Your data matches 108 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001277
(load all 9 compositions to match this statistic)
Mapping
St001277: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
([(0,1),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2 = 1 + 1
([(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(2,5),(3,4),(4,5)],6)
 => 1 = 0 + 1
([(1,2),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => 1 = 0 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1 = 0 + 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => 2 = 1 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => 1 = 0 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1 = 0 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1 = 0 + 1
([(0,5),(1,4),(2,3)],6)
 => 1 = 0 + 1
([(1,5),(2,4),(3,4),(3,5)],6)
 => 1 = 0 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => 1 = 0 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
Description
The degeneracy of a graph. The degeneracy of a graph $G$ is the maximum of the minimum degrees of the (vertex induced) subgraphs of $G$.
Matching statistic: St001358
(load all 4 compositions to match this statistic)
Mapping
St001358: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
([(0,1),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2 = 1 + 1
([(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(2,5),(3,4),(4,5)],6)
 => 1 = 0 + 1
([(1,2),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => 1 = 0 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1 = 0 + 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => 2 = 1 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => 1 = 0 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1 = 0 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1 = 0 + 1
([(0,5),(1,4),(2,3)],6)
 => 1 = 0 + 1
([(1,5),(2,4),(3,4),(3,5)],6)
 => 1 = 0 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => 1 = 0 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
Description
The largest degree of a regular subgraph of a graph. For $k > 2$, it is an NP-complete problem to determine whether a graph has a $k$-regular subgraph, see [1].
Matching statistic: St000256
(load all 2 compositions to match this statistic)
Mapping
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3]
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [1,1,1,1]
 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => 0
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => 0
([(1,5),(2,4),(3,4),(3,5)],6)
 => [1,1,1,1]
 => 0
([(0,1),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => 0
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,1,1]
 => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,3]
 => 1
Description
The number of parts from which one can substract 2 and still get an integer partition.
Matching statistic: St000480
(load all 2 compositions to match this statistic)
Mapping
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
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3]
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [1,1,1,1]
 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => 0
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => 0
([(1,5),(2,4),(3,4),(3,5)],6)
 => [1,1,1,1]
 => 0
([(0,1),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => 0
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,1,1]
 => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,3]
 => 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: St001638
Mapping
Mp00264: Graphs —delete endpoints⟶ Graphs
St001638: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => 0
([(0,3),(1,2),(2,3)],4)
 => ([],1)
 => 0
([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([],2)
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 0
([(1,4),(2,3),(3,4)],5)
 => ([],2)
 => 0
([(0,1),(2,4),(3,4)],5)
 => ([],2)
 => 0
([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([],1)
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 1
([(2,5),(3,5),(4,5)],6)
 => ([],3)
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([],2)
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([],1)
 => 0
([(2,5),(3,4),(4,5)],6)
 => ([],3)
 => 0
([(1,2),(3,5),(4,5)],6)
 => ([],3)
 => 0
([(3,4),(3,5),(4,5)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => ([],2)
 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([],2)
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(3,4)],5)
 => 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([],1)
 => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(2,4),(2,5),(3,4),(3,5)],6)
 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => ([],2)
 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([],1)
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([],1)
 => 0
([(0,5),(1,4),(2,3)],6)
 => ([],3)
 => 0
([(1,5),(2,4),(3,4),(3,5)],6)
 => ([],2)
 => 0
([(0,1),(2,5),(3,4),(4,5)],6)
 => ([],2)
 => 0
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(3,4)],5)
 => 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([],1)
 => 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 1
Description
The book thickness of a graph. The book thickness (or pagenumber, or stacknumber) of a graph is the minimal number of pages required for a book embedding of a graph.
Matching statistic: St000257
(load all 3 compositions to match this statistic)
Mapping
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [3]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [3]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [3]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [4]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [3]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [3]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [4]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [4]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [2,2,2]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [3]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [4]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [5]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [3]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [3]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [4]
 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [4]
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [2,1,1]
 => 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [5]
 => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [3,1,1]
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => [1,1,1,1]
 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [1,1,1,1]
 => [4]
 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => [2,1,1,1]
 => 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [5]
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => [1,1,1,1,1]
 => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [3,1,1]
 => 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [5]
 => 0
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => [3]
 => 0
([(1,5),(2,4),(3,4),(3,5)],6)
 => [1,1,1,1]
 => [4]
 => 0
([(0,1),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [4]
 => 0
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [2,1,1]
 => 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [5]
 => 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,1,1]
 => [3,1,1]
 => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [3,1,1]
 => 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,3]
 => [2,2,2]
 => 1
Description
The number of distinct parts of a partition that occur at least twice. See Section 3.3.1 of [2].
