Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000907
(load all 2 compositions to match this statistic)
Mapping
St000907: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => 1
([],2)
 => 1
([(0,1)],2)
 => 2
([],3)
 => 1
([(1,2)],3)
 => 2
([(0,1),(0,2)],3)
 => 1
([(0,2),(2,1)],3)
 => 3
([(0,2),(1,2)],3)
 => 1
([],4)
 => 1
([(2,3)],4)
 => 2
([(1,2),(1,3)],4)
 => 1
([(0,1),(0,2),(0,3)],4)
 => 1
([(0,2),(0,3),(3,1)],4)
 => 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(1,2),(2,3)],4)
 => 3
([(0,3),(3,1),(3,2)],4)
 => 2
([(1,3),(2,3)],4)
 => 1
([(0,3),(1,3),(3,2)],4)
 => 2
([(0,3),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2)],4)
 => 4
([(0,3),(1,2),(1,3)],4)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,3),(2,1),(3,2)],4)
 => 4
([(0,3),(1,2),(2,3)],4)
 => 1
([],5)
 => 1
([(3,4)],5)
 => 2
([(2,3),(2,4)],5)
 => 1
([(1,2),(1,3),(1,4)],5)
 => 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
([(1,3),(1,4),(4,2)],5)
 => 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3
([(0,3),(0,4),(3,2),(4,1)],5)
 => 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 1
([(2,3),(3,4)],5)
 => 3
([(1,4),(4,2),(4,3)],5)
 => 2
([(0,4),(4,1),(4,2),(4,3)],5)
 => 2
([(2,4),(3,4)],5)
 => 1
([(1,4),(2,4),(4,3)],5)
 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => 1
([(1,4),(2,4),(3,4)],5)
 => 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
([(0,4),(1,4),(2,3)],5)
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => 3
Description
The number of maximal antichains of minimal length in a poset.
Matching statistic: St000160
Mapping
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00251: Graphs —clique sizes⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 54% ●values known / values provided: 64%●distinct values known / distinct values provided: 54%
Values
([],1)
 => ([],1)
 => [1]
 => 1
([],2)
 => ([(0,1)],2)
 => [2]
 => 1
([(0,1)],2)
 => ([],2)
 => [1,1]
 => 2
([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [2,2]
 => 2
([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => 1
([(0,2),(2,1)],3)
 => ([],3)
 => [1,1,1]
 => 3
([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => 1
([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => 2
([(1,2),(1,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => 1
([(0,1),(0,2),(0,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 1
([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => [2,1,1]
 => 2
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => [2,1,1]
 => 2
([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => 1
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => [2,1,1]
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 1
([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => 4
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => 2
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => [1,1,1,1]
 => 4
([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 1
([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,4]
 => 2
([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,3]
 => 1
([(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,2]
 => 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,1]
 => 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 2
([(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => 3
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,1]
 => 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => 1
([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,3]
 => 3
([(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => 2
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 2
([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,3]
 => 1
([(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,2]
 => 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
([(0,4),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,3,2,2]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2,2]
 => 3
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [3,3,3,3,1]
 => ? = 1
([(1,4),(1,5),(4,3),(5,2)],6)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [3,3,3,3,2]
 => ? = 1
([(1,5),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [4,4,3,3]
 => ? = 2
([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,3,3,3]
 => ? = 3
([(0,5),(1,4),(2,4),(2,5),(5,3)],6)
 => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => [3,2,2,2,2,2]
 => ? = 5
([(0,5),(1,5),(2,3),(5,4)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,2,2,2,2]
 => ? = 4
([(0,5),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,2,2]
