- FindStat

Your data matches 30 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001604

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1,2,3,4}}
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
{{1,2,3,4,5}}
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
{{1,2,3,4},{5}}
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
{{1,2,3,5},{4}}
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
{{1,2,3},{4,5}}
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 0
{{1,2,4,5},{3}}
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
{{1,2,4},{3,5}}
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 0
{{1,2,5},{3,4}}
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 0
{{1,2},{3,4,5}}
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 0
{{1,3,4,5},{2}}
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
{{1,3,4},{2,5}}
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 0
{{1,3,5},{2,4}}
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 0
{{1,3},{2,4,5}}
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 0
{{1,4,5},{2,3}}
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 0
{{1,4},{2,3,5}}
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 0
{{1,5},{2,3,4}}
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 0
{{1},{2,3,4,5}}
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
{{1,2,3,4,5,6}}
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
{{1,2,3,4,5},{6}}
 => [5,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
{{1,2,3,4,6},{5}}
 => [5,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
{{1,2,3,4},{5,6}}
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 0
{{1,2,3,4},{5},{6}}
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
{{1,2,3,5,6},{4}}
 => [5,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
{{1,2,3,5},{4,6}}
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 0
{{1,2,3,5},{4},{6}}
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
{{1,2,3,6},{4,5}}
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 0
{{1,2,3},{4,5,6}}
 => [3,3]
 => [2,2,2]
 => [2,2]
 => 1
{{1,2,3},{4,5},{6}}
 => [3,2,1]
 => [3,2,1]
 => [2,1]
 => 0
{{1,2,3,6},{4},{5}}
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
{{1,2,3},{4,6},{5}}
 => [3,2,1]
 => [3,2,1]
 => [2,1]
 => 0
{{1,2,3},{4},{5,6}}
 => [3,2,1]
 => [3,2,1]
 => [2,1]
 => 0
{{1,2,4,5,6},{3}}
 => [5,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
{{1,2,4,5},{3,6}}
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 0
{{1,2,4,5},{3},{6}}
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
{{1,2,4,6},{3,5}}
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 0
{{1,2,4},{3,5,6}}
 => [3,3]
 => [2,2,2]
 => [2,2]
 => 1
{{1,2,4},{3,5},{6}}
 => [3,2,1]
 => [3,2,1]
 => [2,1]
 => 0
{{1,2,4,6},{3},{5}}
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
{{1,2,4},{3,6},{5}}
 => [3,2,1]
 => [3,2,1]
 => [2,1]
 => 0
{{1,2,4},{3},{5,6}}
 => [3,2,1]
 => [3,2,1]
 => [2,1]
 => 0
{{1,2,5,6},{3,4}}
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 0
{{1,2,5},{3,4,6}}
 => [3,3]
 => [2,2,2]
 => [2,2]
 => 1
{{1,2,5},{3,4},{6}}
 => [3,2,1]
 => [3,2,1]
 => [2,1]
 => 0
{{1,2,6},{3,4,5}}
 => [3,3]
 => [2,2,2]
 => [2,2]
 => 1
{{1,2},{3,4,5,6}}
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 0
{{1,2},{3,4,5},{6}}
 => [3,2,1]
 => [3,2,1]
 => [2,1]
 => 0
{{1,2,6},{3,4},{5}}
 => [3,2,1]
 => [3,2,1]
 => [2,1]
 => 0
{{1,2},{3,4,6},{5}}
 => [3,2,1]
 => [3,2,1]
 => [2,1]
 => 0
{{1,2},{3,4},{5,6}}
 => [2,2,2]
 => [3,3]
 => [3]
 => 1
{{1,2,5,6},{3},{4}}
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0

Description

The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.

Matching statistic: St001001

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%

Values

{{1,2,3,4}}
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 0
{{1,2,3,4,5}}
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 0
{{1,2,3,4},{5}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
{{1,2,3,5},{4}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
{{1,2,3},{4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 0
{{1,2,4,5},{3}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
{{1,2,4},{3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 0
{{1,2,5},{3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 0
{{1,2},{3,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 0
{{1,3,4,5},{2}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
{{1,3,4},{2,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 0
{{1,3,5},{2,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 0
{{1,3},{2,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 0
{{1,4,5},{2,3}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 0
{{1,4},{2,3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 0
{{1,5},{2,3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 0
{{1},{2,3,4,5}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
{{1,2,3,4,5,6}}
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 1
{{1,2,3,4,5},{6}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 0
{{1,2,3,4,6},{5}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 0
{{1,2,3,4},{5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0
{{1,2,3,4},{5},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 0
{{1,2,3,5,6},{4}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 0
{{1,2,3,5},{4,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0
{{1,2,3,5},{4},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 0
{{1,2,3,6},{4,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0
{{1,2,3},{4,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 1
{{1,2,3},{4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,2,3,6},{4},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 0
{{1,2,3},{4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,2,3},{4},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,2,4,5,6},{3}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 0
{{1,2,4,5},{3,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0
{{1,2,4,5},{3},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 0
{{1,2,4,6},{3,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0
{{1,2,4},{3,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 1
{{1,2,4},{3,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,2,4,6},{3},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 0
{{1,2,4},{3,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,2,4},{3},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,2,5,6},{3,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0
{{1,2,5},{3,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 1
{{1,2,5},{3,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,2,6},{3,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 1
{{1,2},{3,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0
{{1,2},{3,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,2,6},{3,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,2},{3,4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,2},{3,4},{5,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1
{{1,2,5,6},{3},{4}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 0
{{1,2,5},{3,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,2,5},{3},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,2,6},{3,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,2},{3,5,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,2},{3,5},{4,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1
{{1,2,6},{3},{4,5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,2},{3,6},{4,5}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1
{{1,2},{3},{4,5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,3,4,5,6},{2}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 0
{{1,3,4,5},{2,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0
{{1,3,4,5},{2},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 0
{{1,3,4,6},{2,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0
{{1,3,4},{2,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 1
{{1,3,4},{2,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,3,4,6},{2},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 0
{{1,3,4},{2,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,3,4},{2},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,3,5,6},{2,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0
{{1,3,5},{2,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 1
{{1,3,5},{2,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,3,6},{2,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 1
{{1,3},{2,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0
{{1,3},{2,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,3,6},{2,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,3},{2,4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,3},{2,4},{5,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1
{{1,3,5,6},{2},{4}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 0
{{1,3,5},{2,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,3,5},{2},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,3,6},{2,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,3},{2,5,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,3},{2,5},{4,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1
{{1,3,6},{2},{4,5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,3},{2,6},{4,5}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1
{{1,3},{2},{4,5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,4,5,6},{2,3}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0
{{1,4,5},{2,3,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 1
{{1,4,5},{2,3},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,4,6},{2,3,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 1
{{1,4},{2,3,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0
{{1,4},{2,3,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,4,6},{2,3},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,4},{2,3,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,5},{2,3,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,6},{2,3,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1},{2,3,4},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,5,6},{2,3},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,5},{2,3,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1,6},{2,3,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
{{1},{2,3,5},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0

Description

The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001371

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%

Values

{{1,2,3,4}}
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => ? = 0
{{1,2,3,4,5}}
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 111110000010 => ? = 0
{{1,2,3,4},{5}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
{{1,2,3,5},{4}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
{{1,2,3},{4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0
{{1,2,4,5},{3}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
{{1,2,4},{3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0
{{1,2,5},{3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0
{{1,2},{3,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0
{{1,3,4,5},{2}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
{{1,3,4},{2,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0
{{1,3,5},{2,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0
{{1,3},{2,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0
{{1,4,5},{2,3}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0
{{1,4},{2,3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0
{{1,5},{2,3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0
{{1},{2,3,4,5}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
{{1,2,3,4,5,6}}
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 11111100000010 => ? = 1
{{1,2,3,4,5},{6}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => ? = 0
{{1,2,3,4,6},{5}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => ? = 0
{{1,2,3,4},{5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
{{1,2,3,4},{5},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0
{{1,2,3,5,6},{4}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => ? = 0
{{1,2,3,5},{4,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
{{1,2,3,5},{4},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0
{{1,2,3,6},{4,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
{{1,2,3},{4,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 1
{{1,2,3},{4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2,3,6},{4},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0
{{1,2,3},{4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2,3},{4},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2,4,5,6},{3}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => ? = 0
{{1,2,4,5},{3,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
{{1,2,4,5},{3},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0
{{1,2,4,6},{3,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
{{1,2,4},{3,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 1
{{1,2,4},{3,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2,4,6},{3},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0
{{1,2,4},{3,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2,4},{3},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2,5,6},{3,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
{{1,2,5},{3,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 1
{{1,2,5},{3,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2,6},{3,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 1
{{1,2},{3,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
{{1,2},{3,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2,6},{3,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2},{3,4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2},{3,4},{5,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => ? = 1
{{1,2,5,6},{3},{4}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0
{{1,2,5},{3,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2,5},{3},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2,6},{3,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2},{3,5,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2},{3,5},{4,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => ? = 1
{{1,2,6},{3},{4,5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2},{3,6},{4,5}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => ? = 1
{{1,2},{3},{4,5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,3,4,5,6},{2}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => ? = 0
{{1,3,4,5},{2,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
{{1,3,4,5},{2},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0
{{1,3,4,6},{2,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
{{1,3,4},{2,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 1
{{1,3,4},{2,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,3,4,6},{2},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0
{{1,3,4},{2,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,3,4},{2},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,3,5,6},{2,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
{{1,3,5},{2,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 1
{{1,3,5},{2,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,3,6},{2,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 1
{{1,3},{2,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
{{1,3},{2,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,3,6},{2,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,3},{2,4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,3},{2,4},{5,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => ? = 1
{{1,3,5,6},{2},{4}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0
{{1,3,5},{2,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,3,5},{2},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,3,6},{2,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,3},{2,5,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,3},{2,5},{4,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => ? = 1
{{1,3,6},{2},{4,5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,3},{2,6},{4,5}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => ? = 1
{{1,3},{2},{4,5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,4,5,6},{2,3}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
{{1,4,5},{2,3,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 1
{{1,4,5},{2,3},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,4,6},{2,3,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 1
{{1,4},{2,3,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
{{1,4},{2,3,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,4,6},{2,3},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,4},{2,3,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,5},{2,3,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,6},{2,3,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1},{2,3,4},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,5,6},{2,3},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,5},{2,3,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,6},{2,3,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1},{2,3,5},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0

Description

The length of the longest Yamanouchi prefix of a binary word. This is the largest index

$i$ such that in each of the prefixes

$w_1$ ,

$w_1w_2$ ,

$w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.

Matching statistic: St001556

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001556: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%

Values

{{1,2,3,4}}
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => ? = 0
{{1,2,3,4,5}}
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => ? = 0
{{1,2,3,4},{5}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ? = 0
{{1,2,3,5},{4}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ? = 0
{{1,2,3},{4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0
{{1,2,4,5},{3}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ? = 0
{{1,2,4},{3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0
{{1,2,5},{3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0
{{1,2},{3,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0
{{1,3,4,5},{2}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ? = 0
{{1,3,4},{2,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0
{{1,3,5},{2,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0
{{1,3},{2,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0
{{1,4,5},{2,3}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0
{{1,4},{2,3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0
{{1,5},{2,3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0
{{1},{2,3,4,5}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ? = 0
{{1,2,3,4,5,6}}
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,8,1,7] => ? = 1
{{1,2,3,4,5},{6}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => ? = 0
{{1,2,3,4,6},{5}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => ? = 0
{{1,2,3,4},{5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0
{{1,2,3,4},{5},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0
{{1,2,3,5,6},{4}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => ? = 0
{{1,2,3,5},{4,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0
{{1,2,3,5},{4},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0
{{1,2,3,6},{4,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0
{{1,2,3},{4,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1
{{1,2,3},{4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2,3,6},{4},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0
{{1,2,3},{4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2,3},{4},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2,4,5,6},{3}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => ? = 0
{{1,2,4,5},{3,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0
{{1,2,4,5},{3},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0
{{1,2,4,6},{3,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0
{{1,2,4},{3,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1
{{1,2,4},{3,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2,4,6},{3},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0
{{1,2,4},{3,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2,4},{3},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2,5,6},{3,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0
{{1,2,5},{3,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1
{{1,2,5},{3,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2,6},{3,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1
{{1,2},{3,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0
{{1,2},{3,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2,6},{3,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2},{3,4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2},{3,4},{5,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 1
{{1,2,5,6},{3},{4}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0
{{1,2,5},{3,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2,5},{3},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2,6},{3,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2},{3,5,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2},{3,5},{4,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 1
{{1,2,6},{3},{4,5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2},{3,6},{4,5}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 1
{{1,2},{3},{4,5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,3,4,5,6},{2}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => ? = 0
{{1,3,4,5},{2,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0
{{1,3,4,5},{2},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0
{{1,3,4,6},{2,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0
{{1,3,4},{2,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1
{{1,3,4},{2,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,3,4,6},{2},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0
{{1,3,4},{2,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,3,4},{2},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,3,5,6},{2,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0
{{1,3,5},{2,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1
{{1,3,5},{2,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,3,6},{2,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1
{{1,3},{2,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0
{{1,3},{2,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,3,6},{2,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,3},{2,4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,3},{2,4},{5,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 1
{{1,3,5,6},{2},{4}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0
{{1,3,5},{2,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,3,5},{2},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,3,6},{2,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,3},{2,5,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,3},{2,5},{4,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 1
{{1,3,6},{2},{4,5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,3},{2,6},{4,5}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 1
{{1,3},{2},{4,5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,4,5,6},{2,3}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0
{{1,4,5},{2,3,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1
{{1,4,5},{2,3},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,4,6},{2,3,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1
{{1,4},{2,3,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0
{{1,4},{2,3,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,4,6},{2,3},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,4},{2,3,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,5},{2,3,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,6},{2,3,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1},{2,3,4},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,5,6},{2,3},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,5},{2,3,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,6},{2,3,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1},{2,3,5},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0

Description

The number of inversions of the third entry of a permutation. This is, for a permutation

$\pi$ of length

$n$ ,

$\# \{3 < k \leq n \mid \pi(3) > \pi(k)\}.$ The number of inversions of the first entry is [[St000054]] and the number of inversions of the second entry is [[St001557]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.

Matching statistic: St001730

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%

Values

{{1,2,3,4}}
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => ? = 0
{{1,2,3,4,5}}
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 111110000010 => ? = 0
{{1,2,3,4},{5}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
{{1,2,3,5},{4}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
{{1,2,3},{4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0
{{1,2,4,5},{3}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
{{1,2,4},{3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0
{{1,2,5},{3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0
{{1,2},{3,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0
{{1,3,4,5},{2}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
{{1,3,4},{2,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0
{{1,3,5},{2,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0
{{1,3},{2,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0
{{1,4,5},{2,3}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0
{{1,4},{2,3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0
{{1,5},{2,3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 0
{{1},{2,3,4,5}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
{{1,2,3,4,5,6}}
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 11111100000010 => ? = 1
{{1,2,3,4,5},{6}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => ? = 0
{{1,2,3,4,6},{5}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => ? = 0
{{1,2,3,4},{5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
{{1,2,3,4},{5},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0
{{1,2,3,5,6},{4}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => ? = 0
{{1,2,3,5},{4,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
{{1,2,3,5},{4},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0
{{1,2,3,6},{4,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
{{1,2,3},{4,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 1
{{1,2,3},{4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2,3,6},{4},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0
{{1,2,3},{4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2,3},{4},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2,4,5,6},{3}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => ? = 0
{{1,2,4,5},{3,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
{{1,2,4,5},{3},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0
{{1,2,4,6},{3,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
{{1,2,4},{3,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 1
{{1,2,4},{3,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2,4,6},{3},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0
{{1,2,4},{3,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2,4},{3},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2,5,6},{3,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
{{1,2,5},{3,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 1
{{1,2,5},{3,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2,6},{3,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 1
{{1,2},{3,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
{{1,2},{3,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2,6},{3,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2},{3,4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2},{3,4},{5,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => ? = 1
{{1,2,5,6},{3},{4}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0
{{1,2,5},{3,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2,5},{3},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2,6},{3,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2},{3,5,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2},{3,5},{4,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => ? = 1
{{1,2,6},{3},{4,5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,2},{3,6},{4,5}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => ? = 1
{{1,2},{3},{4,5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,3,4,5,6},{2}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => ? = 0
{{1,3,4,5},{2,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
{{1,3,4,5},{2},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0
{{1,3,4,6},{2,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
{{1,3,4},{2,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 1
{{1,3,4},{2,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,3,4,6},{2},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0
{{1,3,4},{2,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,3,4},{2},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,3,5,6},{2,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
{{1,3,5},{2,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 1
{{1,3,5},{2,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,3,6},{2,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 1
{{1,3},{2,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
{{1,3},{2,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,3,6},{2,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,3},{2,4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,3},{2,4},{5,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => ? = 1
{{1,3,5,6},{2},{4}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0
{{1,3,5},{2,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,3,5},{2},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,3,6},{2,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,3},{2,5,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,3},{2,5},{4,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => ? = 1
{{1,3,6},{2},{4,5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,3},{2,6},{4,5}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => ? = 1
{{1,3},{2},{4,5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,4,5,6},{2,3}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
{{1,4,5},{2,3,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 1
{{1,4,5},{2,3},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,4,6},{2,3,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 1
{{1,4},{2,3,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
{{1,4},{2,3,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,4,6},{2,3},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,4},{2,3,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,5},{2,3,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,6},{2,3,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1},{2,3,4},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,5,6},{2,3},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,5},{2,3,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1,6},{2,3,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
{{1},{2,3,5},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0

Description

The number of times the path corresponding to a binary word crosses the base line. Interpret each

$0$ as a step

$(1,-1)$ and

$1$ as a step

$(1,1)$ . Then this statistic counts the number of times the path crosses the

$x$ -axis.

Matching statistic: St001803

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%

Values

{{1,2,3,4}}
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[1,2,3,4,9],[5,6,7,8,10]]
 => ? = 0
{{1,2,3,4,5}}
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,11],[6,7,8,9,10,12]]
 => ? = 0
{{1,2,3,4},{5}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[1,2,3,5,9],[4,6,7,8,10]]
 => ? = 0
{{1,2,3,5},{4}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[1,2,3,5,9],[4,6,7,8,10]]
 => ? = 0
{{1,2,3},{4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 0
{{1,2,4,5},{3}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[1,2,3,5,9],[4,6,7,8,10]]
 => ? = 0
{{1,2,4},{3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 0
{{1,2,5},{3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 0
{{1,2},{3,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 0
{{1,3,4,5},{2}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[1,2,3,5,9],[4,6,7,8,10]]
 => ? = 0
{{1,3,4},{2,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 0
{{1,3,5},{2,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 0
{{1,3},{2,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 0
{{1,4,5},{2,3}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 0
{{1,4},{2,3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 0
{{1,5},{2,3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 0
{{1},{2,3,4,5}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[1,2,3,5,9],[4,6,7,8,10]]
 => ? = 0
{{1,2,3,4,5,6}}
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,6,13],[7,8,9,10,11,12,14]]
 => ? = 1
{{1,2,3,4,5},{6}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[1,2,3,4,6,11],[5,7,8,9,10,12]]
 => ? = 0
{{1,2,3,4,6},{5}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[1,2,3,4,6,11],[5,7,8,9,10,12]]
 => ? = 0
{{1,2,3,4},{5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => ? = 0
{{1,2,3,4},{5},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => ? = 0
{{1,2,3,5,6},{4}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[1,2,3,4,6,11],[5,7,8,9,10,12]]
 => ? = 0
{{1,2,3,5},{4,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => ? = 0
{{1,2,3,5},{4},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => ? = 0
{{1,2,3,6},{4,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => ? = 0
{{1,2,3},{4,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 1
{{1,2,3},{4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,2,3,6},{4},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => ? = 0
{{1,2,3},{4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,2,3},{4},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,2,4,5,6},{3}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[1,2,3,4,6,11],[5,7,8,9,10,12]]
 => ? = 0
{{1,2,4,5},{3,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => ? = 0
{{1,2,4,5},{3},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => ? = 0
{{1,2,4,6},{3,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => ? = 0
{{1,2,4},{3,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 1
{{1,2,4},{3,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,2,4,6},{3},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => ? = 0
{{1,2,4},{3,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,2,4},{3},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,2,5,6},{3,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => ? = 0
{{1,2,5},{3,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 1
{{1,2,5},{3,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,2,6},{3,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 1
{{1,2},{3,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => ? = 0
{{1,2},{3,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,2,6},{3,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,2},{3,4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,2},{3,4},{5,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 1
{{1,2,5,6},{3},{4}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => ? = 0
{{1,2,5},{3,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,2,5},{3},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,2,6},{3,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,2},{3,5,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,2},{3,5},{4,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 1
{{1,2,6},{3},{4,5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,2},{3,6},{4,5}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 1
{{1,2},{3},{4,5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,3,4,5,6},{2}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[1,2,3,4,6,11],[5,7,8,9,10,12]]
 => ? = 0
{{1,3,4,5},{2,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => ? = 0
{{1,3,4,5},{2},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => ? = 0
{{1,3,4,6},{2,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => ? = 0
{{1,3,4},{2,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 1
{{1,3,4},{2,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,3,4,6},{2},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => ? = 0
{{1,3,4},{2,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,3,4},{2},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,3,5,6},{2,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => ? = 0
{{1,3,5},{2,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 1
{{1,3,5},{2,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,3,6},{2,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 1
{{1,3},{2,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => ? = 0
{{1,3},{2,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,3,6},{2,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,3},{2,4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,3},{2,4},{5,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 1
{{1,3,5,6},{2},{4}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => ? = 0
{{1,3,5},{2,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,3,5},{2},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,3,6},{2,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,3},{2,5,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,3},{2,5},{4,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 1
{{1,3,6},{2},{4,5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,3},{2,6},{4,5}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 1
{{1,3},{2},{4,5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,4,5,6},{2,3}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => ? = 0
{{1,4,5},{2,3,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 1
{{1,4,5},{2,3},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,4,6},{2,3,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 1
{{1,4},{2,3,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => ? = 0
{{1,4},{2,3,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,4,6},{2,3},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,4},{2,3,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,5},{2,3,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,6},{2,3,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1},{2,3,4},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,5,6},{2,3},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,5},{2,3,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1,6},{2,3,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0
{{1},{2,3,5},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0

Description

The maximal overlap of the cylindrical tableau associated with a tableau. A cylindrical tableau associated with a standard Young tableau

$T$ is the skew row-strict tableau obtained by gluing two copies of

$T$ such that the inner shape is a rectangle. The overlap, recorded in this statistic, equals

$\max_C\big(2\ell(T) - \ell(C)\big)$ , where

$\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux. In particular, the statistic equals

$0$ , if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.

Matching statistic: St001960

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%

Values

{{1,2,3,4}}
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => ? = 0
{{1,2,3,4,5}}
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => ? = 0
{{1,2,3,4},{5}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ? = 0
{{1,2,3,5},{4}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ? = 0
{{1,2,3},{4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0
{{1,2,4,5},{3}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ? = 0
{{1,2,4},{3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0
{{1,2,5},{3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0
{{1,2},{3,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0
{{1,3,4,5},{2}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ? = 0
{{1,3,4},{2,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0
{{1,3,5},{2,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0
{{1,3},{2,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0
{{1,4,5},{2,3}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0
{{1,4},{2,3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0
{{1,5},{2,3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0
{{1},{2,3,4,5}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ? = 0
{{1,2,3,4,5,6}}
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,8,1,7] => ? = 1
{{1,2,3,4,5},{6}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => ? = 0
{{1,2,3,4,6},{5}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => ? = 0
{{1,2,3,4},{5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0
{{1,2,3,4},{5},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0
{{1,2,3,5,6},{4}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => ? = 0
{{1,2,3,5},{4,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0
{{1,2,3,5},{4},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0
{{1,2,3,6},{4,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0
{{1,2,3},{4,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1
{{1,2,3},{4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2,3,6},{4},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0
{{1,2,3},{4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2,3},{4},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2,4,5,6},{3}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => ? = 0
{{1,2,4,5},{3,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0
{{1,2,4,5},{3},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0
{{1,2,4,6},{3,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0
{{1,2,4},{3,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1
{{1,2,4},{3,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2,4,6},{3},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0
{{1,2,4},{3,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2,4},{3},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2,5,6},{3,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0
{{1,2,5},{3,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1
{{1,2,5},{3,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2,6},{3,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1
{{1,2},{3,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0
{{1,2},{3,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2,6},{3,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2},{3,4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2},{3,4},{5,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 1
{{1,2,5,6},{3},{4}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0
{{1,2,5},{3,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2,5},{3},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2,6},{3,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2},{3,5,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2},{3,5},{4,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 1
{{1,2,6},{3},{4,5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,2},{3,6},{4,5}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 1
{{1,2},{3},{4,5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,3,4,5,6},{2}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => ? = 0
{{1,3,4,5},{2,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0
{{1,3,4,5},{2},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0
{{1,3,4,6},{2,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0
{{1,3,4},{2,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1
{{1,3,4},{2,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,3,4,6},{2},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0
{{1,3,4},{2,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,3,4},{2},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,3,5,6},{2,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0
{{1,3,5},{2,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1
{{1,3,5},{2,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,3,6},{2,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1
{{1,3},{2,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0
{{1,3},{2,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,3,6},{2,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,3},{2,4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,3},{2,4},{5,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 1
{{1,3,5,6},{2},{4}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0
{{1,3,5},{2,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,3,5},{2},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,3,6},{2,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,3},{2,5,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,3},{2,5},{4,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 1
{{1,3,6},{2},{4,5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,3},{2,6},{4,5}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 1
{{1,3},{2},{4,5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,4,5,6},{2,3}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0
{{1,4,5},{2,3,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1
{{1,4,5},{2,3},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,4,6},{2,3,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1
{{1,4},{2,3,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0
{{1,4},{2,3,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,4,6},{2,3},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,4},{2,3,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,5},{2,3,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,6},{2,3,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1},{2,3,4},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,5,6},{2,3},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,5},{2,3,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1,6},{2,3,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
{{1},{2,3,5},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0

Description

The number of descents of a permutation minus one if its first entry is not one. This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].

Matching statistic: St001195

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%

Values

{{1,2,3,4}}
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 0 + 1
{{1,2,3,4,5}}
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 0 + 1
{{1,2,3,4},{5}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0 + 1
{{1,2,3,5},{4}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0 + 1
{{1,2,3},{4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,2,4,5},{3}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0 + 1
{{1,2,4},{3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,2,5},{3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,2},{3,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,3,4,5},{2}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0 + 1
{{1,3,4},{2,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,3,5},{2,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,3},{2,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,4,5},{2,3}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,4},{2,3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,5},{2,3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1},{2,3,4,5}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0 + 1
{{1,2,3,4,5,6}}
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 1 + 1
{{1,2,3,4,5},{6}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 0 + 1
{{1,2,3,4,6},{5}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 0 + 1
{{1,2,3,4},{5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0 + 1
{{1,2,3,4},{5},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 0 + 1
{{1,2,3,5,6},{4}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 0 + 1
{{1,2,3,5},{4,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0 + 1
{{1,2,3,5},{4},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 0 + 1
{{1,2,3,6},{4,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0 + 1
{{1,2,3},{4,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 1 + 1
{{1,2,3},{4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,2,3,6},{4},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 0 + 1
{{1,2,3},{4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,2,3},{4},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,2,4,5,6},{3}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 0 + 1
{{1,2,4,5},{3,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0 + 1
{{1,2,4,5},{3},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 0 + 1
{{1,2,4,6},{3,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0 + 1
{{1,2,4},{3,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 1 + 1
{{1,2,4},{3,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,2,4,6},{3},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 0 + 1
{{1,2,4},{3,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,2,4},{3},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,2,5,6},{3,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0 + 1
{{1,2,5},{3,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 1 + 1
{{1,2,5},{3,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,2,6},{3,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 1 + 1
{{1,2},{3,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0 + 1
{{1,2},{3,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,2,6},{3,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,2},{3,4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,2},{3,4},{5,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
{{1,2,5,6},{3},{4}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 0 + 1
{{1,2,5},{3,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,2,5},{3},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,2,6},{3,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,2},{3,5,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,2},{3,5},{4,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
{{1,2,6},{3},{4,5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,2},{3,6},{4,5}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
{{1,2},{3},{4,5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,3,4,5,6},{2}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 0 + 1
{{1,3,4,5},{2,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0 + 1
{{1,3,4,5},{2},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 0 + 1
{{1,3,4,6},{2,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0 + 1
{{1,3,4},{2,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 1 + 1
{{1,3,4},{2,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,3,4,6},{2},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 0 + 1
{{1,3,4},{2,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,3,4},{2},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,3,5,6},{2,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0 + 1
{{1,3,5},{2,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 1 + 1
{{1,3,5},{2,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,3,6},{2,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 1 + 1
{{1,3},{2,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0 + 1
{{1,3},{2,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,3,6},{2,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,3},{2,4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,3},{2,4},{5,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
{{1,3,5,6},{2},{4}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 0 + 1
{{1,3,5},{2,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,3,5},{2},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,3,6},{2,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,3},{2,5,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,3},{2,5},{4,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
{{1,3,6},{2},{4,5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,3},{2,6},{4,5}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
{{1,3},{2},{4,5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,4,5,6},{2,3}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0 + 1
{{1,4,5},{2,3,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 1 + 1
{{1,4,5},{2,3},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,4,6},{2,3,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 1 + 1
{{1,4},{2,3,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0 + 1
{{1,4},{2,3,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,4,6},{2,3},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,4},{2,3,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,5},{2,3,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,6},{2,3,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1},{2,3,4},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,5,6},{2,3},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,5},{2,3,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,6},{2,3,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1},{2,3,5},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1

Description

The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ .

Matching statistic: St001208

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%

Values

{{1,2,3,4}}
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => ? = 0 + 1
{{1,2,3,4,5}}
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => ? = 0 + 1
{{1,2,3,4},{5}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ? = 0 + 1
{{1,2,3,5},{4}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ? = 0 + 1
{{1,2,3},{4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1 = 0 + 1
{{1,2,4,5},{3}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ? = 0 + 1
{{1,2,4},{3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1 = 0 + 1
{{1,2,5},{3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1 = 0 + 1
{{1,2},{3,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1 = 0 + 1
{{1,3,4,5},{2}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ? = 0 + 1
{{1,3,4},{2,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1 = 0 + 1
{{1,3,5},{2,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1 = 0 + 1
{{1,3},{2,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1 = 0 + 1
{{1,4,5},{2,3}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1 = 0 + 1
{{1,4},{2,3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1 = 0 + 1
{{1,5},{2,3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1 = 0 + 1
{{1},{2,3,4,5}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ? = 0 + 1
{{1,2,3,4,5,6}}
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,8,1,7] => ? = 1 + 1
{{1,2,3,4,5},{6}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => ? = 0 + 1
{{1,2,3,4,6},{5}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => ? = 0 + 1
{{1,2,3,4},{5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0 + 1
{{1,2,3,4},{5},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0 + 1
{{1,2,3,5,6},{4}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => ? = 0 + 1
{{1,2,3,5},{4,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0 + 1
{{1,2,3,5},{4},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0 + 1
{{1,2,3,6},{4,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0 + 1
{{1,2,3},{4,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1 + 1
{{1,2,3},{4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2,3,6},{4},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0 + 1
{{1,2,3},{4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2,3},{4},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2,4,5,6},{3}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => ? = 0 + 1
{{1,2,4,5},{3,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0 + 1
{{1,2,4,5},{3},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0 + 1
{{1,2,4,6},{3,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0 + 1
{{1,2,4},{3,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1 + 1
{{1,2,4},{3,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2,4,6},{3},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0 + 1
{{1,2,4},{3,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2,4},{3},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2,5,6},{3,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0 + 1
{{1,2,5},{3,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1 + 1
{{1,2,5},{3,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2,6},{3,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1 + 1
{{1,2},{3,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0 + 1
{{1,2},{3,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2,6},{3,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2},{3,4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2},{3,4},{5,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 1 + 1
{{1,2,5,6},{3},{4}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0 + 1
{{1,2,5},{3,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2,5},{3},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2,6},{3,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2},{3,5,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2},{3,5},{4,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 1 + 1
{{1,2,6},{3},{4,5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2},{3,6},{4,5}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 1 + 1
{{1,2},{3},{4,5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,3,4,5,6},{2}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => ? = 0 + 1
{{1,3,4,5},{2,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0 + 1
{{1,3,4,5},{2},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0 + 1
{{1,3,4,6},{2,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0 + 1
{{1,3,4},{2,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1 + 1
{{1,3,4},{2,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,3,4,6},{2},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0 + 1
{{1,3,4},{2,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,3,4},{2},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,3,5,6},{2,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0 + 1
{{1,3,5},{2,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1 + 1
{{1,3,5},{2,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,3,6},{2,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1 + 1
{{1,3},{2,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0 + 1
{{1,3},{2,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,3,6},{2,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,3},{2,4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,3},{2,4},{5,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 1 + 1
{{1,3,5,6},{2},{4}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0 + 1
{{1,3,5},{2,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,3,5},{2},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,3,6},{2,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,3},{2,5,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,3},{2,5},{4,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 1 + 1
{{1,3,6},{2},{4,5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,3},{2,6},{4,5}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 1 + 1
{{1,3},{2},{4,5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,4,5,6},{2,3}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0 + 1
{{1,4,5},{2,3,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1 + 1
{{1,4,5},{2,3},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,4,6},{2,3,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1 + 1
{{1,4},{2,3,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0 + 1
{{1,4},{2,3,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,4,6},{2,3},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,4},{2,3,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,5},{2,3,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,6},{2,3,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1},{2,3,4},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,5,6},{2,3},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,5},{2,3,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,6},{2,3,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1},{2,3,5},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1

Description

The number of connected components of the quiver of

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra

$A$ of

$K[x]/(x^n)$ .

Matching statistic: St001520

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001520: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%

Values

{{1,2,3,4}}
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => ? = 0 + 1
{{1,2,3,4,5}}
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => ? = 0 + 1
{{1,2,3,4},{5}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ? = 0 + 1
{{1,2,3,5},{4}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ? = 0 + 1
{{1,2,3},{4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1 = 0 + 1
{{1,2,4,5},{3}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ? = 0 + 1
{{1,2,4},{3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1 = 0 + 1
{{1,2,5},{3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1 = 0 + 1
{{1,2},{3,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1 = 0 + 1
{{1,3,4,5},{2}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ? = 0 + 1
{{1,3,4},{2,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1 = 0 + 1
{{1,3,5},{2,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1 = 0 + 1
{{1,3},{2,4,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1 = 0 + 1
{{1,4,5},{2,3}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1 = 0 + 1
{{1,4},{2,3,5}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1 = 0 + 1
{{1,5},{2,3,4}}
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1 = 0 + 1
{{1},{2,3,4,5}}
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ? = 0 + 1
{{1,2,3,4,5,6}}
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,8,1,7] => ? = 1 + 1
{{1,2,3,4,5},{6}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => ? = 0 + 1
{{1,2,3,4,6},{5}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => ? = 0 + 1
{{1,2,3,4},{5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0 + 1
{{1,2,3,4},{5},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0 + 1
{{1,2,3,5,6},{4}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => ? = 0 + 1
{{1,2,3,5},{4,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0 + 1
{{1,2,3,5},{4},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0 + 1
{{1,2,3,6},{4,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0 + 1
{{1,2,3},{4,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1 + 1
{{1,2,3},{4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2,3,6},{4},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0 + 1
{{1,2,3},{4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2,3},{4},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2,4,5,6},{3}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => ? = 0 + 1
{{1,2,4,5},{3,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0 + 1
{{1,2,4,5},{3},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0 + 1
{{1,2,4,6},{3,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0 + 1
{{1,2,4},{3,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1 + 1
{{1,2,4},{3,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2,4,6},{3},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0 + 1
{{1,2,4},{3,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2,4},{3},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2,5,6},{3,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0 + 1
{{1,2,5},{3,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1 + 1
{{1,2,5},{3,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2,6},{3,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1 + 1
{{1,2},{3,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0 + 1
{{1,2},{3,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2,6},{3,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2},{3,4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2},{3,4},{5,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 1 + 1
{{1,2,5,6},{3},{4}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0 + 1
{{1,2,5},{3,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2,5},{3},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2,6},{3,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2},{3,5,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2},{3,5},{4,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 1 + 1
{{1,2,6},{3},{4,5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,2},{3,6},{4,5}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 1 + 1
{{1,2},{3},{4,5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,3,4,5,6},{2}}
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => ? = 0 + 1
{{1,3,4,5},{2,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0 + 1
{{1,3,4,5},{2},{6}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0 + 1
{{1,3,4,6},{2,5}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0 + 1
{{1,3,4},{2,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1 + 1
{{1,3,4},{2,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,3,4,6},{2},{5}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0 + 1
{{1,3,4},{2,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,3,4},{2},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,3,5,6},{2,4}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0 + 1
{{1,3,5},{2,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1 + 1
{{1,3,5},{2,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,3,6},{2,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1 + 1
{{1,3},{2,4,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0 + 1
{{1,3},{2,4,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,3,6},{2,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,3},{2,4,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,3},{2,4},{5,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 1 + 1
{{1,3,5,6},{2},{4}}
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 0 + 1
{{1,3,5},{2,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,3,5},{2},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,3,6},{2,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,3},{2,5,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,3},{2,5},{4,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 1 + 1
{{1,3,6},{2},{4,5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,3},{2,6},{4,5}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 1 + 1
{{1,3},{2},{4,5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,4,5,6},{2,3}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0 + 1
{{1,4,5},{2,3,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1 + 1
{{1,4,5},{2,3},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,4,6},{2,3,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 1 + 1
{{1,4},{2,3,5,6}}
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 0 + 1
{{1,4},{2,3,5},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,4,6},{2,3},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,4},{2,3,6},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,5},{2,3,4},{6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,6},{2,3,4},{5}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1},{2,3,4},{5,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,5,6},{2,3},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,5},{2,3,6},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1,6},{2,3,5},{4}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
{{1},{2,3,5},{4,6}}
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1

Description

The number of strict 3-descents. A '''strict 3-descent''' of a permutation

$\pi$ of

$\{1,2, \dots ,n \}$ is a pair

$(i,i+3)$ with

$i+3 \leq n$ and

$\pi(i) > \pi(i+3)$ .

The following 20 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001948The number of augmented double ascents of a permutation. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. 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. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000044The number of vertices of the unicellular map given by a perfect matching. St000017The number of inversions of a standard tableau. St001721The degree of a binary word. St000016The number of attacking pairs of a standard tableau. St000958The number of Bruhat factorizations of a permutation. St001645The pebbling number of a connected graph. St001260The permanent of an alternating sign matrix. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation.