- FindStat

Your data matches 463 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000478

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000478: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [2]
 => 1
([],7)
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(1,6),(2,5),(3,4)],7)
 => [2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 0
([(0,3),(1,2),(4,6),(5,6)],7)
 => [3,2,2]
 => [2,2]
 => [2]
 => 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => [3,2,2]
 => [2,2]
 => [2]
 => 1
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,7),(4,7),(5,7),(6,7)],8)
 => [5,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([],8)
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 0
([(4,7),(5,6)],8)
 => [2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
([(4,7),(5,6),(6,7)],8)
 => [4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(4,6),(4,7),(5,6),(5,7)],8)
 => [4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(2,7),(3,7),(4,6),(5,6)],8)
 => [3,3,1,1]
 => [3,1,1]
 => [1,1]
 => 0
([(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
 => [5,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
 => [3,3,1,1]
 => [3,1,1]
 => [1,1]
 => 0
([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
([(0,7),(1,6),(2,5),(3,4)],8)
 => [2,2,2,2]
 => [2,2,2]
 => [2,2]
 => -1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
 => [4,2,2]
 => [2,2]
 => [2]
 => 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,2,2]
 => [2,2]
 => [2]
 => 1
([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,2,2]
 => [2,2]
 => [2]
 => 1
([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => [6,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => [6,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
 => [6,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
 => [9,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
 => [11,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
 => [11,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
 => [7,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
 => [11,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => [13,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
 => [8,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
 => [12,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0

Description

Another weight of a partition according to Alladi. According to Theorem 3.4 (Alladi 2012) in [1]

$\sum_{\pi\in GG_1(r)} w_1(\pi)$ equals the number of partitions of

$r$ whose odd parts are all distinct.

$GG_1(r)$ is the set of partitions of

$r$ where consecutive entries differ by at least

$2$ , and consecutive even entries differ by at least

$4$ .

Matching statistic: St001119
(load all 18 compositions to match this statistic)

Mapping

Mp00247: Graphs —de-duplicate⟶ Graphs
St001119: Graphs ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 67%

Values

([],4)
 => ([],1)
 => 0
([],5)
 => ([],1)
 => 0
([(3,4)],5)
 => ([(1,2)],3)
 => 0
([],6)
 => ([],1)
 => 0
([(4,5)],6)
 => ([(1,2)],3)
 => 0
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 0
([(2,5),(3,4)],6)
 => ([(1,4),(2,3)],5)
 => 0
([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
([(0,5),(1,4),(2,3)],6)
 => ([(0,5),(1,4),(2,3)],6)
 => 1
([],7)
 => ([],1)
 => 0
([(5,6)],7)
 => ([(1,2)],3)
 => 0
([(4,6),(5,6)],7)
 => ([(1,2)],3)
 => 0
([(3,6),(4,6),(5,6)],7)
 => ([(1,2)],3)
 => 0
([(3,6),(4,5)],7)
 => ([(1,4),(2,3)],5)
 => 0
([(3,6),(4,5),(5,6)],7)
 => ([(1,4),(2,3),(3,4)],5)
 => 0
([(2,3),(4,6),(5,6)],7)
 => ([(1,4),(2,3)],5)
 => 0
([(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,2)],3)
 => 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
([(1,6),(2,5),(3,4)],7)
 => ([(1,6),(2,5),(3,4)],7)
 => 0
([(0,3),(1,2),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3)],6)
 => 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => 0
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => 1
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0
([(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 0
([],8)
 => ?
 => ? = 0
([(4,7),(5,6)],8)
 => ?
 => ? = 0
([(4,7),(5,6),(6,7)],8)
 => ?
 => ? = 0
([(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 0
([(2,7),(3,7),(4,6),(5,6)],8)
 => ?
 => ? = 0
([(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
 => ?
 => ? = 0
([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
 => ?
 => ? = 0
([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0
([(0,7),(1,6),(2,5),(3,4)],8)
 => ([(0,7),(1,6),(2,5),(3,4)],8)
 => ? = -1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
 => ([(0,5),(1,4),(2,3)],6)
 => 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => 1
([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? = 0
([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ? = 0
([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
 => ?
 => ? = 0
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
 => ?
 => ? = 0
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
 => ?
 => ? = 0
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
 => ?
 => ? = 0
([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
 => ?
 => ? = 0
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
 => ?
 => ? = 0
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ?
 => ? = 0
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
 => ?
 => ? = 0
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
 => ?
 => ? = 0
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
 => ?
 => ? = 0
([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
 => ?
 => ? = 0
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
 => ?
 => ? = 0
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
 => ?
 => ? = 0
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
 => ?
 => ? = 0
([(3,8),(4,6),(4,9),(5,7),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
 => ?
 => ? = 0
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
 => ?
 => ? = 0
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
 => ?
 => ? = 0

Description

The length of a shortest maximal path in a graph.

Matching statistic: St001271
(load all 35 compositions to match this statistic)

Mapping

Mp00247: Graphs —de-duplicate⟶ Graphs
St001271: Graphs ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 67%

Values

([],4)
 => ([],1)
 => 0
([],5)
 => ([],1)
 => 0
([(3,4)],5)
 => ([(1,2)],3)
 => 0
([],6)
 => ([],1)
 => 0
([(4,5)],6)
 => ([(1,2)],3)
 => 0
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 0
([(2,5),(3,4)],6)
 => ([(1,4),(2,3)],5)
 => 0
([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
([(0,5),(1,4),(2,3)],6)
 => ([(0,5),(1,4),(2,3)],6)
 => 1
([],7)
 => ([],1)
 => 0
([(5,6)],7)
 => ([(1,2)],3)
 => 0
([(4,6),(5,6)],7)
 => ([(1,2)],3)
 => 0
([(3,6),(4,6),(5,6)],7)
 => ([(1,2)],3)
 => 0
([(3,6),(4,5)],7)
 => ([(1,4),(2,3)],5)
 => 0
([(3,6),(4,5),(5,6)],7)
 => ([(1,4),(2,3),(3,4)],5)
 => 0
([(2,3),(4,6),(5,6)],7)
 => ([(1,4),(2,3)],5)
 => 0
([(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,2)],3)
 => 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
([(1,6),(2,5),(3,4)],7)
 => ([(1,6),(2,5),(3,4)],7)
 => 0
([(0,3),(1,2),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3)],6)
 => 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => 0
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => 1
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0
([(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 0
([],8)
 => ?
 => ? = 0
([(4,7),(5,6)],8)
 => ?
 => ? = 0
([(4,7),(5,6),(6,7)],8)
 => ?
 => ? = 0
([(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 0
([(2,7),(3,7),(4,6),(5,6)],8)
 => ?
 => ? = 0
([(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
 => ?
 => ? = 0
([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
 => ?
 => ? = 0
([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0
([(0,7),(1,6),(2,5),(3,4)],8)
 => ([(0,7),(1,6),(2,5),(3,4)],8)
 => ? = -1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
 => ([(0,5),(1,4),(2,3)],6)
 => 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => 1
([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? = 0
([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ? = 0
([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
 => ?
 => ? = 0
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
 => ?
 => ? = 0
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
 => ?
 => ? = 0
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
 => ?
 => ? = 0
([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
 => ?
 => ? = 0
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
 => ?
 => ? = 0
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ?
 => ? = 0
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
 => ?
 => ? = 0
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
 => ?
 => ? = 0
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
 => ?
 => ? = 0
([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
 => ?
 => ? = 0
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
 => ?
 => ? = 0
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
 => ?
 => ? = 0
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
 => ?
 => ? = 0
([(3,8),(4,6),(4,9),(5,7),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
 => ?
 => ? = 0
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
 => ?
 => ? = 0
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
 => ?
 => ? = 0

Description

The competition number of a graph. The competition graph of a digraph

$D$ is a (simple undirected) graph which has the same vertex set as

$D$ and has an edge between

$x$ and

$y$ if and only if there exists a vertex

$v$ in

$D$ such that

$(x, v)$ and

$(y, v)$ are arcs of

$D$ . For any graph,

$G$ together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number

$k(G)$ is the smallest number of such isolated vertices.

Matching statistic: St001316
(load all 25 compositions to match this statistic)

Mapping

Mp00247: Graphs —de-duplicate⟶ Graphs
St001316: Graphs ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 67%

Values

([],4)
 => ([],1)
 => 1 = 0 + 1
([],5)
 => ([],1)
 => 1 = 0 + 1
([(3,4)],5)
 => ([(1,2)],3)
 => 1 = 0 + 1
([],6)
 => ([],1)
 => 1 = 0 + 1
([(4,5)],6)
 => ([(1,2)],3)
 => 1 = 0 + 1
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 1 = 0 + 1
([(2,5),(3,4)],6)
 => ([(1,4),(2,3)],5)
 => 1 = 0 + 1
([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
([(0,5),(1,4),(2,3)],6)
 => ([(0,5),(1,4),(2,3)],6)
 => 2 = 1 + 1
([],7)
 => ([],1)
 => 1 = 0 + 1
([(5,6)],7)
 => ([(1,2)],3)
 => 1 = 0 + 1
([(4,6),(5,6)],7)
 => ([(1,2)],3)
 => 1 = 0 + 1
([(3,6),(4,6),(5,6)],7)
 => ([(1,2)],3)
 => 1 = 0 + 1
([(3,6),(4,5)],7)
 => ([(1,4),(2,3)],5)
 => 1 = 0 + 1
([(3,6),(4,5),(5,6)],7)
 => ([(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
([(2,3),(4,6),(5,6)],7)
 => ([(1,4),(2,3)],5)
 => 1 = 0 + 1
([(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,2)],3)
 => 1 = 0 + 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
([(1,6),(2,5),(3,4)],7)
 => ([(1,6),(2,5),(3,4)],7)
 => 1 = 0 + 1
([(0,3),(1,2),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3)],6)
 => 2 = 1 + 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0 + 1
([(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 0 + 1
([],8)
 => ?
 => ? = 0 + 1
([(4,7),(5,6)],8)
 => ?
 => ? = 0 + 1
([(4,7),(5,6),(6,7)],8)
 => ?
 => ? = 0 + 1
([(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? = 0 + 1
([(2,7),(3,7),(4,6),(5,6)],8)
 => ?
 => ? = 0 + 1
([(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
 => ?
 => ? = 0 + 1
([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
 => ?
 => ? = 0 + 1
([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 0 + 1
([(0,7),(1,6),(2,5),(3,4)],8)
 => ([(0,7),(1,6),(2,5),(3,4)],8)
 => ? = -1 + 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
 => ([(0,5),(1,4),(2,3)],6)
 => 2 = 1 + 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1 + 1
([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? = 0 + 1
([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ? = 0 + 1
([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
 => ?
 => ? = 0 + 1
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
 => ?
 => ? = 0 + 1
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
 => ?
 => ? = 0 + 1
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
 => ?
 => ? = 0 + 1
([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
 => ?
 => ? = 0 + 1
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
 => ?
 => ? = 0 + 1
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ?
 => ? = 0 + 1
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
 => ?
 => ? = 0 + 1
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
 => ?
 => ? = 0 + 1
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
 => ?
 => ? = 0 + 1
([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
 => ?
 => ? = 0 + 1
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
 => ?
 => ? = 0 + 1
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
 => ?
 => ? = 0 + 1
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
 => ?
 => ? = 0 + 1
([(3,8),(4,6),(4,9),(5,7),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
 => ?
 => ? = 0 + 1
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
 => ?
 => ? = 0 + 1
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
 => ?
 => ? = 0 + 1

Description

The domatic number of a graph. This is the maximal size of a partition of the vertices into dominating sets.

Matching statistic: St000273
(load all 21 compositions to match this statistic)

Mapping

Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000273: Graphs ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 67%

Values

([],4)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
([],5)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
([(3,4)],5)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([],6)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
([(4,5)],6)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(2,5),(3,4)],6)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
([(0,5),(1,4),(2,3)],6)
 => ([(0,5),(1,4),(2,3)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2 = 1 + 1
([],7)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
([(5,6)],7)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(4,6),(5,6)],7)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(3,6),(4,6),(5,6)],7)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(3,6),(4,5)],7)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(3,6),(4,5),(5,6)],7)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(2,3),(4,6),(5,6)],7)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
([(1,6),(2,5),(3,4)],7)
 => ([(1,6),(2,5),(3,4)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
([(0,3),(1,2),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2 = 1 + 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 2 = 1 + 1
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([],8)
 => ?
 => ?
 => ? = 0 + 1
([(4,7),(5,6)],8)
 => ?
 => ?
 => ? = 0 + 1
([(4,7),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([(2,7),(3,7),(4,6),(5,6)],8)
 => ?
 => ?
 => ? = 0 + 1
([(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([(0,7),(1,6),(2,5),(3,4)],8)
 => ([(0,7),(1,6),(2,5),(3,4)],8)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = -1 + 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
 => ([(0,5),(1,4),(2,3)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2 = 1 + 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 2 = 1 + 1
([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 1 + 1
([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ?
 => ? = 0 + 1
([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ?
 => ? = 0 + 1
([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
 => ?
 => ?
 => ? = 0 + 1
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
 => ?
 => ?
 => ? = 0 + 1
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
 => ?
 => ?
 => ? = 0 + 1
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
 => ?
 => ?
 => ? = 0 + 1
([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
 => ?
 => ?
 => ? = 0 + 1
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
 => ?
 => ?
 => ? = 0 + 1
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ?
 => ?
 => ? = 0 + 1
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
 => ?
 => ?
 => ? = 0 + 1
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
 => ?
 => ?
 => ? = 0 + 1
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
 => ?
 => ?
 => ? = 0 + 1
([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
 => ?
 => ?
 => ? = 0 + 1
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
 => ?
 => ?
 => ? = 0 + 1
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
 => ?
 => ?
 => ? = 0 + 1
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
 => ?
 => ?
 => ? = 0 + 1
([(3,8),(4,6),(4,9),(5,7),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
 => ?
 => ?
 => ? = 0 + 1
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
 => ?
 => ?
 => ? = 0 + 1
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
 => ?
 => ?
 => ? = 0 + 1

Description

The domination number of a graph. The domination number of a graph is given by the minimum size of a dominating set of vertices. A dominating set of vertices is a subset of the vertex set of such that every vertex is either in this subset or adjacent to an element of this subset.

Matching statistic: St000544
(load all 21 compositions to match this statistic)

Mapping

Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000544: Graphs ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 67%

Values

([],4)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
([],5)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
([(3,4)],5)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([],6)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
([(4,5)],6)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(2,5),(3,4)],6)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
([(0,5),(1,4),(2,3)],6)
 => ([(0,5),(1,4),(2,3)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2 = 1 + 1
([],7)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
([(5,6)],7)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(4,6),(5,6)],7)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(3,6),(4,6),(5,6)],7)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(3,6),(4,5)],7)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(3,6),(4,5),(5,6)],7)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(2,3),(4,6),(5,6)],7)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
([(1,6),(2,5),(3,4)],7)
 => ([(1,6),(2,5),(3,4)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
([(0,3),(1,2),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2 = 1 + 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 2 = 1 + 1
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([],8)
 => ?
 => ?
 => ? = 0 + 1
([(4,7),(5,6)],8)
 => ?
 => ?
 => ? = 0 + 1
([(4,7),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([(2,7),(3,7),(4,6),(5,6)],8)
 => ?
 => ?
 => ? = 0 + 1
([(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([(0,7),(1,6),(2,5),(3,4)],8)
 => ([(0,7),(1,6),(2,5),(3,4)],8)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = -1 + 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
 => ([(0,5),(1,4),(2,3)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2 = 1 + 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 2 = 1 + 1
([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 1 + 1
([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ?
 => ? = 0 + 1
([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ?
 => ? = 0 + 1
([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
 => ?
 => ?
 => ? = 0 + 1
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
 => ?
 => ?
 => ? = 0 + 1
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
 => ?
 => ?
 => ? = 0 + 1
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
 => ?
 => ?
 => ? = 0 + 1
([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
 => ?
 => ?
 => ? = 0 + 1
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
 => ?
 => ?
 => ? = 0 + 1
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ?
 => ?
 => ? = 0 + 1
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
 => ?
 => ?
 => ? = 0 + 1
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
 => ?
 => ?
 => ? = 0 + 1
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
 => ?
 => ?
 => ? = 0 + 1
([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
 => ?
 => ?
 => ? = 0 + 1
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
 => ?
 => ?
 => ? = 0 + 1
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
 => ?
 => ?
 => ? = 0 + 1
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
 => ?
 => ?
 => ? = 0 + 1
([(3,8),(4,6),(4,9),(5,7),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
 => ?
 => ?
 => ? = 0 + 1
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
 => ?
 => ?
 => ? = 0 + 1
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
 => ?
 => ?
 => ? = 0 + 1

Description

The cop number of a graph. This is the minimal number of cops needed to catch the robber. The algorithm is from [2].

Matching statistic: St000765
(load all 4 compositions to match this statistic)

Mapping

Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St000765: Integer compositions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 67%

Values

([],4)
 => ([],1)
 => [1] => 1 = 0 + 1
([],5)
 => ([],1)
 => [1] => 1 = 0 + 1
([(3,4)],5)
 => ([(1,2)],3)
 => [2,1] => 1 = 0 + 1
([],6)
 => ([],1)
 => [1] => 1 = 0 + 1
([(4,5)],6)
 => ([(1,2)],3)
 => [2,1] => 1 = 0 + 1
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [2,1] => 1 = 0 + 1
([(2,5),(3,4)],6)
 => ([(1,4),(2,3)],5)
 => [3,2] => 1 = 0 + 1
([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 1 = 0 + 1
([(0,5),(1,4),(2,3)],6)
 => ([(0,5),(1,4),(2,3)],6)
 => [3,3] => 2 = 1 + 1
([],7)
 => ([],1)
 => [1] => 1 = 0 + 1
([(5,6)],7)
 => ([(1,2)],3)
 => [2,1] => 1 = 0 + 1
([(4,6),(5,6)],7)
 => ([(1,2)],3)
 => [2,1] => 1 = 0 + 1
([(3,6),(4,6),(5,6)],7)
 => ([(1,2)],3)
 => [2,1] => 1 = 0 + 1
([(3,6),(4,5)],7)
 => ([(1,4),(2,3)],5)
 => [3,2] => 1 = 0 + 1
([(3,6),(4,5),(5,6)],7)
 => ([(1,4),(2,3),(3,4)],5)
 => [3,2] => 1 = 0 + 1
([(2,3),(4,6),(5,6)],7)
 => ([(1,4),(2,3)],5)
 => [3,2] => 1 = 0 + 1
([(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 1 = 0 + 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 1 = 0 + 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,2)],3)
 => [2,1] => 1 = 0 + 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 1 = 0 + 1
([(1,6),(2,5),(3,4)],7)
 => ([(1,6),(2,5),(3,4)],7)
 => [4,3] => 1 = 0 + 1
([(0,3),(1,2),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3)],6)
 => [3,3] => 2 = 1 + 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => 1 = 0 + 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 1 = 0 + 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => [3,3,1] => 2 = 1 + 1
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? => ? = 0 + 1
([(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ? => ? = 0 + 1
([],8)
 => ?
 => ? => ? = 0 + 1
([(4,7),(5,6)],8)
 => ?
 => ? => ? = 0 + 1
([(4,7),(5,6),(6,7)],8)
 => ?
 => ? => ? = 0 + 1
([(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ? => ? = 0 + 1
([(2,7),(3,7),(4,6),(5,6)],8)
 => ?
 => ? => ? = 0 + 1
([(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
 => ?
 => ? => ? = 0 + 1
([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
 => ?
 => ? => ? = 0 + 1
([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? => ? = 0 + 1
([(0,7),(1,6),(2,5),(3,4)],8)
 => ([(0,7),(1,6),(2,5),(3,4)],8)
 => ? => ? = -1 + 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
 => ([(0,5),(1,4),(2,3)],6)
 => [3,3] => 2 = 1 + 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => [3,3,1] => 2 = 1 + 1
([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? => ? = 1 + 1
([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ? => ? = 0 + 1
([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ? => ? = 0 + 1
([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
 => ?
 => ? => ? = 0 + 1
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
 => ?
 => ? => ? = 0 + 1
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
 => ?
 => ? => ? = 0 + 1
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
 => ?
 => ? => ? = 0 + 1
([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
 => ?
 => ? => ? = 0 + 1
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
 => ?
 => ? => ? = 0 + 1
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ?
 => ? => ? = 0 + 1
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
 => ?
 => ? => ? = 0 + 1
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
 => ?
 => ? => ? = 0 + 1
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
 => ?
 => ? => ? = 0 + 1
([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
 => ?
 => ? => ? = 0 + 1
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
 => ?
 => ? => ? = 0 + 1
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
 => ?
 => ? => ? = 0 + 1
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
 => ?
 => ? => ? = 0 + 1
([(3,8),(4,6),(4,9),(5,7),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
 => ?
 => ? => ? = 0 + 1
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
 => ?
 => ? => ? = 0 + 1
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
 => ?
 => ? => ? = 0 + 1

Description

The number of weak records in an integer composition. A weak record is an element

$a_i$ such that

$a_i \geq a_j$ for all

$j < i$ .

Matching statistic: St001829
(load all 21 compositions to match this statistic)

Mapping

Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St001829: Graphs ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 67%

Values

([],4)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
([],5)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
([(3,4)],5)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([],6)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
([(4,5)],6)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(2,5),(3,4)],6)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
([(0,5),(1,4),(2,3)],6)
 => ([(0,5),(1,4),(2,3)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2 = 1 + 1
([],7)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
([(5,6)],7)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(4,6),(5,6)],7)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(3,6),(4,6),(5,6)],7)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(3,6),(4,5)],7)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(3,6),(4,5),(5,6)],7)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(2,3),(4,6),(5,6)],7)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
([(1,6),(2,5),(3,4)],7)
 => ([(1,6),(2,5),(3,4)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
([(0,3),(1,2),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2 = 1 + 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 2 = 1 + 1
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([],8)
 => ?
 => ?
 => ? = 0 + 1
([(4,7),(5,6)],8)
 => ?
 => ?
 => ? = 0 + 1
([(4,7),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([(2,7),(3,7),(4,6),(5,6)],8)
 => ?
 => ?
 => ? = 0 + 1
([(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([(0,7),(1,6),(2,5),(3,4)],8)
 => ([(0,7),(1,6),(2,5),(3,4)],8)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = -1 + 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
 => ([(0,5),(1,4),(2,3)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2 = 1 + 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 2 = 1 + 1
([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 1 + 1
([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ?
 => ? = 0 + 1
([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ?
 => ? = 0 + 1
([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
 => ?
 => ?
 => ? = 0 + 1
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
 => ?
 => ?
 => ? = 0 + 1
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
 => ?
 => ?
 => ? = 0 + 1
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
 => ?
 => ?
 => ? = 0 + 1
([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
 => ?
 => ?
 => ? = 0 + 1
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
 => ?
 => ?
 => ? = 0 + 1
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ?
 => ?
 => ? = 0 + 1
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
 => ?
 => ?
 => ? = 0 + 1
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
 => ?
 => ?
 => ? = 0 + 1
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
 => ?
 => ?
 => ? = 0 + 1
([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
 => ?
 => ?
 => ? = 0 + 1
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
 => ?
 => ?
 => ? = 0 + 1
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
 => ?
 => ?
 => ? = 0 + 1
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
 => ?
 => ?
 => ? = 0 + 1
([(3,8),(4,6),(4,9),(5,7),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
 => ?
 => ?
 => ? = 0 + 1
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
 => ?
 => ?
 => ? = 0 + 1
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
 => ?
 => ?
 => ? = 0 + 1

Description

The common independence number of a graph. The common independence number of a graph

$G$ is the greatest integer

$r$ such that every vertex of

$G$ belongs to some independent set

$X$ of vertices of cardinality at least

$r$ .

Matching statistic: St001217
(load all 8 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001217: Dyck paths ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 67%

Values

([],4)
 => [1,1,1,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
([],5)
 => [1,1,1,1,1]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
([],6)
 => [1,1,1,1,1,1]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0
([(4,5)],6)
 => [2,1,1,1,1]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 0
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 0
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 1
([],7)
 => [1,1,1,1,1,1,1]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 0
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => 0
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 0
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 0
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 0
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 0
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 0
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 0
([(1,6),(2,5),(3,4)],7)
 => [2,2,2,1]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 0
([(0,3),(1,2),(4,6),(5,6)],7)
 => [3,2,2]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 0
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 0
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => [3,2,2]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,1,1,1,1]
 => [5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => ? = 0
([(3,7),(4,7),(5,7),(6,7)],8)
 => [5,1,1,1]
 => [3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => ? = 0
([],8)
 => [1,1,1,1,1,1,1,1]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 0
([(4,7),(5,6)],8)
 => [2,2,1,1,1,1]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => 0
([(4,7),(5,6),(6,7)],8)
 => [4,1,1,1,1]
 => [5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => ? = 0
([(4,6),(4,7),(5,6),(5,7)],8)
 => [4,1,1,1,1]
 => [5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => ? = 0
([(2,7),(3,7),(4,6),(5,6)],8)
 => [3,3,1,1]
 => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 0
([(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
 => [5,1,1,1]
 => [3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => ? = 0
([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
 => [3,3,1,1]
 => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 0
([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,2,1,1]
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0
([(0,7),(1,6),(2,5),(3,4)],8)
 => [2,2,2,2]
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = -1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
 => [4,2,2]
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,2,2]
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,2,2]
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => [6,1,1,1]
 => [3,2,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => [6,1,1,1,1]
 => [4,2,1,1,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
 => [6,1,1,1]
 => [3,2,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
 => [9,1,1,1]
 => [3,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
 => [11,1,1,1]
 => [3,2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
 => [11,1,1,1,1]
 => [4,2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
 => [7,1,1,1]
 => [3,2,1,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
 => [11,1,1,1]
 => [3,2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => [13,1,1,1]
 => [3,2,1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
 => [8,1,1,1]
 => [3,2,1,1,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
 => [12,1,1,1]
 => [3,2,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
 => [10,1,1,1]
 => [3,2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
 => [6,1,1,1,1,1]
 => [4,3,1,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
 => [8,1,1,1]
 => [3,2,1,1,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
 => [10,1,1,1]
 => [3,2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
 => [10,1,1,1,1]
 => [4,2,1,1,1,1,1,1,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
([(3,8),(4,6),(4,9),(5,7),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
 => [7,1,1,1]
 => [3,2,1,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
 => [11,1,1,1]
 => [3,2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
 => [10,1,1,1]
 => [3,2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0

Description

The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1.

Matching statistic: St001322
(load all 21 compositions to match this statistic)

Mapping

Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St001322: Graphs ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 67%

Values

([],4)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
([],5)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
([(3,4)],5)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([],6)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
([(4,5)],6)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(2,5),(3,4)],6)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
([(0,5),(1,4),(2,3)],6)
 => ([(0,5),(1,4),(2,3)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2 = 1 + 1
([],7)
 => ([],1)
 => ([],1)
 => 1 = 0 + 1
([(5,6)],7)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(4,6),(5,6)],7)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(3,6),(4,6),(5,6)],7)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(3,6),(4,5)],7)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(3,6),(4,5),(5,6)],7)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(2,3),(4,6),(5,6)],7)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
([(1,6),(2,5),(3,4)],7)
 => ([(1,6),(2,5),(3,4)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 0 + 1
([(0,3),(1,2),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2 = 1 + 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 2 = 1 + 1
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([],8)
 => ?
 => ?
 => ? = 0 + 1
([(4,7),(5,6)],8)
 => ?
 => ?
 => ? = 0 + 1
([(4,7),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([(2,7),(3,7),(4,6),(5,6)],8)
 => ?
 => ?
 => ? = 0 + 1
([(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ? = 0 + 1
([(0,7),(1,6),(2,5),(3,4)],8)
 => ([(0,7),(1,6),(2,5),(3,4)],8)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = -1 + 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
 => ([(0,5),(1,4),(2,3)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2 = 1 + 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 2 = 1 + 1
([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 1 + 1
([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ?
 => ?
 => ? = 0 + 1
([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => ?
 => ?
 => ? = 0 + 1
([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
 => ?
 => ?
 => ? = 0 + 1
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
 => ?
 => ?
 => ? = 0 + 1
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
 => ?
 => ?
 => ? = 0 + 1
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
 => ?
 => ?
 => ? = 0 + 1
([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
 => ?
 => ?
 => ? = 0 + 1
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
 => ?
 => ?
 => ? = 0 + 1
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => ?
 => ?
 => ? = 0 + 1
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
 => ?
 => ?
 => ? = 0 + 1
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
 => ?
 => ?
 => ? = 0 + 1
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
 => ?
 => ?
 => ? = 0 + 1
([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
 => ?
 => ?
 => ? = 0 + 1
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
 => ?
 => ?
 => ? = 0 + 1
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
 => ?
 => ?
 => ? = 0 + 1
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
 => ?
 => ?
 => ? = 0 + 1
([(3,8),(4,6),(4,9),(5,7),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
 => ?
 => ?
 => ? = 0 + 1
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
 => ?
 => ?
 => ? = 0 + 1
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
 => ?
 => ?
 => ? = 0 + 1

Description

The size of a minimal independent dominating set in a graph.

The following 453 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001339The irredundance number of a graph. St000260The radius of a connected graph. St001672The restrained domination number of a graph. St001139The number of occurrences of hills of size 2 in a Dyck path. St000667The greatest common divisor of the parts of the partition. St000899The maximal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000993The multiplicity of the largest part of an integer partition. St000439The position of the first down step of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000011The number of touch points (or returns) of a Dyck path. St000297The number of leading ones in a binary word. St000877The depth of the binary word interpreted as a path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001075The minimal size of a block of a set partition. St001256Number of simple reflexive modules that are 2-stable reflexive. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001525The number of symmetric hooks on the diagonal of a partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000274The number of perfect matchings of a graph. St000310The minimal degree of a vertex of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000761The number of ascents in an integer composition. St000768The number of peaks in an integer composition. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000097The order of the largest clique of the graph. St000617The number of global maxima of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000758The length of the longest staircase fitting into an integer composition. St000764The number of strong records in an integer composition. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001732The number of peaks visible from the left. St000338The number of pixed points of a permutation. St000674The number of hills of a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001386The number of prime labellings of a graph. St001644The dimension of a graph. St000098The chromatic number of a graph. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000654The first descent of a permutation. St000873The aix statistic of a permutation. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001111The weak 2-dynamic chromatic number of a graph. St000312The number of leaves in a graph. St000313The number of degree 2 vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001481The minimal height of a peak of a Dyck path. St000403The Szeged index minus the Wiener index of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000448The number of pairs of vertices of a graph with distance 2. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St000929The constant term of the character polynomial of an integer partition. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001308The number of induced paths on three vertices in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001374The Padmakar-Ivan index of a graph. St001521Half the total irregularity of a graph. St001522The total irregularity of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001689The number of celebrities in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001708The number of pairs of vertices of different degree in a graph. St001742The difference of the maximal and the minimal degree in a graph. St001764The number of non-convex subsets of vertices in a graph. St001793The difference between the clique number and the chromatic number of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001799The number of proper separations of a graph. St000287The number of connected components of a graph. St000349The number of different adjacency matrices of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000553The number of blocks of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000916The packing number of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001272The number of graphs with the same degree sequence. St001282The number of graphs with the same chromatic polynomial. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001463The number of distinct columns in the nullspace of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001642The Prague dimension of a graph. St001734The lettericity of a graph. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001765The number of connected components of the friends and strangers graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001917The order of toric promotion on the set of labellings of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000091The descent variation of a composition. St000095The number of triangles of a graph. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000272The treewidth of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by

$4$ . St000311The number of vertices of odd degree in a graph. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000350The sum of the vertex degrees of a graph. St000351The determinant of the adjacency matrix of a graph. St000361The second Zagreb index of a graph. St000362The size of a minimal vertex cover of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000387The matching number of a graph. St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St000465The first Zagreb index of a graph. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000571The F-index (or forgotten topological index) of a graph. St000637The length of the longest cycle in a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000915The Ore degree of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001071The beta invariant of the graph. St001117The game chromatic index of a graph. St001120The length of a longest path in a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001309The number of four-cliques in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001333The cardinality of a minimal edge-isolating set of a graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001341The number of edges in the center of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001362The normalized Knill dimension of a graph. St001393The induced matching number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001479The number of bridges of a graph. St001512The minimum rank of a graph. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001638The book thickness of a graph. St001649The length of a longest trail in a graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001736The total number of cycles in a graph. St001743The discrepancy of a graph. St001783The number of odd automorphisms of a graph. St001792The arboricity of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001797The number of overfull subgraphs of a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001812The biclique partition number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001869The maximum cut size of a graph. St001871The number of triconnected components of a graph. St001962The proper pathwidth of a graph. St000086The number of subgraphs. St000172The Grundy number of a graph. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000268The number of strongly connected orientations of a graph. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000286The number of connected components of the complement of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000343The number of spanning subgraphs of a graph. St000344The number of strongly connected outdegree sequences of a graph. St000363The number of minimal vertex covers of a graph. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000468The Hosoya index of a graph. St000722The number of different neighbourhoods in a graph. St000822The Hadwiger number of the graph. St000972The composition number of a graph. St001029The size of the core of a graph. St001072The evaluation of the Tutte polynomial of the graph at x and y equal to 3. St001073The number of nowhere zero 3-flows of a graph. St001093The detour number of a graph. St001108The 2-dynamic chromatic number of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001110The 3-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001116The game chromatic number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001303The number of dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001330The hat guessing number of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001494The Alon-Tarsi number of a graph. St001546The number of monomials in the Tutte polynomial of a graph. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001580The acyclic chromatic number of a graph. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001694The number of maximal dissociation sets in a graph. St001716The 1-improper chromatic number of a graph. St001725The harmonious chromatic number of a graph. St001883The mutual visibility number of a graph. St001957The number of Hasse diagrams with a given underlying undirected graph. St001963The tree-depth of a graph. St000181The number of connected components of the Hasse diagram for the poset. St000096The number of spanning trees of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St001703The villainy of a graph. St000315The number of isolated vertices of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000386The number of factors DDU in a Dyck path. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000629The defect of a binary word. St001371The length of the longest Yamanouchi prefix of a binary word. St000627The exponent of a binary word. St000759The smallest missing part in an integer partition. St001884The number of borders of a binary word. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000699The toughness times the least common multiple of 1,. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000456The monochromatic index of a connected graph. St000475The number of parts equal to 1 in a partition. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000658The number of rises of length 2 of a Dyck path. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001172The number of 1-rises at odd height of a Dyck path. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001204Call 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. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001593This is the number of standard Young tableaux of the given shifted shape. St000335The difference of lower and upper interactions. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n−1}]$ by adding

$c_0$ to

$c_{n−1}$ . St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St000455The second largest eigenvalue of a graph if it is integral. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000379The number of Hamiltonian cycles in a graph. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001890The maximum magnitude of the Möbius function of a poset. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001561The value of the elementary symmetric function evaluated at 1. St001586The number of odd parts smaller than the largest even part in an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000897The number of different multiplicities of parts of an integer partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St000675The number of centered multitunnels of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000781The number of proper colouring schemes of a Ferrers diagram. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001196The global dimension of

$A$ minus the global dimension of

$eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001501The dominant dimension of magnitude 1 Nakayama algebras. St001568The smallest positive integer that does not appear twice in the partition. St001733The number of weak left to right maxima of a Dyck path. St001471The magnitude of a Dyck path. St001570The minimal number of edges to add to make a graph Hamiltonian. St001621The number of atoms of a lattice. St000264The girth of a graph, which is not a tree. St000479The Ramsey number of a graph. St001060The distinguishing index of a graph. St001061The number of indices that are both descents and recoils of a permutation. St001696The natural major index of a standard Young tableau. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001193The dimension of

$Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra

$A$ such that

$eA$ is a minimal faithful projective-injective module. St001691The number of kings in a graph. St000914The sum of the values of the Möbius function of a poset. St000183The side length of the Durfee square of an integer partition. St000258The burning number of a graph. St000766The number of inversions of an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001274The number of indecomposable injective modules with projective dimension equal to two. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001549The number of restricted non-inversions between exceedances. St000314The number of left-to-right-maxima of a permutation. St000805The number of peaks of the associated bargraph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ corresponding to the Dyck path. St001191Number of simple modules

$S$ with

$Ext_A^i(S,A)=0$ for all

$i=0,1,...,g-1$ in the corresponding Nakayama algebra

$A$ with global dimension

$g$ . St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . 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. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001340The cardinality of a minimal non-edge isolating set of a graph. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001631The number of simple modules

$S$ with

$dim Ext^1(S,A)=1$ in the incidence algebra

$A$ of the poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000549The number of odd partial sums of an integer partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000879The number of long braid edges in the graph of braid moves of a permutation. St000951The dimension of

$Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St000259The diameter of a connected graph. St000917The open packing number of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000918The 2-limited packing number of a graph. 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. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001877Number of indecomposable injective modules with projective dimension 2. 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. St001875The number of simple modules with projective dimension at most 1. St001964The interval resolution global dimension of a poset. St000022The number of fixed points of a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000214The number of adjacencies of a permutation. St000223The number of nestings in the permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000534The number of 2-rises of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000352The Elizalde-Pak rank of a permutation. St000665The number of rafts of a permutation. St000834The number of right outer peaks of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000007The number of saliances of the permutation. St000842The breadth of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000891The number of distinct diagonal sums of a permutation matrix. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000636The hull number of a graph. St001654The monophonic hull number of a graph.