 => ? = 2
([(0,5),(1,5),(2,3),(3,4)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,2,2,2]
 => ? = 3
([(2,5),(3,4)],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),(4,5)],6)
 => [4,4,4,4]
 => ? = 4
([(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,3,3]
 => ? = 5
([(0,5),(1,4),(1,5),(4,2),(5,3)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [2,2,2,2,2,2,2]
 => ? = 7
([(1,5),(2,3),(2,4)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [4,4,3,3]
 => ? = 2
([(0,5),(1,3),(1,4),(1,5),(4,2)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,3,2]
 => ? = 1
([(0,4),(1,2),(1,3),(1,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,2,2]
 => ? = 2
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2,2,2,2,1]
 => ? = 1
([(0,4),(1,3),(1,5),(5,2)],6)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,3,3,2,2]
 => ? = 2
([(0,4),(1,2),(1,3),(2,5),(3,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,2,2,2,2]
 => ? = 4
([(1,4),(1,5),(2,3),(2,5)],6)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,3,3,3]
 => ? = 3
([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,2,2,2,2]
 => ? = 4
([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,2,2]
 => ? = 2
([(0,3),(0,5),(1,2),(1,4),(2,5),(3,4)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => [2,2,2,2,2,2,2]
 => ? = 7
([(1,4),(2,3),(2,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,3,3]
 => ? = 5
([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2,2,2,2,2,2]
 => ? = 8
([(0,5),(1,3),(1,4),(5,2)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,2,2,2]
 => ? = 3
([(0,4),(1,3),(1,5),(4,2),(4,5)],6)
 => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => [3,2,2,2,2,2]
 => ? = 5
([(1,3),(2,4),(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)
 => [3,3,3,3,3,3]
 => ? = 6
([(0,4),(1,5),(5,2),(5,3)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,2,2,2,2]
 => ? = 4
([(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [3,3,3,3,2]
 => ? = 1
([(0,5),(1,4),(4,2),(5,3)],6)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => [2,2,2,2,2,2,2,2,2]
 => ? = 9
([(0,3),(1,4),(3,5),(4,2),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [2,2,2,2,2,2,2]
 => ? = 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)
 => [3,3,3,3,3,3,3,3]
 => ? = 8
([(0,5),(1,3),(2,4),(2,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,3,3,3,3]
 => ? = 6
([(0,5),(1,4),(2,3),(2,4),(2,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,3,3,3,3]
 => ? = 5
([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2,2,2,2,2]
 => ? = 7
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,2,2]
 => ? = 2
([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,3,2]
 => ? = 1
([(0,5),(1,4),(1,5),(2,3),(2,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,3,3,3,3]
 => ? = 5
([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,2,2,2,2]
 => ? = 4
([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2,2,2,2,2]
 => ? = 7
([(0,5),(1,4),(2,3),(2,5),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,3,2]
 => ? = 1
([(0,5),(1,3),(4,2),(5,4)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2,2,2,2,2,2]
 => ? = 8
([(0,5),(1,3),(2,4),(4,5)],6)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,3,3,2,2]
 => ? = 2
([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2,2,2,2,1]
 => ? = 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [3,3,3,3,1]
 => ? = 1
([(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,1)],7)
 => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [4,4,3,3,1]
 => ? = 1
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(4,6)],7)
 => ([(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,3,3,3,1]
 => ? = 1
([(0,3),(0,4),(0,5),(0,6),(5,2),(6,1)],7)
 => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,4,4,4,1]
 => ? = 1
([(0,2),(0,3),(0,4),(0,5),(4,6),(5,1),(5,6)],7)
 => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,4,4,1]
 => ? = 1
([(2,4),(2,5),(2,6),(6,3)],7)
 => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,5,3]
 => ? = 1
([(1,3),(1,4),(1,5),(3,6),(4,6),(5,2)],7)
 => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [4,4,3,3,2]
 => ? = 1
Description
The multiplicity of the smallest part of a partition. This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$. The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences \begin{align*} spt(5n+4) &\equiv 0\quad \pmod{5}\\\ spt(7n+5) &\equiv 0\quad \pmod{7}\\\ spt(13n+6) &\equiv 0\quad \pmod{13}, \end{align*} analogous to those of the counting function of partitions, see [1] and [2].