Your data matches 144 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001581
(load all 5 compositions to match this statistic)
Mapping
Mp00198: Posets —incomparability graph⟶ Graphs
St001581: 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
([(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
([(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
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],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)
 => 2
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1
([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],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)
 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,3)],5)
 => 2
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 3
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(1,4),(2,3)],5)
 => 2
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(2,4),(3,4)],5)
 => 2
([(1,4),(3,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(3,4)],5)
 => 2
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 2
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 2
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => ([(2,5),(3,4)],6)
 => 2
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 2
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => ([(2,5),(3,4)],6)
 => 2
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(4,5)],6)
 => 2
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => ([(2,5),(3,4)],6)
 => 2
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 3
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 3
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(3,5),(4,5)],6)
 => 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => 2
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => 2
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => ([(4,5)],6)
 => 2
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 2
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
 => ([(2,5),(3,4)],6)
 => 2
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 3
Description
The achromatic number of a graph. This is the maximal number of colours of a proper colouring, such that for any pair of colours there are two adjacent vertices with these colours.
Matching statistic: St001488
(load all 3 compositions to match this statistic)
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [[1],[]]
 => 1
([],2)
 => [2]
 => [[2],[]]
 => 2
([(0,1)],2)
 => [1]
 => [[1],[]]
 => 1
([(1,2)],3)
 => [3]
 => [[3],[]]
 => 2
([(0,1),(0,2)],3)
 => [2]
 => [[2],[]]
 => 2
([(0,2),(2,1)],3)
 => [1]
 => [[1],[]]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => [[2],[]]
 => 2
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [[3],[]]
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [[2],[]]
 => 2
([(1,2),(2,3)],4)
 => [4]
 => [[4],[]]
 => 2
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [[2],[]]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [[2],[]]
 => 2
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [[3,2],[]]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [[2,2],[]]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [[1],[]]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [[3],[]]
 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [[2],[]]
 => 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [[3,2],[]]
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [[2,2],[]]
 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [[2,2],[]]
 => 2
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [[2],[]]
 => 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => [[3,2],[]]
 => 3
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [[2,2],[]]
 => 2
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [[4],[]]
 => 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [[3],[]]
 => 2
([(1,4),(3,2),(4,3)],5)
 => [5]
 => [[5],[]]
 => 2
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [[2],[]]
 => 2
([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [[3],[]]
 => 2
([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [[3],[]]
 => 2
([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [[1],[]]
 => 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [[4],[]]
 => 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [2]
 => [[2],[]]
 => 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [2,2]
 => [[2,2],[]]
 => 2
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [2,2]
 => [[2,2],[]]
 => 2
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [2,2]
 => [[2,2],[]]
 => 2
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [2]
 => [[2],[]]
 => 2
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => [2,2]
 => [[2,2],[]]
 => 2
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => [3,2]
 => [[3,2],[]]
 => 3
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => [2,2]
 => [[2,2],[]]
 => 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [3,2]
 => [[3,2],[]]
 => 3
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => [3]
 => [[3],[]]
 => 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => [4]
 => [[4],[]]
 => 2
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [2]
 => [[2],[]]
 => 2
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => [5]
 => [[5],[]]
 => 2
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => [2]
 => [[2],[]]
 => 2
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
 => [4]
 => [[4],[]]
 => 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [2]
 => [[2],[]]
 => 2
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [4]
 => [[4],[]]
 => 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
 => [2,2]
 => [[2,2],[]]
 => 2
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => [3,2]
 => [[3,2],[]]
 => 3
Description
The number of corners of a skew partition. This is also known as the number of removable cells of the skew partition.
Matching statistic: St000688
(load all 5 compositions to match this statistic)
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000688: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
([],2)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,1)],2)
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
([(1,2)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,1),(0,2)],3)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(2,1)],3)
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
([(0,2),(1,2)],3)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(1,4),(3,2),(4,3)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
Description
The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. The global dimension is given by [[St000684]] and the dominant dimension is given by [[St000685]]. To every Dyck path there is an LNakayama algebra associated as described in [[St000684]]. Dyck paths for which the global dimension and the dominant dimension of the the LNakayama algebra coincide and both dimensions at least $2$ correspond to the LNakayama algebras that are higher Auslander algebras in the sense of [1].
Matching statistic: St000970
(load all 5 compositions to match this statistic)
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000970: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
([],2)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,1)],2)
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
([(1,2)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,1),(0,2)],3)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(2,1)],3)
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
([(0,2),(1,2)],3)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(1,4),(3,2),(4,3)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
Description
Number of peaks minus the dominant dimension of the corresponding LNakayama algebra.
Matching statistic: St001026
(load all 5 compositions to match this statistic)
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
([],2)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,1)],2)
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
([(1,2)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,1),(0,2)],3)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(2,1)],3)
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
([(0,2),(1,2)],3)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(1,4),(3,2),(4,3)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
Description
The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001183
(load all 6 compositions to match this statistic)
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001183: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [1,0,1,0]
 => 2 = 1 + 1
([],2)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,1)],2)
 => [1]
 => [1,0,1,0]
 => 2 = 1 + 1
([(1,2)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
([(0,1),(0,2)],3)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,2),(2,1)],3)
 => [1]
 => [1,0,1,0]
 => 2 = 1 + 1
([(0,2),(1,2)],3)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1,0,1,0]
 => 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3 = 2 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
([(1,4),(3,2),(4,3)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1,0,1,0]
 => 2 = 1 + 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3 = 2 + 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 3 = 2 + 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
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 6 compositions to match this statistic)
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001258: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [1,0,1,0]
 => 2 = 1 + 1
([],2)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,1)],2)
 => [1]
 => [1,0,1,0]
 => 2 = 1 + 1
([(1,2)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
([(0,1),(0,2)],3)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,2),(2,1)],3)
 => [1]
 => [1,0,1,0]
 => 2 = 1 + 1
([(0,2),(1,2)],3)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1,0,1,0]
 => 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3 = 2 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
([(1,4),(3,2),(4,3)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1,0,1,0]
 => 2 = 1 + 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3 = 2 + 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 3 = 2 + 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [2]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
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: St000318
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [1]
 => []
 => 1
([],2)
 => [2]
 => [1,1]
 => [1]
 => 2
([(0,1)],2)
 => [1]
 => [1]
 => []
 => 1
([(1,2)],3)
 => [3]
 => [1,1,1]
 => [1,1]
 => 2
([(0,1),(0,2)],3)
 => [2]
 => [1,1]
 => [1]
 => 2
([(0,2),(2,1)],3)
 => [1]
 => [1]
 => []
 => 1
([(0,2),(1,2)],3)
 => [2]
 => [1,1]
 => [1]
 => 2
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,1,1]
 => [1,1]
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,1]
 => [1]
 => 2
([(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 2
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,1]
 => [1]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,1]
 => [1]
 => 2
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [2,2]
 => [2]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1]
 => []
 => 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1]
 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [1,1]
 => [1]
 => 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [2,2]
 => [2]
 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [2,2]
 => [2]
 => 2
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,1]
 => [1]
 => 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 3
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [2,2]
 => [2]
 => 2
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,1,1]
 => [1,1]
 => 2
([(1,4),(3,2),(4,3)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 2
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,1]
 => [1]
 => 2
([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,1,1]
 => [1,1]
 => 2
([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,1,1]
 => [1,1]
 => 2
([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1]
 => []
 => 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [2]
 => [1,1]
 => [1]
 => 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [2,2]
 => [2,2]
 => [2]
 => 2
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [2,2]
 => [2,2]
 => [2]
 => 2
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [2,2]
 => [2,2]
 => [2]
 => 2
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [2]
 => [1,1]
 => [1]
 => 2
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => [2,2]
 => [2,2]
 => [2]
 => 2
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 3
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => [2,2]
 => [2,2]
 => [2]
 => 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 3
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => [3]
 => [1,1,1]
 => [1,1]
 => 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 2
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [2]
 => [1,1]
 => [1]
 => 2
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 2
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => [2]
 => [1,1]
 => [1]
 => 2
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [2]
 => [1,1]
 => [1]
 => 2
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
 => [2,2]
 => [2,2]
 => [2]
 => 2
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 3
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St000476
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
([],2)
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
([(0,1)],2)
 => [1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
([(1,2)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
([(0,1),(0,2)],3)
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
([(0,2),(2,1)],3)
 => [1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
([(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
([(1,4),(3,2),(4,3)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 2
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 2
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3
Description
The sum of the semi-lengths of tunnels before a valley of a Dyck path. For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is $$ \sum_v (j_v-i_v)/2. $$
Matching statistic: St001239
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001239: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
([(1,2)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 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
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 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]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 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]
 => [1,0,1,0,1,1,0,0]
 => 2
([(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
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 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
([(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
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 3
([(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
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
([(1,4),(3,2),(4,3)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => [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),(2,5),(4,5),(5,3)],6)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 3
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 3
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 3
Description
The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra.
The following 134 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000159The number of distinct parts of the integer partition. St000340The number of non-final maximal constant sub-paths of length greater than one. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-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. St001584The area statistic between a Dyck path and its bounce path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000172The Grundy number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St000453The number of distinct Laplacian eigenvalues of a graph. St001093The detour number of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000171The degree of the graph. St001117The game chromatic index of a graph. St001315The dissociation number of a graph. St001587Half of the largest even part of an integer partition. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules 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. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001498The normalised height of a Nakayama algebra with magnitude 1. St001432The order dimension of the partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000944The 3-degree of an integer partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001586The number of odd parts smaller than the largest even part in an integer partition. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000388The number of orbits of vertices of a graph under automorphisms. St001118The acyclic chromatic index of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001734The lettericity of a graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001917The order of toric promotion on the set of labellings of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000448The number of pairs of vertices of a graph with distance 2. St000552The number of cut vertices of a graph. St001308The number of induced paths on three vertices in a graph. St001323The independence gap of a graph. St001350Half of the Albertson index of a graph. St001521Half the total irregularity 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. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001689The number of celebrities in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001742The difference of the maximal and the minimal degree in a graph. St001764The number of non-convex subsets of vertices in a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St000454The largest eigenvalue of a graph if it is integral. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000455The second largest eigenvalue of a graph if it is integral. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000299The number of nonisomorphic vertex-induced subtrees. St001512The minimum rank of a graph. St001642The Prague dimension of a graph. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. 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 $eA$. St000699The toughness times the least common multiple of 1,. St001570The minimal number of edges to add to make a graph Hamiltonian. St001335The cardinality of a minimal cycle-isolating set of a graph. St001624The breadth of a lattice. St000272The treewidth of a graph. St000464The Schultz index of a connected graph. St000536The pathwidth of a graph. St000544The cop number of a graph. St001029The size of the core of a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001494The Alon-Tarsi number of a graph. St001545The second Elser number of a connected graph. St001580The acyclic chromatic number of a graph. St001792The arboricity of a graph. St001883The mutual visibility number of a graph. St001962The proper pathwidth of a graph. St000535The rank-width of a graph. St000537The cutwidth of a graph. St001331The size of the minimal feedback vertex set. 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. St001743The discrepancy of a graph. St001746The coalition number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000264The girth of a graph, which is not a tree. St000298The order dimension or Dushnik-Miller dimension of 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. St000640The rank of the largest boolean interval in a poset. St000307The number of rowmotion orbits of a poset. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000822The Hadwiger number of the graph. St001330The hat guessing number of a graph. St001060The distinguishing index of a graph.