- FindStat

Your data matches 95 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001486
(load all 3 compositions to match this statistic)

Mapping

St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => 1 = 0 + 1
[1,1] => 2 = 1 + 1
[2] => 2 = 1 + 1
[1,1,1] => 2 = 1 + 1
[1,2] => 3 = 2 + 1
[2,1] => 3 = 2 + 1
[3] => 2 = 1 + 1
[1,1,1,1] => 2 = 1 + 1
[1,1,2] => 3 = 2 + 1
[1,2,1] => 4 = 3 + 1
[1,3] => 3 = 2 + 1
[2,1,1] => 3 = 2 + 1
[2,2] => 4 = 3 + 1
[3,1] => 3 = 2 + 1
[4] => 2 = 1 + 1
[1,1,1,1,1] => 2 = 1 + 1
[1,1,1,2] => 3 = 2 + 1
[1,1,2,1] => 4 = 3 + 1
[1,1,3] => 3 = 2 + 1
[1,2,1,1] => 4 = 3 + 1
[1,2,2] => 5 = 4 + 1
[1,3,1] => 4 = 3 + 1
[1,4] => 3 = 2 + 1
[2,1,1,1] => 3 = 2 + 1
[2,1,2] => 4 = 3 + 1
[2,2,1] => 5 = 4 + 1
[2,3] => 4 = 3 + 1
[3,1,1] => 3 = 2 + 1
[3,2] => 4 = 3 + 1
[4,1] => 3 = 2 + 1
[5] => 2 = 1 + 1
[1,1,1,1,1,1] => 2 = 1 + 1
[1,1,1,1,2] => 3 = 2 + 1
[1,1,1,2,1] => 4 = 3 + 1
[1,1,1,3] => 3 = 2 + 1
[1,1,2,1,1] => 4 = 3 + 1
[1,1,2,2] => 5 = 4 + 1
[1,1,3,1] => 4 = 3 + 1
[1,1,4] => 3 = 2 + 1
[1,2,1,1,1] => 4 = 3 + 1
[1,2,1,2] => 5 = 4 + 1
[1,2,2,1] => 6 = 5 + 1
[1,2,3] => 5 = 4 + 1
[1,3,1,1] => 4 = 3 + 1
[1,3,2] => 5 = 4 + 1
[1,4,1] => 4 = 3 + 1
[1,5] => 3 = 2 + 1
[2,1,1,1,1] => 3 = 2 + 1
[2,1,1,2] => 4 = 3 + 1
[2,1,2,1] => 5 = 4 + 1

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: St001949

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001949: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => ([],1)
 => 0
[1,1] => ([(0,1)],2)
 => 1
[2] => ([],2)
 => 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2] => ([(1,2)],3)
 => 1
[2,1] => ([(0,2),(1,2)],3)
 => 1
[3] => ([],3)
 => 2
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,3] => ([(2,3)],4)
 => 2
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,2] => ([(1,3),(2,3)],4)
 => 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4] => ([],4)
 => 3
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4] => ([(3,4)],5)
 => 3
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[2,3] => ([(2,4),(3,4)],5)
 => 2
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[5] => ([],5)
 => 4
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 4
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,5] => ([(4,5)],6)
 => 4
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2

Description

The rigidity index of a graph. A base of a permutation group is a set

$B$ such that the pointwise stabilizer of

$B$ is trivial. For example, a base of the symmetric group on

$n$ letters must contain all but one letter. This statistic yields the minimal size of a base for the automorphism group of a graph.

Matching statistic: St001315
(load all 2 compositions to match this statistic)

Mapping

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] => ([],1)
 => 1 = 0 + 1
[1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[2] => ([],2)
 => 2 = 1 + 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,2] => ([(1,2)],3)
 => 3 = 2 + 1
[2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[3] => ([],3)
 => 3 = 2 + 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,3] => ([(2,3)],4)
 => 4 = 3 + 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4] => ([],4)
 => 4 = 3 + 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,4] => ([(3,4)],5)
 => 5 = 4 + 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[2,3] => ([(2,4),(3,4)],5)
 => 4 = 3 + 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5] => ([],5)
 => 5 = 4 + 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
[1,5] => ([(4,5)],6)
 => 6 = 5 + 1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1

Description

The dissociation number of a graph.

Matching statistic: St000288
(load all 9 compositions to match this statistic)

Mapping

Mp00094: Integer compositions —to binary word⟶ 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%

Values

[1] => 1 => 1 => 1 = 0 + 1
[1,1] => 11 => 11 => 2 = 1 + 1
[2] => 10 => 11 => 2 = 1 + 1
[1,1,1] => 111 => 111 => 3 = 2 + 1
[1,2] => 110 => 111 => 3 = 2 + 1
[2,1] => 101 => 110 => 2 = 1 + 1
[3] => 100 => 101 => 2 = 1 + 1
[1,1,1,1] => 1111 => 1111 => 4 = 3 + 1
[1,1,2] => 1110 => 1111 => 4 = 3 + 1
[1,2,1] => 1101 => 1110 => 3 = 2 + 1
[1,3] => 1100 => 1101 => 3 = 2 + 1
[2,1,1] => 1011 => 1101 => 3 = 2 + 1
[2,2] => 1010 => 1101 => 3 = 2 + 1
[3,1] => 1001 => 1010 => 2 = 1 + 1
[4] => 1000 => 1001 => 2 = 1 + 1
[1,1,1,1,1] => 11111 => 11111 => 5 = 4 + 1
[1,1,1,2] => 11110 => 11111 => 5 = 4 + 1
[1,1,2,1] => 11101 => 11110 => 4 = 3 + 1
[1,1,3] => 11100 => 11101 => 4 = 3 + 1
[1,2,1,1] => 11011 => 11101 => 4 = 3 + 1
[1,2,2] => 11010 => 11101 => 4 = 3 + 1
[1,3,1] => 11001 => 11010 => 3 = 2 + 1
[1,4] => 11000 => 11001 => 3 = 2 + 1
[2,1,1,1] => 10111 => 11011 => 4 = 3 + 1
[2,1,2] => 10110 => 11011 => 4 = 3 + 1
[2,2,1] => 10101 => 11010 => 3 = 2 + 1
[2,3] => 10100 => 11001 => 3 = 2 + 1
[3,1,1] => 10011 => 10101 => 3 = 2 + 1
[3,2] => 10010 => 10101 => 3 = 2 + 1
[4,1] => 10001 => 10010 => 2 = 1 + 1
[5] => 10000 => 10001 => 2 = 1 + 1
[1,1,1,1,1,1] => 111111 => 111111 => 6 = 5 + 1
[1,1,1,1,2] => 111110 => 111111 => 6 = 5 + 1
[1,1,1,2,1] => 111101 => 111110 => 5 = 4 + 1
[1,1,1,3] => 111100 => 111101 => 5 = 4 + 1
[1,1,2,1,1] => 111011 => 111101 => 5 = 4 + 1
[1,1,2,2] => 111010 => 111101 => 5 = 4 + 1
[1,1,3,1] => 111001 => 111010 => 4 = 3 + 1
[1,1,4] => 111000 => 111001 => 4 = 3 + 1
[1,2,1,1,1] => 110111 => 111011 => 5 = 4 + 1
[1,2,1,2] => 110110 => 111011 => 5 = 4 + 1
[1,2,2,1] => 110101 => 111010 => 4 = 3 + 1
[1,2,3] => 110100 => 111001 => 4 = 3 + 1
[1,3,1,1] => 110011 => 110101 => 4 = 3 + 1
[1,3,2] => 110010 => 110101 => 4 = 3 + 1
[1,4,1] => 110001 => 110010 => 3 = 2 + 1
[1,5] => 110000 => 110001 => 3 = 2 + 1
[2,1,1,1,1] => 101111 => 110111 => 5 = 4 + 1
[2,1,1,2] => 101110 => 110111 => 5 = 4 + 1
[2,1,2,1] => 101101 => 110110 => 4 = 3 + 1

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

Matching statistic: St000291
(load all 4 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1,0]
 => [[1],[2]]
 => 1 => 0
[1,1] => [1,0,1,0]
 => [[1,3],[2,4]]
 => 101 => 1
[2] => [1,1,0,0]
 => [[1,2],[3,4]]
 => 010 => 1
[1,1,1] => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 10101 => 2
[1,2] => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 10010 => 2
[2,1] => [1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => 01001 => 1
[3] => [1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => 00100 => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 1010101 => 3
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 1010010 => 3
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 1001001 => 2
[1,3] => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 1000100 => 2
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 0100101 => 2
[2,2] => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => 0100010 => 2
[3,1] => [1,1,1,0,0,0,1,0]
 => [[1,2,3,7],[4,5,6,8]]
 => 0010001 => 1
[4] => [1,1,1,1,0,0,0,0]
 => [[1,2,3,4],[5,6,7,8]]
 => 0001000 => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9],[2,4,6,8,10]]
 => 101010101 => 4
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,8],[2,4,6,9,10]]
 => 101010010 => 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [[1,3,5,6,9],[2,4,7,8,10]]
 => 101001001 => 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,6,7],[2,4,8,9,10]]
 => 101000100 => 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [[1,3,4,7,9],[2,5,6,8,10]]
 => 100100101 => 3
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [[1,3,4,7,8],[2,5,6,9,10]]
 => 100100010 => 3
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => 100010001 => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => 100001000 => 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [[1,2,5,7,9],[3,4,6,8,10]]
 => 010010101 => 3
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [[1,2,5,7,8],[3,4,6,9,10]]
 => 010010010 => 3
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [[1,2,5,6,9],[3,4,7,8,10]]
 => 010001001 => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => 010000100 => 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [[1,2,3,7,9],[4,5,6,8,10]]
 => 001000101 => 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => 001000010 => 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [[1,2,3,4,9],[5,6,7,8,10]]
 => 000100001 => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [[1,2,3,4,5],[6,7,8,9,10]]
 => 000010000 => 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9,11],[2,4,6,8,10,12]]
 => 10101010101 => 5
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,9,10],[2,4,6,8,11,12]]
 => 10101010010 => 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [[1,3,5,7,8,11],[2,4,6,9,10,12]]
 => 10101001001 => 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,7,8,9],[2,4,6,10,11,12]]
 => 10101000100 => 4
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [[1,3,5,6,9,11],[2,4,7,8,10,12]]
 => 10100100101 => 4
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[1,3,5,6,9,10],[2,4,7,8,11,12]]
 => 10100100010 => 4
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [[1,3,5,6,7,11],[2,4,8,9,10,12]]
 => 10100010001 => 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [[1,3,5,6,7,8],[2,4,9,10,11,12]]
 => 10100001000 => 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [[1,3,4,7,9,11],[2,5,6,8,10,12]]
 => 10010010101 => 4
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[1,3,4,7,9,10],[2,5,6,8,11,12]]
 => 10010010010 => 4
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [[1,3,4,7,8,11],[2,5,6,9,10,12]]
 => 10010001001 => 3
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [[1,3,4,7,8,9],[2,5,6,10,11,12]]
 => 10010000100 => 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [[1,3,4,5,9,11],[2,6,7,8,10,12]]
 => 10001000101 => 3
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [[1,3,4,5,9,10],[2,6,7,8,11,12]]
 => 10001000010 => 3
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [[1,3,4,5,6,11],[2,7,8,9,10,12]]
 => 10000100001 => 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,3,4,5,6,7],[2,8,9,10,11,12]]
 => 10000010000 => 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [[1,2,5,7,9,11],[3,4,6,8,10,12]]
 => 01001010101 => 4
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [[1,2,5,7,9,10],[3,4,6,8,11,12]]
 => 01001010010 => 4
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [[1,2,5,7,8,11],[3,4,6,9,10,12]]
 => 01001001001 => 3

Description

The number of descents of a binary word.

Matching statistic: St000292
(load all 2 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1,0]
 => [[1],[2]]
 => 1 => 0
[1,1] => [1,0,1,0]
 => [[1,3],[2,4]]
 => 101 => 1
[2] => [1,1,0,0]
 => [[1,2],[3,4]]
 => 010 => 1
[1,1,1] => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 10101 => 2
[1,2] => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 10010 => 1
[2,1] => [1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => 01001 => 2
[3] => [1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => 00100 => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 1010101 => 3
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 1010010 => 2
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 1001001 => 2
[1,3] => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 1000100 => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 0100101 => 3
[2,2] => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => 0100010 => 2
[3,1] => [1,1,1,0,0,0,1,0]
 => [[1,2,3,7],[4,5,6,8]]
 => 0010001 => 2
[4] => [1,1,1,1,0,0,0,0]
 => [[1,2,3,4],[5,6,7,8]]
 => 0001000 => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9],[2,4,6,8,10]]
 => 101010101 => 4
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,8],[2,4,6,9,10]]
 => 101010010 => 3
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [[1,3,5,6,9],[2,4,7,8,10]]
 => 101001001 => 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,6,7],[2,4,8,9,10]]
 => 101000100 => 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [[1,3,4,7,9],[2,5,6,8,10]]
 => 100100101 => 3
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [[1,3,4,7,8],[2,5,6,9,10]]
 => 100100010 => 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => 100010001 => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => 100001000 => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [[1,2,5,7,9],[3,4,6,8,10]]
 => 010010101 => 4
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [[1,2,5,7,8],[3,4,6,9,10]]
 => 010010010 => 3
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [[1,2,5,6,9],[3,4,7,8,10]]
 => 010001001 => 3
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => 010000100 => 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [[1,2,3,7,9],[4,5,6,8,10]]
 => 001000101 => 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => 001000010 => 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [[1,2,3,4,9],[5,6,7,8,10]]
 => 000100001 => 2
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [[1,2,3,4,5],[6,7,8,9,10]]
 => 000010000 => 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9,11],[2,4,6,8,10,12]]
 => 10101010101 => 5
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,9,10],[2,4,6,8,11,12]]
 => 10101010010 => 4
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [[1,3,5,7,8,11],[2,4,6,9,10,12]]
 => 10101001001 => 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,7,8,9],[2,4,6,10,11,12]]
 => 10101000100 => 3
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [[1,3,5,6,9,11],[2,4,7,8,10,12]]
 => 10100100101 => 4
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[1,3,5,6,9,10],[2,4,7,8,11,12]]
 => 10100100010 => 3
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [[1,3,5,6,7,11],[2,4,8,9,10,12]]
 => 10100010001 => 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [[1,3,5,6,7,8],[2,4,9,10,11,12]]
 => 10100001000 => 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [[1,3,4,7,9,11],[2,5,6,8,10,12]]
 => 10010010101 => 4
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[1,3,4,7,9,10],[2,5,6,8,11,12]]
 => 10010010010 => 3
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [[1,3,4,7,8,11],[2,5,6,9,10,12]]
 => 10010001001 => 3
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [[1,3,4,7,8,9],[2,5,6,10,11,12]]
 => 10010000100 => 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [[1,3,4,5,9,11],[2,6,7,8,10,12]]
 => 10001000101 => 3
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [[1,3,4,5,9,10],[2,6,7,8,11,12]]
 => 10001000010 => 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [[1,3,4,5,6,11],[2,7,8,9,10,12]]
 => 10000100001 => 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,3,4,5,6,7],[2,8,9,10,11,12]]
 => 10000010000 => 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [[1,2,5,7,9,11],[3,4,6,8,10,12]]
 => 01001010101 => 5
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [[1,2,5,7,9,10],[3,4,6,8,11,12]]
 => 01001010010 => 4
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [[1,2,5,7,8,11],[3,4,6,9,10,12]]
 => 01001001001 => 4

Description

The number of ascents of a binary word.

Matching statistic: St000552

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000552: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1,0]
 => [2,1] => ([(0,1)],2)
 => 0
[1,1] => [1,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[2] => [1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 2
[2,1] => [1,1,0,0,1,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2
[3] => [1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 3
[1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 3
[3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[4] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 4
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 4
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 2
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,1,2,3,7,4,6] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 3
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => 4
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 3
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => 4
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 5
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,1,5,2,6,7,4] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => 4
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => 3
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => 4
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 3
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,7,1,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 2
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,6,1,3,4,7,5] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 3
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,5,1,3,7,4,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => 4

Description

The number of cut vertices of a graph. A cut vertex is one whose deletion increases the number of connected components.

Matching statistic: St000691
(load all 4 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1,0]
 => [2,1] => 1 => 0
[1,1] => [1,0,1,0]
 => [3,1,2] => 10 => 1
[2] => [1,1,0,0]
 => [2,3,1] => 01 => 1
[1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 100 => 1
[1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 101 => 2
[2,1] => [1,1,0,0,1,0]
 => [2,4,1,3] => 010 => 2
[3] => [1,1,1,0,0,0]
 => [2,3,4,1] => 001 => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1000 => 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1001 => 2
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 1010 => 3
[1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 1001 => 2
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0100 => 2
[2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 0101 => 3
[3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 0010 => 2
[4] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 0001 => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 10000 => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => 10001 => 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => 10010 => 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => 10001 => 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => 10100 => 3
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => 10101 => 4
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => 10010 => 3
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 10001 => 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => 01000 => 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => 01001 => 3
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 01010 => 4
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => 01001 => 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => 00100 => 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 00101 => 3
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 00010 => 2
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 00001 => 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 100000 => 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => 100001 => 2
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,1,2,3,7,4,6] => 100010 => 3
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => 100001 => 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => 100100 => 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,1,2,6,3,7,5] => 100101 => 4
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => 100010 => 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => 100001 => 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => 101000 => 3
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [3,1,6,2,4,7,5] => 101001 => 4
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => 101010 => 5
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,1,5,2,6,7,4] => 101001 => 4
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => 100100 => 3
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5] => 100101 => 4
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => 100010 => 3
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => 100001 => 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,7,1,3,4,5,6] => 010000 => 2
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,6,1,3,4,7,5] => 010001 => 3
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,5,1,3,7,4,6] => 010010 => 4

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: St001027

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001027: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1,0]
 => [1,0]
 => [1,0]
 => 0
[1,1] => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[2] => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 3
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 3
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 3
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 4
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 4
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 3
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 4
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => 3
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 3
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 3
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 3
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 3
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => 2

Description

Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001405
(load all 4 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St001405: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1,0]
 => [1] => [1] => 0
[1,1] => [1,0,1,0]
 => [1,2] => [1,2] => 1
[2] => [1,1,0,0]
 => [2,1] => [2,1] => 1
[1,1,1] => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 2
[1,2] => [1,0,1,1,0,0]
 => [1,3,2] => [3,1,2] => 1
[2,1] => [1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => 1
[3] => [1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => 2
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 3
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [4,1,2,3] => 2
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [3,1,2,4] => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [4,3,1,2] => 2
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => 2
[2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [4,2,1,3] => 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => 2
[4] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => 3
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 4
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [5,1,2,3,4] => 3
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [4,1,2,3,5] => 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [5,4,1,2,3] => 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [3,1,2,4,5] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [5,3,1,2,4] => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [4,3,1,2,5] => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [5,4,3,1,2] => 3
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => 3
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [5,2,1,3,4] => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [4,2,1,3,5] => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [5,4,2,1,3] => 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [3,2,1,4,5] => 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [5,3,2,1,4] => 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => [4,3,2,1,5] => 3
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => [5,4,3,2,1] => 4
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 5
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [6,1,2,3,4,5] => 4
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => [5,1,2,3,4,6] => 3
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => [6,5,1,2,3,4] => 4
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => [4,1,2,3,5,6] => 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => [6,4,1,2,3,5] => 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,5,4,3,6] => [5,4,1,2,3,6] => 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,5,4,3] => [6,5,4,1,2,3] => 4
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6] => [3,1,2,4,5,6] => 3
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,3,2,4,6,5] => [6,3,1,2,4,5] => 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4,6] => [5,3,1,2,4,6] => 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,6,5,4] => [6,5,3,1,2,4] => 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,4,3,2,5,6] => [4,3,1,2,5,6] => 3
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,4,3,2,6,5] => [6,4,3,1,2,5] => 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,5,4,3,2,6] => [5,4,3,1,2,6] => 3
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => [6,5,4,3,1,2] => 4
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 4
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,6,5] => [6,2,1,3,4,5] => 3
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,3,5,4,6] => [5,2,1,3,4,6] => 2

Description

The number of bonds in a permutation. For a permutation

$\pi$ , the pair

$(\pi_i, \pi_{i+1})$ is a bond if

$|\pi_i-\pi_{i+1}| = 1$ .

The following 85 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001692The number of vertices with higher degree than the average degree in a graph. St000203The number of external nodes of a binary tree. St000236The number of cyclical small weak excedances. St000259The diameter of a connected graph. St000390The number of runs of ones in a binary word. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St001120The length of a longest path in a graph. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001093The detour number of a graph. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St000619The number of cyclic descents of a permutation. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000646The number of big ascents of a permutation. St000711The number of big exceedences of a permutation. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St001388The number of non-attacking neighbors of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000741The Colin de Verdière graph invariant. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St001716The 1-improper chromatic number of a graph. St000742The number of big ascents of a permutation after prepending zero. St000670The reversal length of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000642The size of the smallest orbit of antichains under Panyushev complementation. St001488The number of corners of a skew partition. St000831The number of indices that are either descents or recoils. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama 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. St000956The maximal displacement of a permutation. St001649The length of a longest trail in a graph. St001180Number of indecomposable injective modules with projective dimension at most 1. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001960The number of descents of a permutation minus one if its first entry is not one. St001060The distinguishing index of a graph. St001557The number of inversions of the second entry of a permutation. St001118The acyclic chromatic index of a graph. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001720The minimal length of a chain of small intervals in a lattice. St001090The number of pop-stack-sorts needed to sort a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000392The length of the longest run of ones in a binary word. St000628The balance of a binary word. 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. St001626The number of maximal proper sublattices of a lattice. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001875The number of simple modules with projective dimension at most 1. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000834The number of right outer peaks of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000717The number of ordinal summands of a poset. 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. St000454The largest eigenvalue of a graph if it is integral. St000075The orbit size of a standard tableau under promotion. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St000023The number of inner peaks of a permutation. St000264The girth of a graph, which is not a tree. St000668The least common multiple of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000779The tier of a permutation. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001469The holeyness of a permutation. St001520The number of strict 3-descents. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000099The number of valleys of a permutation, including the boundary. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000735The last entry on the main diagonal of a standard tableau. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000260The radius of a connected graph. St001645The pebbling number of a connected graph.