- FindStat

Your data matches 11 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000189
(load all 2 compositions to match this statistic)

Mapping

Mp00282: Posets —Dedekind-MacNeille completion⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000189: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of elements in the poset.

Matching statistic: St001717
(load all 2 compositions to match this statistic)

Mapping

Mp00282: Posets —Dedekind-MacNeille completion⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001717: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest size of an interval in a poset.

Matching statistic: St001300
(load all 2 compositions to match this statistic)

Mapping

Mp00282: Posets —Dedekind-MacNeille completion⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001300: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset.

Matching statistic: St000656
(load all 4 compositions to match this statistic)

Mapping

St000656: Posets ⟶ ℤResult quality: 83% ●values known / values provided: 98%●distinct values known / distinct values provided: 83%

Values

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

Description

The number of cuts of a poset. A cut is a subset $A$ of the poset such that the set of lower bounds of the set of upper bounds of $A$ is exactly $A$.

Matching statistic: St001118

Mapping

Mp00198: Posets —incomparability graph⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St001118: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 38%●distinct values known / distinct values provided: 17%

Values

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

Description

The acyclic chromatic index of a graph. An acyclic edge coloring of a graph is a proper colouring of the edges of a graph such that the union of the edges colored with any two given colours is a forest. The smallest number of colours such that such a colouring exists is the acyclic chromatic index.

Matching statistic: St001198

Mapping

Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 19%●distinct values known / distinct values provided: 17%

Values

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

Description

The 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 $eA$.

Matching statistic: St001200

Mapping

Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 19%●distinct values known / distinct values provided: 17%

Values

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

Description

The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.

Matching statistic: St001206

Mapping

Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 19%●distinct values known / distinct values provided: 17%

Values

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

Description

The 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 $eA$.

Matching statistic: St001199

Mapping

Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 19%●distinct values known / distinct values provided: 17%

Values

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

Description

The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.

Matching statistic: St000264

Mapping

Mp00198: Posets —incomparability graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 17%

Values

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

Description

The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.

The following 1 statistic also match your data. Click on any of them to see the details.

St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.