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Your data matches 16 different statistics following compositions of up to 3 maps.
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Matching statistic: St000681
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The Grundy value of Chomp on Ferrers diagrams.
Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1].
This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.
Matching statistic: St001514
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001514: Dyck paths ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 38%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001514: Dyck paths ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 38%
Values
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 1 + 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 1 + 2
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 1 + 2
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 1 + 2
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 1 + 2
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 1 + 2
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 1 + 2
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 1 + 2
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 1 + 2
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 1 + 2
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 2 + 2
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 2
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,3},{2},{4},{5,6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 2
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
{{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
{{1},{2,3},{4,6},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1},{2,3},{4},{5,6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1},{2,3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 2
{{1,4,5},{2},{3},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
{{1,4},{2,5},{3,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
{{1,4},{2,5},{3},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,4},{2,6},{3,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
{{1,4},{2},{3,5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,4,6},{2},{3},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
{{1,4},{2,6},{3},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,4},{2},{3,6},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,4},{2},{3},{5,6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,4},{2},{3},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 2
{{1,5},{2,4},{3,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
{{1,5},{2,4},{3},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1},{2,4,5},{3},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
{{1,6},{2,4},{3,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
{{1},{2,4},{3,5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1},{2,4,6},{3},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
{{1},{2,4},{3},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 2
{{1},{2},{3,4,5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
{{1},{2},{3,4,6},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
{{1},{2},{3,4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 2
{{1,5,6},{2},{3},{4}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
{{1,5},{2},{3},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 2
{{1},{2,5,6},{3},{4}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
{{1},{2,5},{3},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 2
{{1},{2},{3,5,6},{4}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
{{1},{2},{3,5},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 2
{{1},{2},{3},{4,5,6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
{{1},{2},{3},{4,5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 2
{{1,6},{2},{3},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 2
{{1},{2,6},{3},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 2
{{1},{2},{3,6},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 2
{{1},{2},{3},{4,6},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 2
{{1},{2},{3},{4},{5,6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 2
{{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 2
{{1,2,3,4},{5},{6},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
{{1,2,3,5},{4},{6},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
{{1,2,3},{4,5},{6},{7}}
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
{{1,2,3,6},{4},{5},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
{{1,2,3},{4,6},{5},{7}}
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
{{1,2,3},{4},{5,6},{7}}
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
{{1,2,3,7},{4},{5},{6}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
{{1,2,3},{4,7},{5},{6}}
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
{{1,2,3},{4},{5,7},{6}}
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
{{1,2,3},{4},{5},{6,7}}
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
{{1,2,3},{4},{5},{6},{7}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 2 + 2
{{1,2,4,5},{3},{6},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
{{1,2,4},{3,5},{6},{7}}
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
{{1,2,4,6},{3},{5},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
Description
The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule.
Matching statistic: St000689
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000689: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 25%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000689: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 25%
Values
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,3},{2},{4},{5,6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,3},{4,6},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1},{2,3},{4},{5,6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1},{2,3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2
{{1,4,5},{2},{3},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1,4},{2,5},{3,6}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,4},{2,5},{3},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,4},{2,6},{3,5}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,4},{2},{3,5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,4,6},{2},{3},{5}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1,4},{2,6},{3},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,4},{2},{3,6},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,4},{2},{3},{5,6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,4},{2},{3},{5},{6}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2
{{1,2,3},{4,5},{6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,3},{4,6},{5},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,3},{4},{5,6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,3},{4,7},{5},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,3},{4},{5,7},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,3},{4},{5},{6,7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,4},{3,5},{6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,4},{3,6},{5},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,4},{3},{5,6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,4},{3,7},{5},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,4},{3},{5,7},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,4},{3},{5},{6,7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,5},{3,4},{6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2},{3,4,5},{6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,6},{3,4},{5},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2},{3,4,6},{5},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,7},{3,4},{5},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2},{3,4,7},{5},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,5},{3,6},{4},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,5},{3},{4,6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,5},{3,7},{4},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,5},{3},{4,7},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,5},{3},{4},{6,7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,6},{3,5},{4},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2},{3,5,6},{4},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,7},{3,5},{4},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2},{3,5,7},{4},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,6},{3},{4,5},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2},{3},{4,5,6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,7},{3},{4,5},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2},{3},{4,5,7},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,6},{3,7},{4},{5}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,6},{3},{4,7},{5}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,6},{3},{4},{5,7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,7},{3,6},{4},{5}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2},{3,6,7},{4},{5}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,7},{3},{4,6},{5}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2},{3},{4,6,7},{5}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
Description
The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid.
The correspondence between LNakayama algebras and Dyck paths is explained in [[St000684]]. A module $M$ is $n$-rigid, if $\operatorname{Ext}^i(M,M)=0$ for $1\leq i\leq n$.
This statistic gives the maximal $n$ such that the minimal generator-cogenerator module $A \oplus D(A)$ of the LNakayama algebra $A$ corresponding to a Dyck path is $n$-rigid.
An application is to check for maximal $n$-orthogonal objects in the module category in the sense of [2].
Matching statistic: St001200
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 25%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 25%
Values
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 1 + 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 1 + 2
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 1 + 2
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 1 + 2
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 1 + 2
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 1 + 2
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 1 + 2
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 1 + 2
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 1 + 2
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 1 + 2
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 2 + 2
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 2
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 2
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 2
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 2
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 + 2
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 2
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 2
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 2
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
{{1,3},{2},{4},{5,6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 + 2
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 2
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 2
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 2
{{1},{2,3},{4,6},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
{{1},{2,3},{4},{5,6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
{{1},{2,3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 + 2
{{1,4,5},{2},{3},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 2
{{1,4},{2,5},{3,6}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1,4},{2,5},{3},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
{{1,4},{2,6},{3,5}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1,4},{2},{3,5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
{{1,4,6},{2},{3},{5}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 2
{{1,4},{2,6},{3},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
{{1,4},{2},{3,6},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
{{1,4},{2},{3},{5,6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
{{1,4},{2},{3},{5},{6}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 + 2
{{1,2,3},{4,5},{6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,3},{4,6},{5},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,3},{4},{5,6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,3},{4,7},{5},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,3},{4},{5,7},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,3},{4},{5},{6,7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,4},{3,5},{6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,4},{3,6},{5},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,4},{3},{5,6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,4},{3,7},{5},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,4},{3},{5,7},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,4},{3},{5},{6,7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,5},{3,4},{6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2},{3,4,5},{6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,6},{3,4},{5},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2},{3,4,6},{5},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,7},{3,4},{5},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2},{3,4,7},{5},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,5},{3,6},{4},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,5},{3},{4,6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,5},{3,7},{4},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,5},{3},{4,7},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,5},{3},{4},{6,7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,6},{3,5},{4},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2},{3,5,6},{4},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,7},{3,5},{4},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2},{3,5,7},{4},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,6},{3},{4,5},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2},{3},{4,5,6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,7},{3},{4,5},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2},{3},{4,5,7},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,6},{3,7},{4},{5}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,6},{3},{4,7},{5}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,6},{3},{4},{5,7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,7},{3,6},{4},{5}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2},{3,6,7},{4},{5}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2,7},{3},{4,6},{5}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
{{1,2},{3},{4,6,7},{5}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 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: St000528
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 50%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 50%
Values
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 1 + 3
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 4 = 1 + 3
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 4 = 1 + 3
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 4 = 1 + 3
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 4 = 1 + 3
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 4 = 1 + 3
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 4 = 1 + 3
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 4 = 1 + 3
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 4 = 1 + 3
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 4 = 1 + 3
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 4 = 1 + 3
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 2 + 3
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 4 = 1 + 3
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 4 = 1 + 3
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 1 + 3
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 4 = 1 + 3
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 4 = 1 + 3
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 1 + 3
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 4 = 1 + 3
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 1 + 3
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 4 = 1 + 3
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 4 = 1 + 3
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 4 = 1 + 3
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 4 = 1 + 3
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 4 = 1 + 3
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 5 = 2 + 3
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 4 = 1 + 3
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 1 + 3
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 4 = 1 + 3
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 4 = 1 + 3
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 1 + 3
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 4 = 1 + 3
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 1 + 3
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 4 = 1 + 3
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 4 = 1 + 3
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 4 = 1 + 3
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 4 = 1 + 3
{{1,3},{2},{4},{5,6}}
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 4 = 1 + 3
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 5 = 2 + 3
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 1 + 3
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 4 = 1 + 3
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 4 = 1 + 3
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 1 + 3
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 4 = 1 + 3
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 4 = 1 + 3
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 1 + 3
{{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 4 = 1 + 3
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 4 = 1 + 3
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 4 = 1 + 3
{{1},{2,3},{4,6},{5}}
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 4 = 1 + 3
{{1,2,3},{4,5},{6,7}}
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
{{1,2,3},{4,5},{6},{7}}
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
{{1,2,3},{4,6},{5,7}}
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
{{1,2,3},{4,6},{5},{7}}
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
{{1,2,3},{4,7},{5,6}}
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
{{1,2,3},{4},{5,6},{7}}
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
{{1,2,3},{4,7},{5},{6}}
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
{{1,2,3},{4},{5,7},{6}}
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
{{1,2,3},{4},{5},{6,7}}
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
{{1,2,4},{3,5},{6,7}}
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
{{1,2,4},{3,5},{6},{7}}
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
{{1,2,4},{3,6},{5,7}}
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
{{1,2,4},{3,6},{5},{7}}
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
{{1,2,4},{3,7},{5,6}}
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
{{1,2,4},{3},{5,6},{7}}
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
{{1,2,4},{3,7},{5},{6}}
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
{{1,2,4},{3},{5,7},{6}}
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
{{1,2,4},{3},{5},{6,7}}
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
{{1,2,5},{3,4},{6,7}}
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
{{1,2,5},{3,4},{6},{7}}
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
{{1,2},{3,4,5},{6,7}}
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
{{1,2},{3,4,5},{6},{7}}
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
{{1,2,6},{3,4},{5,7}}
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
{{1,2,6},{3,4},{5},{7}}
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
{{1,2},{3,4,6},{5,7}}
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
{{1,2},{3,4,6},{5},{7}}
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
{{1,2,7},{3,4},{5,6}}
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
{{1,2},{3,4,7},{5,6}}
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
{{1,2},{3,4},{5,6,7}}
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
{{1,2},{3,4},{5,6},{7}}
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ? = 0 + 3
{{1,2,7},{3,4},{5},{6}}
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
{{1,2},{3,4,7},{5},{6}}
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
{{1,2},{3,4},{5,7},{6}}
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ? = 0 + 3
{{1,2},{3,4},{5},{6,7}}
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ? = 0 + 3
{{1,2},{3,4},{5},{6},{7}}
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ? = 2 + 3
{{1,2,5},{3,6},{4,7}}
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
{{1,2,5},{3,6},{4},{7}}
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
{{1,2,5},{3,7},{4,6}}
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
{{1,2,5},{3},{4,6},{7}}
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
{{1,2,5},{3,7},{4},{6}}
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
{{1,2,5},{3},{4,7},{6}}
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
{{1,2,5},{3},{4},{6,7}}
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
{{1,2,6},{3,5},{4,7}}
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
{{1,2,6},{3,5},{4},{7}}
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
{{1,2},{3,5,6},{4,7}}
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
{{1,2},{3,5,6},{4},{7}}
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
{{1,2,7},{3,5},{4,6}}
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
{{1,2},{3,5,7},{4,6}}
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
{{1,2},{3,5},{4,6,7}}
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
{{1,2},{3,5},{4,6},{7}}
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ? = 0 + 3
Description
The height of a poset.
This equals the rank of the poset [[St000080]] plus one.
Matching statistic: St001232
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 12%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 12%
Values
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 2
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 2
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
{{1,3},{2},{4},{5,6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 2
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1},{2,3},{4,6},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
{{1},{2,3},{4},{5,6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
{{1},{2,3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 2
{{1,4,5},{2},{3},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1,4},{2,5},{3,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,4},{2,5},{3},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
{{1,4},{2,6},{3,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,4},{2},{3,5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
{{1,4,6},{2},{3},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
{{1,4},{2,6},{3},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
{{1,4},{2},{3,6},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
{{1,4},{2},{3},{5,6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
{{1,5},{2,4},{3,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,6},{2,4},{3,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,5},{2,6},{3,4}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,6},{2,5},{3,4}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,3},{4,5},{6,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,3},{4,6},{5,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,3},{4,7},{5,6}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,4},{3,5},{6,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,4},{3,6},{5,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,4},{3,7},{5,6}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,5},{3,4},{6,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3,4,5},{6,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,6},{3,4},{5,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3,4,6},{5,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,7},{3,4},{5,6}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3,4,7},{5,6}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3,4},{5,6,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,5},{3,6},{4,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,5},{3,7},{4,6}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,6},{3,5},{4,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3,5,6},{4,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,7},{3,5},{4,6}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3,5,7},{4,6}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3,5},{4,6,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,6},{3,7},{4,5}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,7},{3,6},{4,5}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3,6,7},{4,5}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3,6},{4,5,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3,7},{4,5,6}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,3,4},{2,5},{6,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,3,4},{2,6},{5,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,3,4},{2,7},{5,6}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,3,5},{2,4},{6,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,3},{2,4,5},{6,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,3,6},{2,4},{5,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,3},{2,4,6},{5,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,3,7},{2,4},{5,6}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,3},{2,4,7},{5,6}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,3},{2,4},{5,6,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001330
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 62%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 62%
Values
{{1},{2},{3},{4}}
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,4},{2},{3},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
{{1},{2,4},{3},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1},{2},{3,4},{5}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,5},{2},{3},{4}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
{{1},{2,5},{3},{4}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1},{2},{3,5},{4}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1},{2},{3},{4,5}}
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 2 + 3
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
{{1,2},{3,4},{5,6}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
{{1,2},{3,5},{4,6}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,2},{3,6},{4,5}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,2},{3},{4,5},{6}}
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,2},{3},{4,6},{5}}
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,2},{3},{4},{5,6}}
=> [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 2 + 3
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
{{1,3},{2,4},{5,6}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
{{1,3},{2,5},{4,6}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,3},{2,6},{4,5}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,3},{2},{4,5},{6}}
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,3},{2},{4,6},{5}}
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,3},{2},{4},{5,6}}
=> [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 2 + 3
{{1,4},{2,3},{5,6}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1},{2,3,4},{5},{6}}
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,5},{2,3},{4,6}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1},{2,3,5},{4},{6}}
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,6},{2,3},{4,5}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1},{2,3},{4,5},{6}}
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 3
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1},{2,3,6},{4},{5}}
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1},{2,3},{4,6},{5}}
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 3
{{1},{2,3},{4},{5,6}}
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 3
{{1},{2,3},{4},{5},{6}}
=> [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 3
{{1,4,5},{2},{3},{6}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
{{1,4},{2,5},{3,6}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,4},{2,5},{3},{6}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,4},{2,6},{3,5}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,4},{2},{3,5},{6}}
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,4,6},{2},{3},{5}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
{{1,4},{2,6},{3},{5}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,4},{2},{3,6},{5}}
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,4},{2},{3},{5,6}}
=> [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
{{1,4},{2},{3},{5},{6}}
=> [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 2 + 3
{{1,5},{2,4},{3,6}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,5},{2,4},{3},{6}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1},{2,4,5},{3},{6}}
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1,6},{2,4},{3,5}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1},{2,4},{3,5},{6}}
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 3
{{1,6},{2,4},{3},{5}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1},{2,4,6},{3},{5}}
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
{{1},{2,4},{3,6},{5}}
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 3
{{1},{2,4},{3},{5,6}}
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 3
{{1},{2,4},{3},{5},{6}}
=> [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 3
{{1,5,6},{2},{3},{4}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
{{1,5},{2},{3},{4,6}}
=> [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
{{1,5},{2},{3},{4},{6}}
=> [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 2 + 3
{{1,6},{2},{3},{4,5}}
=> [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
{{1},{2},{3},{4,5,6}}
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
{{1,6},{2},{3},{4},{5}}
=> [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 2 + 3
{{1},{2},{3},{4},{5,6}}
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 2 + 3
{{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 3 + 3
{{1,2,3,4},{5},{6},{7}}
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
{{1,2,3,5},{4},{6},{7}}
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
{{1,2,3,6},{4},{5},{7}}
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
{{1,2,3,7},{4},{5},{6}}
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
{{1,2,3},{4},{5},{6,7}}
=> [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
{{1,2,3},{4},{5},{6},{7}}
=> [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 2 + 3
{{1,2,4,5},{3},{6},{7}}
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
{{1,2,4,6},{3},{5},{7}}
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
{{1,2,4,7},{3},{5},{6}}
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
{{1,2,4},{3},{5},{6,7}}
=> [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
{{1,2,4},{3},{5},{6},{7}}
=> [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 2 + 3
{{1,2,5,6},{3},{4},{7}}
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
{{1,2,5,7},{3},{4},{6}}
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
{{1,2,5},{3},{4},{6,7}}
=> [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
{{1,2,5},{3},{4},{6},{7}}
=> [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 2 + 3
{{1,2,6,7},{3},{4},{5}}
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
{{1,2,6},{3},{4},{5,7}}
=> [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
{{1,2,6},{3},{4},{5},{7}}
=> [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 2 + 3
{{1,2,7},{3},{4},{5,6}}
=> [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
{{1,2},{3},{4},{5,6,7}}
=> [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000454
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Values
{{1},{2},{3},{4}}
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,4},{2},{3},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
{{1},{2,4},{3},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1},{2},{3,4},{5}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,5},{2},{3},{4}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
{{1},{2,5},{3},{4}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1},{2},{3,5},{4}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1},{2},{3},{4,5}}
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
{{1,2},{3,4},{5,6}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
{{1,2},{3,5},{4,6}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,2},{3,6},{4,5}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,2},{3},{4,5},{6}}
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,2},{3},{4,6},{5}}
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,2},{3},{4},{5,6}}
=> [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
{{1,3},{2,4},{5,6}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
{{1,3},{2,5},{4,6}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,3},{2,6},{4,5}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,3},{2},{4,5},{6}}
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,3},{2},{4,6},{5}}
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,3},{2},{4},{5,6}}
=> [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
{{1,4},{2,3},{5,6}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1},{2,3,4},{5},{6}}
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,5},{2,3},{4,6}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1},{2,3,5},{4},{6}}
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,6},{2,3},{4,5}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1},{2,3},{4,5},{6}}
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1},{2,3,6},{4},{5}}
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1},{2,3},{4,6},{5}}
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
{{1},{2,3},{4},{5,6}}
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
{{1},{2,3},{4},{5},{6}}
=> [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
{{1,4,5},{2},{3},{6}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
{{1,4},{2,5},{3,6}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,4},{2,5},{3},{6}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,4},{2,6},{3,5}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,4},{2},{3,5},{6}}
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,4,6},{2},{3},{5}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
{{1,4},{2,6},{3},{5}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,4},{2},{3,6},{5}}
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,4},{2},{3},{5,6}}
=> [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
{{1,4},{2},{3},{5},{6}}
=> [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
{{1,5},{2,4},{3,6}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,5},{2,4},{3},{6}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1},{2,4,5},{3},{6}}
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1,6},{2,4},{3,5}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1},{2,4},{3,5},{6}}
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
{{1,6},{2,4},{3},{5}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1},{2,4,6},{3},{5}}
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
{{1},{2,4},{3,6},{5}}
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
{{1},{2,4},{3},{5,6}}
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
{{1},{2,4},{3},{5},{6}}
=> [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
{{1,5,6},{2},{3},{4}}
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
{{1,5},{2},{3},{4,6}}
=> [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
{{1,5},{2},{3},{4},{6}}
=> [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
{{1,6},{2},{3},{4,5}}
=> [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
{{1},{2},{3},{4,5,6}}
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
{{1,6},{2},{3},{4},{5}}
=> [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
{{1},{2},{3},{4},{5,6}}
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
{{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 3 + 2
{{1,2,3,4},{5},{6},{7}}
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
{{1,2,3,5},{4},{6},{7}}
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
{{1,2,3,6},{4},{5},{7}}
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
{{1,2,3,7},{4},{5},{6}}
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
{{1,2,3},{4},{5},{6,7}}
=> [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
{{1,2,3},{4},{5},{6},{7}}
=> [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
{{1,2,4,5},{3},{6},{7}}
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
{{1,2,4,6},{3},{5},{7}}
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
{{1,2,4,7},{3},{5},{6}}
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
{{1,2,4},{3},{5},{6,7}}
=> [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
{{1,2,4},{3},{5},{6},{7}}
=> [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
{{1,2,5,6},{3},{4},{7}}
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
{{1,2,5,7},{3},{4},{6}}
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
{{1,2,5},{3},{4},{6,7}}
=> [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
{{1,2,5},{3},{4},{6},{7}}
=> [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
{{1,2,6,7},{3},{4},{5}}
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
{{1,2,6},{3},{4},{5,7}}
=> [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
{{1,2,6},{3},{4},{5},{7}}
=> [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
{{1,2,7},{3},{4},{5,6}}
=> [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
{{1,2},{3},{4},{5,6,7}}
=> [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000365
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000365: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000365: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Values
{{1},{2},{3},{4}}
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 1 + 1
{{1,2},{3},{4},{5}}
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> [[1,3],[2],[4],[5]]
=> [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 2 = 1 + 1
{{1},{2,3},{4},{5}}
=> [[1,3],[2],[4],[5]]
=> [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 2 = 1 + 1
{{1,4},{2},{3},{5}}
=> [[1,4],[2],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 2 = 1 + 1
{{1},{2,4},{3},{5}}
=> [[1,4],[2],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 2 = 1 + 1
{{1},{2},{3,4},{5}}
=> [[1,4],[2],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 2 = 1 + 1
{{1,5},{2},{3},{4}}
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 2 = 1 + 1
{{1},{2,5},{3},{4}}
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 2 = 1 + 1
{{1},{2},{3,5},{4}}
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 2 = 1 + 1
{{1},{2},{3},{4,5}}
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 2 = 1 + 1
{{1},{2},{3},{4},{5}}
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 3 = 2 + 1
{{1,2,3},{4},{5},{6}}
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => 2 = 1 + 1
{{1,2,4},{3},{5},{6}}
=> [[1,2,4],[3],[5],[6]]
=> [[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => 2 = 1 + 1
{{1,2},{3,4},{5,6}}
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => 2 = 1 + 1
{{1,2},{3,4},{5},{6}}
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => 2 = 1 + 1
{{1,2,5},{3},{4},{6}}
=> [[1,2,5],[3],[4],[6]]
=> [[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => 2 = 1 + 1
{{1,2},{3,5},{4,6}}
=> [[1,2],[3,5],[4,6]]
=> [[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => 2 = 1 + 1
{{1,2},{3,5},{4},{6}}
=> [[1,2],[3,5],[4],[6]]
=> [[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => 2 = 1 + 1
{{1,2},{3,6},{4,5}}
=> [[1,2],[3,5],[4,6]]
=> [[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => 2 = 1 + 1
{{1,2},{3},{4,5},{6}}
=> [[1,2],[3,5],[4],[6]]
=> [[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => 2 = 1 + 1
{{1,2,6},{3},{4},{5}}
=> [[1,2,6],[3],[4],[5]]
=> [[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => 2 = 1 + 1
{{1,2},{3,6},{4},{5}}
=> [[1,2],[3,6],[4],[5]]
=> [[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => 2 = 1 + 1
{{1,2},{3},{4,6},{5}}
=> [[1,2],[3,6],[4],[5]]
=> [[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => 2 = 1 + 1
{{1,2},{3},{4},{5,6}}
=> [[1,2],[3,6],[4],[5]]
=> [[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => 2 = 1 + 1
{{1,2},{3},{4},{5},{6}}
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => 3 = 2 + 1
{{1,3,4},{2},{5},{6}}
=> [[1,3,4],[2],[5],[6]]
=> [[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => 2 = 1 + 1
{{1,3},{2,4},{5,6}}
=> [[1,3],[2,4],[5,6]]
=> [[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => 2 = 1 + 1
{{1,3},{2,4},{5},{6}}
=> [[1,3],[2,4],[5],[6]]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => 2 = 1 + 1
{{1,3,5},{2},{4},{6}}
=> [[1,3,5],[2],[4],[6]]
=> [[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => 2 = 1 + 1
{{1,3},{2,5},{4,6}}
=> [[1,3],[2,5],[4,6]]
=> [[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => 2 = 1 + 1
{{1,3},{2,5},{4},{6}}
=> [[1,3],[2,5],[4],[6]]
=> [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => 2 = 1 + 1
{{1,3},{2,6},{4,5}}
=> [[1,3],[2,5],[4,6]]
=> [[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => 2 = 1 + 1
{{1,3},{2},{4,5},{6}}
=> [[1,3],[2,5],[4],[6]]
=> [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => 2 = 1 + 1
{{1,3,6},{2},{4},{5}}
=> [[1,3,6],[2],[4],[5]]
=> [[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => 2 = 1 + 1
{{1,3},{2,6},{4},{5}}
=> [[1,3],[2,6],[4],[5]]
=> [[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => 2 = 1 + 1
{{1,3},{2},{4,6},{5}}
=> [[1,3],[2,6],[4],[5]]
=> [[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => 2 = 1 + 1
{{1,3},{2},{4},{5,6}}
=> [[1,3],[2,6],[4],[5]]
=> [[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => 2 = 1 + 1
{{1,3},{2},{4},{5},{6}}
=> [[1,3],[2],[4],[5],[6]]
=> [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => 3 = 2 + 1
{{1,4},{2,3},{5,6}}
=> [[1,3],[2,4],[5,6]]
=> [[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => 2 = 1 + 1
{{1,4},{2,3},{5},{6}}
=> [[1,3],[2,4],[5],[6]]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => 2 = 1 + 1
{{1},{2,3,4},{5},{6}}
=> [[1,3,4],[2],[5],[6]]
=> [[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => 2 = 1 + 1
{{1,5},{2,3},{4,6}}
=> [[1,3],[2,5],[4,6]]
=> [[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => 2 = 1 + 1
{{1,5},{2,3},{4},{6}}
=> [[1,3],[2,5],[4],[6]]
=> [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => 2 = 1 + 1
{{1},{2,3,5},{4},{6}}
=> [[1,3,5],[2],[4],[6]]
=> [[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => 2 = 1 + 1
{{1,6},{2,3},{4,5}}
=> [[1,3],[2,5],[4,6]]
=> [[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => 2 = 1 + 1
{{1},{2,3},{4,5},{6}}
=> [[1,3],[2,5],[4],[6]]
=> [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => 2 = 1 + 1
{{1,6},{2,3},{4},{5}}
=> [[1,3],[2,6],[4],[5]]
=> [[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => 2 = 1 + 1
{{1},{2,3,6},{4},{5}}
=> [[1,3,6],[2],[4],[5]]
=> [[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => 2 = 1 + 1
{{1},{2,3},{4,6},{5}}
=> [[1,3],[2,6],[4],[5]]
=> [[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => 2 = 1 + 1
{{1,2,3,4},{5},{6},{7}}
=> [[1,2,3,4],[5],[6],[7]]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 1 + 1
{{1,2,3,5},{4},{6},{7}}
=> [[1,2,3,5],[4],[6],[7]]
=> [[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => ? = 1 + 1
{{1,2,3},{4,5},{6,7}}
=> [[1,2,3],[4,5],[6,7]]
=> [[1,4,6],[2,5,7],[3]]
=> [3,2,5,7,1,4,6] => ? = 1 + 1
{{1,2,3},{4,5},{6},{7}}
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,4,6,7],[2,5],[3]]
=> [3,2,5,1,4,6,7] => ? = 1 + 1
{{1,2,3,6},{4},{5},{7}}
=> [[1,2,3,6],[4],[5],[7]]
=> [[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => ? = 1 + 1
{{1,2,3},{4,6},{5,7}}
=> [[1,2,3],[4,6],[5,7]]
=> [[1,4,5],[2,6,7],[3]]
=> [3,2,6,7,1,4,5] => ? = 1 + 1
{{1,2,3},{4,6},{5},{7}}
=> [[1,2,3],[4,6],[5],[7]]
=> [[1,4,5,7],[2,6],[3]]
=> [3,2,6,1,4,5,7] => ? = 1 + 1
{{1,2,3},{4,7},{5,6}}
=> [[1,2,3],[4,6],[5,7]]
=> [[1,4,5],[2,6,7],[3]]
=> [3,2,6,7,1,4,5] => ? = 1 + 1
{{1,2,3},{4},{5,6},{7}}
=> [[1,2,3],[4,6],[5],[7]]
=> [[1,4,5,7],[2,6],[3]]
=> [3,2,6,1,4,5,7] => ? = 1 + 1
{{1,2,3,7},{4},{5},{6}}
=> [[1,2,3,7],[4],[5],[6]]
=> [[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => ? = 1 + 1
{{1,2,3},{4,7},{5},{6}}
=> [[1,2,3],[4,7],[5],[6]]
=> [[1,4,5,6],[2,7],[3]]
=> [3,2,7,1,4,5,6] => ? = 1 + 1
{{1,2,3},{4},{5,7},{6}}
=> [[1,2,3],[4,7],[5],[6]]
=> [[1,4,5,6],[2,7],[3]]
=> [3,2,7,1,4,5,6] => ? = 1 + 1
{{1,2,3},{4},{5},{6,7}}
=> [[1,2,3],[4,7],[5],[6]]
=> [[1,4,5,6],[2,7],[3]]
=> [3,2,7,1,4,5,6] => ? = 1 + 1
{{1,2,3},{4},{5},{6},{7}}
=> [[1,2,3],[4],[5],[6],[7]]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 2 + 1
{{1,2,4,5},{3},{6},{7}}
=> [[1,2,4,5],[3],[6],[7]]
=> [[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => ? = 1 + 1
{{1,2,4},{3,5},{6,7}}
=> [[1,2,4],[3,5],[6,7]]
=> [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ? = 1 + 1
{{1,2,4},{3,5},{6},{7}}
=> [[1,2,4],[3,5],[6],[7]]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 1 + 1
{{1,2,4,6},{3},{5},{7}}
=> [[1,2,4,6],[3],[5],[7]]
=> [[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => ? = 1 + 1
{{1,2,4},{3,6},{5,7}}
=> [[1,2,4],[3,6],[5,7]]
=> [[1,3,5],[2,6,7],[4]]
=> [4,2,6,7,1,3,5] => ? = 1 + 1
{{1,2,4},{3,6},{5},{7}}
=> [[1,2,4],[3,6],[5],[7]]
=> [[1,3,5,7],[2,6],[4]]
=> [4,2,6,1,3,5,7] => ? = 1 + 1
{{1,2,4},{3,7},{5,6}}
=> [[1,2,4],[3,6],[5,7]]
=> [[1,3,5],[2,6,7],[4]]
=> [4,2,6,7,1,3,5] => ? = 1 + 1
{{1,2,4},{3},{5,6},{7}}
=> [[1,2,4],[3,6],[5],[7]]
=> [[1,3,5,7],[2,6],[4]]
=> [4,2,6,1,3,5,7] => ? = 1 + 1
{{1,2,4,7},{3},{5},{6}}
=> [[1,2,4,7],[3],[5],[6]]
=> [[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => ? = 1 + 1
{{1,2,4},{3,7},{5},{6}}
=> [[1,2,4],[3,7],[5],[6]]
=> [[1,3,5,6],[2,7],[4]]
=> [4,2,7,1,3,5,6] => ? = 1 + 1
{{1,2,4},{3},{5,7},{6}}
=> [[1,2,4],[3,7],[5],[6]]
=> [[1,3,5,6],[2,7],[4]]
=> [4,2,7,1,3,5,6] => ? = 1 + 1
{{1,2,4},{3},{5},{6,7}}
=> [[1,2,4],[3,7],[5],[6]]
=> [[1,3,5,6],[2,7],[4]]
=> [4,2,7,1,3,5,6] => ? = 1 + 1
{{1,2,4},{3},{5},{6},{7}}
=> [[1,2,4],[3],[5],[6],[7]]
=> [[1,3,5,6,7],[2],[4]]
=> [4,2,1,3,5,6,7] => ? = 2 + 1
{{1,2,5},{3,4},{6,7}}
=> [[1,2,5],[3,4],[6,7]]
=> [[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => ? = 1 + 1
{{1,2,5},{3,4},{6},{7}}
=> [[1,2,5],[3,4],[6],[7]]
=> [[1,3,6,7],[2,4],[5]]
=> [5,2,4,1,3,6,7] => ? = 1 + 1
{{1,2},{3,4,5},{6,7}}
=> [[1,2,5],[3,4],[6,7]]
=> [[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => ? = 1 + 1
{{1,2},{3,4,5},{6},{7}}
=> [[1,2,5],[3,4],[6],[7]]
=> [[1,3,6,7],[2,4],[5]]
=> [5,2,4,1,3,6,7] => ? = 1 + 1
{{1,2,6},{3,4},{5,7}}
=> [[1,2,6],[3,4],[5,7]]
=> [[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => ? = 1 + 1
{{1,2,6},{3,4},{5},{7}}
=> [[1,2,6],[3,4],[5],[7]]
=> [[1,3,5,7],[2,4],[6]]
=> [6,2,4,1,3,5,7] => ? = 1 + 1
{{1,2},{3,4,6},{5,7}}
=> [[1,2,6],[3,4],[5,7]]
=> [[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => ? = 1 + 1
{{1,2},{3,4,6},{5},{7}}
=> [[1,2,6],[3,4],[5],[7]]
=> [[1,3,5,7],[2,4],[6]]
=> [6,2,4,1,3,5,7] => ? = 1 + 1
{{1,2,7},{3,4},{5,6}}
=> [[1,2,7],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7]]
=> [7,2,4,6,1,3,5] => ? = 1 + 1
{{1,2},{3,4,7},{5,6}}
=> [[1,2,7],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7]]
=> [7,2,4,6,1,3,5] => ? = 1 + 1
{{1,2},{3,4},{5,6,7}}
=> [[1,2,7],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7]]
=> [7,2,4,6,1,3,5] => ? = 1 + 1
{{1,2},{3,4},{5,6},{7}}
=> [[1,2],[3,4],[5,6],[7]]
=> [[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => ? = 0 + 1
{{1,2,7},{3,4},{5},{6}}
=> [[1,2,7],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4],[7]]
=> [7,2,4,1,3,5,6] => ? = 1 + 1
{{1,2},{3,4,7},{5},{6}}
=> [[1,2,7],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4],[7]]
=> [7,2,4,1,3,5,6] => ? = 1 + 1
{{1,2},{3,4},{5,7},{6}}
=> [[1,2],[3,4],[5,7],[6]]
=> [[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => ? = 0 + 1
{{1,2},{3,4},{5},{6,7}}
=> [[1,2],[3,4],[5,7],[6]]
=> [[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => ? = 0 + 1
{{1,2},{3,4},{5},{6},{7}}
=> [[1,2],[3,4],[5],[6],[7]]
=> [[1,3,5,6,7],[2,4]]
=> [2,4,1,3,5,6,7] => ? = 2 + 1
{{1,2,5,6},{3},{4},{7}}
=> [[1,2,5,6],[3],[4],[7]]
=> [[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => ? = 1 + 1
{{1,2,5},{3,6},{4,7}}
=> [[1,2,5],[3,6],[4,7]]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => ? = 1 + 1
{{1,2,5},{3,6},{4},{7}}
=> [[1,2,5],[3,6],[4],[7]]
=> [[1,3,4,7],[2,6],[5]]
=> [5,2,6,1,3,4,7] => ? = 1 + 1
{{1,2,5},{3,7},{4,6}}
=> [[1,2,5],[3,6],[4,7]]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => ? = 1 + 1
{{1,2,5},{3},{4,6},{7}}
=> [[1,2,5],[3,6],[4],[7]]
=> [[1,3,4,7],[2,6],[5]]
=> [5,2,6,1,3,4,7] => ? = 1 + 1
{{1,2,5,7},{3},{4},{6}}
=> [[1,2,5,7],[3],[4],[6]]
=> [[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => ? = 1 + 1
Description
The number of double ascents of a permutation.
A double ascent of a permutation $\pi$ is a position $i$ such that $\pi(i) < \pi(i+1) < \pi(i+2)$.
Matching statistic: St000923
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000923: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000923: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Values
{{1},{2},{3},{4}}
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> [1,2,3,4] => 4 = 1 + 3
{{1,2},{3},{4},{5}}
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 4 = 1 + 3
{{1,3},{2},{4},{5}}
=> [[1,3],[2],[4],[5]]
=> [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 4 = 1 + 3
{{1},{2,3},{4},{5}}
=> [[1,3],[2],[4],[5]]
=> [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 4 = 1 + 3
{{1,4},{2},{3},{5}}
=> [[1,4],[2],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 4 = 1 + 3
{{1},{2,4},{3},{5}}
=> [[1,4],[2],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 4 = 1 + 3
{{1},{2},{3,4},{5}}
=> [[1,4],[2],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 4 = 1 + 3
{{1,5},{2},{3},{4}}
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 4 = 1 + 3
{{1},{2,5},{3},{4}}
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 4 = 1 + 3
{{1},{2},{3,5},{4}}
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 4 = 1 + 3
{{1},{2},{3},{4,5}}
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 4 = 1 + 3
{{1},{2},{3},{4},{5}}
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 5 = 2 + 3
{{1,2,3},{4},{5},{6}}
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => 4 = 1 + 3
{{1,2,4},{3},{5},{6}}
=> [[1,2,4],[3],[5],[6]]
=> [[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => 4 = 1 + 3
{{1,2},{3,4},{5,6}}
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => 4 = 1 + 3
{{1,2},{3,4},{5},{6}}
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => 4 = 1 + 3
{{1,2,5},{3},{4},{6}}
=> [[1,2,5],[3],[4],[6]]
=> [[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => 4 = 1 + 3
{{1,2},{3,5},{4,6}}
=> [[1,2],[3,5],[4,6]]
=> [[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => 4 = 1 + 3
{{1,2},{3,5},{4},{6}}
=> [[1,2],[3,5],[4],[6]]
=> [[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => 4 = 1 + 3
{{1,2},{3,6},{4,5}}
=> [[1,2],[3,5],[4,6]]
=> [[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => 4 = 1 + 3
{{1,2},{3},{4,5},{6}}
=> [[1,2],[3,5],[4],[6]]
=> [[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => 4 = 1 + 3
{{1,2,6},{3},{4},{5}}
=> [[1,2,6],[3],[4],[5]]
=> [[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => 4 = 1 + 3
{{1,2},{3,6},{4},{5}}
=> [[1,2],[3,6],[4],[5]]
=> [[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => 4 = 1 + 3
{{1,2},{3},{4,6},{5}}
=> [[1,2],[3,6],[4],[5]]
=> [[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => 4 = 1 + 3
{{1,2},{3},{4},{5,6}}
=> [[1,2],[3,6],[4],[5]]
=> [[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => 4 = 1 + 3
{{1,2},{3},{4},{5},{6}}
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => 5 = 2 + 3
{{1,3,4},{2},{5},{6}}
=> [[1,3,4],[2],[5],[6]]
=> [[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => 4 = 1 + 3
{{1,3},{2,4},{5,6}}
=> [[1,3],[2,4],[5,6]]
=> [[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => 4 = 1 + 3
{{1,3},{2,4},{5},{6}}
=> [[1,3],[2,4],[5],[6]]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => 4 = 1 + 3
{{1,3,5},{2},{4},{6}}
=> [[1,3,5],[2],[4],[6]]
=> [[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => 4 = 1 + 3
{{1,3},{2,5},{4,6}}
=> [[1,3],[2,5],[4,6]]
=> [[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => 4 = 1 + 3
{{1,3},{2,5},{4},{6}}
=> [[1,3],[2,5],[4],[6]]
=> [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => 4 = 1 + 3
{{1,3},{2,6},{4,5}}
=> [[1,3],[2,5],[4,6]]
=> [[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => 4 = 1 + 3
{{1,3},{2},{4,5},{6}}
=> [[1,3],[2,5],[4],[6]]
=> [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => 4 = 1 + 3
{{1,3,6},{2},{4},{5}}
=> [[1,3,6],[2],[4],[5]]
=> [[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => 4 = 1 + 3
{{1,3},{2,6},{4},{5}}
=> [[1,3],[2,6],[4],[5]]
=> [[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => 4 = 1 + 3
{{1,3},{2},{4,6},{5}}
=> [[1,3],[2,6],[4],[5]]
=> [[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => 4 = 1 + 3
{{1,3},{2},{4},{5,6}}
=> [[1,3],[2,6],[4],[5]]
=> [[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => 4 = 1 + 3
{{1,3},{2},{4},{5},{6}}
=> [[1,3],[2],[4],[5],[6]]
=> [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => 5 = 2 + 3
{{1,4},{2,3},{5,6}}
=> [[1,3],[2,4],[5,6]]
=> [[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => 4 = 1 + 3
{{1,4},{2,3},{5},{6}}
=> [[1,3],[2,4],[5],[6]]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => 4 = 1 + 3
{{1},{2,3,4},{5},{6}}
=> [[1,3,4],[2],[5],[6]]
=> [[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => 4 = 1 + 3
{{1,5},{2,3},{4,6}}
=> [[1,3],[2,5],[4,6]]
=> [[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => 4 = 1 + 3
{{1,5},{2,3},{4},{6}}
=> [[1,3],[2,5],[4],[6]]
=> [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => 4 = 1 + 3
{{1},{2,3,5},{4},{6}}
=> [[1,3,5],[2],[4],[6]]
=> [[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => 4 = 1 + 3
{{1,6},{2,3},{4,5}}
=> [[1,3],[2,5],[4,6]]
=> [[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => 4 = 1 + 3
{{1},{2,3},{4,5},{6}}
=> [[1,3],[2,5],[4],[6]]
=> [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => 4 = 1 + 3
{{1,6},{2,3},{4},{5}}
=> [[1,3],[2,6],[4],[5]]
=> [[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => 4 = 1 + 3
{{1},{2,3,6},{4},{5}}
=> [[1,3,6],[2],[4],[5]]
=> [[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => 4 = 1 + 3
{{1},{2,3},{4,6},{5}}
=> [[1,3],[2,6],[4],[5]]
=> [[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => 4 = 1 + 3
{{1,2,3,4},{5},{6},{7}}
=> [[1,2,3,4],[5],[6],[7]]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 1 + 3
{{1,2,3,5},{4},{6},{7}}
=> [[1,2,3,5],[4],[6],[7]]
=> [[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => ? = 1 + 3
{{1,2,3},{4,5},{6,7}}
=> [[1,2,3],[4,5],[6,7]]
=> [[1,4,6],[2,5,7],[3]]
=> [3,2,5,7,1,4,6] => ? = 1 + 3
{{1,2,3},{4,5},{6},{7}}
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,4,6,7],[2,5],[3]]
=> [3,2,5,1,4,6,7] => ? = 1 + 3
{{1,2,3,6},{4},{5},{7}}
=> [[1,2,3,6],[4],[5],[7]]
=> [[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => ? = 1 + 3
{{1,2,3},{4,6},{5,7}}
=> [[1,2,3],[4,6],[5,7]]
=> [[1,4,5],[2,6,7],[3]]
=> [3,2,6,7,1,4,5] => ? = 1 + 3
{{1,2,3},{4,6},{5},{7}}
=> [[1,2,3],[4,6],[5],[7]]
=> [[1,4,5,7],[2,6],[3]]
=> [3,2,6,1,4,5,7] => ? = 1 + 3
{{1,2,3},{4,7},{5,6}}
=> [[1,2,3],[4,6],[5,7]]
=> [[1,4,5],[2,6,7],[3]]
=> [3,2,6,7,1,4,5] => ? = 1 + 3
{{1,2,3},{4},{5,6},{7}}
=> [[1,2,3],[4,6],[5],[7]]
=> [[1,4,5,7],[2,6],[3]]
=> [3,2,6,1,4,5,7] => ? = 1 + 3
{{1,2,3,7},{4},{5},{6}}
=> [[1,2,3,7],[4],[5],[6]]
=> [[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => ? = 1 + 3
{{1,2,3},{4,7},{5},{6}}
=> [[1,2,3],[4,7],[5],[6]]
=> [[1,4,5,6],[2,7],[3]]
=> [3,2,7,1,4,5,6] => ? = 1 + 3
{{1,2,3},{4},{5,7},{6}}
=> [[1,2,3],[4,7],[5],[6]]
=> [[1,4,5,6],[2,7],[3]]
=> [3,2,7,1,4,5,6] => ? = 1 + 3
{{1,2,3},{4},{5},{6,7}}
=> [[1,2,3],[4,7],[5],[6]]
=> [[1,4,5,6],[2,7],[3]]
=> [3,2,7,1,4,5,6] => ? = 1 + 3
{{1,2,3},{4},{5},{6},{7}}
=> [[1,2,3],[4],[5],[6],[7]]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 2 + 3
{{1,2,4,5},{3},{6},{7}}
=> [[1,2,4,5],[3],[6],[7]]
=> [[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => ? = 1 + 3
{{1,2,4},{3,5},{6,7}}
=> [[1,2,4],[3,5],[6,7]]
=> [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => ? = 1 + 3
{{1,2,4},{3,5},{6},{7}}
=> [[1,2,4],[3,5],[6],[7]]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 1 + 3
{{1,2,4,6},{3},{5},{7}}
=> [[1,2,4,6],[3],[5],[7]]
=> [[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => ? = 1 + 3
{{1,2,4},{3,6},{5,7}}
=> [[1,2,4],[3,6],[5,7]]
=> [[1,3,5],[2,6,7],[4]]
=> [4,2,6,7,1,3,5] => ? = 1 + 3
{{1,2,4},{3,6},{5},{7}}
=> [[1,2,4],[3,6],[5],[7]]
=> [[1,3,5,7],[2,6],[4]]
=> [4,2,6,1,3,5,7] => ? = 1 + 3
{{1,2,4},{3,7},{5,6}}
=> [[1,2,4],[3,6],[5,7]]
=> [[1,3,5],[2,6,7],[4]]
=> [4,2,6,7,1,3,5] => ? = 1 + 3
{{1,2,4},{3},{5,6},{7}}
=> [[1,2,4],[3,6],[5],[7]]
=> [[1,3,5,7],[2,6],[4]]
=> [4,2,6,1,3,5,7] => ? = 1 + 3
{{1,2,4,7},{3},{5},{6}}
=> [[1,2,4,7],[3],[5],[6]]
=> [[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => ? = 1 + 3
{{1,2,4},{3,7},{5},{6}}
=> [[1,2,4],[3,7],[5],[6]]
=> [[1,3,5,6],[2,7],[4]]
=> [4,2,7,1,3,5,6] => ? = 1 + 3
{{1,2,4},{3},{5,7},{6}}
=> [[1,2,4],[3,7],[5],[6]]
=> [[1,3,5,6],[2,7],[4]]
=> [4,2,7,1,3,5,6] => ? = 1 + 3
{{1,2,4},{3},{5},{6,7}}
=> [[1,2,4],[3,7],[5],[6]]
=> [[1,3,5,6],[2,7],[4]]
=> [4,2,7,1,3,5,6] => ? = 1 + 3
{{1,2,4},{3},{5},{6},{7}}
=> [[1,2,4],[3],[5],[6],[7]]
=> [[1,3,5,6,7],[2],[4]]
=> [4,2,1,3,5,6,7] => ? = 2 + 3
{{1,2,5},{3,4},{6,7}}
=> [[1,2,5],[3,4],[6,7]]
=> [[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => ? = 1 + 3
{{1,2,5},{3,4},{6},{7}}
=> [[1,2,5],[3,4],[6],[7]]
=> [[1,3,6,7],[2,4],[5]]
=> [5,2,4,1,3,6,7] => ? = 1 + 3
{{1,2},{3,4,5},{6,7}}
=> [[1,2,5],[3,4],[6,7]]
=> [[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => ? = 1 + 3
{{1,2},{3,4,5},{6},{7}}
=> [[1,2,5],[3,4],[6],[7]]
=> [[1,3,6,7],[2,4],[5]]
=> [5,2,4,1,3,6,7] => ? = 1 + 3
{{1,2,6},{3,4},{5,7}}
=> [[1,2,6],[3,4],[5,7]]
=> [[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => ? = 1 + 3
{{1,2,6},{3,4},{5},{7}}
=> [[1,2,6],[3,4],[5],[7]]
=> [[1,3,5,7],[2,4],[6]]
=> [6,2,4,1,3,5,7] => ? = 1 + 3
{{1,2},{3,4,6},{5,7}}
=> [[1,2,6],[3,4],[5,7]]
=> [[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => ? = 1 + 3
{{1,2},{3,4,6},{5},{7}}
=> [[1,2,6],[3,4],[5],[7]]
=> [[1,3,5,7],[2,4],[6]]
=> [6,2,4,1,3,5,7] => ? = 1 + 3
{{1,2,7},{3,4},{5,6}}
=> [[1,2,7],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7]]
=> [7,2,4,6,1,3,5] => ? = 1 + 3
{{1,2},{3,4,7},{5,6}}
=> [[1,2,7],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7]]
=> [7,2,4,6,1,3,5] => ? = 1 + 3
{{1,2},{3,4},{5,6,7}}
=> [[1,2,7],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7]]
=> [7,2,4,6,1,3,5] => ? = 1 + 3
{{1,2},{3,4},{5,6},{7}}
=> [[1,2],[3,4],[5,6],[7]]
=> [[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => ? = 0 + 3
{{1,2,7},{3,4},{5},{6}}
=> [[1,2,7],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4],[7]]
=> [7,2,4,1,3,5,6] => ? = 1 + 3
{{1,2},{3,4,7},{5},{6}}
=> [[1,2,7],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4],[7]]
=> [7,2,4,1,3,5,6] => ? = 1 + 3
{{1,2},{3,4},{5,7},{6}}
=> [[1,2],[3,4],[5,7],[6]]
=> [[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => ? = 0 + 3
{{1,2},{3,4},{5},{6,7}}
=> [[1,2],[3,4],[5,7],[6]]
=> [[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => ? = 0 + 3
{{1,2},{3,4},{5},{6},{7}}
=> [[1,2],[3,4],[5],[6],[7]]
=> [[1,3,5,6,7],[2,4]]
=> [2,4,1,3,5,6,7] => ? = 2 + 3
{{1,2,5,6},{3},{4},{7}}
=> [[1,2,5,6],[3],[4],[7]]
=> [[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => ? = 1 + 3
{{1,2,5},{3,6},{4,7}}
=> [[1,2,5],[3,6],[4,7]]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => ? = 1 + 3
{{1,2,5},{3,6},{4},{7}}
=> [[1,2,5],[3,6],[4],[7]]
=> [[1,3,4,7],[2,6],[5]]
=> [5,2,6,1,3,4,7] => ? = 1 + 3
{{1,2,5},{3,7},{4,6}}
=> [[1,2,5],[3,6],[4,7]]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => ? = 1 + 3
{{1,2,5},{3},{4,6},{7}}
=> [[1,2,5],[3,6],[4],[7]]
=> [[1,3,4,7],[2,6],[5]]
=> [5,2,6,1,3,4,7] => ? = 1 + 3
{{1,2,5,7},{3},{4},{6}}
=> [[1,2,5,7],[3],[4],[6]]
=> [[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => ? = 1 + 3
Description
The minimal number with no two order isomorphic substrings of this length in a permutation.
For example, the length $3$ substrings of the permutation $12435$ are $124$, $243$ and $435$, whereas its length $2$ substrings are $12$, $24$, $43$ and $35$.
No two sequences among $124$, $243$ and $435$ are order isomorphic, but $12$ and $24$ are, so the statistic on $12435$ is $3$.
This is inspired by [[St000922]].
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000080The rank of the poset. St000864The number of circled entries of the shifted recording tableau of a permutation. St001566The length of the longest arithmetic progression in a permutation. St000863The length of the first row of the shifted shape of a permutation. St001644The dimension of a graph. St001645The pebbling number of a connected graph.
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