Matching statistic: St000386
(load all 4 compositions to match this statistic)
Mapping
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 0
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
([(1,5),(2,4),(3,4),(3,5)],6)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
([(0,1),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000481
(load all 3 compositions to match this statistic)
Mapping
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [3]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [3]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [3]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [4]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [3]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [3]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [4]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [4]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [2,2,2]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [3]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [4]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [5]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [3]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [3]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [4]
 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [4]
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [2,1,1]
 => 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [5]
 => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [3,1,1]
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => [1,1,1,1]
 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [1,1,1,1]
 => [4]
 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => [2,1,1,1]
 => 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [5]
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => [1,1,1,1,1]
 => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [3,1,1]
 => 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [5]
 => 0
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => [3]
 => 0
([(1,5),(2,4),(3,4),(3,5)],6)
 => [1,1,1,1]
 => [4]
 => 0
([(0,1),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [4]
 => 0
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [2,1,1]
 => 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [5]
 => 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,1,1]
 => [3,1,1]
 => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [3,1,1]
 => 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,3]
 => [2,2,2]
 => 1
Description
The number of upper covers of a partition in dominance order.
Matching statistic: St001092
Mapping
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [2,1]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1,1]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1,1]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [2,1]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [2,2]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [2,2,1]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,1]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,1,1,1]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1,1]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,1]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [2,1]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,1,1,1]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [2,1,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [2,2]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [3,2]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [2,2,1]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [2,1,1,1]
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,1,1,1]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [2,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [3,2,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [2,2,1]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,1]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1,1]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,1,1]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,1]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [2,1]
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1,1]
 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1,1]
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [2,1,1]
 => 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [2,1,1,1]
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => [2,2]
 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [1,1,1,1]
 => [1,1,1,1]
 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => [3,2]
 => 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => [2,2,1]
 => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [2,1,1,1]
 => 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => [1,1,1]
 => 0
([(1,5),(2,4),(3,4),(3,5)],6)
 => [1,1,1,1]
 => [1,1,1,1]
 => 0
([(0,1),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [1,1,1,1]
 => 0
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [2,1,1]
 => 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,1,1]
 => [2,1,1,1]
 => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [2,1,1,1]
 => 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,3]
 => [3,2,1]
 => 1
Description
The number of distinct even parts of a partition. See Section 3.3.1 of [1].
Matching statistic: St000659
(load all 3 compositions to match this statistic)
Mapping
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(1,5),(2,4),(3,4),(3,5)],6)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(0,1),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,3]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 1
Description
The number of rises of length at least 2 of a Dyck path.
The following 98 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000660The number of rises of length at least 3 of a Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000758The length of the longest staircase fitting into an integer composition. St000920The logarithmic height of a Dyck path. St000272The treewidth of a graph. St000536The pathwidth of a graph. St001792The arboricity of a graph. St001580The acyclic chromatic number of a graph. St001716The 1-improper chromatic number of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001071The beta invariant of the graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001393The induced matching number of a graph. St000741The Colin de Verdière graph invariant. St000455The second largest eigenvalue of a graph if it is integral. 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. St001593This is the number of standard Young tableaux of the given shifted shape. St000264The girth of a graph, which is not a tree. St000822The Hadwiger number of the graph. St000671The maximin edge-connectivity for choosing a subgraph. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000944The 3-degree of an integer partition. St001673The degree of asymmetry of an integer composition. St001871The number of triconnected components of a graph. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000767The number of runs in an integer composition. St000903The number of different parts of an integer composition. St001112The 3-weak dynamic number of a graph. St001651The Frankl number of a lattice. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. 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. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000284The Plancherel distribution on integer partitions. St000379The number of Hamiltonian cycles in a graph. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000997The even-odd crank of an integer partition. St001118The acyclic chromatic index of a graph. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000934The 2-degree of an integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000454The largest eigenvalue of a graph if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St001331The size of the minimal feedback vertex set. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001626The number of maximal proper sublattices of a lattice. St000260The radius of a connected graph. St001845The number of join irreducibles minus the rank of a lattice. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001644The dimension of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice.