Processing math: 12%

Your data matches 180 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000071
(load all 2 compositions to match this statistic)
Mapping
St000071: 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)
 => 3
([(1,2)],3)
 => 2
([(0,1),(0,2)],3)
 => 2
([(0,2),(2,1)],3)
 => 1
([(0,2),(1,2)],3)
 => 2
([],4)
 => 4
([(2,3)],4)
 => 3
([(1,2),(1,3)],4)
 => 3
([(0,1),(0,2),(0,3)],4)
 => 3
([(0,2),(0,3),(3,1)],4)
 => 2
([(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)
 => 3
([(0,3),(1,2)],4)
 => 2
([(0,3),(1,2),(1,3)],4)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
([(0,3),(2,1),(3,2)],4)
 => 1
([(0,3),(1,2),(2,3)],4)
 => 2
([],5)
 => 5
([(3,4)],5)
 => 4
([(2,3),(2,4)],5)
 => 4
([(1,2),(1,3),(1,4)],5)
 => 4
([(0,1),(0,2),(0,3),(0,4)],5)
 => 4
([(0,2),(0,3),(0,4),(4,1)],5)
 => 3
([(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)
 => 3
([(1,3),(1,4),(4,2)],5)
 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
([(2,3),(3,4)],5)
 => 3
([(1,4),(4,2),(4,3)],5)
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => 3
([(2,4),(3,4)],5)
 => 4
([(1,4),(2,4),(4,3)],5)
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => 4
([(1,4),(2,4),(3,4)],5)
 => 4
([(0,4),(1,4),(2,4),(4,3)],5)
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
([(0,4),(1,4),(2,3)],5)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => 4
Description
The number of maximal chains in a poset.
Matching statistic: St001304
(load all 9 compositions to match this statistic)
Mapping
Mp00198: Posets —incomparability graph⟶ Graphs
St001304: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => 1
([],2)
 => ([(0,1)],2)
 => 2
([(0,1)],2)
 => ([],2)
 => 1
([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => 2
([(0,2),(2,1)],3)
 => ([],3)
 => 1
([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 2
([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(1,2),(1,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,1),(0,2),(0,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 2
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 2
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => 2
([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => 4
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1
([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
([(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3)],5)
 => 4
([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,3)],5)
 => 4
([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,4),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 4
Description
The number of maximally independent sets of vertices of a graph. An '''independent set''' of vertices of a graph is a set of vertices no two of which are adjacent. If a set of vertices is independent then so is every subset. This statistic counts the number of maximally independent sets of vertices.
Matching statistic: St000010
Mapping
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
Mp00251: Graphs —clique sizes⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([],2)
 => ([(0,1)],2)
 => ([],2)
 => [1,1]
 => 2
([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => [2]
 => 1
([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => [1,1,1]
 => 3
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => 2
([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [2,2]
 => 2
([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [2,2]
 => 2
([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => [1,1,1,1]
 => 4
([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => [2,1,1]
 => 3
([(1,2),(1,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 3
([(0,1),(0,2),(0,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => 2
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 2
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => 2
([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 3
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => 2
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],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)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => 4
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => 2
([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => [1,1,1,1,1]
 => 5
([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => 4
([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 4
([(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 4
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,3]
 => 3
([(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,1]
 => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(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
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,3]
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [3,3,3,3]
 => 4
([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 3
([(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,1]
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,3]
 => 3
([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 4
([(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,1]
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [3,3,3,3]
 => 4
([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 4
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,3]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 4
([(0,4),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,2,2]
 => 4
Description
The length of the partition.
Matching statistic: St000741
Mapping
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00157: Graphs —connected complement⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 83%
Values
([],1)
 => ([],1)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
([],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,1)],2)
 => ([],2)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2} - 1
([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2} - 1
([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(1,2),(1,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,3,4} - 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,3,4} - 1
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,3,4} - 1
([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,3,4} - 1
([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? ∊ {2,2,2,2,3,4} - 1
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,3,4} - 1
([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 1 = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,4),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 2 - 1
([(0,4),(1,2),(1,3)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(3,4)],5)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
Description
The Colin de Verdière graph invariant.
Matching statistic: St001232
(load all 9 compositions to match this statistic)
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 83%
Values
([],1)
 => [1]
 => [1,0]
 => [1,0]
 => 0 = 1 - 1
([],2)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
([],3)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
([(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,1),(0,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([],4)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 4 - 1
([(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
([(1,2),(1,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
([(0,3),(1,2)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ? ∊ {2,3,4} - 1
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ? ∊ {2,3,4} - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ? ∊ {2,3,4} - 1
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([],5)
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 5 - 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 4 - 1
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 4 - 1
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 4 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 4 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 4 - 1
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 4 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 4 - 1
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(1,4),(2,3),(2,4)],5)
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,4),(1,2),(1,4),(2,3)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,4),(1,2),(1,4),(4,3)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,4),(1,2),(1,3)],5)
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,4),(1,2),(1,3),(1,4)],5)
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4)],5)
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,3),(1,2),(1,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(1,4),(3,2),(4,3)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(1,4),(4,2)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,4),(1,2),(2,3),(2,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
([(0,3),(1,2),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6} - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001183
(load all 3 compositions to match this statistic)
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001183: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [1,0]
 => 1
([],2)
 => [2]
 => [1,0,1,0]
 => 2
([(0,1)],2)
 => [1]
 => [1,0]
 => 1
([],3)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 2
([(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(0,1),(0,2)],3)
 => [2]
 => [1,0,1,0]
 => 2
([(0,2),(2,1)],3)
 => [1]
 => [1,0]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => [1,0,1,0]
 => 2
([],4)
 => [4,4,4,4,4,4]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => ? ∊ {3,3,3}
([(2,3)],4)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 2
([(1,2),(1,3)],4)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3}
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 2
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,0,1,0]
 => 2
([(1,2),(2,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,0,1,0]
 => 2
([(1,3),(2,3)],4)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3}
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,0,1,0]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 2
([(0,3),(1,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 4
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1,0]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
 => [1,1,1,1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(3,4)],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => [1,1,1,1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(2,3),(2,4)],5)
 => [10,10,10,10]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => [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]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,1),(0,2),(0,3),(0,4)],5)
 => [4,4,4,4,4,4]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 2
([(1,3),(1,4),(4,2)],5)
 => [15]
 => [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]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,3),(0,4),(4,1),(4,2)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [1,0,1,0]
 => 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
([(2,3),(3,4)],5)
 => [5,5,5,5]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 4
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 2
([(2,4),(3,4)],5)
 => [10,10,10,10]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(1,4),(2,4),(4,3)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 4
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
([(1,4),(2,4),(3,4)],5)
 => [15,15]
 => [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]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,4,4,4,4,4]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,3),(2,3),(2,4)],5)
 => [12,4]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,3),(1,4),(2,3),(2,4)],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]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6,6]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,0,1,0]
 => 2
([(0,4),(1,3),(2,3),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,4),(2,3),(2,4)],5)
 => [10,4,4]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 2
([(1,4),(2,3)],5)
 => [5,5,5,5,5,5]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,2),(1,4),(2,3)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 4
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
([(0,4),(1,2),(1,4),(4,3)],5)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,2),(1,3),(1,4)],5)
 => [10,4,4]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([(0,4),(1,2),(1,3),(3,4)],5)
 => [4,4,3]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,3),(0,4),(1,2),(1,4)],5)
 => [12,4]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,3),(0,4),(1,2),(1,3),(1,4)],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]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [6,6]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 4
([(0,3),(1,2),(1,4),(3,4)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 4
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
([(1,4),(3,2),(4,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,0,1,0]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [15]
 => [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]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 4
([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(0,4),(1,2),(2,3),(2,4)],5)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
Description
The maximum of projdim(S)+injdim(S) over all simple modules in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001258
(load all 3 compositions to match this statistic)
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001258: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [1,0]
 => 1
([],2)
 => [2]
 => [1,0,1,0]
 => 2
([(0,1)],2)
 => [1]
 => [1,0]
 => 1
([],3)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 2
([(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(0,1),(0,2)],3)
 => [2]
 => [1,0,1,0]
 => 2
([(0,2),(2,1)],3)
 => [1]
 => [1,0]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => [1,0,1,0]
 => 2
([],4)
 => [4,4,4,4,4,4]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => ? ∊ {3,3,3}
([(2,3)],4)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 2
([(1,2),(1,3)],4)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3}
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 2
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,0,1,0]
 => 2
([(1,2),(2,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,0,1,0]
 => 2
([(1,3),(2,3)],4)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3}
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,0,1,0]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 2
([(0,3),(1,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 4
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1,0]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
 => [1,1,1,1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(3,4)],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => [1,1,1,1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(2,3),(2,4)],5)
 => [10,10,10,10]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => [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]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,1),(0,2),(0,3),(0,4)],5)
 => [4,4,4,4,4,4]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 2
([(1,3),(1,4),(4,2)],5)
 => [15]
 => [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]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,3),(0,4),(4,1),(4,2)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [1,0,1,0]
 => 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
([(2,3),(3,4)],5)
 => [5,5,5,5]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 4
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 2
([(2,4),(3,4)],5)
 => [10,10,10,10]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(1,4),(2,4),(4,3)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 4
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
([(1,4),(2,4),(3,4)],5)
 => [15,15]
 => [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]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,4,4,4,4,4]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,3),(2,3),(2,4)],5)
 => [12,4]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,3),(1,4),(2,3),(2,4)],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]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6,6]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,0,1,0]
 => 2
([(0,4),(1,3),(2,3),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,4),(2,3),(2,4)],5)
 => [10,4,4]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 2
([(1,4),(2,3)],5)
 => [5,5,5,5,5,5]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,2),(1,4),(2,3)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 4
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
([(0,4),(1,2),(1,4),(4,3)],5)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,2),(1,3),(1,4)],5)
 => [10,4,4]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([(0,4),(1,2),(1,3),(3,4)],5)
 => [4,4,3]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,3),(0,4),(1,2),(1,4)],5)
 => [12,4]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,3),(0,4),(1,2),(1,3),(1,4)],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]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [6,6]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 4
([(0,3),(1,2),(1,4),(3,4)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 4
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
([(1,4),(3,2),(4,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,0,1,0]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [15]
 => [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]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 4
([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(0,4),(1,2),(2,3),(2,4)],5)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5}
Description
Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. For at most 6 simple modules this statistic coincides with the injective dimension of the regular module as a bimodule.
Matching statistic: St000676
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 67%
Values
([],1)
 => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
([],2)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,1)],2)
 => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
([],3)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3
([(1,2)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
([(0,1),(0,2)],3)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,2),(2,1)],3)
 => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([],4)
 => [4,4,4,4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
 => ? ∊ {2,3,4}
([(2,3)],4)
 => [4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => 4
([(1,2),(1,3)],4)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {2,3,4}
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(1,3),(2,3)],4)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {2,3,4}
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3
([(0,3),(1,2)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
([],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(3,4)],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(2,3),(2,4)],5)
 => [10,10,10,10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,1),(0,2),(0,3),(0,4)],5)
 => [4,4,4,4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3
([(1,3),(1,4),(4,2)],5)
 => [15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,3),(0,4),(4,1),(4,2)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
([(2,3),(3,4)],5)
 => [5,5,5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => 4
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3
([(2,4),(3,4)],5)
 => [10,10,10,10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(1,4),(2,4),(4,3)],5)
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => 4
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [15,15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,4,4,4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,3),(2,3),(2,4)],5)
 => [12,4]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,4),(1,3),(2,3),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,4),(2,3),(2,4)],5)
 => [10,4,4]
 => [1,1,1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => 4
([(1,4),(2,3)],5)
 => [5,5,5,5,5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,2),(1,4),(2,3)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 4
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => 4
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
([(0,4),(1,2),(1,4),(4,3)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,2),(1,3),(1,4)],5)
 => [10,4,4]
 => [1,1,1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3
([(0,4),(1,2),(1,3),(3,4)],5)
 => [4,4,3]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,3),(0,4),(1,2),(1,4)],5)
 => [12,4]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => 4
([(0,3),(1,2),(1,4),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 4
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => 4
([(1,4),(3,2),(4,3)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 3
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => 4
([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
([(0,4),(1,2),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
Description
The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength n with k up steps in odd positions and k returns to the main diagonal are counted by the binomial coefficient \binom{n-1}{k-1} [3,4].
Matching statistic: St001039
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 67%
Values
([],1)
 => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
([],2)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,1)],2)
 => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
([],3)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3
([(1,2)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
([(0,1),(0,2)],3)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,2),(2,1)],3)
 => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([],4)
 => [4,4,4,4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
 => ? ∊ {3,3,4}
([(2,3)],4)
 => [4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => 4
([(1,2),(1,3)],4)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {3,3,4}
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(1,3),(2,3)],4)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {3,3,4}
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3
([(0,3),(1,2)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
([],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(3,4)],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(2,3),(2,4)],5)
 => [10,10,10,10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,1),(0,2),(0,3),(0,4)],5)
 => [4,4,4,4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3
([(1,3),(1,4),(4,2)],5)
 => [15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,3),(0,4),(4,1),(4,2)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
([(2,3),(3,4)],5)
 => [5,5,5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => 4
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3
([(2,4),(3,4)],5)
 => [10,10,10,10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(1,4),(2,4),(4,3)],5)
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => 4
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
([(1,4),(2,4),(3,4)],5)
 => [15,15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,4,4,4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,3),(2,3),(2,4)],5)
 => [12,4]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,4),(1,3),(2,3),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,4),(2,3),(2,4)],5)
 => [10,4,4]
 => [1,1,1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => 4
([(1,4),(2,3)],5)
 => [5,5,5,5,5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,2),(1,4),(2,3)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 4
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => 4
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
([(0,4),(1,2),(1,4),(4,3)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,2),(1,3),(1,4)],5)
 => [10,4,4]
 => [1,1,1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3
([(0,4),(1,2),(1,3),(3,4)],5)
 => [4,4,3]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,3),(0,4),(1,2),(1,4)],5)
 => [12,4]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => 4
([(0,3),(1,2),(1,4),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 4
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => 4
([(1,4),(3,2),(4,3)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 3
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => 4
([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
([(0,4),(1,2),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ? ∊ {2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6}
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St001068
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 67%
Values
([],1)
 => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
([],2)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,1)],2)
 => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
([],3)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3
([(1,2)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
([(0,1),(0,2)],3)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,2),(2,1)],3)
 => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([],4)
 => [4,4,4,4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
 => ? ∊ {2,3,4}
([(2,3)],4)
 => [4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => 4
([(1,2),(1,3)],4)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {2,3,4}
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(1,3),(2,3)],4)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {2,3,4}
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3
([(0,3),(1,2)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
([],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(3,4)],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(2,3),(2,4)],5)
 => [10,10,10,10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,1),(0,2),(0,3),(0,4)],5)
 => [4,4,4,4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3
([(1,3),(1,4),(4,2)],5)
 => [15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,3),(0,4),(4,1),(4,2)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
([(2,3),(3,4)],5)
 => [5,5,5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => 4
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3
([(2,4),(3,4)],5)
 => [10,10,10,10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(1,4),(2,4),(4,3)],5)
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => 4
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [15,15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,4,4,4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,1,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,3),(2,3),(2,4)],5)
 => [12,4]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,1,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,4),(1,3),(2,3),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,4),(2,3),(2,4)],5)
 => [10,4,4]
 => [1,1,1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,1,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => 4
([(1,4),(2,3)],5)
 => [5,5,5,5,5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0,1,1,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,2),(1,4),(2,3)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 3
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => 4
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
([(0,4),(1,2),(1,4),(4,3)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,1,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,2),(1,3),(1,4)],5)
 => [10,4,4]
 => [1,1,1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,1,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
([(0,4),(1,2),(1,3),(3,4)],5)
 => [4,4,3]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,3),(0,4),(1,2),(1,4)],5)
 => [12,4]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,1,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => 4
([(0,3),(1,2),(1,4),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 3
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => 4
([(1,4),(3,2),(4,3)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 3
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => 4
([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
([(0,4),(1,2),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6}
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
The following 170 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001471The magnitude of a Dyck path. St000053The number of valleys of the Dyck path. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between e_i J and e_j J (the radical of the indecomposable projective modules). St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000259The diameter of a connected graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St000015The number of peaks of a Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St000120The number of left tunnels of a Dyck path. St000144The pyramid weight of the Dyck path. St000331The number of upper interactions of a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules S with grade \inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \} at least two in the Nakayama algebra A corresponding to the Dyck path. St001194The injective dimension of A/AfA in the corresponding Nakayama algebra A when Af is the minimal faithful projective-injective left A-module St001205The number of non-simple indecomposable projective-injective modules of the algebra eAe in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000456The monochromatic index of a connected graph. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001613The binary logarithm of the size of the center of a lattice. St001617The dimension of the space of valuations of a lattice. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001480The number of simple summands of the module J^2/J^3. St001180Number of indecomposable injective modules with projective dimension at most 1. St001720The minimal length of a chain of small intervals in a lattice. St001060The distinguishing index of a graph. St001199The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001200The number of simple modules in eAe with projective dimension at most 2 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St000264The girth of a graph, which is not a tree. St001488The number of corners of a skew partition. St000098The chromatic number of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000271The chromatic index of a graph. St001198The number of simple modules in the algebra eAe with projective dimension at most 1 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001206The maximal dimension of an indecomposable projective eAe-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module eA. St000455The second largest eigenvalue of a graph if it is integral. St001875The number of simple modules with projective dimension at most 1. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000656The number of cuts of a poset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001570The minimal number of edges to add to make a graph Hamiltonian. St001118The acyclic chromatic index of a graph. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000172The Grundy number of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000363The number of minimal vertex covers of a graph. St000387The matching number of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000482The (zero)-forcing number of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000778The metric dimension of a graph. St000785The number of distinct colouring schemes of a graph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001270The bandwidth of a graph. St001271The competition number of a graph. St001277The degeneracy of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001725The harmonious chromatic number of a graph. St001792The arboricity of a graph. St001829The common independence number of a graph. St001883The mutual visibility number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001962The proper pathwidth of a graph. St001963The tree-depth of a graph. St000171The degree of the graph. St000469The distinguishing number of a graph. St000535The rank-width of a graph. St000552The number of cut vertices of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001305The number of induced cycles on four vertices in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001333The cardinality of a minimal edge-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001366The maximal multiplicity of a degree of a vertex of a graph. St001393The induced matching number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001638The book thickness of a graph. St001644The dimension of a graph. St001689The number of celebrities in a graph. St001691The number of kings in a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001743The discrepancy of a graph. St001746The coalition number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000632The jump number of the poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000907The number of maximal antichains of minimal length in a poset. St001195The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St000307The number of rowmotion orbits of a poset. St000450The number of edges minus the number of vertices plus 2 of a graph. St000822The Hadwiger number of the graph. St001330The hat guessing number of a graph. St001734The lettericity of a graph. St000315The number of isolated vertices of a graph. St001117The game chromatic index of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001642The Prague dimension of a graph. St001742The difference of the maximal and the minimal degree in a graph. St001812The biclique partition number of a graph. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001568The smallest positive integer that does not appear twice in the partition.