Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St000509
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000509: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => -1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [[3,3],[2]]
 => [2]
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => -1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => -1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => -1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [[4,3],[2]]
 => [2]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => -1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => -3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [[4,4],[3]]
 => [3]
 => 3
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,3,2] => [[4,3,1],[2]]
 => [2]
 => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,4,1] => [[4,4,1],[3]]
 => [3]
 => 3
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => -1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => -1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => -1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => -1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => -1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => 2
([(0,5),(1,4),(2,3)],6)
 => [3,3] => [[5,3],[2]]
 => [2]
 => 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => -1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => -1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => -1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => 0
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 0
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => -2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,3,2] => [[4,3,1],[2]]
 => [2]
 => 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 0
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => -1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => -3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 0
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => -3
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => -3
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3] => [[5,3],[2]]
 => [2]
 => 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => -1
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => 0
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => -1
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => -3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => 0
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => -2
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => -3
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 0
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => -1
Description
The diagonal index (content) of a partition. The '''diagonal index''' of the cell at row $r$ and column $c$ of a partition is $c - r$; this is sometimes called the '''content''' of the cell. The '''diagonal index of a partition''' is the sum of the diagonal index of each cell of the partition.