Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001498
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],2)
 => [2] => [1,1] => [1,0,1,0]
 => 1
([],3)
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 1
([(1,2)],3)
 => [1,2] => [1,2] => [1,0,1,1,0,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [2,1] => [1,1,0,0,1,0]
 => 2
([],4)
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
([(2,3)],4)
 => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
([(1,3),(2,3)],4)
 => [1,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
([(0,3),(1,2)],4)
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
([],5)
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
([(3,4)],5)
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
([(2,4),(3,4)],5)
 => [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
([(1,4),(2,3)],5)
 => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2
([],6)
 => [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1
([(4,5)],6)
 => [1,5] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
([(3,5),(4,5)],6)
 => [1,1,4] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [1,2,3] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,3,2] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 2
([(2,5),(3,4)],6)
 => [2,4] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 2
([(2,5),(3,4),(4,5)],6)
 => [1,1,1,3] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 1
([(1,2),(3,5),(4,5)],6)
 => [1,1,1,3] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 1
([(3,4),(3,5),(4,5)],6)
 => [2,4] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 2
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,2] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,2,2] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,3] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 1
Description
The normalised height of a Nakayama algebra with magnitude 1. We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St001933
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 83%
Values
([],2)
 => [2] => [[2],[]]
 => []
 => ? = 1 - 1
([],3)
 => [3] => [[3],[]]
 => []
 => ? = 1 - 1
([(1,2)],3)
 => [1,2] => [[2,1],[]]
 => []
 => ? = 1 - 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [[2,2],[1]]
 => [1]
 => 1 = 2 - 1
([],4)
 => [4] => [[4],[]]
 => []
 => ? = 1 - 1
([(2,3)],4)
 => [1,3] => [[3,1],[]]
 => []
 => ? = 1 - 1
([(1,3),(2,3)],4)
 => [1,1,2] => [[2,1,1],[]]
 => []
 => ? = 1 - 1
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => [2,2] => [[3,2],[1]]
 => [1]
 => 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [[3,2],[1]]
 => [1]
 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [[3,3],[2]]
 => [2]
 => 1 = 2 - 1
([],5)
 => [5] => [[5],[]]
 => []
 => ? = 1 - 1
([(3,4)],5)
 => [1,4] => [[4,1],[]]
 => []
 => ? = 1 - 1
([(2,4),(3,4)],5)
 => [1,1,3] => [[3,1,1],[]]
 => []
 => ? = 1 - 1
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 1 = 2 - 1
([(1,4),(2,3)],5)
 => [2,3] => [[4,2],[1]]
 => [1]
 => 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ? = 1 - 1
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ? = 1 - 1
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [[4,2],[1]]
 => [1]
 => 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ? = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [[4,3],[2]]
 => [2]
 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 3 = 4 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 2 = 3 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [[4,4],[3]]
 => [3]
 => 1 = 2 - 1
([],6)
 => [6] => [[6],[]]
 => []
 => ? = 1 - 1
([(4,5)],6)
 => [1,5] => [[5,1],[]]
 => []
 => ? = 1 - 1
([(3,5),(4,5)],6)
 => [1,1,4] => [[4,1,1],[]]
 => []
 => ? = 1 - 1
([(2,5),(3,5),(4,5)],6)
 => [1,2,3] => [[4,2,1],[1]]
 => [1]
 => 1 = 2 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,3,2] => [[4,3,1],[2]]
 => [2]
 => 1 = 2 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,4,1] => [[4,4,1],[3]]
 => [3]
 => 1 = 2 - 1
([(2,5),(3,4)],6)
 => [2,4] => [[5,2],[1]]
 => [1]
 => 1 = 2 - 1
([(2,5),(3,4),(4,5)],6)
 => [1,1,1,3] => [[3,1,1,1],[]]
 => []
 => ? = 1 - 1
([(1,2),(3,5),(4,5)],6)
 => [1,1,1,3] => [[3,1,1,1],[]]
 => []
 => ? = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => [2,4] => [[5,2],[1]]
 => [1]
 => 1 = 2 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ? = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => 1 = 2 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,3] => [[3,1,1,1],[]]
 => []
 => ? = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 2 = 3 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => 1 = 2 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => 1 = 2 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,3] => [[4,2,1],[1]]
 => [1]
 => 1 = 2 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 1 = 2 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ? = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 2 = 3 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ? = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 2 = 3 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ? = 1 - 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => 1 = 2 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => 1 = 2 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 2 = 3 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 1 = 2 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 2 = 3 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => 1 = 2 - 1
([(0,5),(1,4),(2,3)],6)
 => [3,3] => [[5,3],[2]]
 => [2]
 => 1 = 2 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ? = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => 2 = 3 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 2 = 3 - 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ? = 1 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ? = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => 2 = 3 - 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => 1 = 2 - 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ? = 1 - 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ? = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ? = 1 - 1
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ? = 1 - 1
([],7)
 => [7] => [[7],[]]
 => []
 => ? = 1 - 1
([(5,6)],7)
 => [1,6] => [[6,1],[]]
 => []
 => ? = 1 - 1
([(4,6),(5,6)],7)
 => [1,1,5] => [[5,1,1],[]]
 => []
 => ? = 1 - 1
([(3,6),(4,5),(5,6)],7)
 => [1,1,1,4] => [[4,1,1,1],[]]
 => []
 => ? = 1 - 1
([(2,3),(4,6),(5,6)],7)
 => [1,1,1,4] => [[4,1,1,1],[]]
 => []
 => ? = 1 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => [1,1,1,1,3] => [[3,1,1,1,1],[]]
 => []
 => ? = 1 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,4] => [[4,1,1,1],[]]
 => []
 => ? = 1 - 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => [1,1,1,1,3] => [[3,1,1,1,1],[]]
 => []
 => ? = 1 - 1
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => [1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
 => []
 => ? = 1 - 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,3] => [[3,1,1,1,1],[]]
 => []
 => ? = 1 - 1
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
 => []
 => ? = 1 - 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,3] => [[3,1,1,1,1],[]]
 => []
 => ? = 1 - 1
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
 => []
 => ? = 1 - 1
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => [1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
 => []
 => ? = 1 - 1
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
 => []
 => ? = 1 - 1
([(2,6),(3,5),(4,5),(4,6)],7)
 => [1,1,1,1,3] => [[3,1,1,1,1],[]]
 => []
 => ? = 1 - 1
([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
 => []
 => ? = 1 - 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
 => [1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
 => []
 => ? = 1 - 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => [1,1,1,1,3] => [[3,1,1,1,1],[]]
 => []
 => ? = 1 - 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,3] => [[3,1,1,1,1],[]]
 => []
 => ? = 1 - 1
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
 => []
 => ? = 1 - 1
Description
The largest multiplicity of a part in an integer partition.
Matching statistic: St001194
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001194: Dyck paths ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 83%
Values
([],2)
 => [2] => [1,1,0,0]
 => 1
([],3)
 => [3] => [1,1,1,0,0,0]
 => 1
([(1,2)],3)
 => [1,2] => [1,0,1,1,0,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => 2
([],4)
 => [4] => [1,1,1,1,0,0,0,0]
 => 1
([(2,3)],4)
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
([(1,3),(2,3)],4)
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
([(0,3),(1,2)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
([],5)
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
([(3,4)],5)
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
([(2,4),(3,4)],5)
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
([(1,4),(2,3)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2
([],6)
 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
([(4,5)],6)
 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
([(3,5),(4,5)],6)
 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 2
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2
([(2,5),(3,4)],6)
 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2
([(2,5),(3,4),(4,5)],6)
 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1
([(1,2),(3,5),(4,5)],6)
 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1
([(3,4),(3,5),(4,5)],6)
 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1
([],7)
 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
([(5,6)],7)
 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
([(4,6),(5,6)],7)
 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 1
([(3,6),(4,6),(5,6)],7)
 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
([(2,6),(3,6),(4,6),(5,6)],7)
 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 2
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 2
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2
([(3,6),(4,5)],7)
 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
([(3,6),(4,5),(5,6)],7)
 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1
([(2,3),(4,6),(5,6)],7)
 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1
([(4,5),(4,6),(5,6)],7)
 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
([(2,6),(3,6),(4,5),(5,6)],7)
 => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2
([(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2
([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 2
([(3,5),(3,6),(4,5),(4,6)],7)
 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
([(1,6),(2,6),(3,5),(4,5)],7)
 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1
([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3
Description
The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module