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Your data matches 95 different statistics following compositions of up to 3 maps.
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Matching statistic: St001335
(load all 56 compositions to match this statistic)
(load all 56 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001335: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001335: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 0
[1,1] => ([(0,1)],2)
=> 0
[2] => ([],2)
=> 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2] => ([(1,2)],3)
=> 0
[2,1] => ([(0,2),(1,2)],3)
=> 0
[3] => ([],3)
=> 0
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,3] => ([(2,3)],4)
=> 0
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,2] => ([(1,3),(2,3)],4)
=> 0
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[4] => ([],4)
=> 0
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4] => ([(3,4)],5)
=> 0
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,3] => ([(2,4),(3,4)],5)
=> 0
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[5] => ([],5)
=> 0
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,5] => ([(4,5)],6)
=> 0
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 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: St000480
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> []
=> 0
[1,1] => ([(0,1)],2)
=> [1]
=> 0
[2] => ([],2)
=> []
=> 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,2] => ([(1,2)],3)
=> [1]
=> 0
[2,1] => ([(0,2),(1,2)],3)
=> [1,1]
=> 0
[3] => ([],3)
=> []
=> 0
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1
[1,3] => ([(2,3)],4)
=> [1]
=> 0
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 1
[2,2] => ([(1,3),(2,3)],4)
=> [1,1]
=> 0
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 0
[4] => ([],4)
=> []
=> 0
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[1,4] => ([(3,4)],5)
=> [1]
=> 0
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> 1
[2,3] => ([(2,4),(3,4)],5)
=> [1,1]
=> 0
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 0
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 0
[5] => ([],5)
=> []
=> 0
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10,1]
=> 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [12]
=> 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6,1]
=> 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6,1,1]
=> 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> [3]
=> 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8,1]
=> 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 1
[1,5] => ([(4,5)],6)
=> [1]
=> 0
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9,1]
=> 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> 1
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 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: St000481
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> []
=> 0
[1,1] => [1,1]
=> [1]
=> 0
[2] => [2]
=> []
=> 0
[1,1,1] => [1,1,1]
=> [1,1]
=> 1
[1,2] => [2,1]
=> [1]
=> 0
[2,1] => [2,1]
=> [1]
=> 0
[3] => [3]
=> []
=> 0
[1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,1,2] => [2,1,1]
=> [1,1]
=> 1
[1,2,1] => [2,1,1]
=> [1,1]
=> 1
[1,3] => [3,1]
=> [1]
=> 0
[2,1,1] => [2,1,1]
=> [1,1]
=> 1
[2,2] => [2,2]
=> [2]
=> 0
[3,1] => [3,1]
=> [1]
=> 0
[4] => [4]
=> []
=> 0
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,1,1,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,2,1] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,3] => [3,1,1]
=> [1,1]
=> 1
[1,2,1,1] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,2,2] => [2,2,1]
=> [2,1]
=> 1
[1,3,1] => [3,1,1]
=> [1,1]
=> 1
[1,4] => [4,1]
=> [1]
=> 0
[2,1,1,1] => [2,1,1,1]
=> [1,1,1]
=> 1
[2,1,2] => [2,2,1]
=> [2,1]
=> 1
[2,2,1] => [2,2,1]
=> [2,1]
=> 1
[2,3] => [3,2]
=> [2]
=> 0
[3,1,1] => [3,1,1]
=> [1,1]
=> 1
[3,2] => [3,2]
=> [2]
=> 0
[4,1] => [4,1]
=> [1]
=> 0
[5] => [5]
=> []
=> 0
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,1,1,3] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,1,2,2] => [2,2,1,1]
=> [2,1,1]
=> 1
[1,1,3,1] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,1,4] => [4,1,1]
=> [1,1]
=> 1
[1,2,1,2] => [2,2,1,1]
=> [2,1,1]
=> 1
[1,2,2,1] => [2,2,1,1]
=> [2,1,1]
=> 1
[1,2,3] => [3,2,1]
=> [2,1]
=> 1
[1,3,1,1] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,3,2] => [3,2,1]
=> [2,1]
=> 1
[1,4,1] => [4,1,1]
=> [1,1]
=> 1
[1,5] => [5,1]
=> [1]
=> 0
[2,1,1,2] => [2,2,1,1]
=> [2,1,1]
=> 1
[2,1,2,1] => [2,2,1,1]
=> [2,1,1]
=> 1
[2,1,3] => [3,2,1]
=> [2,1]
=> 1
[2,2,1,1] => [2,2,1,1]
=> [2,1,1]
=> 1
[2,2,2] => [2,2,2]
=> [2,2]
=> 1
Description
The number of upper covers of a partition in dominance order.
Matching statistic: St000535
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 0
[1,1] => ([(0,1)],2)
=> ([],1)
=> 0
[2] => ([],2)
=> ([],2)
=> 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2] => ([(1,2)],3)
=> ([],2)
=> 0
[2,1] => ([(0,2),(1,2)],3)
=> ([],1)
=> 0
[3] => ([],3)
=> ([],3)
=> 0
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3] => ([(2,3)],4)
=> ([],3)
=> 0
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,2] => ([(1,3),(2,3)],4)
=> ([],2)
=> 0
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> 0
[4] => ([],4)
=> ([],4)
=> 0
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,4] => ([(3,4)],5)
=> ([],4)
=> 0
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,3] => ([(2,4),(3,4)],5)
=> ([],3)
=> 0
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> 0
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> 0
[5] => ([],5)
=> ([],5)
=> 0
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,5] => ([(4,5)],6)
=> ([],5)
=> 0
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
Description
The rank-width of a graph.
Matching statistic: St000660
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 0
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 1
Description
The number of rises of length at least 3 of a Dyck path.
The number of Dyck paths without such rises are counted by the Motzkin numbers [1].
Matching statistic: St001333
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 0
[1,1] => ([(0,1)],2)
=> ([],1)
=> 0
[2] => ([],2)
=> ([],2)
=> 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2] => ([(1,2)],3)
=> ([],2)
=> 0
[2,1] => ([(0,2),(1,2)],3)
=> ([],1)
=> 0
[3] => ([],3)
=> ([],3)
=> 0
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3] => ([(2,3)],4)
=> ([],3)
=> 0
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,2] => ([(1,3),(2,3)],4)
=> ([],2)
=> 0
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> 0
[4] => ([],4)
=> ([],4)
=> 0
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,4] => ([(3,4)],5)
=> ([],4)
=> 0
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,3] => ([(2,4),(3,4)],5)
=> ([],3)
=> 0
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> 0
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> 0
[5] => ([],5)
=> ([],5)
=> 0
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,5] => ([(4,5)],6)
=> ([],5)
=> 0
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
Description
The cardinality of a minimal edge-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 only the graph with one edge.
Matching statistic: St001393
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 0
[1,1] => ([(0,1)],2)
=> ([],1)
=> 0
[2] => ([],2)
=> ([],2)
=> 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2] => ([(1,2)],3)
=> ([],2)
=> 0
[2,1] => ([(0,2),(1,2)],3)
=> ([],1)
=> 0
[3] => ([],3)
=> ([],3)
=> 0
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3] => ([(2,3)],4)
=> ([],3)
=> 0
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,2] => ([(1,3),(2,3)],4)
=> ([],2)
=> 0
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> 0
[4] => ([],4)
=> ([],4)
=> 0
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,4] => ([(3,4)],5)
=> ([],4)
=> 0
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,3] => ([(2,4),(3,4)],5)
=> ([],3)
=> 0
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> 0
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> 0
[5] => ([],5)
=> ([],5)
=> 0
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,5] => ([(4,5)],6)
=> ([],5)
=> 0
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
Description
The induced matching number of a graph.
An induced matching of a graph is a set of independent edges which is an induced subgraph. This statistic records the maximal number of edges in an induced matching.
Matching statistic: St000920
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 1 = 0 + 1
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[2] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 2 = 1 + 1
Description
The logarithmic height of a Dyck path.
This is the floor of the binary logarithm of the usual height increased by one:
$$
\lfloor\log_2(1+height(D))\rfloor
$$
Matching statistic: St001261
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,1] => ([(0,1)],2)
=> ([],1)
=> 1 = 0 + 1
[2] => ([],2)
=> ([],2)
=> 1 = 0 + 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2] => ([(1,2)],3)
=> ([],2)
=> 1 = 0 + 1
[2,1] => ([(0,2),(1,2)],3)
=> ([],1)
=> 1 = 0 + 1
[3] => ([],3)
=> ([],3)
=> 1 = 0 + 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,3] => ([(2,3)],4)
=> ([],3)
=> 1 = 0 + 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,2] => ([(1,3),(2,3)],4)
=> ([],2)
=> 1 = 0 + 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> 1 = 0 + 1
[4] => ([],4)
=> ([],4)
=> 1 = 0 + 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,4] => ([(3,4)],5)
=> ([],4)
=> 1 = 0 + 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3] => ([(2,4),(3,4)],5)
=> ([],3)
=> 1 = 0 + 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> 1 = 0 + 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> 1 = 0 + 1
[5] => ([],5)
=> ([],5)
=> 1 = 0 + 1
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,5] => ([(4,5)],6)
=> ([],5)
=> 1 = 0 + 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
Description
The Castelnuovo-Mumford regularity of a graph.
Matching statistic: St000386
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(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1,0]
=> 0
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 0
[3] => [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 1
Description
The number of factors DDU in a Dyck path.
The following 85 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001842The major index of a set partition. St000396The register function (or Horton-Strahler number) of a binary tree. St000679The pruning number of an ordered tree. St000701The protection number of a binary tree. St001732The number of peaks visible from the left. St000257The number of distinct parts of a partition that occur at least twice. St001280The number of parts of an integer partition that are at least two. St000183The side length of the Durfee square of an integer partition. St000897The number of different multiplicities of parts of an integer partition. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St000256The number of parts from which one can substract 2 and still get an integer partition. St000143The largest repeated part of a partition. St000353The number of inner valleys of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001568The smallest positive integer that does not appear twice in the partition. St000264The girth of a graph, which is not a tree. 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. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. 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. St000219The number of occurrences of the pattern 231 in a permutation. St001175The size of a partition minus the hook length of the base cell. St000647The number of big descents of a permutation. St000862The number of parts of the shifted shape of a permutation. St000779The tier of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000486The number of cycles of length at least 3 of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000646The number of big ascents of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000023The number of inner peaks of a permutation. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000944The 3-degree of an integer partition. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001638The book thickness of a graph. St001673The degree of asymmetry of an integer composition. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001871The number of triconnected components of a graph. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000767The number of runs in an integer composition. St000820The number of compositions obtained by rotating the composition. St000903The number of different parts of an integer composition. St001644The dimension of a graph. St001962The proper pathwidth of a graph. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000260The radius of a connected graph. St001060The distinguishing index of a graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000782The indicator function of whether a given perfect matching is an L & P matching. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001845The number of join irreducibles minus the rank of a lattice. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001651The Frankl number of a lattice. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001875The number of simple modules with projective dimension at most 1.
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