Your data matches 44 different statistics following compositions of up to 3 maps.
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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
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,2,2] => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,2,2] => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => [1,2,2] => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,2,2] => 0
([(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
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,2,2] => 0
([(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
([(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
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [1,1,2,1,1] => [1,2,1,1,1] => 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => [1,2,1,1,1] => 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => [1,2,1,1,1] => 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0
([(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
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0
([(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
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0
([(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
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0
([(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
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0
([(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
([(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
([(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
([(1,4),(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
([(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
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [3,3] => [1,1,2,1,1] => [1,2,1,1,1] => 0
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3] => [1,1,2,1,1] => [1,2,1,1,1] => 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 0
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
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: 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)
=> 0
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
([(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
([(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
([(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
([(1,4),(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
([(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
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
([(1,2),(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
([(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
([(1,5),(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
([(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
([(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
([(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
([(1,4),(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
([(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
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
([(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
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Mp00324: Graphs chromatic difference sequenceInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00192: Skew partitions dominating sublatticeLattices
St001877: 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)
=> 0
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
([(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
([(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
([(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
([(1,4),(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
([(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
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
([(1,2),(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
([(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
([(1,5),(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
([(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
([(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
([(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
([(1,4),(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
([(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
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
([(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
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
Description
Number of indecomposable injective modules with projective dimension 2.
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 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,2,2] => 1 = 0 + 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 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => [1,2,2] => 1 = 0 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => [1,2,2] => 1 = 0 + 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 + 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 + 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 + 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 + 1
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [1,1,2,1,1] => [1,2,1,1,1] => 1 = 0 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => [1,2,1,1,1] => 1 = 0 + 1
([(1,2),(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),(2,3),(3,5),(4,5)],6)
=> [3,3] => [1,1,2,1,1] => [1,2,1,1,1] => 1 = 0 + 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
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,2,2] => [1,1,2,2] => 1 = 0 + 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 = 0 + 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 + 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 = 0 + 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 + 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 = 0 + 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 + 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 = 0 + 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 + 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 + 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 + 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 = 0 + 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 + 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 + 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
([(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 + 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 + 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 + 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 + 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
([(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 + 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 + 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 + 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 + 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
([(0,1),(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 + 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
([(0,1),(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 + 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 + 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 + 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
([(0,1),(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 + 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 + 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 + 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 + 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 = 0 + 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 + 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 + 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 + 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
([(0,1),(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 + 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 + 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 + 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 + 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 + 1
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 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 + 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 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
([(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 + 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 = 0 + 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 + 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 = 0 + 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 + 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 = 0 + 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 + 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 = 0 + 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 + 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 + 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 + 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 = 0 + 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 + 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 + 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
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 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 + 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 + 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 + 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
([(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 + 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 + 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 + 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 + 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
([(0,1),(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 + 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
([(0,1),(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 + 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 + 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 + 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
([(0,1),(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 + 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 + 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 + 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 + 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 = 0 + 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: St001624
Mp00324: Graphs chromatic difference sequenceInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00192: Skew partitions dominating sublatticeLattices
St001624: 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 + 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 + 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 + 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
([(0,1),(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 + 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 + 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 + 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 + 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 + 1
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 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 + 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 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
([(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 + 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 = 0 + 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 + 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 = 0 + 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 + 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 = 0 + 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 + 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 = 0 + 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 + 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 + 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 + 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 = 0 + 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 + 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 + 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
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 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 + 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 + 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 + 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
([(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 + 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 + 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 + 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 + 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
([(0,1),(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 + 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
([(0,1),(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 + 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 + 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 + 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
([(0,1),(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 + 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 + 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 + 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 + 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 = 0 + 1
Description
The breadth of a lattice. The '''breadth''' of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.
Mp00324: Graphs chromatic difference sequenceInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00192: Skew partitions dominating sublatticeLattices
St001630: 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 + 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 + 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 + 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
([(0,1),(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 + 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 + 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 + 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 + 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 + 1
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 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 + 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 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
([(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 + 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 = 0 + 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 + 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 = 0 + 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 + 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 = 0 + 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 + 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 = 0 + 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 + 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 + 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 + 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 = 0 + 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 + 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 + 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
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 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 + 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 + 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 + 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
([(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 + 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 + 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 + 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 + 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
([(0,1),(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 + 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
([(0,1),(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 + 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 + 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 + 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
([(0,1),(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 + 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 + 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 + 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 + 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 = 0 + 1
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
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 + 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 + 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 + 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
([(0,1),(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 + 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 + 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 + 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 + 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 + 1
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 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 + 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 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
([(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 + 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 = 0 + 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 + 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 = 0 + 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 + 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 = 0 + 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 + 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 = 0 + 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 + 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 + 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 + 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 = 0 + 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 + 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 + 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
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 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 + 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 + 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 + 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
([(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 + 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 + 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 + 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 + 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
([(0,1),(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 + 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
([(0,1),(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 + 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 + 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 + 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
([(0,1),(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 + 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 + 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 + 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 + 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 = 0 + 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 + 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 + 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 + 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
([(0,1),(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 + 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 + 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 + 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 + 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 + 1
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 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 + 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 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
([(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 + 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 = 0 + 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 + 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 = 0 + 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 + 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 = 0 + 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 + 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 = 0 + 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 + 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 + 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 + 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 = 0 + 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 + 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 + 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
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 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 + 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 + 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 + 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
([(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 + 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 + 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 + 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 + 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
([(0,1),(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 + 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
([(0,1),(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 + 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 + 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 + 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
([(0,1),(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 + 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 + 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 + 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 + 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 = 0 + 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Mp00324: Graphs chromatic difference sequenceInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
St001232: Dyck paths ⟶ ℤResult quality: 73% values known / values provided: 73%distinct values known / distinct values provided: 100%
Values
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,5),(1,4),(2,3)],6)
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 0 + 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(0,6),(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(1,4),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(0,5),(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
([(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] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 3
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
The following 34 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000481The number of upper covers of a partition in dominance order. St000781The number of proper colouring schemes of a Ferrers diagram. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000159The number of distinct parts of the integer partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001645The pebbling number of a connected graph. St000260The radius of a connected graph. 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 $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001578The minimal number of edges to add or remove to make a graph a line graph. 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 $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ 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. 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. St000455The second largest eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St001875The number of simple modules with projective dimension at most 1. St001570The minimal number of edges to add to make a graph Hamiltonian. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation.