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Your data matches 275 different statistics following compositions of up to 3 maps.
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Matching statistic: St001605

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of colourings of a cycle such that the multiplicities of colours are given by a partition. Two colourings are considered equal, if they are obtained by an action of the cyclic group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.

Matching statistic: St000308

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 39%●distinct values known / distinct values provided: 12%

Values

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

Description

The height of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices

$\{0,1,2,\ldots,n\}$ and root

$0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1]. The statistic is given by the height of this tree. See also [[St000325]] for the width of this tree.

Matching statistic: St001195
(load all 23 compositions to match this statistic)

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 39%●distinct values known / distinct values provided: 6%

Values

{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
{{1,2},{3},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 2
{{1,3},{2},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 2
{{1},{2,3},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 2
{{1,4},{2},{3},{5},{6}}
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 2
{{1},{2,4},{3},{5},{6}}
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 2
{{1},{2},{3,4},{5},{6}}
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 2
{{1,5},{2},{3},{4},{6}}
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 2
{{1},{2,5},{3},{4},{6}}
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 2
{{1},{2},{3,5},{4},{6}}
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 2
{{1},{2},{3},{4,5},{6}}
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 2
{{1,6},{2},{3},{4},{5}}
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 2
{{1},{2,6},{3},{4},{5}}
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 2
{{1},{2},{3,6},{4},{5}}
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 2
{{1},{2},{3},{4,6},{5}}
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 2
{{1},{2},{3},{4},{5,6}}
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 2
{{1},{2},{3},{4},{5},{6}}
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
{{1,2,3},{4},{5},{6},{7}}
 => [3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => ? = 2
{{1,2,4},{3},{5},{6},{7}}
 => [3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => ? = 2
{{1,2},{3,4},{5,6},{7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2},{3,4},{5,7},{6}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2},{3,4},{5},{6,7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2},{3,4},{5},{6},{7}}
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 2
{{1,2,5},{3},{4},{6},{7}}
 => [3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => ? = 2
{{1,2},{3,5},{4,6},{7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2},{3,5},{4,7},{6}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2},{3,5},{4},{6,7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2},{3,5},{4},{6},{7}}
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 2
{{1,2},{3,6},{4,5},{7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2},{3,7},{4,5},{6}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2},{3},{4,5},{6,7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2},{3},{4,5},{6},{7}}
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 2
{{1,2,6},{3},{4},{5},{7}}
 => [3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => ? = 2
{{1,2},{3,6},{4,7},{5}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2},{3,6},{4},{5,7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2},{3,6},{4},{5},{7}}
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 2
{{1,2},{3,7},{4,6},{5}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2},{3},{4,6},{5,7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2},{3},{4,6},{5},{7}}
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 2
{{1,2},{3,7},{4},{5,6}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2},{3},{4,7},{5,6}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2},{3},{4},{5,6},{7}}
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 2
{{1,2,7},{3},{4},{5},{6}}
 => [3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => ? = 2
{{1,2},{3,7},{4},{5},{6}}
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 2
{{1,2},{3},{4,7},{5},{6}}
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 2
{{1,2},{3},{4},{5,7},{6}}
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 2
{{1,2},{3},{4},{5},{6,7}}
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 2
{{1,2},{3},{4},{5},{6},{7}}
 => [2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 6
{{1,3,4},{2},{5},{6},{7}}
 => [3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => ? = 2
{{1,3},{2,4},{5,6},{7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2,4},{5,7},{6}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2,4},{5},{6,7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2,4},{5},{6},{7}}
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 2
{{1,3,5},{2},{4},{6},{7}}
 => [3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => ? = 2
{{1,3},{2,5},{4,6},{7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2,5},{4,7},{6}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2,5},{4},{6,7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2,5},{4},{6},{7}}
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 2
{{1,3},{2,6},{4,5},{7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2,7},{4,5},{6}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2},{4,5},{6,7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2},{4,5},{6},{7}}
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 2
{{1,3,6},{2},{4},{5},{7}}
 => [3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => ? = 2
{{1,3},{2,6},{4,7},{5}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2,6},{4},{5,7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2,6},{4},{5},{7}}
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 2
{{1,3},{2,7},{4,6},{5}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2},{4,6},{5,7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2},{4,6},{5},{7}}
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 2
{{1,3},{2,7},{4},{5,6}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2},{4,7},{5,6}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2},{4},{5,6},{7}}
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 2
{{1,3,7},{2},{4},{5},{6}}
 => [3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => ? = 2
{{1,3},{2,7},{4},{5},{6}}
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 2
{{1,3},{2},{4,7},{5},{6}}
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 2
{{1,3},{2},{4},{5,7},{6}}
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 2
{{1,3},{2},{4},{5},{6,7}}
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 2
{{1,3},{2},{4},{5},{6},{7}}
 => [2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 6
{{1,4},{2,3},{5,6},{7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,4},{2,3},{5,7},{6}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,4},{2,3},{5},{6,7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,4},{2,3},{5},{6},{7}}
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 2
{{1},{2,3,4},{5},{6},{7}}
 => [3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => ? = 2
{{1,5},{2,3},{4,6},{7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,5},{2,3},{4,7},{6}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,5},{2,3},{4},{6,7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,6},{2,3},{4,5},{7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,7},{2,3},{4,5},{6}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1},{2,3},{4,5},{6,7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,6},{2,3},{4,7},{5}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,6},{2,3},{4},{5,7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,7},{2,3},{4,6},{5}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1},{2,3},{4,6},{5,7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,7},{2,3},{4},{5,6}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1},{2,3},{4,7},{5,6}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,4},{2,5},{3,6},{7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,4},{2,5},{3,7},{6}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,4},{2,5},{3},{6,7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,4},{2,6},{3,5},{7}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,4},{2,7},{3,5},{6}}
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 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: St001001
(load all 25 compositions to match this statistic)

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 39%●distinct values known / distinct values provided: 6%

Values

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

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: St001200
(load all 14 compositions to match this statistic)

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 39%●distinct values known / distinct values provided: 6%

Values

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

Description

The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ .

Matching statistic: St001514
(load all 15 compositions to match this statistic)

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001514: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 39%●distinct values known / distinct values provided: 6%

Values

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

Description

The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule.

Matching statistic: St001526
(load all 15 compositions to match this statistic)

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001526: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 39%●distinct values known / distinct values provided: 6%

Values

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

Description

The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001515
(load all 28 compositions to match this statistic)

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001515: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 39%●distinct values known / distinct values provided: 6%

Values

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

Description

The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule).

Matching statistic: St000023

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 39%●distinct values known / distinct values provided: 6%

Values

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

Description

The number of inner peaks of a permutation. The number of peaks including the boundary is [[St000092]].

Matching statistic: St000031

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 39%●distinct values known / distinct values provided: 6%

Values

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

Description

The number of cycles in the cycle decomposition of a permutation.

The following 265 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000056The decomposition (or block) number of a permutation. St000117The number of centered tunnels of a Dyck path. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000352The Elizalde-Pak rank of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000570The Edelman-Greene number of a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000654The first descent of a permutation. St000664The number of right ropes of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000694The number of affine bounded permutations that project to a given permutation. St000711The number of big exceedences of a permutation. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000871The number of very big ascents of a permutation. St000873The aix statistic of a permutation. St000883The number of longest increasing subsequences of a permutation. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n−1}]$ by adding

$c_0$ to

$c_{n−1}$ . St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001081The number of minimal length factorizations of a permutation into star transpositions. St001083The number of boxed occurrences of 132 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001162The minimum jump of a permutation. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001191Number of simple modules

$S$ with

$Ext_A^i(S,A)=0$ for all

$i=0,1,...,g-1$ in the corresponding Nakayama algebra

$A$ with global dimension

$g$ . St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001208The 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

$K[x]/(x^n)$ . St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001256Number of simple reflexive modules that are 2-stable reflexive. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001344The neighbouring number of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001537The number of cyclic crossings of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001557The number of inversions of the second entry of a permutation. St001590The crossing number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000021The number of descents of a permutation. St000022The number of fixed points of a permutation. St000035The number of left outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000120The number of left tunnels of a Dyck path. St000153The number of adjacent cycles of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000221The number of strong fixed points of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000338The number of pixed points of a permutation. St000360The number of occurrences of the pattern 32-1. St000366The number of double descents of a permutation. St000367The number of simsun double descents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000441The number of successions of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000534The number of 2-rises of a permutation. St000542The number of left-to-right-minima of a permutation. St000619The number of cyclic descents of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000647The number of big descents of a permutation. St000648The number of 2-excedences of a permutation. St000662The staircase size of the code of a permutation. St000663The number of right floats of a permutation. St000665The number of rafts of a permutation. St000732The number of double deficiencies of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St000842The breadth of a permutation. St000862The number of parts of the shifted shape of a permutation. St000884The number of isolated descents of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000958The number of Bruhat factorizations of a permutation. St000990The first ascent of a permutation. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001096The size of the overlap set of a permutation. St001130The number of two successive successions in a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ corresponding to the Dyck path. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001381The fertility of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001405The number of bonds in a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001481The minimal height of a peak of a Dyck path. St001489The maximum of the number of descents and the number of inverse descents. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001520The number of strict 3-descents. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001556The number of inversions of the third entry of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001569The maximal modular displacement of a permutation. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001684The reduced word complexity of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001729The number of visible descents of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001807The lower middle entry of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001836The number of occurrences of a 213 pattern in the restricted growth word of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001928The number of non-overlapping descents in a permutation. St001948The number of augmented double ascents of a permutation. St000037The sign of a permutation. St000245The number of ascents of a permutation. St000325The width of the tree associated to a permutation. St000446The disorder of a permutation. St000462The major index minus the number of excedences of a permutation. St000470The number of runs in a permutation. St000651The maximal size of a rise in a permutation. St000652The maximal difference between successive positions of a permutation. St000696The number of cycles in the breakpoint graph of a permutation. St000831The number of indices that are either descents or recoils. St000891The number of distinct diagonal sums of a permutation matrix. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001220The width of a permutation. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001298The number of repeated entries in the Lehmer code of a permutation. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001388The number of non-attacking neighbors of a permutation. St001517The length of a longest pair of twins in a permutation. St001530The depth of a Dyck path. St001667The maximal size of a pair of weak twins for a permutation. St001726The number of visible inversions of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000064The number of one-box pattern of a permutation. St000154The sum of the descent bottoms of a permutation. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000220The number of occurrences of the pattern 132 in a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000670The reversal length of a permutation. St000677The standardized bi-alternating inversion number of a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000863The length of the first row of the shifted shape of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001375The pancake length of a permutation. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001760The number of prefix or suffix reversals needed to sort a permutation. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000019The cardinality of the support of a permutation. St000209Maximum difference of elements in cycles. St000210Minimum over maximum difference of elements in cycles. St000216The absolute length of a permutation. St000809The reduced reflection length of the permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001480The number of simple summands of the module J^2/J^3. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001806The upper middle entry of a permutation. St001958The degree of the polynomial interpolating the values of a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000673The number of non-fixed points of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001468The smallest fixpoint of a permutation. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St000134The size of the orbit of an alternating sign matrix under gyration. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001074The number of inversions of the cyclic embedding of a permutation. St000304The load of a permutation. St000545The number of parabolic double cosets with minimal element being the given permutation. St001731The factorization defect of a permutation. St000079The number of alternating sign matrices for a given Dyck path. St000033The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St000690The size of the conjugacy class of a permutation. St001560The product of the cardinalities of the lower order ideal and upper order ideal generated by a permutation in weak order. St001528The number of permutations such that the product with the permutation has the same number of fixed points. St000324The shape of the tree associated to a permutation. St001243The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path. St001242The toal dimension of certain Sn modules determined by LLT polynomials associated with a Dyck path. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001488The number of corners of a skew partition. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000954Number of times the corresponding LNakayama algebra has

$Ext^i(D(A),A)=0$ for

$i>0$ . St001812The biclique partition number of a graph. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St000479The Ramsey number of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001703The villainy of a graph. St001834The number of non-isomorphic minors of a graph. St001545The second Elser number of a connected graph.