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Your data matches 40 different statistics following compositions of up to 3 maps.
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Matching statistic: St000805
Mp00324: Graphs chromatic difference sequenceInteger compositions
Mp00039: Integer compositions complementInteger compositions
Mp00314: Integer compositions Foata bijectionInteger compositions
St000805: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,2,2] => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,2,2] => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,2,2] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => [1,2,2] => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,2,2] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,2,2] => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,2,2] => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [1,2,2] => [1,2,2] => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,2,2] => 1
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [1,1,2,1,1] => [1,2,1,1,1] => 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => [1,2,1,1,1] => 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => [1,2,1,1,1] => 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => [1,2,1,1,1] => 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [3,3] => [1,1,2,1,1] => [1,2,1,1,1] => 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3] => [1,1,2,1,1] => [1,2,1,1,1] => 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => [1,2,1,1,1] => 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => [1,2,1,1,1] => 1
Description
The number of peaks of the associated bargraph. Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of contiguous subsequences consisting of an up step, a sequence of horizontal steps, and a down step.
Matching statistic: St001621
Mp00324: Graphs chromatic difference sequenceInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00192: Skew partitions dominating sublatticeLattices
St001621: Lattices ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
Description
The number of atoms of a lattice. An element of a lattice is an '''atom''' if it covers the least element.
Matching statistic: St001681
Mp00324: Graphs chromatic difference sequenceInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00192: Skew partitions dominating sublatticeLattices
St001681: Lattices ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
Description
The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. For example, the pentagon lattice has three such sets: the bottom element, and the two antichains of size two. The cube is the smallest lattice which has such sets of three different sizes: the bottom element, six antichains of size two and one antichain of size three.
Mp00324: Graphs chromatic difference sequenceInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00192: Skew partitions dominating sublatticeLattices
St001878: Lattices ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St000807
Mp00324: Graphs chromatic difference sequenceInteger compositions
Mp00039: Integer compositions complementInteger compositions
Mp00314: Integer compositions Foata bijectionInteger compositions
St000807: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,2,2] => 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,2,2] => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,2,2] => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => [1,2,2] => 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,2,2] => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,2,2] => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,2,2] => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [1,2,2] => [1,2,2] => 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,2,2] => 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [1,1,2,1,1] => [1,2,1,1,1] => 0 = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => [1,2,1,1,1] => 0 = 1 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => [1,2,1,1,1] => 0 = 1 - 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => [1,2,1,1,1] => 0 = 1 - 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [3,3] => [1,1,2,1,1] => [1,2,1,1,1] => 0 = 1 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3] => [1,1,2,1,1] => [1,2,1,1,1] => 0 = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => [1,2,1,1,1] => 0 = 1 - 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0 = 1 - 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => [1,2,1,1,1] => 0 = 1 - 1
Description
The sum of the heights of the valleys of the associated bargraph. Interpret the composition as the sequence of heights of the bars of a bargraph. A valley is a contiguous subsequence consisting of an up step, a sequence of horizontal steps, and a down step. This statistic is the sum of the heights of the valleys.
Matching statistic: St001876
Mp00324: Graphs chromatic difference sequenceInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00192: Skew partitions dominating sublatticeLattices
St001876: Lattices ⟶ ℤResult quality: 93% values known / values provided: 93%distinct values known / distinct values provided: 100%
Values
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,4),(0,5),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,3),(0,4),(1,2),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,1),(0,2),(0,6),(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,3),(0,5),(1,2),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,1),(0,3),(0,6),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,3),(0,4),(1,2),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,2),(0,6),(1,3),(1,4),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,1),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,2),(0,6),(1,3),(1,4),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,1),(0,2),(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,1),(0,4),(0,6),(1,3),(1,4),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,4),(0,5),(1,2),(1,3),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,1),(0,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,1),(0,6),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,1),(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,1),(0,2),(0,6),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,1),(0,5),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,1),(0,4),(0,6),(1,3),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,3),(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,2),(0,3),(0,4),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,4),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,3),(0,6),(1,2),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,2),(0,3),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,3),(0,5),(0,6),(1,2),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,1),(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,1),(0,4),(0,6),(1,4),(1,5),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,3),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,1),(0,4),(0,6),(1,3),(1,5),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1 - 1
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St000481
Mp00324: Graphs chromatic difference sequenceInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00182: Skew partitions outer shapeInteger partitions
St000481: Integer partitions ⟶ ℤResult quality: 33% values known / values provided: 33%distinct values known / distinct values provided: 100%
Values
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 1
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [[5,3],[2]]
=> [5,3]
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> [5,3]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> [5,3]
=> 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> [5,3]
=> 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [3,3] => [[5,3],[2]]
=> [5,3]
=> 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3] => [[5,3],[2]]
=> [5,3]
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> [5,3]
=> 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> [5,3]
=> 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(0,6),(1,6),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(0,6),(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(0,6),(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(0,6),(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(0,6),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(0,5),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(0,6),(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(0,6),(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(0,6),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(0,6),(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
([(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1
Description
The number of upper covers of a partition in dominance order.
Matching statistic: St000159
Mp00324: Graphs chromatic difference sequenceInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00182: Skew partitions outer shapeInteger partitions
St000159: Integer partitions ⟶ ℤResult quality: 33% values known / values provided: 33%distinct values known / distinct values provided: 100%
Values
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 2 = 1 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 2 = 1 + 1
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [[5,3],[2]]
=> [5,3]
=> 2 = 1 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> [5,3]
=> 2 = 1 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> [5,3]
=> 2 = 1 + 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> [5,3]
=> 2 = 1 + 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [3,3] => [[5,3],[2]]
=> [5,3]
=> 2 = 1 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3] => [[5,3],[2]]
=> [5,3]
=> 2 = 1 + 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> [5,3]
=> 2 = 1 + 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> 2 = 1 + 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> [5,3]
=> 2 = 1 + 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(0,6),(1,6),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(0,6),(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(0,6),(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(0,6),(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(0,6),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(0,5),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(0,6),(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(0,6),(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(0,6),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(0,6),(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
([(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> [4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> ? = 1 + 1
Description
The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Mp00324: Graphs chromatic difference sequenceInteger compositions
Mp00041: Integer compositions conjugateInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St001645: Graphs ⟶ ℤResult quality: 26% values known / values provided: 26%distinct values known / distinct values provided: 50%
Values
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [1,1,2,1,1] => ([(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)
=> ? = 1 + 6
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => ([(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)
=> ? = 1 + 6
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => ([(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)
=> ? = 1 + 6
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => ([(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)
=> ? = 1 + 6
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [3,3] => [1,1,2,1,1] => ([(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)
=> ? = 1 + 6
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3] => [1,1,2,1,1] => ([(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)
=> ? = 1 + 6
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => ([(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)
=> ? = 1 + 6
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [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)
=> ? = 1 + 6
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => ([(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)
=> ? = 1 + 6
([(1,6),(2,5),(3,4)],7)
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(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)
=> 7 = 1 + 6
([(1,2),(3,6),(4,5),(5,6)],7)
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(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)
=> 7 = 1 + 6
([(0,3),(1,2),(4,6),(5,6)],7)
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(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)
=> 7 = 1 + 6
([(2,3),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(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)
=> 7 = 1 + 6
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(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)
=> 7 = 1 + 6
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(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)
=> 7 = 1 + 6
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(0,6),(1,6),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(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)
=> 7 = 1 + 6
([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(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)
=> 7 = 1 + 6
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(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)
=> 7 = 1 + 6
([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(0,6),(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(0,6),(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(0,6),(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(0,6),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(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)
=> 7 = 1 + 6
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(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)
=> 7 = 1 + 6
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(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)
=> 7 = 1 + 6
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(1,5),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(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)
=> 7 = 1 + 6
([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(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)
=> 7 = 1 + 6
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(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)
=> 7 = 1 + 6
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(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)
=> 7 = 1 + 6
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(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)
=> 7 = 1 + 6
([(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
([(0,5),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => [2,2,1,1,1] => ([(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)
=> 7 = 1 + 6
Description
The pebbling number of a connected graph.
Matching statistic: St000260
Mp00324: Graphs chromatic difference sequenceInteger compositions
Mp00172: Integer compositions rotate back to frontInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000260: Graphs ⟶ ℤResult quality: 25% values known / values provided: 25%distinct values known / distinct values provided: 50%
Values
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,6),(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,4),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,6),(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,6),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,5),(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,2),(1,5),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,6),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,1),(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,4),(0,6),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,6),(1,3),(1,5),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,5),(1,2),(1,4),(1,6),(2,3),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
Description
The radius of a connected graph. This is the minimum eccentricity of any vertex.
The following 30 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000422The energy of a graph, if it is integral. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between eiJ and ejJ (the radical of the indecomposable projective modules). St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St001188The number of simple modules S with grade inf at least two in the Nakayama algebra A corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001578The minimal number of edges to add or remove to make a graph a line graph. St000299The number of nonisomorphic vertex-induced subtrees. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001964The interval resolution global dimension of a poset. St000741The Colin de Verdière graph invariant. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000455The second largest eigenvalue of a graph if it is integral. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St001570The minimal number of edges to add to make a graph Hamiltonian. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation.