- FindStat

Your data matches 81 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000987
(load all 33 compositions to match this statistic)

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of positive eigenvalues of the Laplacian matrix of the graph. This is the number of vertices minus the number of connected components of the graph.

Matching statistic: St000024

Mapping

Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] =>  => [1] => [1,0]
 => 0
[1,2] => 1 => [1,1] => [1,0,1,0]
 => 0
[2,1] => 0 => [2] => [1,1,0,0]
 => 1
[1,2,3] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 0
[1,3,2] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,3] => 01 => [2,1] => [1,1,0,0,1,0]
 => 1
[2,3,1] => 00 => [3] => [1,1,1,0,0,0]
 => 2
[3,1,2] => 00 => [3] => [1,1,1,0,0,0]
 => 2
[3,2,1] => 00 => [3] => [1,1,1,0,0,0]
 => 2
[1,2,3,4] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[1,2,4,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[1,3,2,4] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[1,3,4,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[1,4,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[1,4,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[2,1,3,4] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[2,1,4,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[2,3,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[2,3,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[2,4,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[2,4,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[3,1,2,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[3,1,4,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[3,2,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[3,2,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[3,4,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[3,4,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[4,1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[4,1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[4,2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[4,2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[4,3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[4,3,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,2,3,5,4] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,2,4,3,5] => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,2,4,5,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,2,5,3,4] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,2,5,4,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,3,2,4,5] => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,3,2,5,4] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,3,4,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,3,4,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3
[1,3,5,2,4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3
[1,3,5,4,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3
[1,4,2,3,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,4,2,5,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3
[1,4,3,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,4,3,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3
[1,4,5,2,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3

Description

The number of double up and double down steps of a Dyck path. In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.

Matching statistic: St000377

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The dinv defect of an integer partition. This is the number of cells

$c$ in the diagram of an integer partition

$\lambda$ for which

$\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$ .

Matching statistic: St000394
(load all 2 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The sum of the heights of the peaks of a Dyck path minus the number of peaks.

Matching statistic: St001176

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.

Matching statistic: St001189

Mapping

Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] =>  => [1] => [1,0]
 => 0
[1,2] => 1 => [1,1] => [1,0,1,0]
 => 0
[2,1] => 0 => [2] => [1,1,0,0]
 => 1
[1,2,3] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 0
[1,3,2] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,3] => 01 => [2,1] => [1,1,0,0,1,0]
 => 1
[2,3,1] => 00 => [3] => [1,1,1,0,0,0]
 => 2
[3,1,2] => 00 => [3] => [1,1,1,0,0,0]
 => 2
[3,2,1] => 00 => [3] => [1,1,1,0,0,0]
 => 2
[1,2,3,4] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[1,2,4,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[1,3,2,4] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[1,3,4,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[1,4,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[1,4,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[2,1,3,4] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[2,1,4,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[2,3,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[2,3,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[2,4,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[2,4,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[3,1,2,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[3,1,4,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[3,2,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[3,2,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[3,4,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[3,4,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[4,1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[4,1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[4,2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[4,2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[4,3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[4,3,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,2,3,5,4] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,2,4,3,5] => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,2,4,5,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,2,5,3,4] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,2,5,4,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,3,2,4,5] => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,3,2,5,4] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,3,4,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,3,4,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3
[1,3,5,2,4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3
[1,3,5,4,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3
[1,4,2,3,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,4,2,5,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3
[1,4,3,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,4,3,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3
[1,4,5,2,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3

Description

The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St000093

Mapping

Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] =>  => [1] => ([],1)
 => 1 = 0 + 1
[1,2] => 1 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[2,1] => 0 => [2] => ([],2)
 => 2 = 1 + 1
[1,2,3] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,3,2] => 10 => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[2,1,3] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[2,3,1] => 00 => [3] => ([],3)
 => 3 = 2 + 1
[3,1,2] => 00 => [3] => ([],3)
 => 3 = 2 + 1
[3,2,1] => 00 => [3] => ([],3)
 => 3 = 2 + 1
[1,2,3,4] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,2,4,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,3,2,4] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,3,4,2] => 100 => [1,3] => ([(2,3)],4)
 => 3 = 2 + 1
[1,4,2,3] => 100 => [1,3] => ([(2,3)],4)
 => 3 = 2 + 1
[1,4,3,2] => 100 => [1,3] => ([(2,3)],4)
 => 3 = 2 + 1
[2,1,3,4] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,1,4,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,3,1,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,3,4,1] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[2,4,1,3] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[2,4,3,1] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[3,1,2,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1,4,2] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[3,2,1,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,2,4,1] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[3,4,1,2] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[3,4,2,1] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[4,1,2,3] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[4,1,3,2] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[4,2,1,3] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[4,2,3,1] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[4,3,1,2] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[4,3,2,1] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,2,3,5,4] => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,4,3,5] => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,4,5,3] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,5,3,4] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,5,4,3] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,2,4,5] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,2,5,4] => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,4,2,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,4,5,2] => 1000 => [1,4] => ([(3,4)],5)
 => 4 = 3 + 1
[1,3,5,2,4] => 1000 => [1,4] => ([(3,4)],5)
 => 4 = 3 + 1
[1,3,5,4,2] => 1000 => [1,4] => ([(3,4)],5)
 => 4 = 3 + 1
[1,4,2,3,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,2,5,3] => 1000 => [1,4] => ([(3,4)],5)
 => 4 = 3 + 1
[1,4,3,2,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,3,5,2] => 1000 => [1,4] => ([(3,4)],5)
 => 4 = 3 + 1
[1,4,5,2,3] => 1000 => [1,4] => ([(3,4)],5)
 => 4 = 3 + 1

Description

The cardinality of a maximal independent set of vertices of a graph. An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number

$\alpha(G)$ of

$G$ .

Matching statistic: St000507
(load all 2 compositions to match this statistic)

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of ascents of a standard tableau. Entry

$i$ of a standard Young tableau is an '''ascent''' if

$i+1$ appears to the right or above

$i$ in the tableau (with respect to the English notation for tableaux).

Matching statistic: St000786

Mapping

Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] =>  => [1] => ([],1)
 => 1 = 0 + 1
[1,2] => 1 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[2,1] => 0 => [2] => ([],2)
 => 2 = 1 + 1
[1,2,3] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,3,2] => 10 => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[2,1,3] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[2,3,1] => 00 => [3] => ([],3)
 => 3 = 2 + 1
[3,1,2] => 00 => [3] => ([],3)
 => 3 = 2 + 1
[3,2,1] => 00 => [3] => ([],3)
 => 3 = 2 + 1
[1,2,3,4] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,2,4,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,3,2,4] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,3,4,2] => 100 => [1,3] => ([(2,3)],4)
 => 3 = 2 + 1
[1,4,2,3] => 100 => [1,3] => ([(2,3)],4)
 => 3 = 2 + 1
[1,4,3,2] => 100 => [1,3] => ([(2,3)],4)
 => 3 = 2 + 1
[2,1,3,4] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,1,4,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,3,1,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,3,4,1] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[2,4,1,3] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[2,4,3,1] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[3,1,2,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1,4,2] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[3,2,1,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,2,4,1] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[3,4,1,2] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[3,4,2,1] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[4,1,2,3] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[4,1,3,2] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[4,2,1,3] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[4,2,3,1] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[4,3,1,2] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[4,3,2,1] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,2,3,5,4] => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,4,3,5] => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,4,5,3] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,5,3,4] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,5,4,3] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,2,4,5] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,2,5,4] => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,4,2,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,4,5,2] => 1000 => [1,4] => ([(3,4)],5)
 => 4 = 3 + 1
[1,3,5,2,4] => 1000 => [1,4] => ([(3,4)],5)
 => 4 = 3 + 1
[1,3,5,4,2] => 1000 => [1,4] => ([(3,4)],5)
 => 4 = 3 + 1
[1,4,2,3,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,2,5,3] => 1000 => [1,4] => ([(3,4)],5)
 => 4 = 3 + 1
[1,4,3,2,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,3,5,2] => 1000 => [1,4] => ([(3,4)],5)
 => 4 = 3 + 1
[1,4,5,2,3] => 1000 => [1,4] => ([(3,4)],5)
 => 4 = 3 + 1

Description

The maximal number of occurrences of a colour in a proper colouring of a graph. To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions. For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes,

$[2,2,2]$ and

$[3,2,1]$ . Therefore, the statistic on this graph is

$3$ .

Matching statistic: St001007

Mapping

Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] =>  => [1] => [1,0]
 => 1 = 0 + 1
[1,2] => 1 => [1,1] => [1,0,1,0]
 => 1 = 0 + 1
[2,1] => 0 => [2] => [1,1,0,0]
 => 2 = 1 + 1
[1,2,3] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[1,3,2] => 10 => [1,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
[2,1,3] => 01 => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[2,3,1] => 00 => [3] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[3,1,2] => 00 => [3] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[3,2,1] => 00 => [3] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[1,2,3,4] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,2,4,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,3,2,4] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,3,4,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,4,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,4,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[2,1,3,4] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[2,1,4,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[2,3,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[2,3,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[2,4,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[2,4,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[3,1,2,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[3,1,4,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[3,2,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[3,2,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[3,4,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[3,4,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[4,1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[4,1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[4,2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[4,2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[4,3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[4,3,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,2,3,5,4] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,2,4,3,5] => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,2,4,5,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,2,5,3,4] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,2,5,4,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,3,2,4,5] => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,3,2,5,4] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,3,4,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,3,4,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,3,5,2,4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,3,5,4,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,4,2,3,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,4,2,5,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,4,3,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,4,3,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,4,5,2,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1

Description

Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.

The following 71 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000288The number of ones in a binary word. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000171The degree of the graph. St000019The cardinality of the support of a permutation. St000214The number of adjacencies of a permutation. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001497The position of the largest weak excedence of a permutation. St000147The largest part of an integer partition. St000010The length of the partition. St000740The last entry of a permutation. St000454The largest eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000054The first entry of the permutation. St000141The maximum drop size of a permutation. St000653The last descent of a permutation. St000067The inversion number of the alternating sign matrix. St001330The hat guessing number of a graph. St000957The number of Bruhat lower covers of a permutation. St001725The harmonious chromatic number of a graph. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000831The number of indices that are either descents or recoils. St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000795The mad of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001061The number of indices that are both descents and recoils of a permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000240The number of indices that are not small excedances. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000021The number of descents of a permutation. St000030The sum of the descent differences of a permutations. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. 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. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000840The number of closers smaller than the largest opener in a perfect matching. St001812The biclique partition number of a graph. St001082The number of boxed occurrences of 123 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000327The number of cover relations in a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001866The nesting alignments of a signed permutation. 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. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St001896The number of right descents of a signed permutations. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001946The number of descents in a parking function. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001877Number of indecomposable injective modules with projective dimension 2.