Your data matches 71 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001335
(load all 3 compositions to match this statistic)
Mapping
Mp00193: Lattices —to poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001335: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => ([],1)
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 0
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 1
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 0
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
 => ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 1
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5)],7)
 => 1
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
 => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
 => ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 1
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
 => ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
 => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 1
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => ([(0,6),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => 1
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
 => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
 => ([(0,6),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => 1
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
 => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
 => ([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 1
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
 => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
 => ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 1
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
 => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
 => ([(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(5,6)],7)
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
 => ([(0,1),(0,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => 1
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => 1
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
 => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
 => ([(0,6),(1,2),(1,3),(2,5),(3,4),(4,6),(5,6)],7)
 => 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => 1
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
 => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
 => ([(0,1),(0,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => 1
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
 => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
 => ([(0,1),(0,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => 1
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
 => ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
 => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5)],7)
 => 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
 => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
 => ([(0,1),(0,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,6),(5,6)],7)
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
 => ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
 => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
 => 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
 => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 1
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
 => ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
 => ([(0,1),(0,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
 => 1
Description
The cardinality of a minimal cycle-isolating set of a graph. Let $\mathcal F$ be a set of graphs. A set of vertices $S$ is $\mathcal F$-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of $S$ does not contain any graph in $\mathcal F$. This statistic returns the cardinality of the smallest isolating set when $\mathcal F$ contains all cycles.
Matching statistic: St000897
(load all 2 compositions to match this statistic)
Mapping
Mp00193: Lattices —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000897: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => [1]
 => []
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => []
 => 0
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => [3]
 => []
 => 0
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => []
 => 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [4,1]
 => [1]
 => 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [1]
 => 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => []
 => 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [1]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => [4,1,1]
 => [1,1]
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => [4,1,1]
 => [1,1]
 => 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => [4,1,1]
 => [1,1]
 => 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => [5,1]
 => [1]
 => 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [1]
 => 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => [5,1]
 => [1]
 => 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [5,1]
 => [1]
 => 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [5,1]
 => [1]
 => 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [4,1,1]
 => [1,1]
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => [4,2]
 => [2]
 => 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => [2]
 => 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => []
 => 0
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => [5,1]
 => [1]
 => 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 1
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 1
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
 => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 1
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
 => ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 1
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => [5,1,1]
 => [1,1]
 => 1
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
 => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 1
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
 => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
 => [5,1,1]
 => [1,1]
 => 1
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
 => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
 => [5,1,1]
 => [1,1]
 => 1
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
 => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
 => [5,1,1]
 => [1,1]
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 1
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => [6,1]
 => [1]
 => 1
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
 => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
 => [5,2]
 => [2]
 => 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [2]
 => 1
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
 => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
 => [5,1,1]
 => [1,1]
 => 1
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
 => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
 => [5,1,1]
 => [1,1]
 => 1
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
 => ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
 => [5,1,1]
 => [1,1]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
 => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
 => [5,1,1]
 => [1,1]
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => [5,2]
 => [2]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
 => ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
 => [5,1,1]
 => [1,1]
 => 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
 => ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => 1
Description
The number of different multiplicities of parts of an integer partition.
Matching statistic: St001803
Mapping
Mp00193: Lattices —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => [1]
 => [[1]]
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => [[1,2]]
 => 0
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => [3]
 => [[1,2,3]]
 => 0
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => 1
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [[1,2,3,4]]
 => 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [[1,2,3,4,5,6]]
 => 0
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
 => [4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => 1
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => [4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => 1
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
 => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
 => [4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => 1
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
 => ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
 => [4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => 1
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => [5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => 1
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
 => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
 => [5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => 1
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
 => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
 => [5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => 1
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
 => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
 => [5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => 1
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
 => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
 => [5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
 => [5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => 1
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => [6,1]
 => [[1,2,3,4,5,6],[7]]
 => 1
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
 => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
 => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 1
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
 => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
 => [5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => 1
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
 => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
 => [5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => 1
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
 => ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
 => [5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
 => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
 => [5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => [4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
 => ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
 => [5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
 => [4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => 1
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
 => ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
 => [4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => [4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => 1
Description
The maximal overlap of the cylindrical tableau associated with a tableau. A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle. The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux. In particular, the statistic equals $0$, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
Matching statistic: St000306
Mapping
Mp00193: Lattices —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 1 = 0 + 1
([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 2 = 1 + 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 2 = 1 + 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 2 = 1 + 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 2 = 1 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 1 + 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 1 + 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1 = 0 + 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
 => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
 => ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
 => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
 => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
 => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
 => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
 => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
 => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
 => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
 => ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
 => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
 => ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
 => ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2 = 1 + 1
Description
The bounce count of a Dyck path. For a Dyck path $D$ of length $2n$, this is the number of points $(i,i)$ for $1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of $D$.
Matching statistic: St001732
Mapping
Mp00193: Lattices —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001732: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => [1]
 => [1,0]
 => 1 = 0 + 1
([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 1 = 0 + 1
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => 2 = 1 + 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
 => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
 => ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
 => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
 => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
 => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
 => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
 => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
 => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
 => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
 => ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
 => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
 => ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
 => ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
Description
The number of peaks visible from the left. This is, the number of left-to-right maxima of the heights of the peaks of a Dyck path.
Matching statistic: St000298
(load all 7 compositions to match this statistic)
Mapping
Mp00263: Lattices —join irreducibles⟶ Posets
St000298: Posets ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],0)
 => ? = 0 + 1
([(0,1)],2)
 => ([],1)
 => 1 = 0 + 1
([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1 = 0 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2 = 1 + 1
([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],3)
 => 2 = 1 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(1,2)],3)
 => 2 = 1 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,1),(0,2)],3)
 => 2 = 1 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([],4)
 => 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => ([(2,3)],4)
 => 2 = 1 + 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => ([(1,2),(1,3)],4)
 => 2 = 1 + 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => ([(0,1),(0,2),(0,3)],4)
 => 2 = 1 + 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => ([(0,2),(0,3),(3,1)],4)
 => 2 = 1 + 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ([(1,2),(2,3)],4)
 => 2 = 1 + 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,3),(3,1),(3,2)],4)
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([],3)
 => 2 = 1 + 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ([(0,3),(1,2)],4)
 => 2 = 1 + 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(1,2)],3)
 => 2 = 1 + 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([],5)
 => 2 = 1 + 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
 => ([(3,4)],5)
 => 2 = 1 + 1
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => ([(2,3),(2,4)],5)
 => 2 = 1 + 1
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
 => ([(1,2),(1,3),(1,4)],5)
 => 2 = 1 + 1
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 2 = 1 + 1
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => ([(0,2),(0,3),(0,4),(4,1)],5)
 => 2 = 1 + 1
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
 => ([(0,1),(0,2),(0,3)],4)
 => 2 = 1 + 1
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
 => ([(1,3),(1,4),(4,2)],5)
 => 2 = 1 + 1
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => 2 = 1 + 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
 => ([(1,2),(1,3)],4)
 => 2 = 1 + 1
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2 = 1 + 1
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 2 = 1 + 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,2),(0,3),(3,1)],4)
 => 2 = 1 + 1
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
 => ([(2,3),(3,4)],5)
 => 2 = 1 + 1
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
 => ([(1,4),(4,2),(4,3)],5)
 => 2 = 1 + 1
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
 => ([(0,4),(4,1),(4,2),(4,3)],5)
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([],4)
 => 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
 => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([],4)
 => 2 = 1 + 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => 2 = 1 + 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
 => ([(2,3)],4)
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([],3)
 => 2 = 1 + 1
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2 = 1 + 1
Description
The order dimension or Dushnik-Miller dimension of a poset. This is the minimal number of linear orderings whose intersection is the given poset.
Matching statistic: St000481
(load all 2 compositions to match this statistic)
Mapping
Mp00263: Lattices —join irreducibles⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],0)
 => ?
 => ? = 0
([(0,1)],2)
 => ([],1)
 => [1]
 => 0
([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => [2]
 => 0
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => [1,1]
 => 1
([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => [3]
 => 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],3)
 => [1,1,1]
 => 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(1,2)],3)
 => [2,1]
 => 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,1),(0,2)],3)
 => [2,1]
 => 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([],4)
 => [1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => ([(2,3)],4)
 => [2,1,1]
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => ([(1,2),(1,3)],4)
 => [2,1,1]
 => 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => ([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => ([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ([(1,2),(2,3)],4)
 => [3,1]
 => 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([],3)
 => [1,1,1]
 => 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 0
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([],5)
 => [1,1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
 => ([(3,4)],5)
 => [2,1,1,1]
 => 1
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => ([(2,3),(2,4)],5)
 => [2,1,1,1]
 => 1
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
 => ([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => 1
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => 1
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => ([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => 1
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
 => ([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => 1
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => 1
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
 => ([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => 1
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
 => ([(1,2),(1,3)],4)
 => [2,1,1]
 => 1
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => 1
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => 1
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
 => ([(2,3),(3,4)],5)
 => [3,1,1]
 => 1
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
 => ([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => 1
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
 => ([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([],4)
 => [1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
 => ([(1,3),(2,3)],4)
 => [2,1,1]
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([],4)
 => [1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
 => ([(2,3)],4)
 => [2,1,1]
 => 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([],3)
 => [1,1,1]
 => 1
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => 1
Description
The number of upper covers of a partition in dominance order.
Matching statistic: St000535
(load all 4 compositions to match this statistic)
Mapping
Mp00263: Lattices —join irreducibles⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000535: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],0)
 => ?
 => ? = 0
([(0,1)],2)
 => ([],1)
 => ([],1)
 => 0
([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => ([],2)
 => 0
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => ([(0,1)],2)
 => 1
([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => ([(1,2),(1,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => ([(0,1),(0,2),(0,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => ([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => ([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
 => ([(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
 => ([(0,1),(0,2),(0,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 1
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
 => ([(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
 => ([(1,2),(1,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => 1
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => 1
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
 => ([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
 => ([(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
 => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => 1
Description
The rank-width of a graph.
Matching statistic: St000480
(load all 2 compositions to match this statistic)
Mapping
Mp00263: Lattices —join irreducibles⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],0)
 => ?
 => ?
 => ? = 0
([(0,1)],2)
 => ([],1)
 => [1]
 => [1]
 => 0
([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => [2]
 => [1,1]
 => 0
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => [1,1]
 => [2]
 => 1
([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1]
 => 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],3)
 => [1,1,1]
 => [3]
 => 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(1,2)],3)
 => [2,1]
 => [2,1]
 => 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,1),(0,2)],3)
 => [2,1]
 => [2,1]
 => 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1]
 => 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [2,1]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([],4)
 => [1,1,1,1]
 => [4]
 => 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => ([(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => ([(1,2),(1,3)],4)
 => [2,1,1]
 => [3,1]
 => 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => ([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => [3,1]
 => 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => ([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [2,1,1]
 => 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ([(1,2),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [2,1,1]
 => 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([],3)
 => [1,1,1]
 => [3]
 => 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [2,1,1]
 => 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2,2]
 => 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(1,2)],3)
 => [2,1]
 => [2,1]
 => 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1]
 => 0
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([],5)
 => [1,1,1,1,1]
 => [5]
 => 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
 => ([(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 1
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => ([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 1
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
 => ([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 1
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 1
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => ([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [3,1,1]
 => 1
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
 => ([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => [3,1]
 => 1
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 1
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
 => ([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [3,1,1]
 => 1
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [3,1,1]
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
 => ([(1,2),(1,3)],4)
 => [2,1,1]
 => [3,1]
 => 1
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [2,1,1,1]
 => 1
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => [2,2,1]
 => 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [2,1,1]
 => 1
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
 => ([(2,3),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 1
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
 => ([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => 1
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
 => ([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([],4)
 => [1,1,1,1]
 => [4]
 => 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
 => ([(1,3),(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [2,2]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([],4)
 => [1,1,1,1]
 => [4]
 => 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 1
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
 => ([(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([],3)
 => [1,1,1]
 => [3]
 => 1
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [2,1,1,1]
 => 1
Description
The number of lower covers of a partition in dominance order. According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is $$ \frac{1}{2}(\sqrt{1+8n}-3) $$ and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.
Matching statistic: St000659
Mapping
Mp00193: Lattices —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => [1]
 => [1,0]
 => ? = 0
([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 0
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => 1
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => 1
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
 => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => 1
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
 => ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => 1
([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
 => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
 => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
 => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
 => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
 => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
 => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
 => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
 => ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
 => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
 => ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => 1
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
 => ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 1
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
Description
The number of rises of length at least 2 of a Dyck path.
The following 61 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001624The breadth of a lattice. St000544The cop number of a graph. St001673The degree of asymmetry of an integer composition. St000159The number of distinct parts of the integer partition. St000783The side length of the largest staircase partition fitting into a partition. St001432The order dimension of the partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000379The number of Hamiltonian cycles in a graph. St000699The toughness times the least common multiple of 1,. St001281The normalized isoperimetric number of a graph. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000264The girth of a graph, which is not a tree. St000929The constant term of the character polynomial of an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St001568The smallest positive integer that does not appear twice in the partition. St001734The lettericity of a graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000671The maximin edge-connectivity for choosing a subgraph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001271The competition number of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001512The minimum rank of a graph. St000258The burning number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000918The 2-limited packing number of a graph. St001093The detour number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001792The arboricity of a graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001871The number of triconnected components of a graph. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000632The jump number of the poset. St000100The number of linear extensions of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset.