- FindStat

Your data matches 65 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000024
(load all 2 compositions to match this statistic)

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ 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
([],2)
 => [2] => [1,1,0,0]
 => 1
([(0,1)],2)
 => [1,1] => [1,0,1,0]
 => 0
([],3)
 => [3] => [1,1,1,0,0,0]
 => 2
([(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => 1
([(0,2),(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [1,0,1,0,1,0]
 => 0
([],4)
 => [4] => [1,1,1,1,0,0,0,0]
 => 3
([(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
([(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
([(0,3),(1,2)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
([],5)
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 4
([(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 3
([(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 3
([(1,4),(2,3)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2

Description

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

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

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St001189
(load all 2 compositions to match this statistic)

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ 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
([],2)
 => [2] => [1,1,0,0]
 => 1
([(0,1)],2)
 => [1,1] => [1,0,1,0]
 => 0
([],3)
 => [3] => [1,1,1,0,0,0]
 => 2
([(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => 1
([(0,2),(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [1,0,1,0,1,0]
 => 0
([],4)
 => [4] => [1,1,1,1,0,0,0,0]
 => 3
([(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
([(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
([(0,3),(1,2)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
([],5)
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 4
([(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 3
([(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 3
([(1,4),(2,3)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2

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
(load all 6 compositions to match this statistic)

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ 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
([],2)
 => [2] => ([],2)
 => 2 = 1 + 1
([(0,1)],2)
 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
([],3)
 => [3] => ([],3)
 => 3 = 2 + 1
([(1,2)],3)
 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,2),(1,2)],3)
 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 0 + 1
([],4)
 => [4] => ([],4)
 => 4 = 3 + 1
([(2,3)],4)
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
([(1,3),(2,3)],4)
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
([],5)
 => [5] => ([],5)
 => 5 = 4 + 1
([(3,4)],5)
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(2,4),(3,4)],5)
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(1,4),(2,4),(3,4)],5)
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(1,4),(2,3)],5)
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(1,4),(2,3),(3,4)],5)
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(0,1),(2,4),(3,4)],5)
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [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
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [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
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [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
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 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: St000786
(load all 6 compositions to match this statistic)

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ 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
([],2)
 => [2] => ([],2)
 => 2 = 1 + 1
([(0,1)],2)
 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
([],3)
 => [3] => ([],3)
 => 3 = 2 + 1
([(1,2)],3)
 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,2),(1,2)],3)
 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 0 + 1
([],4)
 => [4] => ([],4)
 => 4 = 3 + 1
([(2,3)],4)
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
([(1,3),(2,3)],4)
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
([],5)
 => [5] => ([],5)
 => 5 = 4 + 1
([(3,4)],5)
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(2,4),(3,4)],5)
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(1,4),(2,4),(3,4)],5)
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(1,4),(2,3)],5)
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(1,4),(2,3),(3,4)],5)
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(0,1),(2,4),(3,4)],5)
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [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
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [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
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [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
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 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
(load all 2 compositions to match this statistic)

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ 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
([],2)
 => [2] => [1,1,0,0]
 => 2 = 1 + 1
([(0,1)],2)
 => [1,1] => [1,0,1,0]
 => 1 = 0 + 1
([],3)
 => [3] => [1,1,1,0,0,0]
 => 3 = 2 + 1
([(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
([(0,2),(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [1,0,1,0,1,0]
 => 1 = 0 + 1
([],4)
 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
([(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
([(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
([],5)
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
([(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
([(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
([(1,4),(2,3)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1

Description

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

Matching statistic: St001088
(load all 2 compositions to match this statistic)

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001088: 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
([],2)
 => [2] => [1,1,0,0]
 => 2 = 1 + 1
([(0,1)],2)
 => [1,1] => [1,0,1,0]
 => 1 = 0 + 1
([],3)
 => [3] => [1,1,1,0,0,0]
 => 3 = 2 + 1
([(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
([(0,2),(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [1,0,1,0,1,0]
 => 1 = 0 + 1
([],4)
 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
([(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
([(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
([],5)
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
([(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
([(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
([(1,4),(2,3)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1

Description

Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra.

Matching statistic: St000053
(load all 2 compositions to match this statistic)

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of valleys of the Dyck path.

Matching statistic: St000272
(load all 2 compositions to match this statistic)

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000272: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The treewidth of a graph. A graph has treewidth zero if and only if it has no edges. A connected graph has treewidth at most one if and only if it is a tree. A connected graph has treewidth at most two if and only if it is a series-parallel graph.

Matching statistic: St000288
(load all 3 compositions to match this statistic)

Mapping

Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1] => 1 => 0 => 0
([],2)
 => [2] => 10 => 01 => 1
([(0,1)],2)
 => [1,1] => 11 => 00 => 0
([],3)
 => [3] => 100 => 011 => 2
([(1,2)],3)
 => [2,1] => 101 => 010 => 1
([(0,2),(1,2)],3)
 => [2,1] => 101 => 010 => 1
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => 111 => 000 => 0
([],4)
 => [4] => 1000 => 0111 => 3
([(2,3)],4)
 => [3,1] => 1001 => 0110 => 2
([(1,3),(2,3)],4)
 => [3,1] => 1001 => 0110 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,1] => 1001 => 0110 => 2
([(0,3),(1,2)],4)
 => [2,2] => 1010 => 0101 => 2
([(0,3),(1,2),(2,3)],4)
 => [2,2] => 1010 => 0101 => 2
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 1011 => 0100 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 1011 => 0100 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => 1010 => 0101 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 1011 => 0100 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => 1111 => 0000 => 0
([],5)
 => [5] => 10000 => 01111 => 4
([(3,4)],5)
 => [4,1] => 10001 => 01110 => 3
([(2,4),(3,4)],5)
 => [4,1] => 10001 => 01110 => 3
([(1,4),(2,4),(3,4)],5)
 => [4,1] => 10001 => 01110 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => 10001 => 01110 => 3
([(1,4),(2,3)],5)
 => [3,2] => 10010 => 01101 => 3
([(1,4),(2,3),(3,4)],5)
 => [3,2] => 10010 => 01101 => 3
([(0,1),(2,4),(3,4)],5)
 => [3,2] => 10010 => 01101 => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 10011 => 01100 => 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => 10010 => 01101 => 3
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 10011 => 01100 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 10011 => 01100 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => 10010 => 01101 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => 10010 => 01101 => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 10011 => 01100 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 10011 => 01100 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 10011 => 01100 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => 10010 => 01101 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 10011 => 01100 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => 10010 => 01101 => 3
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 01010 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 01010 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 01010 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => 10101 => 01010 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 01010 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 01010 => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 01010 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 10111 => 01000 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 10111 => 01000 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 10111 => 01000 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => 10101 => 01010 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 01010 => 2

Description

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

The following 55 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001176The size of a partition minus its first part. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001644The dimension of a graph. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000172The Grundy number of a graph. St000507The number of ascents of a standard tableau. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. 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: 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. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001962The proper pathwidth of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St000306The bounce count of a Dyck path. 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. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000820The number of compositions obtained by rotating the composition. St000822The Hadwiger number of the graph. 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001480The number of simple summands of the module J^2/J^3. St001812The biclique partition number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.