- FindStat

Your data matches 160 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000205

Mapping

St000205: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1]
 => 0
[2]
 => 0
[1,1]
 => 0
[3]
 => 0
[2,1]
 => 0
[1,1,1]
 => 0
[4]
 => 0
[3,1]
 => 0
[2,2]
 => 0
[2,1,1]
 => 0
[1,1,1,1]
 => 0
[5]
 => 0
[4,1]
 => 0
[3,2]
 => 1
[3,1,1]
 => 0
[2,2,1]
 => 1
[2,1,1,1]
 => 0
[1,1,1,1,1]
 => 0
[6]
 => 0
[5,1]
 => 0
[4,2]
 => 1
[4,1,1]
 => 0
[3,3]
 => 1
[3,2,1]
 => 2
[3,1,1,1]
 => 0
[2,2,2]
 => 1
[2,2,1,1]
 => 1
[2,1,1,1,1]
 => 0
[1,1,1,1,1,1]
 => 0
[7]
 => 0
[6,1]
 => 0
[5,2]
 => 1
[5,1,1]
 => 0
[4,3]
 => 3
[4,2,1]
 => 4
[4,1,1,1]
 => 0
[3,3,1]
 => 3
[3,2,2]
 => 2
[3,2,1,1]
 => 2
[3,1,1,1,1]
 => 0
[2,2,2,1]
 => 2
[2,2,1,1,1]
 => 1
[2,1,1,1,1,1]
 => 0
[1,1,1,1,1,1,1]
 => 0
[8]
 => 0
[7,1]
 => 0
[6,2]
 => 1
[6,1,1]
 => 0
[5,3]
 => 3
[5,2,1]
 => 7

Description

Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. Given

$\lambda$ count how many ''integer partitions''

$w$ (weight) there are, such that

$P_{\lambda,w}$ is non-integral, i.e.,

$w$ such that the Gelfand-Tsetlin polytope

$P_{\lambda,w}$ has at least one non-integral vertex.

Matching statistic: St001578

Mapping

Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001578: Graphs ⟶ ℤResult quality: 43% ●values known / values provided: 44%●distinct values known / distinct values provided: 43%

Values

[1]
 => [[1],[]]
 => ([],1)
 => ([],1)
 => 0
[2]
 => [[2],[]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[3]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 0
[2,1]
 => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
[1,1,1]
 => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 0
[4]
 => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[3,1]
 => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[2,2]
 => [[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 0
[2,1,1]
 => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[1,1,1,1]
 => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[5]
 => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[4,1]
 => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[3,2]
 => [[3,2],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
[3,1,1]
 => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[2,2,1]
 => [[2,2,1],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
[2,1,1,1]
 => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[6]
 => [[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 0
[5,1]
 => [[5,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 0
[4,2]
 => [[4,2],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => 1
[4,1,1]
 => [[4,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 0
[3,3]
 => [[3,3],[]]
 => ([(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
[3,2,1]
 => [[3,2,1],[]]
 => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => 2
[3,1,1,1]
 => [[3,1,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 0
[2,2,2]
 => [[2,2,2],[]]
 => ([(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
[2,2,1,1]
 => [[2,2,1,1],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => 1
[2,1,1,1,1]
 => [[2,1,1,1,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 0
[1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 0
[7]
 => [[7],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => ? = 0
[6,1]
 => [[6,1],[]]
 => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => ? = 0
[5,2]
 => [[5,2],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
 => ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
 => ? = 1
[5,1,1]
 => [[5,1,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => ? = 0
[4,3]
 => [[4,3],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 3
[4,2,1]
 => [[4,2,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ? = 4
[4,1,1,1]
 => [[4,1,1,1],[]]
 => ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => ? = 0
[3,3,1]
 => [[3,3,1],[]]
 => ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 3
[3,2,2]
 => [[3,2,2],[]]
 => ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 2
[3,2,1,1]
 => [[3,2,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ? = 2
[3,1,1,1,1]
 => [[3,1,1,1,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => ? = 0
[2,2,2,1]
 => [[2,2,2,1],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 2
[2,2,1,1,1]
 => [[2,2,1,1,1],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
 => ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
 => ? = 1
[2,1,1,1,1,1]
 => [[2,1,1,1,1,1],[]]
 => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => ? = 0
[1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => ? = 0
[8]
 => [[8],[]]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
 => ? = 0
[7,1]
 => [[7,1],[]]
 => ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
 => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
 => ? = 0
[6,2]
 => [[6,2],[]]
 => ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
 => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,7),(4,7),(5,7)],8)
 => ? = 1
[6,1,1]
 => [[6,1,1],[]]
 => ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
 => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
 => ? = 0
[5,3]
 => [[5,3],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
 => ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
 => ? = 3
[5,2,1]
 => [[5,2,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
 => ([(0,7),(1,5),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6)],8)
 => ? = 7
[5,1,1,1]
 => [[5,1,1,1],[]]
 => ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
 => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
 => ? = 0
[4,4]
 => [[4,4],[]]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 3
[4,3,1]
 => [[4,3,1],[]]
 => ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
 => ? = 5
[4,2,2]
 => [[4,2,2],[]]
 => ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
 => ? = 4
[4,2,1,1]
 => [[4,2,1,1],[]]
 => ([(0,5),(0,6),(3,2),(4,1),(5,3),(5,7),(6,4),(6,7)],8)
 => ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
 => ? = 4
[4,1,1,1,1]
 => [[4,1,1,1,1],[]]
 => ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
 => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
 => ? = 0
[3,3,2]
 => [[3,3,2],[]]
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7)],8)
 => ([(0,3),(0,7),(1,2),(1,7),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 5
[3,3,1,1]
 => [[3,3,1,1],[]]
 => ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
 => ? = 4
[3,2,2,1]
 => [[3,2,2,1],[]]
 => ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
 => ? = 5
[3,2,1,1,1]
 => [[3,2,1,1,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
 => ([(0,7),(1,5),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6)],8)
 => ? = 2
[3,1,1,1,1,1]
 => [[3,1,1,1,1,1],[]]
 => ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
 => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
 => ? = 0
[2,2,2,2]
 => [[2,2,2,2],[]]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 2
[2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
 => ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
 => ? = 2
[2,2,1,1,1,1]
 => [[2,2,1,1,1,1],[]]
 => ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
 => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,7),(4,7),(5,7)],8)
 => ? = 1
[2,1,1,1,1,1,1]
 => [[2,1,1,1,1,1,1],[]]
 => ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
 => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
 => ? = 0
[1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
 => ? = 0

Description

The minimal number of edges to add or remove to make a graph a line graph.

