Your data matches 35 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000307
(load all 4 compositions to match this statistic)
Mapping
St000307: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => 1
([],2)
 => 2
([(0,1)],2)
 => 1
([],3)
 => 4
([(1,2)],3)
 => 1
([(0,1),(0,2)],3)
 => 2
([(0,2),(2,1)],3)
 => 1
([(0,2),(1,2)],3)
 => 2
([],4)
 => 8
([(2,3)],4)
 => 2
([(1,2),(1,3)],4)
 => 3
([(0,1),(0,2),(0,3)],4)
 => 4
([(0,2),(0,3),(3,1)],4)
 => 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(1,2),(2,3)],4)
 => 2
([(0,3),(3,1),(3,2)],4)
 => 2
([(1,3),(2,3)],4)
 => 3
([(0,3),(1,3),(3,2)],4)
 => 2
([(0,3),(1,3),(2,3)],4)
 => 4
([(0,3),(1,2)],4)
 => 3
([(0,3),(1,2),(1,3)],4)
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => 3
([(0,3),(2,1),(3,2)],4)
 => 1
([(0,3),(1,2),(2,3)],4)
 => 1
([],5)
 => 16
([(3,4)],5)
 => 4
([(2,3),(2,4)],5)
 => 6
([(1,2),(1,3),(1,4)],5)
 => 7
([(0,1),(0,2),(0,3),(0,4)],5)
 => 8
([(0,2),(0,3),(0,4),(4,1)],5)
 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 4
([(1,3),(1,4),(4,2)],5)
 => 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
([(2,3),(3,4)],5)
 => 4
([(1,4),(4,2),(4,3)],5)
 => 4
([(0,4),(4,1),(4,2),(4,3)],5)
 => 4
([(2,4),(3,4)],5)
 => 6
([(1,4),(2,4),(4,3)],5)
 => 4
([(0,4),(1,4),(4,2),(4,3)],5)
 => 3
([(1,4),(2,4),(3,4)],5)
 => 7
([(0,4),(1,4),(2,4),(4,3)],5)
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => 8
([(0,4),(1,4),(2,3)],5)
 => 4
([(0,4),(1,3),(2,3),(2,4)],5)
 => 3
Description
The number of rowmotion orbits of a poset. Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
Matching statistic: St000010
(load all 2 compositions to match this statistic)
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 24% ●values known / values provided: 52%●distinct values known / distinct values provided: 24%
Values
([],1)
 => [2]
 => []
 => 0 = 1 - 1
([],2)
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(0,1)],2)
 => [3]
 => []
 => 0 = 1 - 1
([],3)
 => [2,2,2,2]
 => [2,2,2]
 => 3 = 4 - 1
([(1,2)],3)
 => [6]
 => []
 => 0 = 1 - 1
([(0,1),(0,2)],3)
 => [3,2]
 => [2]
 => 1 = 2 - 1
([(0,2),(2,1)],3)
 => [4]
 => []
 => 0 = 1 - 1
([(0,2),(1,2)],3)
 => [3,2]
 => [2]
 => 1 = 2 - 1
([],4)
 => [2,2,2,2,2,2,2,2]
 => [2,2,2,2,2,2,2]
 => 7 = 8 - 1
([(2,3)],4)
 => [6,6]
 => [6]
 => 1 = 2 - 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => [2,2]
 => 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [2,2,2]
 => 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => []
 => 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(1,2),(2,3)],4)
 => [4,4]
 => [4]
 => 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => [2,2]
 => 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [2,2,2]
 => 3 = 4 - 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => [3,3]
 => 2 = 3 - 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [3]
 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [2,2]
 => 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => []
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => []
 => 0 = 1 - 1
([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? = 16 - 1
([(3,4)],5)
 => [6,6,6,6]
 => [6,6,6]
 => ? = 4 - 1
([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => [6,2,2,2,2]
 => 5 = 6 - 1
([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => [2,2,2,2,2,2]
 => 6 = 7 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [2,2,2,2,2,2,2]
 => 7 = 8 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => [6]
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [2,2]
 => 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [2,2,2]
 => 3 = 4 - 1
([(1,3),(1,4),(4,2)],5)
 => [14]
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => [2,2]
 => 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [4,2,2]
 => 3 = 4 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [2]
 => 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [3,3]
 => 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [4]
 => 1 = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [2,2]
 => 2 = 3 - 1
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => [4,4,4]
 => 3 = 4 - 1
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [4,2,2]
 => 3 = 4 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [2,2,2]
 => 3 = 4 - 1
([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => [6,2,2,2,2]
 => 5 = 6 - 1
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [4,2,2]
 => 3 = 4 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [2,2]
 => 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => [2,2,2,2,2,2]
 => 6 = 7 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [2,2,2]
 => 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [2,2,2,2,2,2,2]
 => 7 = 8 - 1
([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => [3,3,3]
 => 3 = 4 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => [3,2]
 => 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [3,2,2]
 => 3 = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => [2,2,2,2]
 => 4 = 5 - 1
([],6)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 32 - 1
([(4,5)],6)
 => [6,6,6,6,6,6,6,6]
 => ?
 => ? = 8 - 1
([(3,4),(3,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ?
 => ? = 12 - 1
([(2,3),(2,4),(2,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 14 - 1
([(1,2),(1,3),(1,4),(1,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 15 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 16 - 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => [7,6,6,6]
 => ?
 => ? = 4 - 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => [7,6,2,2,2,2]
 => ?
 => ? = 6 - 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => [4,2,2,2,2,2,2,2]
 => ?
 => ? = 8 - 1
([(1,3),(1,4),(1,5),(5,2)],6)
 => [14,6,6]
 => ?
 => ? = 3 - 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => [7,6,2,2,2,2]
 => ?
 => ? = 6 - 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => [14,2,2,2,2]
 => ?
 => ? = 5 - 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ? = 8 - 1
([(2,3),(2,4),(4,5)],6)
 => [14,14]
 => ?
 => ? = 2 - 1
([(1,4),(1,5),(5,2),(5,3)],6)
 => [14,2,2,2,2]
 => ?
 => ? = 5 - 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ? = 7 - 1
([(2,3),(2,4),(3,5),(4,5)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ? = 8 - 1
([(3,4),(4,5)],6)
 => [4,4,4,4,4,4,4,4]
 => ?
 => ? = 8 - 1
([(2,3),(3,4),(3,5)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ? = 8 - 1
([(1,5),(5,2),(5,3),(5,4)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ? = 8 - 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
 => [4,2,2,2,2,2,2,2]
 => ?
 => ? = 8 - 1
([(3,5),(4,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ?
 => ? = 12 - 1
([(2,5),(3,5),(5,4)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ? = 8 - 1
([(2,5),(3,5),(4,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 14 - 1
([(1,5),(2,5),(3,5),(5,4)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ? = 8 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 15 - 1
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
 => [4,2,2,2,2,2,2,2]
 => ?
 => ? = 8 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 16 - 1
([(0,5),(1,5),(2,5),(3,4)],6)
 => [6,6,6,3,3,3]
 => ?
 => ? = 6 - 1
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ? = 7 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [7,6,6,6]
 => ?
 => ? = 4 - 1
([(1,5),(2,5),(3,4)],6)
 => [6,6,6,6,6]
 => [6,6,6,6]
 => ? = 5 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
 => [8,8,6,2,2]
 => ?
 => ? = 5 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [10,6,2,2,2,2]
 => ?
 => ? = 6 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,2,2,2,2,2,2,2,2]
 => ?
 => ? = 9 - 1
([(1,5),(2,4),(3,4),(4,5)],6)
 => [14,2,2,2,2]
 => ?
 => ? = 5 - 1
([(0,5),(1,5),(2,3),(5,4)],6)
 => [12,6]
 => ?
 => ? = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [7,6,2,2,2,2]
 => ?
 => ? = 6 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [14,6,6]
 => ?
 => ? = 3 - 1
([(0,5),(1,5),(2,3),(3,4)],6)
 => [12,4,4]
 => ?
 => ? = 3 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [6,6,3,3,3,2,2]
 => ?
 => ? = 7 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5)],6)
 => [8,6,3,2,2,2]
 => ?
 => ? = 6 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [8,3,2,2,2,2,2]
 => ?
 => ? = 7 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,3,2,2,2,2,2,2]
 => ?
 => ? = 8 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,2,2,2,2,2,2,2,2]
 => ?
 => ? = 9 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [7,6,2,2,2,2]
 => ?
 => ? = 6 - 1
([(2,5),(3,4)],6)
 => [6,6,6,6,6,6]
 => ?
 => ? = 6 - 1
Description
The length of the partition.
Matching statistic: St000147
(load all 3 compositions to match this statistic)
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 24% ●values known / values provided: 52%●distinct values known / distinct values provided: 24%
Values
([],1)
 => [2]
 => []
 => []
 => 0 = 1 - 1
([],2)
 => [2,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(0,1)],2)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([],3)
 => [2,2,2,2]
 => [2,2,2]
 => [3,3]
 => 3 = 4 - 1
([(1,2)],3)
 => [6]
 => []
 => []
 => 0 = 1 - 1
([(0,1),(0,2)],3)
 => [3,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(0,2),(2,1)],3)
 => [4]
 => []
 => []
 => 0 = 1 - 1
([(0,2),(1,2)],3)
 => [3,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([],4)
 => [2,2,2,2,2,2,2,2]
 => [2,2,2,2,2,2,2]
 => [7,7]
 => 7 = 8 - 1
([(2,3)],4)
 => [6,6]
 => [6]
 => [1,1,1,1,1,1]
 => 1 = 2 - 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => [2,2]
 => [2,2]
 => 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [2,2,2]
 => [3,3]
 => 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => []
 => []
 => 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(1,2),(2,3)],4)
 => [4,4]
 => [4]
 => [1,1,1,1]
 => 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => [2,2]
 => [2,2]
 => 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [2,2,2]
 => [3,3]
 => 3 = 4 - 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => [3,3]
 => [2,2,2]
 => 2 = 3 - 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [3]
 => [1,1,1]
 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [2,2]
 => [2,2]
 => 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => []
 => []
 => 0 = 1 - 1
([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [15,15]
 => ? = 16 - 1
([(3,4)],5)
 => [6,6,6,6]
 => [6,6,6]
 => [3,3,3,3,3,3]
 => ? = 4 - 1
([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => [6,2,2,2,2]
 => [5,5,1,1,1,1]
 => 5 = 6 - 1
([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => [2,2,2,2,2,2]
 => [6,6]
 => 6 = 7 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [2,2,2,2,2,2,2]
 => [7,7]
 => 7 = 8 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => [6]
 => [1,1,1,1,1,1]
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [2,2]
 => [2,2]
 => 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [2,2,2]
 => [3,3]
 => 3 = 4 - 1
([(1,3),(1,4),(4,2)],5)
 => [14]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => [2,2]
 => [2,2]
 => 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [4,2,2]
 => [3,3,1,1]
 => 3 = 4 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [3,3]
 => [2,2,2]
 => 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [4]
 => [1,1,1,1]
 => 1 = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [2,2]
 => [2,2]
 => 2 = 3 - 1
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => [4,4,4]
 => [3,3,3,3]
 => 3 = 4 - 1
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [4,2,2]
 => [3,3,1,1]
 => 3 = 4 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [2,2,2]
 => [3,3]
 => 3 = 4 - 1
([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => [6,2,2,2,2]
 => [5,5,1,1,1,1]
 => 5 = 6 - 1
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [4,2,2]
 => [3,3,1,1]
 => 3 = 4 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [2,2]
 => [2,2]
 => 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => [2,2,2,2,2,2]
 => [6,6]
 => 6 = 7 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [2,2,2]
 => [3,3]
 => 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [2,2,2,2,2,2,2]
 => [7,7]
 => 7 = 8 - 1
([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => [3,3,3]
 => [3,3,3]
 => 3 = 4 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => [3,2]
 => [2,2,1]
 => 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [3,2,2]
 => [3,3,1]
 => 3 = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => [2,2,2,2]
 => [4,4]
 => 4 = 5 - 1
([],6)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 32 - 1
([(4,5)],6)
 => [6,6,6,6,6,6,6,6]
 => ?
 => ?
 => ? = 8 - 1
([(3,4),(3,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 12 - 1
([(2,3),(2,4),(2,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 14 - 1
([(1,2),(1,3),(1,4),(1,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 15 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 16 - 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => [7,6,6,6]
 => ?
 => ?
 => ? = 4 - 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => [7,6,2,2,2,2]
 => ?
 => ?
 => ? = 6 - 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => [4,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(1,3),(1,4),(1,5),(5,2)],6)
 => [14,6,6]
 => ?
 => ?
 => ? = 3 - 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => [7,6,2,2,2,2]
 => ?
 => ?
 => ? = 6 - 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => [14,2,2,2,2]
 => ?
 => ?
 => ? = 5 - 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(2,3),(2,4),(4,5)],6)
 => [14,14]
 => ?
 => ?
 => ? = 2 - 1
([(1,4),(1,5),(5,2),(5,3)],6)
 => [14,2,2,2,2]
 => ?
 => ?
 => ? = 5 - 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 7 - 1
([(2,3),(2,4),(3,5),(4,5)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(3,4),(4,5)],6)
 => [4,4,4,4,4,4,4,4]
 => ?
 => ?
 => ? = 8 - 1
([(2,3),(3,4),(3,5)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(1,5),(5,2),(5,3),(5,4)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
 => [4,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(3,5),(4,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 12 - 1
([(2,5),(3,5),(5,4)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(2,5),(3,5),(4,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 14 - 1
([(1,5),(2,5),(3,5),(5,4)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 15 - 1
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
 => [4,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 16 - 1
([(0,5),(1,5),(2,5),(3,4)],6)
 => [6,6,6,3,3,3]
 => ?
 => ?
 => ? = 6 - 1
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 7 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [7,6,6,6]
 => ?
 => ?
 => ? = 4 - 1
([(1,5),(2,5),(3,4)],6)
 => [6,6,6,6,6]
 => [6,6,6,6]
 => [4,4,4,4,4,4]
 => ? = 5 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
 => [8,8,6,2,2]
 => ?
 => ?
 => ? = 5 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [10,6,2,2,2,2]
 => ?
 => ?
 => ? = 6 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 9 - 1
([(1,5),(2,4),(3,4),(4,5)],6)
 => [14,2,2,2,2]
 => ?
 => ?
 => ? = 5 - 1
([(0,5),(1,5),(2,3),(5,4)],6)
 => [12,6]
 => ?
 => ?
 => ? = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [7,6,2,2,2,2]
 => ?
 => ?
 => ? = 6 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [14,6,6]
 => ?
 => ?
 => ? = 3 - 1
([(0,5),(1,5),(2,3),(3,4)],6)
 => [12,4,4]
 => ?
 => ?
 => ? = 3 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [6,6,3,3,3,2,2]
 => ?
 => ?
 => ? = 7 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5)],6)
 => [8,6,3,2,2,2]
 => ?
 => ?
 => ? = 6 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [8,3,2,2,2,2,2]
 => ?
 => ?
 => ? = 7 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,3,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 9 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [7,6,2,2,2,2]
 => ?
 => ?
 => ? = 6 - 1
([(2,5),(3,4)],6)
 => [6,6,6,6,6,6]
 => ?
 => ?
 => ? = 6 - 1
Description
The largest part of an integer partition.
Matching statistic: St000473
(load all 2 compositions to match this statistic)
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000473: Integer partitions ⟶ ℤResult quality: 21% ●values known / values provided: 51%●distinct values known / distinct values provided: 21%
Values
([],1)
 => [2]
 => []
 => 0 = 1 - 1
([],2)
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(0,1)],2)
 => [3]
 => []
 => 0 = 1 - 1
([],3)
 => [2,2,2,2]
 => [2,2,2]
 => 3 = 4 - 1
([(1,2)],3)
 => [6]
 => []
 => 0 = 1 - 1
([(0,1),(0,2)],3)
 => [3,2]
 => [2]
 => 1 = 2 - 1
([(0,2),(2,1)],3)
 => [4]
 => []
 => 0 = 1 - 1
([(0,2),(1,2)],3)
 => [3,2]
 => [2]
 => 1 = 2 - 1
([],4)
 => [2,2,2,2,2,2,2,2]
 => [2,2,2,2,2,2,2]
 => ? = 8 - 1
([(2,3)],4)
 => [6,6]
 => [6]
 => 1 = 2 - 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => [2,2]
 => 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [2,2,2]
 => 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => []
 => 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(1,2),(2,3)],4)
 => [4,4]
 => [4]
 => 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => [2,2]
 => 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [2,2,2]
 => 3 = 4 - 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => [3,3]
 => 2 = 3 - 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [3]
 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [2,2]
 => 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => []
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => []
 => 0 = 1 - 1
([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? = 16 - 1
([(3,4)],5)
 => [6,6,6,6]
 => [6,6,6]
 => ? = 4 - 1
([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => [6,2,2,2,2]
 => ? = 6 - 1
([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => [2,2,2,2,2,2]
 => 6 = 7 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [2,2,2,2,2,2,2]
 => ? = 8 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => [6]
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [2,2]
 => 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [2,2,2]
 => 3 = 4 - 1
([(1,3),(1,4),(4,2)],5)
 => [14]
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => [2,2]
 => 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [4,2,2]
 => 3 = 4 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [2]
 => 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [3,3]
 => 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [4]
 => 1 = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [2,2]
 => 2 = 3 - 1
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => [4,4,4]
 => 3 = 4 - 1
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [4,2,2]
 => 3 = 4 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [2,2,2]
 => 3 = 4 - 1
([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => [6,2,2,2,2]
 => ? = 6 - 1
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [4,2,2]
 => 3 = 4 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [2,2]
 => 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => [2,2,2,2,2,2]
 => 6 = 7 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [2,2,2]
 => 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [2,2,2,2,2,2,2]
 => ? = 8 - 1
([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => [3,3,3]
 => 3 = 4 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => [3,2]
 => 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [3,2,2]
 => 3 = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => [2,2,2,2]
 => 4 = 5 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [2]
 => 1 = 2 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => [2,2]
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => [5,3]
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => [6]
 => 1 = 2 - 1
([(1,4),(2,3)],5)
 => [6,6,6]
 => [6,6]
 => 2 = 3 - 1
([],6)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 32 - 1
([(4,5)],6)
 => [6,6,6,6,6,6,6,6]
 => ?
 => ? = 8 - 1
([(3,4),(3,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ?
 => ? = 12 - 1
([(2,3),(2,4),(2,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 14 - 1
([(1,2),(1,3),(1,4),(1,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 15 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 16 - 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => [7,6,6,6]
 => ?
 => ? = 4 - 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => [7,6,2,2,2,2]
 => ?
 => ? = 6 - 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => [4,2,2,2,2,2,2,2]
 => ?
 => ? = 8 - 1
([(1,3),(1,4),(1,5),(5,2)],6)
 => [14,6,6]
 => ?
 => ? = 3 - 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => [7,6,2,2,2,2]
 => ?
 => ? = 6 - 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => [14,2,2,2,2]
 => ?
 => ? = 5 - 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ? = 8 - 1
([(2,3),(2,4),(4,5)],6)
 => [14,14]
 => ?
 => ? = 2 - 1
([(1,4),(1,5),(5,2),(5,3)],6)
 => [14,2,2,2,2]
 => ?
 => ? = 5 - 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ? = 7 - 1
([(2,3),(2,4),(3,5),(4,5)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ? = 8 - 1
([(3,4),(4,5)],6)
 => [4,4,4,4,4,4,4,4]
 => ?
 => ? = 8 - 1
([(2,3),(3,4),(3,5)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ? = 8 - 1
([(1,5),(5,2),(5,3),(5,4)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ? = 8 - 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
 => [4,2,2,2,2,2,2,2]
 => ?
 => ? = 8 - 1
([(3,5),(4,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ?
 => ? = 12 - 1
([(2,5),(3,5),(5,4)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ? = 8 - 1
([(2,5),(3,5),(4,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 14 - 1
([(1,5),(2,5),(3,5),(5,4)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ? = 8 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 15 - 1
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
 => [4,2,2,2,2,2,2,2]
 => ?
 => ? = 8 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 16 - 1
([(0,5),(1,5),(2,5),(3,4)],6)
 => [6,6,6,3,3,3]
 => ?
 => ? = 6 - 1
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ? = 7 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [7,6,6,6]
 => ?
 => ? = 4 - 1
([(1,5),(2,5),(3,4)],6)
 => [6,6,6,6,6]
 => [6,6,6,6]
 => ? = 5 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
 => [8,8,6,2,2]
 => ?
 => ? = 5 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [10,6,2,2,2,2]
 => ?
 => ? = 6 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,2,2,2,2,2,2,2,2]
 => ?
 => ? = 9 - 1
([(1,5),(2,4),(3,4),(4,5)],6)
 => [14,2,2,2,2]
 => ?
 => ? = 5 - 1
([(0,5),(1,5),(2,3),(5,4)],6)
 => [12,6]
 => ?
 => ? = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [7,6,2,2,2,2]
 => ?
 => ? = 6 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [14,6,6]
 => ?
 => ? = 3 - 1
([(0,5),(1,5),(2,3),(3,4)],6)
 => [12,4,4]
 => ?
 => ? = 3 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [6,6,3,3,3,2,2]
 => ?
 => ? = 7 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5)],6)
 => [8,6,3,2,2,2]
 => ?
 => ? = 6 - 1
Description
The number of parts of a partition that are strictly bigger than the number of ones. This is part of the definition of Dyson's crank of a partition, see [[St000474]].
Matching statistic: St000146
(load all 2 compositions to match this statistic)
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000146: Integer partitions ⟶ ℤResult quality: 21% ●values known / values provided: 51%●distinct values known / distinct values provided: 21%
Values
([],1)
 => [2]
 => []
 => 0 = 1 - 1
([],2)
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(0,1)],2)
 => [3]
 => []
 => 0 = 1 - 1
([],3)
 => [2,2,2,2]
 => [2,2,2]
 => 3 = 4 - 1
([(1,2)],3)
 => [6]
 => []
 => 0 = 1 - 1
([(0,1),(0,2)],3)
 => [3,2]
 => [2]
 => 1 = 2 - 1
([(0,2),(2,1)],3)
 => [4]
 => []
 => 0 = 1 - 1
([(0,2),(1,2)],3)
 => [3,2]
 => [2]
 => 1 = 2 - 1
([],4)
 => [2,2,2,2,2,2,2,2]
 => [2,2,2,2,2,2,2]
 => ? = 8 - 1
([(2,3)],4)
 => [6,6]
 => [6]
 => 1 = 2 - 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => [2,2]
 => 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [2,2,2]
 => 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => []
 => 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(1,2),(2,3)],4)
 => [4,4]
 => [4]
 => 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => [2,2]
 => 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [2,2,2]
 => 3 = 4 - 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => [3,3]
 => 2 = 3 - 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [3]
 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [2,2]
 => 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => []
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => []
 => 0 = 1 - 1
([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? = 16 - 1
([(3,4)],5)
 => [6,6,6,6]
 => [6,6,6]
 => ? = 4 - 1
([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => [6,2,2,2,2]
 => 5 = 6 - 1
([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => [2,2,2,2,2,2]
 => 6 = 7 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [2,2,2,2,2,2,2]
 => ? = 8 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => [6]
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [2,2]
 => 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [2,2,2]
 => 3 = 4 - 1
([(1,3),(1,4),(4,2)],5)
 => [14]
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => [2,2]
 => 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [4,2,2]
 => 3 = 4 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [2]
 => 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [3,3]
 => 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [4]
 => 1 = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [2,2]
 => 2 = 3 - 1
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => [4,4,4]
 => 3 = 4 - 1
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [4,2,2]
 => 3 = 4 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [2,2,2]
 => 3 = 4 - 1
([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => [6,2,2,2,2]
 => 5 = 6 - 1
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [4,2,2]
 => 3 = 4 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [2,2]
 => 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => [2,2,2,2,2,2]
 => 6 = 7 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [2,2,2]
 => 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [2,2,2,2,2,2,2]
 => ? = 8 - 1
([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => [3,3,3]
 => 3 = 4 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => [3,2]
 => 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [3,2,2]
 => 3 = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => [2,2,2,2]
 => 4 = 5 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [2]
 => 1 = 2 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => [2,2]
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => [5,3]
 => 2 = 3 - 1
([],6)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 32 - 1
([(4,5)],6)
 => [6,6,6,6,6,6,6,6]
 => ?
 => ? = 8 - 1
([(3,4),(3,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ?
 => ? = 12 - 1
([(2,3),(2,4),(2,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 14 - 1
([(1,2),(1,3),(1,4),(1,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 15 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 16 - 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => [7,6,6,6]
 => ?
 => ? = 4 - 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => [7,6,2,2,2,2]
 => ?
 => ? = 6 - 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => [4,2,2,2,2,2,2,2]
 => ?
 => ? = 8 - 1
([(1,3),(1,4),(1,5),(5,2)],6)
 => [14,6,6]
 => ?
 => ? = 3 - 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => [7,6,2,2,2,2]
 => ?
 => ? = 6 - 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => [14,2,2,2,2]
 => ?
 => ? = 5 - 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ? = 8 - 1
([(2,3),(2,4),(4,5)],6)
 => [14,14]
 => ?
 => ? = 2 - 1
([(1,4),(1,5),(5,2),(5,3)],6)
 => [14,2,2,2,2]
 => ?
 => ? = 5 - 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ? = 7 - 1
([(2,3),(2,4),(3,5),(4,5)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ? = 8 - 1
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,4,2,2,2,2]
 => [4,2,2,2,2]
 => ? = 6 - 1
([(3,4),(4,5)],6)
 => [4,4,4,4,4,4,4,4]
 => ?
 => ? = 8 - 1
([(2,3),(3,4),(3,5)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ? = 8 - 1
([(1,5),(5,2),(5,3),(5,4)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ? = 8 - 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
 => [4,2,2,2,2,2,2,2]
 => ?
 => ? = 8 - 1
([(3,5),(4,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ?
 => ? = 12 - 1
([(2,5),(3,5),(5,4)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ? = 8 - 1
([(1,5),(2,5),(5,3),(5,4)],6)
 => [4,4,2,2,2,2]
 => [4,2,2,2,2]
 => ? = 6 - 1
([(2,5),(3,5),(4,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 14 - 1
([(1,5),(2,5),(3,5),(5,4)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ? = 8 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 15 - 1
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
 => [4,2,2,2,2,2,2,2]
 => ?
 => ? = 8 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 16 - 1
([(0,5),(1,5),(2,5),(3,4)],6)
 => [6,6,6,3,3,3]
 => ?
 => ? = 6 - 1
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ? = 7 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [7,6,6,6]
 => ?
 => ? = 4 - 1
([(1,5),(2,5),(3,4)],6)
 => [6,6,6,6,6]
 => [6,6,6,6]
 => ? = 5 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
 => [8,8,6,2,2]
 => ?
 => ? = 5 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [10,6,2,2,2,2]
 => ?
 => ? = 6 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,2,2,2,2,2,2,2,2]
 => ?
 => ? = 9 - 1
([(1,5),(2,4),(3,4),(4,5)],6)
 => [14,2,2,2,2]
 => ?
 => ? = 5 - 1
([(0,5),(1,5),(2,3),(5,4)],6)
 => [12,6]
 => ?
 => ? = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [7,6,2,2,2,2]
 => ?
 => ? = 6 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [14,6,6]
 => ?
 => ? = 3 - 1
([(0,5),(1,5),(2,3),(3,4)],6)
 => [12,4,4]
 => ?
 => ? = 3 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [6,6,3,3,3,2,2]
 => ?
 => ? = 7 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5)],6)
 => [8,6,3,2,2,2]
 => ?
 => ? = 6 - 1
Description
The Andrews-Garvan crank of a partition. If $\pi$ is a partition, let $l(\pi)$ be its length (number of parts), $\omega(\pi)$ be the number of parts equal to 1, and $\mu(\pi)$ be the number of parts larger than $\omega(\pi)$. The crank is then defined by $$ c(\pi) = \begin{cases} l(\pi) &\text{if \(\omega(\pi)=0\)}\\ \mu(\pi) - \omega(\pi) &\text{otherwise}. \end{cases} $$ This statistic was defined in [1] to explain Ramanujan's partition congruence $$p(11n+6) \equiv 0 \pmod{11}$$ in the same way as the Dyson rank ([[St000145]]) explains the congruences $$p(5n+4) \equiv 0 \pmod{5}$$ and $$p(7n+5) \equiv 0 \pmod{7}.$$
Matching statistic: St001280
(load all 2 compositions to match this statistic)
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 24% ●values known / values provided: 49%●distinct values known / distinct values provided: 24%
Values
([],1)
 => [2]
 => 1
([],2)
 => [2,2]
 => 2
([(0,1)],2)
 => [3]
 => 1
([],3)
 => [2,2,2,2]
 => 4
([(1,2)],3)
 => [6]
 => 1
([(0,1),(0,2)],3)
 => [3,2]
 => 2
([(0,2),(2,1)],3)
 => [4]
 => 1
([(0,2),(1,2)],3)
 => [3,2]
 => 2
([],4)
 => [2,2,2,2,2,2,2,2]
 => 8
([(2,3)],4)
 => [6,6]
 => 2
([(1,2),(1,3)],4)
 => [6,2,2]
 => 3
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => 4
([(0,2),(0,3),(3,1)],4)
 => [7]
 => 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => 2
([(1,2),(2,3)],4)
 => [4,4]
 => 2
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => 2
([(1,3),(2,3)],4)
 => [6,2,2]
 => 3
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => 4
([(0,3),(1,2)],4)
 => [3,3,3]
 => 3
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => 3
([(0,3),(2,1),(3,2)],4)
 => [5]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => 1
([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? = 16
([(3,4)],5)
 => [6,6,6,6]
 => ? = 4
([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => ? = 6
([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => ? = 7
([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => 8
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => 4
([(1,3),(1,4),(4,2)],5)
 => [14]
 => 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => 3
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => 4
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => 4
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => 4
([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => ? = 6
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => 4
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => ? = 7
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => 8
([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => 4
([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => 5
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => 2
([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => 3
([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => 2
([(1,4),(2,3)],5)
 => [6,6,6]
 => ? = 3
([],6)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? = 32
([(4,5)],6)
 => [6,6,6,6,6,6,6,6]
 => ? = 8
([(3,4),(3,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ? = 12
([(2,3),(2,4),(2,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? = 14
([(1,2),(1,3),(1,4),(1,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? = 15
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? = 16
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => [7,6,6,6]
 => ? = 4
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => [7,6,2,2,2,2]
 => ? = 6
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [7,2,2,2,2,2,2]
 => ? = 7
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => [4,2,2,2,2,2,2,2]
 => ? = 8
([(1,3),(1,4),(1,5),(5,2)],6)
 => [14,6,6]
 => ? = 3
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => [7,6,2,2,2,2]
 => ? = 6
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => [14,2,2,2,2]
 => ? = 5
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => [4,4,2,2,2,2,2,2]
 => ? = 8
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => [7,6,6]
 => ? = 3
([(2,3),(2,4),(4,5)],6)
 => [14,14]
 => ? = 2
([(1,4),(1,5),(5,2),(5,3)],6)
 => [14,2,2,2,2]
 => ? = 5
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => [7,2,2,2,2,2,2]
 => ? = 7
([(2,3),(2,4),(3,5),(4,5)],6)
 => [4,4,4,4,2,2,2,2]
 => ? = 8
([(3,4),(4,5)],6)
 => [4,4,4,4,4,4,4,4]
 => ? = 8
([(2,3),(3,4),(3,5)],6)
 => [4,4,4,4,2,2,2,2]
 => ? = 8
([(1,5),(5,2),(5,3),(5,4)],6)
 => [4,4,2,2,2,2,2,2]
 => ? = 8
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
 => [4,2,2,2,2,2,2,2]
 => ? = 8
([(2,3),(3,5),(5,4)],6)
 => [10,10]
 => ? = 2
([(3,5),(4,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ? = 12
([(2,5),(3,5),(5,4)],6)
 => [4,4,4,4,2,2,2,2]
 => ? = 8
([(2,5),(3,5),(4,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? = 14
([(1,5),(2,5),(3,5),(5,4)],6)
 => [4,4,2,2,2,2,2,2]
 => ? = 8
([(1,5),(2,5),(3,5),(4,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? = 15
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
 => [4,2,2,2,2,2,2,2]
 => ? = 8
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? = 16
([(0,5),(1,5),(2,5),(3,4)],6)
 => [6,6,6,3,3,3]
 => ? = 6
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => [7,2,2,2,2,2,2]
 => ? = 7
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [7,6,6,6]
 => ? = 4
([(1,5),(2,5),(3,4)],6)
 => [6,6,6,6,6]
 => ? = 5
([(1,5),(2,4),(3,4),(3,5)],6)
 => [8,8,6,2,2]
 => ? = 5
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [10,6,2,2,2,2]
 => ? = 6
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,2,2,2,2,2,2,2,2]
 => ? = 9
([(1,5),(2,4),(3,4),(4,5)],6)
 => [14,2,2,2,2]
 => ? = 5
([(0,5),(1,5),(2,3),(5,4)],6)
 => [12,6]
 => ? = 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [7,6,2,2,2,2]
 => ? = 6
([(1,5),(2,5),(3,4),(4,5)],6)
 => [14,6,6]
 => ? = 3
([(0,5),(1,5),(2,3),(3,4)],6)
 => [12,4,4]
 => ? = 3
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000143
(load all 2 compositions to match this statistic)
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000143: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 48%●distinct values known / distinct values provided: 15%
Values
([],1)
 => [2]
 => []
 => []
 => 0 = 1 - 1
([],2)
 => [2,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(0,1)],2)
 => [3]
 => []
 => []
 => 0 = 1 - 1
([],3)
 => [2,2,2,2]
 => [2,2,2]
 => [3,3]
 => 3 = 4 - 1
([(1,2)],3)
 => [6]
 => []
 => []
 => 0 = 1 - 1
([(0,1),(0,2)],3)
 => [3,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(0,2),(2,1)],3)
 => [4]
 => []
 => []
 => 0 = 1 - 1
([(0,2),(1,2)],3)
 => [3,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([],4)
 => [2,2,2,2,2,2,2,2]
 => [2,2,2,2,2,2,2]
 => [7,7]
 => ? = 8 - 1
([(2,3)],4)
 => [6,6]
 => [6]
 => [1,1,1,1,1,1]
 => 1 = 2 - 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => [2,2]
 => [2,2]
 => 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [2,2,2]
 => [3,3]
 => 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => []
 => []
 => 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(1,2),(2,3)],4)
 => [4,4]
 => [4]
 => [1,1,1,1]
 => 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => [2,2]
 => [2,2]
 => 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [2,2,2]
 => [3,3]
 => 3 = 4 - 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => [3,3]
 => [2,2,2]
 => 2 = 3 - 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [3]
 => [1,1,1]
 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [2,2]
 => [2,2]
 => 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => []
 => []
 => 0 = 1 - 1
([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [15,15]
 => ? = 16 - 1
([(3,4)],5)
 => [6,6,6,6]
 => [6,6,6]
 => [3,3,3,3,3,3]
 => ? = 4 - 1
([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => [6,2,2,2,2]
 => [5,5,1,1,1,1]
 => ? = 6 - 1
([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => [2,2,2,2,2,2]
 => [6,6]
 => ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [2,2,2,2,2,2,2]
 => [7,7]
 => ? = 8 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => [6]
 => [1,1,1,1,1,1]
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [2,2]
 => [2,2]
 => 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [2,2,2]
 => [3,3]
 => 3 = 4 - 1
([(1,3),(1,4),(4,2)],5)
 => [14]
 => []
 => []
 => 0 = 1 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => [2,2]
 => [2,2]
 => 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [4,2,2]
 => [3,3,1,1]
 => 3 = 4 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [3,3]
 => [2,2,2]
 => 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [4]
 => [1,1,1,1]
 => 1 = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [2,2]
 => [2,2]
 => 2 = 3 - 1
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => [4,4,4]
 => [3,3,3,3]
 => ? = 4 - 1
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [4,2,2]
 => [3,3,1,1]
 => 3 = 4 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [2,2,2]
 => [3,3]
 => 3 = 4 - 1
([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => [6,2,2,2,2]
 => [5,5,1,1,1,1]
 => ? = 6 - 1
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [4,2,2]
 => [3,3,1,1]
 => 3 = 4 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [2,2]
 => [2,2]
 => 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => [2,2,2,2,2,2]
 => [6,6]
 => ? = 7 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [2,2,2]
 => [3,3]
 => 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [2,2,2,2,2,2,2]
 => [7,7]
 => ? = 8 - 1
([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => [3,3,3]
 => [3,3,3]
 => 3 = 4 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => [3,2]
 => [2,2,1]
 => 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [3,2,2]
 => [3,3,1]
 => 3 = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => [2,2,2,2]
 => [4,4]
 => 4 = 5 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => [2,2]
 => [2,2]
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => [5,3]
 => [2,2,2,1,1]
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => [6]
 => [1,1,1,1,1,1]
 => 1 = 2 - 1
([(1,4),(2,3)],5)
 => [6,6,6]
 => [6,6]
 => [2,2,2,2,2,2]
 => ? = 3 - 1
([(1,4),(2,3),(2,4)],5)
 => [10,6]
 => [6]
 => [1,1,1,1,1,1]
 => 1 = 2 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
 => [8,3]
 => [3]
 => [1,1,1]
 => 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [4]
 => [1,1,1,1]
 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [6,2,2,2,2]
 => [2,2,2,2]
 => [4,4]
 => 4 = 5 - 1
([],6)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 32 - 1
([(4,5)],6)
 => [6,6,6,6,6,6,6,6]
 => ?
 => ?
 => ? = 8 - 1
([(3,4),(3,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 12 - 1
([(2,3),(2,4),(2,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 14 - 1
([(1,2),(1,3),(1,4),(1,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 15 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 16 - 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => [7,6,6,6]
 => ?
 => ?
 => ? = 4 - 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => [7,6,2,2,2,2]
 => ?
 => ?
 => ? = 6 - 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => [4,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(1,3),(1,4),(1,5),(5,2)],6)
 => [14,6,6]
 => ?
 => ?
 => ? = 3 - 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => [7,6,2,2,2,2]
 => ?
 => ?
 => ? = 6 - 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => [14,2,2,2,2]
 => ?
 => ?
 => ? = 5 - 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => [7,6,6]
 => [6,6]
 => [2,2,2,2,2,2]
 => ? = 3 - 1
([(2,3),(2,4),(4,5)],6)
 => [14,14]
 => ?
 => ?
 => ? = 2 - 1
([(1,4),(1,5),(5,2),(5,3)],6)
 => [14,2,2,2,2]
 => ?
 => ?
 => ? = 5 - 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 7 - 1
([(2,3),(2,4),(3,5),(4,5)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,4,2,2,2,2]
 => [4,2,2,2,2]
 => [5,5,1,1]
 => ? = 6 - 1
([(3,4),(4,5)],6)
 => [4,4,4,4,4,4,4,4]
 => ?
 => ?
 => ? = 8 - 1
([(2,3),(3,4),(3,5)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(1,5),(5,2),(5,3),(5,4)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
 => [4,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(3,5),(4,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 12 - 1
([(2,5),(3,5),(5,4)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(1,5),(2,5),(5,3),(5,4)],6)
 => [4,4,2,2,2,2]
 => [4,2,2,2,2]
 => [5,5,1,1]
 => ? = 6 - 1
([(2,5),(3,5),(4,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 14 - 1
([(1,5),(2,5),(3,5),(5,4)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 15 - 1
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
 => [4,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 16 - 1
([(0,5),(1,5),(2,5),(3,4)],6)
 => [6,6,6,3,3,3]
 => ?
 => ?
 => ? = 6 - 1
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 7 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [7,6,6,6]
 => ?
 => ?
 => ? = 4 - 1
([(1,5),(2,5),(3,4)],6)
 => [6,6,6,6,6]
 => [6,6,6,6]
 => [4,4,4,4,4,4]
 => ? = 5 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
 => [8,8,6,2,2]
 => ?
 => ?
 => ? = 5 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [10,6,2,2,2,2]
 => ?
 => ?
 => ? = 6 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 9 - 1
Description
The largest repeated part of a partition. If the parts of the partition are all distinct, the value of the statistic is defined to be zero.
Matching statistic: St000378
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 18% ●values known / values provided: 37%●distinct values known / distinct values provided: 18%
Values
([],1)
 => [2]
 => [1,1]
 => 1
([],2)
 => [2,2]
 => [4]
 => 2
([(0,1)],2)
 => [3]
 => [1,1,1]
 => 1
([],3)
 => [2,2,2,2]
 => [5,1,1,1]
 => 4
([(1,2)],3)
 => [6]
 => [1,1,1,1,1,1]
 => 1
([(0,1),(0,2)],3)
 => [3,2]
 => [5]
 => 2
([(0,2),(2,1)],3)
 => [4]
 => [1,1,1,1]
 => 1
([(0,2),(1,2)],3)
 => [3,2]
 => [5]
 => 2
([],4)
 => [2,2,2,2,2,2,2,2]
 => [7,5,1,1,1,1]
 => ? = 8
([(2,3)],4)
 => [6,6]
 => [12]
 => 2
([(1,2),(1,3)],4)
 => [6,2,2]
 => [5,1,1,1,1,1]
 => 3
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [5,1,1,1,1]
 => 4
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,1,1,1,1,1,1]
 => 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [2,2,1,1]
 => 2
([(1,2),(2,3)],4)
 => [4,4]
 => [8]
 => 2
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => 2
([(1,3),(2,3)],4)
 => [6,2,2]
 => [5,1,1,1,1,1]
 => 3
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [5,1,1,1,1]
 => 4
([(0,3),(1,2)],4)
 => [3,3,3]
 => [2,2,2,2,1]
 => 3
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [2,2,2,1,1]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [2,2,2,1]
 => 3
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,1,1,1,1]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,1,1,1,1,1,1]
 => 1
([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [9,4,4,4,4,4,2,1]
 => ? = 16
([(3,4)],5)
 => [6,6,6,6]
 => [13,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 4
([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => [12,5,1,1,1]
 => ? = 6
([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => [7,7,4]
 => ? = 7
([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [7,6,1,1,1,1]
 => ? = 8
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => [13]
 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [5,1,1,1,1,1,1]
 => 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [5,5]
 => 4
([(1,3),(1,4),(4,2)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => [5,1,1,1,1,1,1]
 => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [8,4]
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [2,2,1,1,1]
 => 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [2,2,2,2,2]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [9]
 => 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [6,1,1]
 => 3
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => [9,1,1,1,1,1,1,1]
 => ? = 4
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [8,4]
 => 4
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [5,5]
 => 4
([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => [12,5,1,1,1]
 => ? = 6
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [8,4]
 => 4
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [6,1,1]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => [7,7,4]
 => ? = 7
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [5,5]
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [7,6,1,1,1,1]
 => ? = 8
([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => [9,6]
 => 4
([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => [6,1,1,1,1,1,1,1]
 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [6,6]
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => [6,5]
 => 5
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => 2
([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => [5,1,1,1,1,1,1]
 => 3
([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => [11,1,1,1]
 => ? = 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => [13]
 => 2
([(1,4),(2,3)],5)
 => [6,6,6]
 => [2,2,2,2,2,2,2,2,2]
 => 3
([(1,4),(2,3),(2,4)],5)
 => [10,6]
 => [2,2,2,2,2,2,1,1,1,1]
 => 2
([(0,4),(1,2),(1,4),(2,3)],5)
 => [8,3]
 => [2,2,2,1,1,1,1,1]
 => 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [9]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [6,2,2,2,2]
 => [7,2,2,2,1]
 => ? = 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => [2,2,1,1,1,1,1]
 => 2
([(0,4),(1,2),(1,3),(1,4)],5)
 => [6,5,3]
 => [11,1,1,1]
 => ? = 3
([],6)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 32
([(4,5)],6)
 => [6,6,6,6,6,6,6,6]
 => ?
 => ? = 8
([(3,4),(3,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ?
 => ? = 12
([(2,3),(2,4),(2,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 14
([(1,2),(1,3),(1,4),(1,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 15
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 16
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => [7,6,6,6]
 => ?
 => ? = 4
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => [7,6,2,2,2,2]
 => ?
 => ? = 6
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ? = 7
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => [4,2,2,2,2,2,2,2]
 => ?
 => ? = 8
([(1,3),(1,4),(1,5),(5,2)],6)
 => [14,6,6]
 => ?
 => ? = 3
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => [7,6,2,2,2,2]
 => ?
 => ? = 6
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => [14,2,2,2,2]
 => ?
 => ? = 5
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ? = 8
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,4,2,2]
 => [9,4]
 => ? = 4
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => [7,6,6]
 => ?
 => ? = 3
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7,2,2,2,2]
 => [6,2,2,2,1,1,1]
 => ? = 5
([(2,3),(2,4),(4,5)],6)
 => [14,14]
 => ?
 => ? = 2
([(1,4),(1,5),(5,2),(5,3)],6)
 => [14,2,2,2,2]
 => ?
 => ? = 5
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ? = 7
([(2,3),(2,4),(3,5),(4,5)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ? = 8
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
 => [10,2,2]
 => [5,1,1,1,1,1,1,1,1,1]
 => ? = 3
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,4,2,2,2,2]
 => [7,6,1,1,1]
 => ? = 6
([(3,4),(4,5)],6)
 => [4,4,4,4,4,4,4,4]
 => ?
 => ? = 8
([(2,3),(3,4),(3,5)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ? = 8
([(1,5),(5,2),(5,3),(5,4)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ? = 8
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
 => [4,2,2,2,2,2,2,2]
 => ?
 => ? = 8
([(1,4),(4,5),(5,2),(5,3)],6)
 => [10,2,2]
 => [5,1,1,1,1,1,1,1,1,1]
 => ? = 3
([(3,5),(4,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ?
 => ? = 12
([(2,5),(3,5),(5,4)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ? = 8
([(1,5),(2,5),(5,3),(5,4)],6)
 => [4,4,2,2,2,2]
 => [7,6,1,1,1]
 => ? = 6
([(2,5),(3,5),(4,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 14
([(1,5),(2,5),(3,5),(5,4)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ? = 8
([(1,5),(2,5),(3,5),(4,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 15
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
 => [4,2,2,2,2,2,2,2]
 => ?
 => ? = 8
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ? = 16
([(0,5),(1,5),(2,5),(3,4)],6)
 => [6,6,6,3,3,3]
 => ?
 => ? = 6
Description
The diagonal inversion number of an integer partition. The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$. See also exercise 3.19 of [2]. This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000445
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 21% ●values known / values provided: 35%●distinct values known / distinct values provided: 21%
Values
([],1)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
([],2)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
([(0,1)],2)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
([],3)
 => [2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 4 - 1
([(1,2)],3)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
([(0,1),(0,2)],3)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
([(0,2),(2,1)],3)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
([(0,2),(1,2)],3)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
([],4)
 => [2,2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 7 = 8 - 1
([(2,3)],4)
 => [6,6]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 1 = 2 - 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 2 - 1
([(1,2),(2,3)],4)
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 3 = 4 - 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2 = 3 - 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0 = 1 - 1
([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 16 - 1
([(3,4)],5)
 => [6,6,6,6]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => ? = 4 - 1
([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => [1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 - 1
([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 7 = 8 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
 => 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => 3 = 4 - 1
([(1,3),(1,4),(4,2)],5)
 => [14]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? = 1 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
 => 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => 3 = 4 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 1 = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 2 = 3 - 1
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => 3 = 4 - 1
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => 3 = 4 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => 3 = 4 - 1
([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => [1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 - 1
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => 3 = 4 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 7 = 8 - 1
([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 4 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0,0]
 => ? = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
 => ? = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => 4 = 5 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 1 = 2 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => 1 = 2 - 1
([(1,4),(2,3)],5)
 => [6,6,6]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => 2 = 3 - 1
([(1,4),(2,3),(2,4)],5)
 => [10,6]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0,0]
 => ? = 2 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
 => [8,3]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
 => 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [6,2,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => 1 = 2 - 1
([(0,4),(1,2),(1,3)],5)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 4 - 1
([(0,3),(0,4),(1,2),(1,4)],5)
 => [8,3,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0,0]
 => ? = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
 => ? = 4 - 1
([(1,4),(2,3),(3,4)],5)
 => [14]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? = 1 - 1
([(0,3),(1,4),(4,2)],5)
 => [12]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? = 1 - 1
([],6)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 32 - 1
([(4,5)],6)
 => [6,6,6,6,6,6,6,6]
 => ?
 => ?
 => ? = 8 - 1
([(3,4),(3,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 12 - 1
([(2,3),(2,4),(2,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 14 - 1
([(1,2),(1,3),(1,4),(1,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 15 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 16 - 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => [7,6,6,6]
 => ?
 => ?
 => ? = 4 - 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => [7,6,2,2,2,2]
 => ?
 => ?
 => ? = 6 - 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => [4,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(1,3),(1,4),(1,5),(5,2)],6)
 => [14,6,6]
 => ?
 => ?
 => ? = 3 - 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => [7,6,2,2,2,2]
 => ?
 => ?
 => ? = 6 - 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => [14,2,2,2,2]
 => ?
 => ?
 => ? = 5 - 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 3 - 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,4,2,2]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
 => ? = 4 - 1
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7,2,2,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => ?
 => ? = 5 - 1
([(2,3),(2,4),(4,5)],6)
 => [14,14]
 => ?
 => ?
 => ? = 2 - 1
([(1,4),(1,5),(5,2),(5,3)],6)
 => [14,2,2,2,2]
 => ?
 => ?
 => ? = 5 - 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 7 - 1
([(2,3),(2,4),(3,5),(4,5)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
 => [10,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ?
 => ? = 3 - 1
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,4,2,2,2,2]
 => [1,1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 - 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
 => [6,4,3,3]
 => [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ?
 => ? = 4 - 1
([(3,4),(4,5)],6)
 => [4,4,4,4,4,4,4,4]
 => ?
 => ?
 => ? = 8 - 1
([(2,3),(3,4),(3,5)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(1,5),(5,2),(5,3),(5,4)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
 => [4,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(1,4),(4,5),(5,2),(5,3)],6)
 => [10,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ?
 => ? = 3 - 1
([(3,5),(4,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 12 - 1
([(2,5),(3,5),(5,4)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(1,5),(2,5),(5,3),(5,4)],6)
 => [4,4,2,2,2,2]
 => [1,1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 - 1
([(2,5),(3,5),(4,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 14 - 1
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St000052
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 21% ●values known / values provided: 27%●distinct values known / distinct values provided: 21%
Values
([],1)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
([],2)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
([(0,1)],2)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
([],3)
 => [2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 4 - 1
([(1,2)],3)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
([(0,1),(0,2)],3)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
([(0,2),(2,1)],3)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
([(0,2),(1,2)],3)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
([],4)
 => [2,2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 7 = 8 - 1
([(2,3)],4)
 => [6,6]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 1 = 2 - 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 3 - 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 2 - 1
([(1,2),(2,3)],4)
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 3 - 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 3 = 4 - 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2 = 3 - 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0 = 1 - 1
([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 16 - 1
([(3,4)],5)
 => [6,6,6,6]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => ? = 4 - 1
([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => [1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 - 1
([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 7 = 8 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => 3 = 4 - 1
([(1,3),(1,4),(4,2)],5)
 => [14]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? = 1 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => 3 = 4 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 1 = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 2 = 3 - 1
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => 3 = 4 - 1
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => 3 = 4 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => 3 = 4 - 1
([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => [1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 - 1
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => 3 = 4 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 7 = 8 - 1
([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 4 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0,0]
 => ? = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
 => ? = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => 4 = 5 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 1 = 2 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 3 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
 => ? = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => 1 = 2 - 1
([(1,4),(2,3)],5)
 => [6,6,6]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => 2 = 3 - 1
([(1,4),(2,3),(2,4)],5)
 => [10,6]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0,0]
 => ? = 2 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
 => [8,3]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
 => 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [6,2,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 2 = 3 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => 0 = 1 - 1
([(0,4),(1,2),(1,3)],5)
 => [6,3,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 4 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
 => [6,5,3]
 => [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
 => ? = 3 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(3,4)],5)
 => [10,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
 => 1 = 2 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 3 - 1
([(0,3),(0,4),(1,2),(1,4)],5)
 => [8,3,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0,0]
 => ? = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
 => ? = 4 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [3,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => 4 = 5 - 1
([(1,4),(2,3),(3,4)],5)
 => [14]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? = 1 - 1
([(0,3),(1,4),(4,2)],5)
 => [12]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? = 1 - 1
([],6)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 32 - 1
([(4,5)],6)
 => [6,6,6,6,6,6,6,6]
 => ?
 => ?
 => ? = 8 - 1
([(3,4),(3,5)],6)
 => [6,6,6,6,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 12 - 1
([(2,3),(2,4),(2,5)],6)
 => [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 14 - 1
([(1,2),(1,3),(1,4),(1,5)],6)
 => [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 15 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 16 - 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => [7,6,6,6]
 => ?
 => ?
 => ? = 4 - 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => [7,6,2,2,2,2]
 => ?
 => ?
 => ? = 6 - 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => [4,2,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(1,3),(1,4),(1,5),(5,2)],6)
 => [14,6,6]
 => ?
 => ?
 => ? = 3 - 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => [7,6,2,2,2,2]
 => ?
 => ?
 => ? = 6 - 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => [14,2,2,2,2]
 => ?
 => ?
 => ? = 5 - 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => [4,4,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [5,2,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
 => ? = 4 - 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => [8,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
 => ? = 3 - 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,4,2,2]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
 => ? = 4 - 1
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => [7,6,6]
 => [1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
 => ? = 3 - 1
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7,2,2,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => ?
 => ? = 5 - 1
([(2,3),(2,4),(4,5)],6)
 => [14,14]
 => ?
 => ?
 => ? = 2 - 1
([(1,4),(1,5),(5,2),(5,3)],6)
 => [14,2,2,2,2]
 => ?
 => ?
 => ? = 5 - 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => [7,2,2,2,2,2,2]
 => ?
 => ?
 => ? = 7 - 1
([(2,3),(2,4),(3,5),(4,5)],6)
 => [4,4,4,4,2,2,2,2]
 => ?
 => ?
 => ? = 8 - 1
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
 => [10,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ?
 => ? = 3 - 1
Description
The number of valleys of a Dyck path not on the x-axis. That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
The following 25 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000288The number of ones in a binary word. St000734The last entry in the first row of a standard tableau. St000733The row containing the largest entry of a standard tableau. St000157The number of descents of a standard tableau. St000507The number of ascents of a standard tableau. St000676The number of odd rises of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000053The number of valleys of the Dyck path. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000015The number of peaks of a Dyck path. St001462The number of factors of a standard tableaux under concatenation. St000331The number of upper interactions of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001712The number of natural descents of a standard Young tableau. St001596The number of two-by-two squares inside a skew partition. St001354The number of series nodes in the modular decomposition of a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1).