,2,3] => 00 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[1,3,2] => 01 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[2,1,3] => 10 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[2,3,1] => 01 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[3,1,2] => 10 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[3,2,1] => 11 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[1,2,3,4] => 000 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,4,3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,3,2,4] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,3,4,2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,2,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,4,3,2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,1,3,4] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,1,4,3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[2,3,1,4] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[2,3,4,1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,4,1,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[2,4,3,1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[3,1,2,4] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[3,1,4,2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[3,2,1,4] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[3,2,4,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[3,4,1,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[3,4,2,1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[4,1,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[4,1,3,2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[4,2,1,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[4,2,3,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[4,3,1,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[4,3,2,1] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,3,4,5] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,3,5,4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,2,4,3,5] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,2,4,5,3] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,2,5,3,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,2,5,4,3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,3,2,4,5] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,3,2,5,4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,3,4,2,5] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,3,4,5,2] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,3,5,2,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,3,5,4,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,4,2,3,5] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,4,2,5,3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,4,3,2,5] => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,4,3,5,2] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,4,5,2,3] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,4,5,3,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000015
Mp00109: Permutations —descent word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Values
[1] => => [] => ?
=> ? = 1 - 1
[1,2] => 0 => [1] => [1,0]
=> 1 = 2 - 1
[2,1] => 1 => [1] => [1,0]
=> 1 = 2 - 1
[1,2,3] => 00 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[1,3,2] => 01 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[2,1,3] => 10 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[2,3,1] => 01 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[3,1,2] => 10 => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[3,2,1] => 11 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[1,2,3,4] => 000 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,4,3] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,3,2,4] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,3,4,2] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,2,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,4,3,2] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,1,3,4] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,1,4,3] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[2,3,1,4] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[2,3,4,1] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,4,1,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[2,4,3,1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[3,1,2,4] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[3,1,4,2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[3,2,1,4] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[3,2,4,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[3,4,1,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[3,4,2,1] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[4,1,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[4,1,3,2] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[4,2,1,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[4,2,3,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[4,3,1,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[4,3,2,1] => 111 => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,3,4,5] => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,3,5,4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,2,4,3,5] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,2,4,5,3] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,2,5,3,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,2,5,4,3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,3,2,4,5] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,3,2,5,4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,3,4,2,5] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,3,4,5,2] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,3,5,2,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,3,5,4,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,4,2,3,5] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,4,2,5,3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,4,3,2,5] => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,4,3,5,2] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,4,5,2,3] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,4,5,3,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
Description
The number of peaks of a Dyck path.
Matching statistic: St000097
Mp00109: Permutations —descent word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Values
[1] => => [] => ?
=> ? = 1 - 1
[1,2] => 0 => [1] => ([],1)
=> 1 = 2 - 1
[2,1] => 1 => [1] => ([],1)
=> 1 = 2 - 1
[1,2,3] => 00 => [2] => ([],2)
=> 1 = 2 - 1
[1,3,2] => 01 => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[2,1,3] => 10 => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[2,3,1] => 01 => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[3,1,2] => 10 => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[3,2,1] => 11 => [2] => ([],2)
=> 1 = 2 - 1
[1,2,3,4] => 000 => [3] => ([],3)
=> 1 = 2 - 1
[1,2,4,3] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,3,2,4] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,3,4,2] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,4,2,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,4,3,2] => 011 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[2,1,3,4] => 100 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[2,1,4,3] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[2,3,1,4] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[2,3,4,1] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,4,1,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[2,4,3,1] => 011 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[3,1,2,4] => 100 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[3,1,4,2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[3,2,1,4] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[3,2,4,1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[3,4,1,2] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[3,4,2,1] => 011 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[4,1,2,3] => 100 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[4,1,3,2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[4,2,1,3] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[4,2,3,1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[4,3,1,2] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[4,3,2,1] => 111 => [3] => ([],3)
=> 1 = 2 - 1
[1,2,3,4,5] => 0000 => [4] => ([],4)
=> 1 = 2 - 1
[1,2,3,5,4] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,2,4,3,5] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,2,4,5,3] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,2,5,3,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,2,5,4,3] => 0011 => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,3,2,4,5] => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,3,2,5,4] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,3,4,2,5] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,3,4,5,2] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,3,5,2,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,3,5,4,2] => 0011 => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,4,2,3,5] => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,4,2,5,3] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,4,3,2,5] => 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,4,3,5,2] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,4,5,2,3] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,4,5,3,2] => 0011 => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000098
Mp00109: Permutations —descent word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Values
[1] => => [] => ?
=> ? = 1 - 1
[1,2] => 0 => [1] => ([],1)
=> 1 = 2 - 1
[2,1] => 1 => [1] => ([],1)
=> 1 = 2 - 1
[1,2,3] => 00 => [2] => ([],2)
=> 1 = 2 - 1
[1,3,2] => 01 => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[2,1,3] => 10 => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[2,3,1] => 01 => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[3,1,2] => 10 => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[3,2,1] => 11 => [2] => ([],2)
=> 1 = 2 - 1
[1,2,3,4] => 000 => [3] => ([],3)
=> 1 = 2 - 1
[1,2,4,3] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,3,2,4] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,3,4,2] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,4,2,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,4,3,2] => 011 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[2,1,3,4] => 100 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[2,1,4,3] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[2,3,1,4] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[2,3,4,1] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,4,1,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[2,4,3,1] => 011 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[3,1,2,4] => 100 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[3,1,4,2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[3,2,1,4] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[3,2,4,1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[3,4,1,2] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[3,4,2,1] => 011 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[4,1,2,3] => 100 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[4,1,3,2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[4,2,1,3] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[4,2,3,1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[4,3,1,2] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[4,3,2,1] => 111 => [3] => ([],3)
=> 1 = 2 - 1
[1,2,3,4,5] => 0000 => [4] => ([],4)
=> 1 = 2 - 1
[1,2,3,5,4] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,2,4,3,5] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,2,4,5,3] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,2,5,3,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,2,5,4,3] => 0011 => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,3,2,4,5] => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,3,2,5,4] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,3,4,2,5] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,3,4,5,2] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,3,5,2,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,3,5,4,2] => 0011 => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,4,2,3,5] => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,4,2,5,3] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,4,3,2,5] => 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,4,3,5,2] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,4,5,2,3] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,4,5,3,2] => 0011 => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St000172
Mp00109: Permutations —descent word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000172: Graphs ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000172: Graphs ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Values
[1] => => [] => ?
=> ? = 1 - 1
[1,2] => 0 => [1] => ([],1)
=> 1 = 2 - 1
[2,1] => 1 => [1] => ([],1)
=> 1 = 2 - 1
[1,2,3] => 00 => [2] => ([],2)
=> 1 = 2 - 1
[1,3,2] => 01 => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[2,1,3] => 10 => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[2,3,1] => 01 => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[3,1,2] => 10 => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[3,2,1] => 11 => [2] => ([],2)
=> 1 = 2 - 1
[1,2,3,4] => 000 => [3] => ([],3)
=> 1 = 2 - 1
[1,2,4,3] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,3,2,4] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,3,4,2] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,4,2,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,4,3,2] => 011 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[2,1,3,4] => 100 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[2,1,4,3] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[2,3,1,4] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[2,3,4,1] => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,4,1,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[2,4,3,1] => 011 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[3,1,2,4] => 100 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[3,1,4,2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[3,2,1,4] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[3,2,4,1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[3,4,1,2] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[3,4,2,1] => 011 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[4,1,2,3] => 100 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[4,1,3,2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[4,2,1,3] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[4,2,3,1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[4,3,1,2] => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[4,3,2,1] => 111 => [3] => ([],3)
=> 1 = 2 - 1
[1,2,3,4,5] => 0000 => [4] => ([],4)
=> 1 = 2 - 1
[1,2,3,5,4] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,2,4,3,5] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,2,4,5,3] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,2,5,3,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,2,5,4,3] => 0011 => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,3,2,4,5] => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,3,2,5,4] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,3,4,2,5] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,3,4,5,2] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,3,5,2,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,3,5,4,2] => 0011 => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,4,2,3,5] => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,4,2,5,3] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,4,3,2,5] => 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,4,3,5,2] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,4,5,2,3] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,4,5,3,2] => 0011 => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
Description
The Grundy number of a graph.
The Grundy number $\Gamma(G)$ is defined to be the largest $k$ such that $G$ admits a greedy $k$-coloring. Any order of the vertices of $G$ induces a greedy coloring by assigning to the $i$-th vertex in this order the smallest positive integer such that the partial coloring remains a proper coloring.
In particular, we have that $\chi(G) \leq \Gamma(G) \leq \Delta(G) + 1$, where $\chi(G)$ is the chromatic number of $G$ ([[St000098]]), and where $\Delta(G)$ is the maximal degree of a vertex of $G$ ([[St000171]]).
The following 67 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000288The number of ones in a binary word. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000053The number of valleys of the Dyck path. St000272The treewidth of a graph. St000306The bounce count of a Dyck path. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001358The largest degree of a regular subgraph of a graph. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001971The number of negative eigenvalues of the adjacency matrix of the graph. St001812The biclique partition number of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000259The diameter of a connected graph. 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. St000264The girth of a graph, which is not a tree. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St001488The number of corners of a skew partition. St000831The number of indices that are either descents or recoils. St000528The height of a poset. St000080The rank of the poset. St001875The number of simple modules with projective dimension at most 1. St001668The number of points of the poset minus the width of the poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000307The number of rowmotion orbits of a poset. St000632The jump number of the poset. St001746The coalition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St000778The metric dimension of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001270The bandwidth of a graph. St001962The proper pathwidth of a graph. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001769The reflection length of a signed permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001642The Prague dimension of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000640The rank of the largest boolean interval in a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree.
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