Matching statistic: St000243

Mapping

Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000243: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 44%●distinct values known / distinct values provided: 43%

Values

[1]
 => [[1]]
 => [1] => [1] => ? = 0 + 1
[2]
 => [[1,2]]
 => [1,2] => [1,2] => 1 = 0 + 1
[1,1]
 => [[1],[2]]
 => [2,1] => [2,1] => 1 = 0 + 1
[3]
 => [[1,2,3]]
 => [1,2,3] => [1,3,2] => 1 = 0 + 1
[2,1]
 => [[1,3],[2]]
 => [2,1,3] => [2,1,3] => 1 = 0 + 1
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => [3,2,1] => 1 = 0 + 1
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => [1,4,3,2] => 1 = 0 + 1
[3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => [2,1,4,3] => 1 = 0 + 1
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => [3,4,1,2] => 1 = 0 + 1
[2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => 1 = 0 + 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,5,4,3,2] => 1 = 0 + 1
[4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => [2,1,5,4,3] => 1 = 0 + 1
[3,2]
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => [3,5,1,4,2] => 2 = 1 + 1
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [3,2,1,5,4] => 1 = 0 + 1
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [4,2,5,1,3] => 2 = 1 + 1
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [4,3,2,1,5] => 1 = 0 + 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5,4,3,2,1] => 1 = 0 + 1
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,6,5,4,3,2] => 1 = 0 + 1
[5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [2,1,6,5,4,3] => 1 = 0 + 1
[4,2]
 => [[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => [3,6,1,5,4,2] => 2 = 1 + 1
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => [3,2,1,6,5,4] => 1 = 0 + 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => [4,6,5,1,3,2] => 2 = 1 + 1
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [4,2,5,1,3,6] => [4,2,6,1,5,3] => 3 = 2 + 1
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => [4,3,2,1,6,5] => 1 = 0 + 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => [5,6,3,4,1,2] => 2 = 1 + 1
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [5,3,2,6,1,4] => [5,3,2,6,1,4] => 2 = 1 + 1
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [5,4,3,2,1,6] => [5,4,3,2,1,6] => 1 = 0 + 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1 = 0 + 1
[7]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [1,7,6,5,4,3,2] => ? = 0 + 1
[6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => [2,1,7,6,5,4,3] => ? = 0 + 1
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => [3,7,1,6,5,4,2] => ? = 1 + 1
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => [3,2,1,7,6,5,4] => ? = 0 + 1
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => [4,7,6,1,5,3,2] => ? = 3 + 1
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [4,2,5,1,3,6,7] => [4,2,7,1,6,5,3] => ? = 4 + 1
[4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => [4,3,2,1,5,6,7] => [4,3,2,1,7,6,5] => ? = 0 + 1
[3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [5,2,6,7,1,3,4] => [5,2,7,6,1,4,3] => ? = 3 + 1
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [5,6,3,4,1,2,7] => [5,7,3,6,1,4,2] => ? = 2 + 1
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => [5,3,2,6,1,4,7] => [5,3,2,7,1,6,4] => ? = 2 + 1
[3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => [5,4,3,2,1,6,7] => [5,4,3,2,1,7,6] => ? = 0 + 1
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3] => [6,4,7,2,5,1,3] => ? = 2 + 1
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => [6,4,3,2,7,1,5] => [6,4,3,2,7,1,5] => ? = 1 + 1
[2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ? = 0 + 1
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0 + 1
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [1,2,3,4,5,6,7,8] => [1,8,7,6,5,4,3,2] => ? = 0 + 1
[7,1]
 => [[1,3,4,5,6,7,8],[2]]
 => [2,1,3,4,5,6,7,8] => [2,1,8,7,6,5,4,3] => ? = 0 + 1
[6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [3,4,1,2,5,6,7,8] => [3,8,1,7,6,5,4,2] => ? = 1 + 1
[6,1,1]
 => [[1,4,5,6,7,8],[2],[3]]
 => [3,2,1,4,5,6,7,8] => [3,2,1,8,7,6,5,4] => ? = 0 + 1
[5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => [4,5,6,1,2,3,7,8] => [4,8,7,1,6,5,3,2] => ? = 3 + 1
[5,2,1]
 => [[1,3,6,7,8],[2,5],[4]]
 => [4,2,5,1,3,6,7,8] => [4,2,8,1,7,6,5,3] => ? = 7 + 1
[5,1,1,1]
 => [[1,5,6,7,8],[2],[3],[4]]
 => [4,3,2,1,5,6,7,8] => [4,3,2,1,8,7,6,5] => ? = 0 + 1
[4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => [5,8,7,6,1,4,3,2] => ? = 3 + 1
[4,3,1]
 => [[1,3,4,8],[2,6,7],[5]]
 => [5,2,6,7,1,3,4,8] => [5,2,8,7,1,6,4,3] => ? = 5 + 1
[4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => [5,6,3,4,1,2,7,8] => [5,8,3,7,1,6,4,2] => ? = 4 + 1
[4,2,1,1]
 => [[1,4,7,8],[2,6],[3],[5]]
 => [5,3,2,6,1,4,7,8] => [5,3,2,8,1,7,6,4] => ? = 4 + 1
[4,1,1,1,1]
 => [[1,6,7,8],[2],[3],[4],[5]]
 => [5,4,3,2,1,6,7,8] => [5,4,3,2,1,8,7,6] => ? = 0 + 1
[3,3,2]
 => [[1,2,5],[3,4,8],[6,7]]
 => [6,7,3,4,8,1,2,5] => [6,8,3,7,5,1,4,2] => ? = 5 + 1
[3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => [6,3,2,7,8,1,4,5] => [6,3,2,8,7,1,5,4] => ? = 4 + 1
[3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3,8] => [6,4,8,2,7,1,5,3] => ? = 5 + 1
[3,2,1,1,1]
 => [[1,5,8],[2,7],[3],[4],[6]]
 => [6,4,3,2,7,1,5,8] => [6,4,3,2,8,1,7,5] => ? = 2 + 1
[3,1,1,1,1,1]
 => [[1,7,8],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,8,7] => ? = 0 + 1
[2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2] => [7,8,5,6,3,4,1,2] => 3 = 2 + 1
[2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5],[7]]
 => [7,5,3,8,2,6,1,4] => [7,5,3,8,2,6,1,4] => ? = 2 + 1
[2,2,1,1,1,1]
 => [[1,6],[2,8],[3],[4],[5],[7]]
 => [7,5,4,3,2,8,1,6] => [7,5,4,3,2,8,1,6] => ? = 1 + 1
[2,1,1,1,1,1,1]
 => [[1,8],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => ? = 0 + 1
[1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => ? = 0 + 1

Description

The number of cyclic valleys and cyclic peaks of a permutation. This is given by the number of indices

$i$ such that

$\pi_{i-1} > \pi_i < \pi_{i+1}$ with indices considered cyclically. Equivalently, this is the number of indices

$i$ such that

$\pi_{i-1} < \pi_i > \pi_{i+1}$ with indices considered cyclically.

Matching statistic: St001964
(load all 5 compositions to match this statistic)

Mapping

Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 43%

Values

[1]
 => [[1],[]]
 => [[1],[]]
 => ([],1)
 => 0
[2]
 => [[2],[]]
 => [[2],[]]
 => ([(0,1)],2)
 => 0
[1,1]
 => [[1,1],[]]
 => [[1,1],[]]
 => ([(0,1)],2)
 => 0
[3]
 => [[3],[]]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 0
[2,1]
 => [[2,1],[]]
 => [[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => 0
[1,1,1]
 => [[1,1,1],[]]
 => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 0
[4]
 => [[4],[]]
 => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[3,1]
 => [[3,1],[]]
 => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[2,2]
 => [[2,2],[]]
 => [[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[2,1,1]
 => [[2,1,1],[]]
 => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => 0
[1,1,1,1]
 => [[1,1,1,1],[]]
 => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[5]
 => [[5],[]]
 => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[4,1]
 => [[4,1],[]]
 => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 0
[3,2]
 => [[3,2],[]]
 => [[3,3],[1]]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
[3,1,1]
 => [[3,1,1],[]]
 => [[3,3,3],[2,2]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 0
[2,2,1]
 => [[2,2,1],[]]
 => [[2,2,2],[1]]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
[2,1,1,1]
 => [[2,1,1,1],[]]
 => [[2,2,2,2],[1,1,1]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 0
[1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[6]
 => [[6],[]]
 => [[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[5,1]
 => [[5,1],[]]
 => [[5,5],[4]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 0
[4,2]
 => [[4,2],[]]
 => [[4,4],[2]]
 => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
 => ? = 1
[4,1,1]
 => [[4,1,1],[]]
 => [[4,4,4],[3,3]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => 0
[3,3]
 => [[3,3],[]]
 => [[3,3],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1
[3,2,1]
 => [[3,2,1],[]]
 => [[3,3,3],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => 2
[3,1,1,1]
 => [[3,1,1,1],[]]
 => [[3,3,3,3],[2,2,2]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => 0
[2,2,2]
 => [[2,2,2],[]]
 => [[2,2,2],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1
[2,2,1,1]
 => [[2,2,1,1],[]]
 => [[2,2,2,2],[1,1]]
 => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
 => ? = 1
[2,1,1,1,1]
 => [[2,1,1,1,1],[]]
 => [[2,2,2,2,2],[1,1,1,1]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 0
[1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => [[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[7]
 => [[7],[]]
 => [[7],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ? = 0
[6,1]
 => [[6,1],[]]
 => [[6,6],[5]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => ? = 0
[5,2]
 => [[5,2],[]]
 => [[5,5],[3]]
 => ([(0,3),(0,6),(1,4),(2,6),(3,5),(4,2),(6,5)],7)
 => ? = 1
[5,1,1]
 => [[5,1,1],[]]
 => [[5,5,5],[4,4]]
 => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
 => ? = 0
[4,3]
 => [[4,3],[]]
 => [[4,4],[1]]
 => ([(0,6),(1,3),(1,6),(2,4),(3,2),(3,5),(5,4),(6,5)],7)
 => ? = 3
[4,2,1]
 => [[4,2,1],[]]
 => [[4,4,4],[3,2]]
 => ([(0,4),(1,4),(1,6),(2,3),(3,6),(4,5),(6,5)],7)
 => ? = 4
[4,1,1,1]
 => [[4,1,1,1],[]]
 => [[4,4,4,4],[3,3,3]]
 => ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
 => ? = 0
[3,3,1]
 => [[3,3,1],[]]
 => [[3,3,3],[2]]
 => ([(0,2),(0,3),(1,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
 => ? = 3
[3,2,2]
 => [[3,2,2],[]]
 => [[3,3,3],[1,1]]
 => ([(0,2),(0,3),(1,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
 => ? = 2
[3,2,1,1]
 => [[3,2,1,1],[]]
 => [[3,3,3,3],[2,2,1]]
 => ([(0,4),(1,4),(1,6),(2,3),(3,6),(4,5),(6,5)],7)
 => ? = 2
[3,1,1,1,1]
 => [[3,1,1,1,1],[]]
 => [[3,3,3,3,3],[2,2,2,2]]
 => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
 => ? = 0
[2,2,2,1]
 => [[2,2,2,1],[]]
 => [[2,2,2,2],[1]]
 => ([(0,6),(1,3),(1,6),(2,4),(3,2),(3,5),(5,4),(6,5)],7)
 => ? = 2
[2,2,1,1,1]
 => [[2,2,1,1,1],[]]
 => [[2,2,2,2,2],[1,1,1]]
 => ([(0,3),(0,6),(1,4),(2,6),(3,5),(4,2),(6,5)],7)
 => ? = 1
[2,1,1,1,1,1]
 => [[2,1,1,1,1,1],[]]
 => [[2,2,2,2,2,2],[1,1,1,1,1]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => ? = 0
[1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1],[]]
 => [[1,1,1,1,1,1,1],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ? = 0
[8]
 => [[8],[]]
 => [[8],[]]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ? = 0
[7,1]
 => [[7,1],[]]
 => [[7,7],[6]]
 => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
 => ? = 0
[6,2]
 => [[6,2],[]]
 => [[6,6],[4]]
 => ([(0,5),(1,3),(1,7),(2,7),(3,6),(4,2),(5,4),(7,6)],8)
 => ? = 1
[6,1,1]
 => [[6,1,1],[]]
 => [[6,6,6],[5,5]]
 => ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
 => ? = 0
[5,3]
 => [[5,3],[]]
 => [[5,5],[2]]
 => ([(0,3),(1,4),(1,7),(2,6),(3,7),(4,2),(4,5),(5,6),(7,5)],8)
 => ? = 3
[5,2,1]
 => [[5,2,1],[]]
 => [[5,5,5],[4,3]]
 => ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6),(5,7),(6,7)],8)
 => ? = 7
[5,1,1,1]
 => [[5,1,1,1],[]]
 => [[5,5,5,5],[4,4,4]]
 => ([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
 => ? = 0
[4,4]
 => [[4,4],[]]
 => [[4,4],[]]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 3
[4,3,1]
 => [[4,3,1],[]]
 => [[4,4,4],[3,1]]
 => ([(0,7),(1,6),(2,3),(2,7),(3,5),(3,6),(5,4),(6,4),(7,5)],8)
 => ? = 5
[4,2,2]
 => [[4,2,2],[]]
 => [[4,4,4],[2,2]]
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
 => ? = 4
[4,2,1,1]
 => [[4,2,1,1],[]]
 => [[4,4,4,4],[3,3,2]]
 => ([(0,6),(0,7),(1,4),(2,3),(3,6),(4,7),(6,5),(7,5)],8)
 => ? = 4
[4,1,1,1,1]
 => [[4,1,1,1,1],[]]
 => [[4,4,4,4,4],[3,3,3,3]]
 => ([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
 => ? = 0
[3,3,2]
 => [[3,3,2],[]]
 => [[3,3,3],[1]]
 => ([(0,3),(0,7),(1,2),(1,7),(2,5),(3,6),(5,4),(6,4),(7,5),(7,6)],8)
 => ? = 5
[3,3,1,1]
 => [[3,3,1,1],[]]
 => [[3,3,3,3],[2,2]]
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
 => ? = 4
[3,2,2,1]
 => [[3,2,2,1],[]]
 => [[3,3,3,3],[2,1,1]]
 => ([(0,7),(1,6),(2,3),(2,7),(3,5),(3,6),(5,4),(6,4),(7,5)],8)
 => ? = 5
[3,2,1,1,1]
 => [[3,2,1,1,1],[]]
 => [[3,3,3,3,3],[2,2,2,1]]
 => ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6),(5,7),(6,7)],8)
 => ? = 2
[3,1,1,1,1,1]
 => [[3,1,1,1,1,1],[]]
 => [[3,3,3,3,3,3],[2,2,2,2,2]]
 => ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
 => ? = 0
[2,2,2,2]
 => [[2,2,2,2],[]]
 => [[2,2,2,2],[]]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => [[2,2,2,2,2],[1,1]]
 => ([(0,3),(1,4),(1,7),(2,6),(3,7),(4,2),(4,5),(5,6),(7,5)],8)
 => ? = 2
[2,2,1,1,1,1]
 => [[2,2,1,1,1,1],[]]
 => [[2,2,2,2,2,2],[1,1,1,1]]
 => ([(0,5),(1,3),(1,7),(2,7),(3,6),(4,2),(5,4),(7,6)],8)
 => ? = 1
[2,1,1,1,1,1,1]
 => [[2,1,1,1,1,1,1],[]]
 => [[2,2,2,2,2,2,2],[1,1,1,1,1,1]]
 => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
 => ? = 0
[1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => [[1,1,1,1,1,1,1,1],[]]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ? = 0

Description

The interval resolution global dimension of a poset. This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.

Matching statistic: St000455

Mapping

Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 24%●distinct values known / distinct values provided: 14%

Values

[1]
 => [[1],[]]
 => ([],1)
 => ([],1)
 => ? = 0
[2]
 => [[2],[]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 0
[1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 0
[3]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 0
[2,1]
 => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => 0
[1,1,1]
 => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 0
[4]
 => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 0
[3,1]
 => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => 0
[2,2]
 => [[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 0
[2,1,1]
 => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,1,1,1]
 => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 0
[5]
 => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 0
[4,1]
 => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0
[3,2]
 => [[3,2],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[3,1,1]
 => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 0
[2,2,1]
 => [[2,2,1],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[2,1,1,1]
 => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 0
[6]
 => [[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 0
[5,1]
 => [[5,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0
[4,2]
 => [[4,2],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 1
[4,1,1]
 => [[4,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 0
[3,3]
 => [[3,3],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ? = 1
[3,2,1]
 => [[3,2,1],[]]
 => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 2
[3,1,1,1]
 => [[3,1,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 0
[2,2,2]
 => [[2,2,2],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ? = 1
[2,2,1,1]
 => [[2,2,1,1],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 1
[2,1,1,1,1]
 => [[2,1,1,1,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0
[1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 0
[7]
 => [[7],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => ? = 0
[6,1]
 => [[6,1],[]]
 => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
 => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0
[5,2]
 => [[5,2],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
 => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 1
[5,1,1]
 => [[5,1,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
 => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 0
[4,3]
 => [[4,3],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
 => ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 3
[4,2,1]
 => [[4,2,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
 => ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 4
[4,1,1,1]
 => [[4,1,1,1],[]]
 => ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
 => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => 0
[3,3,1]
 => [[3,3,1],[]]
 => ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
 => ([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 3
[3,2,2]
 => [[3,2,2],[]]
 => ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
 => ([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 2
[3,2,1,1]
 => [[3,2,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
 => ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 2
[3,1,1,1,1]
 => [[3,1,1,1,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
 => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 0
[2,2,2,1]
 => [[2,2,2,1],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
 => ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2
[2,2,1,1,1]
 => [[2,2,1,1,1],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
 => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 1
[2,1,1,1,1,1]
 => [[2,1,1,1,1,1],[]]
 => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
 => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0
[1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => ? = 0
[8]
 => [[8],[]]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ([],8)
 => ? = 0
[7,1]
 => [[7,1],[]]
 => ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
 => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 0
[6,2]
 => [[6,2],[]]
 => ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 1
[6,1,1]
 => [[6,1,1],[]]
 => ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
 => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 0
[5,3]
 => [[5,3],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
 => ([(1,7),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 3
[5,2,1]
 => [[5,2,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
 => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7)],8)
 => ? = 7
[5,1,1,1]
 => [[5,1,1,1],[]]
 => ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
 => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
 => ? = 0
[4,4]
 => [[4,4],[]]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 3
[4,3,1]
 => [[4,3,1],[]]
 => ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ([(1,5),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(6,7)],8)
 => ? = 5
[4,2,2]
 => [[4,2,2],[]]
 => ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ([(1,6),(1,7),(2,4),(2,5),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 4
[4,2,1,1]
 => [[4,2,1,1],[]]
 => ([(0,5),(0,6),(3,2),(4,1),(5,3),(5,7),(6,4),(6,7)],8)
 => ([(1,2),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 4
[4,1,1,1,1]
 => [[4,1,1,1,1],[]]
 => ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
 => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
 => ? = 0
[3,3,2]
 => [[3,3,2],[]]
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7)],8)
 => ([(1,4),(1,7),(2,3),(2,7),(3,6),(4,6),(5,6),(5,7),(6,7)],8)
 => ? = 5
[3,3,1,1]
 => [[3,3,1,1],[]]
 => ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ([(1,6),(1,7),(2,4),(2,5),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 4
[3,2,2,1]
 => [[3,2,2,1],[]]
 => ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ([(1,5),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(6,7)],8)
 => ? = 5
[3,2,1,1,1]
 => [[3,2,1,1,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
 => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7)],8)
 => ? = 2
[3,1,1,1,1,1]
 => [[3,1,1,1,1,1],[]]
 => ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
 => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 0
[2,2,2,2]
 => [[2,2,2,2],[]]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 2
[2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
 => ([(1,7),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 2
[2,2,1,1,1,1]
 => [[2,2,1,1,1,1],[]]
 => ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 1
[2,1,1,1,1,1,1]
 => [[2,1,1,1,1,1,1],[]]
 => ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
 => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 0
[1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ([],8)
 => ? = 0

Description

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.

Matching statistic: St001371
(load all 3 compositions to match this statistic)

Mapping

Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 14% ●values known / values provided: 23%●distinct values known / distinct values provided: 14%

Values

[1]
 => [1]
 => [1,0]
 => 10 => 0
[2]
 => [1,1]
 => [1,1,0,0]
 => 1100 => 0
[1,1]
 => [2]
 => [1,0,1,0]
 => 1010 => 0
[3]
 => [3]
 => [1,0,1,0,1,0]
 => 101010 => 0
[2,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 110100 => 0
[1,1,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 0
[4]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => 0
[3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0
[2,2]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
[2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0
[1,1,1,1]
 => [2,2]
 => [1,1,1,0,0,0]
 => 111000 => 0
[5]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? = 0
[4,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => ? = 0
[3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => ? = 1
[3,1,1]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0
[2,2,1]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 1
[2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => ? = 0
[1,1,1,1,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => 0
[6]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 11101000 => 0
[5,1]
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 101010101100 => ? = 0
[4,2]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 110101010100 => ? = 1
[4,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 101101010100 => ? = 0
[3,3]
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 101010101010 => ? = 1
[3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 101011010100 => ? = 2
[3,1,1,1]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
[2,2,2]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 101010110100 => ? = 1
[2,2,1,1]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => ? = 1
[2,1,1,1,1]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => ? = 0
[1,1,1,1,1,1]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => 0
[7]
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 10101010101010 => ? = 0
[6,1]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => ? = 0
[5,2]
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 10101010110100 => ? = 1
[5,1,1]
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => ? = 0
[4,3]
 => [3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => 10101101010100 => ? = 3
[4,2,1]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 11010101010100 => ? = 4
[4,1,1,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => 10110101010100 => ? = 0
[3,3,1]
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 10101010101100 => ? = 3
[3,2,2]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 2
[3,2,1,1]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 101110010100 => ? = 2
[3,1,1,1,1]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? = 0
[2,2,2,1]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => 10101011010100 => ? = 2
[2,2,1,1,1]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 101011100100 => ? = 1
[2,1,1,1,1,1]
 => [2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 111001010100 => ? = 0
[1,1,1,1,1,1,1]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => ? = 0
[8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 1101010101010100 => ? = 0
[7,1]
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 1010101010101100 => ? = 0
[6,2]
 => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 111010010100 => ? = 1
[6,1,1]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => ? = 0
[5,3]
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 101011101000 => ? = 3
[5,2,1]
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => 1010101011010100 => ? = 7
[5,1,1,1]
 => [5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => 10101011100100 => ? = 0
[4,4]
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 1010101010101010 => ? = 3
[4,3,1]
 => [3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => 1010110101010100 => ? = 5
[4,2,2]
 => [4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => 1010101101010100 => ? = 4
[4,2,1,1]
 => [2,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 1011010101010100 => ? = 4
[4,1,1,1,1]
 => [2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => 11100101010100 => ? = 0
[3,3,2]
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 1010101010110100 => ? = 5
[3,3,1,1]
 => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => 10101010111000 => ? = 4
[3,2,2,1]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 101110100100 => ? = 5
[3,2,1,1,1]
 => [3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => 10111001010100 => ? = 2
[3,1,1,1,1,1]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 101111000100 => ? = 0
[2,2,2,2]
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => ? = 2
[2,2,2,1,1]
 => [4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => 10101110010100 => ? = 2
[2,2,1,1,1,1]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 101011110000 => ? = 1
[2,1,1,1,1,1,1]
 => [2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 111100010100 => ? = 0

Description

The length of the longest Yamanouchi prefix of a binary word. This is the largest index

$i$ such that in each of the prefixes

$w_1$ ,

$w_1w_2$ ,

$w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.

Matching statistic: St001730
(load all 3 compositions to match this statistic)

Mapping

Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 14% ●values known / values provided: 23%●distinct values known / distinct values provided: 14%

Values

[1]
 => [1]
 => [1,0]
 => 10 => 0
[2]
 => [1,1]
 => [1,1,0,0]
 => 1100 => 0
[1,1]
 => [2]
 => [1,0,1,0]
 => 1010 => 0
[3]
 => [3]
 => [1,0,1,0,1,0]
 => 101010 => 0
[2,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 110100 => 0
[1,1,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 101100 => 0
[4]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => 0
[3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0
[2,2]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
[2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0
[1,1,1,1]
 => [2,2]
 => [1,1,1,0,0,0]
 => 111000 => 0
[5]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? = 0
[4,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => ? = 0
[3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => ? = 1
[3,1,1]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0
[2,2,1]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 1
[2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => ? = 0
[1,1,1,1,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => 0
[6]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 11101000 => 0
[5,1]
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 101010101100 => ? = 0
[4,2]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 110101010100 => ? = 1
[4,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 101101010100 => ? = 0
[3,3]
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 101010101010 => ? = 1
[3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 101011010100 => ? = 2
[3,1,1,1]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
[2,2,2]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 101010110100 => ? = 1
[2,2,1,1]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => ? = 1
[2,1,1,1,1]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => ? = 0
[1,1,1,1,1,1]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => 0
[7]
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 10101010101010 => ? = 0
[6,1]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => ? = 0
[5,2]
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 10101010110100 => ? = 1
[5,1,1]
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => ? = 0
[4,3]
 => [3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => 10101101010100 => ? = 3
[4,2,1]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 11010101010100 => ? = 4
[4,1,1,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => 10110101010100 => ? = 0
[3,3,1]
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 10101010101100 => ? = 3
[3,2,2]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 2
[3,2,1,1]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 101110010100 => ? = 2
[3,1,1,1,1]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? = 0
[2,2,2,1]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => 10101011010100 => ? = 2
[2,2,1,1,1]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 101011100100 => ? = 1
[2,1,1,1,1,1]
 => [2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 111001010100 => ? = 0
[1,1,1,1,1,1,1]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => ? = 0
[8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 1101010101010100 => ? = 0
[7,1]
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 1010101010101100 => ? = 0
[6,2]
 => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 111010010100 => ? = 1
[6,1,1]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => ? = 0
[5,3]
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 101011101000 => ? = 3
[5,2,1]
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => 1010101011010100 => ? = 7
[5,1,1,1]
 => [5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => 10101011100100 => ? = 0
[4,4]
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 1010101010101010 => ? = 3
[4,3,1]
 => [3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => 1010110101010100 => ? = 5
[4,2,2]
 => [4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => 1010101101010100 => ? = 4
[4,2,1,1]
 => [2,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 1011010101010100 => ? = 4
[4,1,1,1,1]
 => [2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => 11100101010100 => ? = 0
[3,3,2]
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 1010101010110100 => ? = 5
[3,3,1,1]
 => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => 10101010111000 => ? = 4
[3,2,2,1]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 101110100100 => ? = 5
[3,2,1,1,1]
 => [3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => 10111001010100 => ? = 2
[3,1,1,1,1,1]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 101111000100 => ? = 0
[2,2,2,2]
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => ? = 2
[2,2,2,1,1]
 => [4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => 10101110010100 => ? = 2
[2,2,1,1,1,1]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 101011110000 => ? = 1
[2,1,1,1,1,1,1]
 => [2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 111100010100 => ? = 0

Description

The number of times the path corresponding to a binary word crosses the base line. Interpret each

$0$ as a step

$(1,-1)$ and

$1$ as a step

$(1,1)$ . Then this statistic counts the number of times the path crosses the

$x$ -axis.

Matching statistic: St001208
(load all 3 compositions to match this statistic)

Mapping

Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 23%●distinct values known / distinct values provided: 14%

Values

[1]
 => [1]
 => [1,0]
 => [2,1] => 1 = 0 + 1
[2]
 => [1,1]
 => [1,1,0,0]
 => [2,3,1] => 1 = 0 + 1
[1,1]
 => [2]
 => [1,0,1,0]
 => [3,1,2] => 1 = 0 + 1
[3]
 => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 1 = 0 + 1
[2,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => 1 = 0 + 1
[1,1,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 1 = 0 + 1
[4]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => 1 = 0 + 1
[3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1 = 0 + 1
[2,2]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 0 + 1
[2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 1 = 0 + 1
[1,1,1,1]
 => [2,2]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => 1 = 0 + 1
[5]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ? = 0 + 1
[4,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => ? = 0 + 1
[3,2]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => ? = 1 + 1
[3,1,1]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 1 = 0 + 1
[2,2,1]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ? = 1 + 1
[2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => ? = 0 + 1
[1,1,1,1,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => 1 = 0 + 1
[6]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => 1 = 0 + 1
[5,1]
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => ? = 0 + 1
[4,2]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,1,2,3,4,5] => ? = 1 + 1
[4,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,1,7,2,3,4,5] => ? = 0 + 1
[3,3]
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 1 + 1
[3,2,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,1,2,6,3,4,5] => ? = 2 + 1
[3,1,1,1]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ? = 0 + 1
[2,2,2]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [7,1,2,3,6,4,5] => ? = 1 + 1
[2,2,1,1]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ? = 1 + 1
[2,1,1,1,1]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => ? = 0 + 1
[1,1,1,1,1,1]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1 = 0 + 1
[7]
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [8,1,2,3,4,5,6,7] => ? = 0 + 1
[6,1]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => ? = 0 + 1
[5,2]
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [8,1,2,3,4,7,5,6] => ? = 1 + 1
[5,1,1]
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ? = 0 + 1
[4,3]
 => [3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [7,1,2,8,3,4,5,6] => ? = 3 + 1
[4,2,1]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [8,7,1,2,3,4,5,6] => ? = 4 + 1
[4,1,1,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [8,1,7,2,3,4,5,6] => ? = 0 + 1
[3,3,1]
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [7,1,2,3,4,5,8,6] => ? = 3 + 1
[3,2,2]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 2 + 1
[3,2,1,1]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => ? = 2 + 1
[3,1,1,1,1]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 0 + 1
[2,2,2,1]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [8,1,2,3,7,4,5,6] => ? = 2 + 1
[2,2,1,1,1]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,1,2,7,6,3,5] => ? = 1 + 1
[2,1,1,1,1,1]
 => [2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,6,7,1,3,4,5] => ? = 0 + 1
[1,1,1,1,1,1,1]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [2,3,6,5,1,4] => ? = 0 + 1
[8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [8,9,1,2,3,4,5,6,7] => ? = 0 + 1
[7,1]
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [8,1,2,3,4,5,6,9,7] => ? = 0 + 1
[6,2]
 => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [6,3,7,1,2,4,5] => ? = 1 + 1
[6,1,1]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5,3,4,1,6,2] => ? = 0 + 1
[5,3]
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [7,1,2,5,6,3,4] => ? = 3 + 1
[5,2,1]
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [9,1,2,3,4,8,5,6,7] => ? = 7 + 1
[5,1,1,1]
 => [5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,1,2,3,8,7,4,6] => ? = 0 + 1
[4,4]
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [9,1,2,3,4,5,6,7,8] => ? = 3 + 1
[4,3,1]
 => [3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [9,1,2,8,3,4,5,6,7] => ? = 5 + 1
[4,2,2]
 => [4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [8,1,2,3,9,4,5,6,7] => ? = 4 + 1
[4,2,1,1]
 => [2,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [9,1,8,2,3,4,5,6,7] => ? = 4 + 1
[4,1,1,1,1]
 => [2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [2,8,7,1,3,4,5,6] => ? = 0 + 1
[3,3,2]
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [9,1,2,3,4,5,8,6,7] => ? = 5 + 1
[3,3,1,1]
 => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 4 + 1
[3,2,2,1]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => ? = 5 + 1
[3,2,1,1,1]
 => [3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,7,8,2,4,5,6] => ? = 2 + 1
[3,1,1,1,1,1]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => ? = 0 + 1
[2,2,2,2]
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => ? = 2 + 1
[2,2,2,1,1]
 => [4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [4,1,2,8,7,3,5,6] => ? = 2 + 1
[2,2,1,1,1,1]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => ? = 1 + 1
[2,1,1,1,1,1,1]
 => [2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [2,3,7,6,1,4,5] => ? = 0 + 1

Description

The number of connected components of the quiver of

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra

$A$ of

$K[x]/(x^n)$ .

Matching statistic: St001236
(load all 4 compositions to match this statistic)

Mapping

Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001236: Integer compositions ⟶ ℤResult quality: 14% ●values known / values provided: 23%●distinct values known / distinct values provided: 14%

Values

[1]
 => [1]
 => 10 => [1,2] => 1 = 0 + 1
[2]
 => [1,1]
 => 110 => [1,1,2] => 1 = 0 + 1
[1,1]
 => [2]
 => 100 => [1,3] => 1 = 0 + 1
[3]
 => [3]
 => 1000 => [1,4] => 1 = 0 + 1
[2,1]
 => [1,1,1]
 => 1110 => [1,1,1,2] => 1 = 0 + 1
[1,1,1]
 => [2,1]
 => 1010 => [1,2,2] => 1 = 0 + 1
[4]
 => [2,2]
 => 1100 => [1,1,3] => 1 = 0 + 1
[3,1]
 => [3,1]
 => 10010 => [1,3,2] => 1 = 0 + 1
[2,2]
 => [1,1,1,1]
 => 11110 => [1,1,1,1,2] => 1 = 0 + 1
[2,1,1]
 => [2,1,1]
 => 10110 => [1,2,1,2] => 1 = 0 + 1
[1,1,1,1]
 => [4]
 => 10000 => [1,5] => 1 = 0 + 1
[5]
 => [5]
 => 100000 => [1,6] => ? = 0 + 1
[4,1]
 => [2,2,1]
 => 11010 => [1,1,2,2] => 1 = 0 + 1
[3,2]
 => [3,1,1]
 => 100110 => [1,3,1,2] => ? = 1 + 1
[3,1,1]
 => [3,2]
 => 10100 => [1,2,3] => 1 = 0 + 1
[2,2,1]
 => [1,1,1,1,1]
 => 111110 => [1,1,1,1,1,2] => ? = 1 + 1
[2,1,1,1]
 => [2,1,1,1]
 => 101110 => [1,2,1,1,2] => ? = 0 + 1
[1,1,1,1,1]
 => [4,1]
 => 100010 => [1,4,2] => ? = 0 + 1
[6]
 => [3,3]
 => 11000 => [1,1,4] => 1 = 0 + 1
[5,1]
 => [5,1]
 => 1000010 => [1,5,2] => ? = 0 + 1
[4,2]
 => [2,2,1,1]
 => 110110 => [1,1,2,1,2] => ? = 1 + 1
[4,1,1]
 => [2,2,2]
 => 11100 => [1,1,1,3] => 1 = 0 + 1
[3,3]
 => [6]
 => 1000000 => [1,7] => ? = 1 + 1
[3,2,1]
 => [3,1,1,1]
 => 1001110 => [1,3,1,1,2] => ? = 2 + 1
[3,1,1,1]
 => [3,2,1]
 => 101010 => [1,2,2,2] => ? = 0 + 1
[2,2,2]
 => [1,1,1,1,1,1]
 => 1111110 => [1,1,1,1,1,1,2] => ? = 1 + 1
[2,2,1,1]
 => [2,1,1,1,1]
 => 1011110 => [1,2,1,1,1,2] => ? = 1 + 1
[2,1,1,1,1]
 => [4,1,1]
 => 1000110 => [1,4,1,2] => ? = 0 + 1
[1,1,1,1,1,1]
 => [4,2]
 => 100100 => [1,3,3] => ? = 0 + 1
[7]
 => [7]
 => 10000000 => [1,8] => ? = 0 + 1
[6,1]
 => [3,3,1]
 => 110010 => [1,1,3,2] => ? = 0 + 1
[5,2]
 => [5,1,1]
 => 10000110 => [1,5,1,2] => ? = 1 + 1
[5,1,1]
 => [5,2]
 => 1000100 => [1,4,3] => ? = 0 + 1
[4,3]
 => [3,2,2]
 => 101100 => [1,2,1,3] => ? = 3 + 1
[4,2,1]
 => [2,2,1,1,1]
 => 1101110 => [1,1,2,1,1,2] => ? = 4 + 1
[4,1,1,1]
 => [2,2,2,1]
 => 111010 => [1,1,1,2,2] => ? = 0 + 1
[3,3,1]
 => [6,1]
 => 10000010 => [1,6,2] => ? = 3 + 1
[3,2,2]
 => [3,1,1,1,1]
 => 10011110 => [1,3,1,1,1,2] => ? = 2 + 1
[3,2,1,1]
 => [3,2,1,1]
 => 1010110 => [1,2,2,1,2] => ? = 2 + 1
[3,1,1,1,1]
 => [4,3]
 => 101000 => [1,2,4] => ? = 0 + 1
[2,2,2,1]
 => [1,1,1,1,1,1,1]
 => 11111110 => [1,1,1,1,1,1,1,2] => ? = 2 + 1
[2,2,1,1,1]
 => [2,1,1,1,1,1]
 => 10111110 => [1,2,1,1,1,1,2] => ? = 1 + 1
[2,1,1,1,1,1]
 => [4,1,1,1]
 => 10001110 => [1,4,1,1,2] => ? = 0 + 1
[1,1,1,1,1,1,1]
 => [4,2,1]
 => 1001010 => [1,3,2,2] => ? = 0 + 1
[8]
 => [4,4]
 => 110000 => [1,1,5] => ? = 0 + 1
[7,1]
 => [7,1]
 => 100000010 => [1,7,2] => ? = 0 + 1
[6,2]
 => [3,3,1,1]
 => 1100110 => [1,1,3,1,2] => ? = 1 + 1
[6,1,1]
 => [3,3,2]
 => 110100 => [1,1,2,3] => ? = 0 + 1
[5,3]
 => [5,3]
 => 1001000 => [1,3,4] => ? = 3 + 1
[5,2,1]
 => [5,1,1,1]
 => 100001110 => [1,5,1,1,2] => ? = 7 + 1
[5,1,1,1]
 => [5,2,1]
 => 10001010 => [1,4,2,2] => ? = 0 + 1
[4,4]
 => [2,2,2,2]
 => 111100 => [1,1,1,1,3] => ? = 3 + 1
[4,3,1]
 => [3,2,2,1]
 => 1011010 => [1,2,1,2,2] => ? = 5 + 1
[4,2,2]
 => [2,2,1,1,1,1]
 => 11011110 => [1,1,2,1,1,1,2] => ? = 4 + 1
[4,2,1,1]
 => [2,2,2,1,1]
 => 1110110 => [1,1,1,2,1,2] => ? = 4 + 1
[4,1,1,1,1]
 => [4,2,2]
 => 1001100 => [1,3,1,3] => ? = 0 + 1
[3,3,2]
 => [6,1,1]
 => 100000110 => [1,6,1,2] => ? = 5 + 1
[3,3,1,1]
 => [6,2]
 => 10000100 => [1,5,3] => ? = 4 + 1
[3,2,2,1]
 => [3,1,1,1,1,1]
 => 100111110 => [1,3,1,1,1,1,2] => ? = 5 + 1
[3,2,1,1,1]
 => [3,2,1,1,1]
 => 10101110 => [1,2,2,1,1,2] => ? = 2 + 1
[3,1,1,1,1,1]
 => [4,3,1]
 => 1010010 => [1,2,3,2] => ? = 0 + 1
[2,2,2,2]
 => [1,1,1,1,1,1,1,1]
 => 111111110 => [1,1,1,1,1,1,1,1,2] => ? = 2 + 1
[2,2,2,1,1]
 => [2,1,1,1,1,1,1]
 => 101111110 => [1,2,1,1,1,1,1,2] => ? = 2 + 1
[2,2,1,1,1,1]
 => [4,1,1,1,1]
 => 100011110 => [1,4,1,1,1,2] => ? = 1 + 1
[2,1,1,1,1,1,1]
 => [4,2,1,1]
 => 10010110 => [1,3,2,1,2] => ? = 0 + 1

Description

The dominant dimension of the corresponding Comp-Nakayama algebra.

Matching statistic: St001621

Mapping

Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 14% ●values known / values provided: 23%●distinct values known / distinct values provided: 14%

Values

[1]
 => [[1],[]]
 => ([],1)
 => ([(0,1)],2)
 => 1 = 0 + 1
[2]
 => [[2],[]]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[3]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,1]
 => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1 = 0 + 1
[1,1,1]
 => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[4]
 => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,1]
 => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 1 = 0 + 1
[2,2]
 => [[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 1 = 0 + 1
[2,1,1]
 => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 1 = 0 + 1
[1,1,1,1]
 => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[5]
 => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[4,1]
 => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? = 0 + 1
[3,2]
 => [[3,2],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? = 1 + 1
[3,1,1]
 => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 0 + 1
[2,2,1]
 => [[2,2,1],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? = 1 + 1
[2,1,1,1]
 => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? = 0 + 1
[1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[6]
 => [[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 0 + 1
[5,1]
 => [[5,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
 => ? = 0 + 1
[4,2]
 => [[4,2],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => ? = 1 + 1
[4,1,1]
 => [[4,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => ? = 0 + 1
[3,3]
 => [[3,3],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 1 + 1
[3,2,1]
 => [[3,2,1],[]]
 => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
 => ? = 2 + 1
[3,1,1,1]
 => [[3,1,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => ? = 0 + 1
[2,2,2]
 => [[2,2,2],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 1 + 1
[2,2,1,1]
 => [[2,2,1,1],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => ? = 1 + 1
[2,1,1,1,1]
 => [[2,1,1,1,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
 => ? = 0 + 1
[1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 0 + 1
[7]
 => [[7],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ? = 0 + 1
[6,1]
 => [[6,1],[]]
 => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
 => ([(0,7),(1,8),(2,9),(3,5),(3,8),(4,6),(4,10),(5,4),(5,12),(6,2),(6,11),(7,1),(7,3),(8,12),(10,11),(11,9),(12,10)],13)
 => ? = 0 + 1
[5,2]
 => [[5,2],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
 => ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
 => ? = 1 + 1
[5,1,1]
 => [[5,1,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
 => ([(0,1),(1,2),(1,3),(2,5),(2,13),(3,7),(3,13),(4,12),(5,11),(6,4),(6,15),(7,6),(7,14),(9,10),(10,8),(11,9),(12,8),(13,11),(13,14),(14,9),(14,15),(15,10),(15,12)],16)
 => ? = 0 + 1
[4,3]
 => [[4,3],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => ? = 3 + 1
[4,2,1]
 => [[4,2,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
 => ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
 => ? = 4 + 1
[4,1,1,1]
 => [[4,1,1,1],[]]
 => ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
 => ([(0,1),(1,2),(1,3),(2,7),(2,14),(3,6),(3,14),(4,11),(5,12),(6,4),(6,15),(7,5),(7,16),(9,8),(10,8),(11,9),(12,10),(13,9),(13,10),(14,15),(14,16),(15,11),(15,13),(16,12),(16,13)],17)
 => ? = 0 + 1
[3,3,1]
 => [[3,3,1],[]]
 => ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ? = 3 + 1
[3,2,2]
 => [[3,2,2],[]]
 => ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => ? = 2 + 1
[3,2,1,1]
 => [[3,2,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
 => ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
 => ? = 2 + 1
[3,1,1,1,1]
 => [[3,1,1,1,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
 => ([(0,1),(1,2),(1,3),(2,5),(2,13),(3,7),(3,13),(4,12),(5,11),(6,4),(6,15),(7,6),(7,14),(9,10),(10,8),(11,9),(12,8),(13,11),(13,14),(14,9),(14,15),(15,10),(15,12)],16)
 => ? = 0 + 1
[2,2,2,1]
 => [[2,2,2,1],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => ? = 2 + 1
[2,2,1,1,1]
 => [[2,2,1,1,1],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
 => ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
 => ? = 1 + 1
[2,1,1,1,1,1]
 => [[2,1,1,1,1,1],[]]
 => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
 => ([(0,7),(1,8),(2,9),(3,5),(3,8),(4,6),(4,10),(5,4),(5,12),(6,2),(6,11),(7,1),(7,3),(8,12),(10,11),(11,9),(12,10)],13)
 => ? = 0 + 1
[1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1],[]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ? = 0 + 1
[8]
 => [[8],[]]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
 => ? = 0 + 1
[7,1]
 => [[7,1],[]]
 => ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
 => ([(0,8),(1,9),(2,10),(3,6),(3,9),(4,5),(4,12),(5,7),(5,11),(6,4),(6,14),(7,2),(7,13),(8,1),(8,3),(9,14),(11,13),(12,11),(13,10),(14,12)],15)
 => ? = 0 + 1
[6,2]
 => [[6,2],[]]
 => ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
 => ([(0,1),(1,3),(1,4),(2,12),(3,10),(4,6),(4,10),(5,14),(6,7),(6,15),(7,8),(7,17),(8,5),(8,16),(10,2),(10,15),(11,13),(12,11),(13,9),(14,9),(15,12),(15,17),(16,13),(16,14),(17,11),(17,16)],18)
 => ? = 1 + 1
[6,1,1]
 => [[6,1,1],[]]
 => ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
 => ([(0,1),(1,2),(1,3),(2,5),(2,15),(3,6),(3,15),(4,14),(5,13),(6,7),(6,16),(7,8),(7,18),(8,4),(8,17),(10,12),(11,10),(12,9),(13,11),(14,9),(15,13),(15,16),(16,11),(16,18),(17,12),(17,14),(18,10),(18,17)],19)
 => ? = 0 + 1
[5,3]
 => [[5,3],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
 => ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
 => ? = 3 + 1
[5,2,1]
 => [[5,2,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
 => ?
 => ? = 7 + 1
[5,1,1,1]
 => [[5,1,1,1],[]]
 => ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
 => ?
 => ? = 0 + 1
[4,4]
 => [[4,4],[]]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ? = 3 + 1
[4,3,1]
 => [[4,3,1],[]]
 => ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ?
 => ? = 5 + 1
[4,2,2]
 => [[4,2,2],[]]
 => ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ?
 => ? = 4 + 1
[4,2,1,1]
 => [[4,2,1,1],[]]
 => ([(0,5),(0,6),(3,2),(4,1),(5,3),(5,7),(6,4),(6,7)],8)
 => ?
 => ? = 4 + 1
[4,1,1,1,1]
 => [[4,1,1,1,1],[]]
 => ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
 => ?
 => ? = 0 + 1
[3,3,2]
 => [[3,3,2],[]]
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7)],8)
 => ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
 => ? = 5 + 1
[3,3,1,1]
 => [[3,3,1,1],[]]
 => ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ?
 => ? = 4 + 1
[3,2,2,1]
 => [[3,2,2,1],[]]
 => ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ?
 => ? = 5 + 1
[3,2,1,1,1]
 => [[3,2,1,1,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
 => ?
 => ? = 2 + 1
[3,1,1,1,1,1]
 => [[3,1,1,1,1,1],[]]
 => ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
 => ([(0,1),(1,2),(1,3),(2,5),(2,15),(3,6),(3,15),(4,14),(5,13),(6,7),(6,16),(7,8),(7,18),(8,4),(8,17),(10,12),(11,10),(12,9),(13,11),(14,9),(15,13),(15,16),(16,11),(16,18),(17,12),(17,14),(18,10),(18,17)],19)
 => ? = 0 + 1
[2,2,2,2]
 => [[2,2,2,2],[]]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ? = 2 + 1
[2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
 => ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
 => ? = 2 + 1
[2,2,1,1,1,1]
 => [[2,2,1,1,1,1],[]]
 => ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
 => ([(0,1),(1,3),(1,4),(2,12),(3,10),(4,6),(4,10),(5,14),(6,7),(6,15),(7,8),(7,17),(8,5),(8,16),(10,2),(10,15),(11,13),(12,11),(13,9),(14,9),(15,12),(15,17),(16,13),(16,14),(17,11),(17,16)],18)
 => ? = 1 + 1
[2,1,1,1,1,1,1]
 => [[2,1,1,1,1,1,1],[]]
 => ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
 => ([(0,8),(1,9),(2,10),(3,6),(3,9),(4,5),(4,12),(5,7),(5,11),(6,4),(6,14),(7,2),(7,13),(8,1),(8,3),(9,14),(11,13),(12,11),(13,10),(14,12)],15)
 => ? = 0 + 1

Description

The number of atoms of a lattice. An element of a lattice is an '''atom''' if it covers the least element.

The following 150 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001712The number of natural descents of a standard Young tableau. 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

$eA$ . St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation. St001410The minimal entry of a semistandard tableau. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000039The number of crossings of a permutation. St000091The descent variation of a composition. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000295The length of the border of a binary word. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000462The major index minus the number of excedences of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000496The rcs statistic of a set partition. St000516The number of stretching pairs of a permutation. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000623The number of occurrences of the pattern 52341 in a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000962The 3-shifted major index of a permutation. St000989The number of final rises of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001381The fertility of a permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St001810The number of fixed points of a permutation smaller than its largest moved point. St001847The number of occurrences of the pattern 1432 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001903The number of fixed points of a parking function. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000021The number of descents of a permutation. St000056The decomposition (or block) number of a permutation. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000253The crossing number of a set partition. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000570The Edelman-Greene number of a permutation. St000654The first descent of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000729The minimal arc length of a set partition. St000864The number of circled entries of the shifted recording tableau of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001490The number of connected components of a skew partition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001806The upper middle entry of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001884The number of borders of a binary word. St001889The size of the connectivity set of a signed permutation. St001928The number of non-overlapping descents in a permutation. St000084The number of subtrees. St000105The number of blocks in the set partition. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000504The cardinality of the first block of a set partition. St000542The number of left-to-right-minima of a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000823The number of unsplittable factors of the set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau.