- FindStat

Your data matches 34 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000011
(load all 18 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => [1,0]
 => 1 = 0 + 1
{{1,2}}
 => [2,1] => [1,1,0,0]
 => 1 = 0 + 1
{{1},{2}}
 => [1,2] => [1,0,1,0]
 => 2 = 1 + 1
{{1,2,3}}
 => [2,3,1] => [1,1,0,1,0,0]
 => 1 = 0 + 1
{{1,2},{3}}
 => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,3},{2}}
 => [3,2,1] => [1,1,1,0,0,0]
 => 1 = 0 + 1
{{1},{2,3}}
 => [1,3,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
{{1},{2},{3}}
 => [1,2,3] => [1,0,1,0,1,0]
 => 3 = 2 + 1
{{1,2,3,4}}
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,2,3},{4}}
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
{{1,2,4},{3}}
 => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,3,4},{2}}
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
{{1,3},{2,4}}
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
{{1,4},{2,3}}
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => 2 = 1 + 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => 1 = 0 + 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => 1 = 0 + 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => 1 = 0 + 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 1 + 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 1 = 0 + 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
 => 1 = 0 + 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 1 + 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => 1 = 0 + 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 0 + 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 0 + 1

Description

The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.

Matching statistic: St000234
(load all 265 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => [1] => [1] => 0
{{1,2}}
 => [2,1] => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 1
{{1,2,3}}
 => [2,3,1] => [3,2,1] => [3,2,1] => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1,3] => 1
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => [3,2,1] => 0
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [1,3,2] => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,2,3] => 2
{{1,2,3,4}}
 => [2,3,4,1] => [4,2,3,1] => [4,3,2,1] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [4,3,2,1] => [4,3,2,1] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
{{1,3,4},{2}}
 => [3,2,4,1] => [4,2,3,1] => [4,3,2,1] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
{{1,4},{2},{3}}
 => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 3
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,2,3,4,1] => [5,4,3,2,1] => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,2,3,1,5] => [4,3,2,1,5] => 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [5,2,4,3,1] => [5,4,3,2,1] => 0
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 2
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [5,3,2,4,1] => [5,4,3,2,1] => 0
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,5,3,1,2] => [5,4,3,2,1] => 0
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => 0
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => 0
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 3
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [5,2,3,4,1] => [5,4,3,2,1] => 0
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => 0
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [4,2,3,1,5] => [4,3,2,1,5] => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => 0
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [5,3,2,4,1] => [5,4,3,2,1] => 0
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [5,2,4,3,1] => [5,4,3,2,1] => 0
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => 0
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 2
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [5,3,2,4,1] => [5,4,3,2,1] => 0
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => 0

Description

The number of global ascents of a permutation. The global ascents are the integers

$i$ such that

$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$ Equivalently, by the pigeonhole principle,

$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$ For

$n > 1$ it can also be described as an occurrence of the mesh pattern

$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$ or equivalently

$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$ see [3]. According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives [[oeis:A003319]].

Matching statistic: St000010

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => ([],1)
 => [1]
 => 1 = 0 + 1
{{1,2}}
 => [2,1] => ([(0,1)],2)
 => [2]
 => 1 = 0 + 1
{{1},{2}}
 => [1,2] => ([],2)
 => [1,1]
 => 2 = 1 + 1
{{1,2,3}}
 => [2,3,1] => ([(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
{{1,2},{3}}
 => [2,1,3] => ([(1,2)],3)
 => [2,1]
 => 2 = 1 + 1
{{1,3},{2}}
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
{{1},{2,3}}
 => [1,3,2] => ([(1,2)],3)
 => [2,1]
 => 2 = 1 + 1
{{1},{2},{3}}
 => [1,2,3] => ([],3)
 => [1,1,1]
 => 3 = 2 + 1
{{1,2,3,4}}
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
{{1,2,3},{4}}
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => [3,1]
 => 2 = 1 + 1
{{1,2,4},{3}}
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
{{1,2},{3,4}}
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => [2,2]
 => 2 = 1 + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => ([(2,3)],4)
 => [2,1,1]
 => 3 = 2 + 1
{{1,3,4},{2}}
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
{{1,3},{2,4}}
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 1 = 0 + 1
{{1,3},{2},{4}}
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 2 = 1 + 1
{{1,4},{2,3}}
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
{{1},{2,3,4}}
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => 2 = 1 + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => ([(2,3)],4)
 => [2,1,1]
 => 3 = 2 + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
{{1},{2,4},{3}}
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 2 = 1 + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => ([(2,3)],4)
 => [2,1,1]
 => 3 = 2 + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => ([],4)
 => [1,1,1,1]
 => 4 = 3 + 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 2 = 1 + 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => 2 = 1 + 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 3 = 2 + 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 2 = 1 + 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => 2 = 1 + 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => 3 = 2 + 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => 2 = 1 + 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => 3 = 2 + 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => ([(3,4)],5)
 => [2,1,1,1]
 => 4 = 3 + 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => 1 = 0 + 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 2 = 1 + 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => 2 = 1 + 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => 2 = 1 + 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 3 = 2 + 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => 1 = 0 + 1

Description

The length of the partition.

Matching statistic: St000025

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => [1,0]
 => [1,0]
 => 1 = 0 + 1
{{1,2}}
 => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
{{1},{2}}
 => [1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
{{1,2,3}}
 => [2,3,1] => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
{{1,2},{3}}
 => [2,1,3] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,3},{2}}
 => [3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
{{1},{2,3}}
 => [1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
{{1},{2},{3}}
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
{{1,2,3,4}}
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
{{1,2,3},{4}}
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
{{1,2,4},{3}}
 => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
{{1,3,4},{2}}
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
{{1,3},{2,4}}
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
{{1,4},{2,3}}
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2 = 1 + 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3 = 2 + 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2 = 1 + 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 0 + 1

Description

The number of initial rises of a Dyck path. In other words, this is the height of the first peak of

$D$ .

Matching statistic: St000383

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => ([],1)
 => [1] => 1 = 0 + 1
{{1,2}}
 => [2,1] => ([(0,1)],2)
 => [1,1] => 1 = 0 + 1
{{1},{2}}
 => [1,2] => ([],2)
 => [2] => 2 = 1 + 1
{{1,2,3}}
 => [2,3,1] => ([(0,2),(1,2)],3)
 => [1,1,1] => 1 = 0 + 1
{{1,2},{3}}
 => [2,1,3] => ([(1,2)],3)
 => [1,2] => 2 = 1 + 1
{{1,3},{2}}
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 1 = 0 + 1
{{1},{2,3}}
 => [1,3,2] => ([(1,2)],3)
 => [1,2] => 2 = 1 + 1
{{1},{2},{3}}
 => [1,2,3] => ([],3)
 => [3] => 3 = 2 + 1
{{1,2,3,4}}
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => 1 = 0 + 1
{{1,2,3},{4}}
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => [1,1,2] => 2 = 1 + 1
{{1,2,4},{3}}
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => 1 = 0 + 1
{{1,2},{3,4}}
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => [2,2] => 2 = 1 + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => ([(2,3)],4)
 => [1,3] => 3 = 2 + 1
{{1,3,4},{2}}
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => 1 = 0 + 1
{{1,3},{2,4}}
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => 1 = 0 + 1
{{1,3},{2},{4}}
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [2,2] => 2 = 1 + 1
{{1,4},{2,3}}
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => 1 = 0 + 1
{{1},{2,3,4}}
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => [1,1,2] => 2 = 1 + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => ([(2,3)],4)
 => [1,3] => 3 = 2 + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 1 = 0 + 1
{{1},{2,4},{3}}
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => [2,2] => 2 = 1 + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => ([(2,3)],4)
 => [1,3] => 3 = 2 + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => ([],4)
 => [4] => 4 = 3 + 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => 1 = 0 + 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => 2 = 1 + 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => 1 = 0 + 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => 2 = 1 + 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => [1,1,3] => 3 = 2 + 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => 1 = 0 + 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1 = 0 + 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => 2 = 1 + 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => 1 = 0 + 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => 2 = 1 + 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => [2,3] => 3 = 2 + 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1 = 0 + 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => 2 = 1 + 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => [2,3] => 3 = 2 + 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => ([(3,4)],5)
 => [1,4] => 4 = 3 + 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => 1 = 0 + 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => 1 = 0 + 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => 2 = 1 + 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => 1 = 0 + 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1 = 0 + 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => 2 = 1 + 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => 1 = 0 + 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1 = 0 + 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => 2 = 1 + 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => [2,3] => 3 = 2 + 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => 1 = 0 + 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => 1 = 0 + 1

Description

The last part of an integer composition.

Matching statistic: St001462
(load all 2 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => [1] => [[1]]
 => 1 = 0 + 1
{{1,2}}
 => [2,1] => [2,1] => [[1],[2]]
 => 1 = 0 + 1
{{1},{2}}
 => [1,2] => [1,2] => [[1,2]]
 => 2 = 1 + 1
{{1,2,3}}
 => [2,3,1] => [3,2,1] => [[1],[2],[3]]
 => 1 = 0 + 1
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [[1,3],[2]]
 => 2 = 1 + 1
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => [[1],[2],[3]]
 => 1 = 0 + 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [[1,2],[3]]
 => 2 = 1 + 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [[1,2,3]]
 => 3 = 2 + 1
{{1,2,3,4}}
 => [2,3,4,1] => [4,2,3,1] => [[1,3],[2],[4]]
 => 1 = 0 + 1
{{1,2,3},{4}}
 => [2,3,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
 => 2 = 1 + 1
{{1,2,4},{3}}
 => [2,4,3,1] => [4,3,2,1] => [[1],[2],[3],[4]]
 => 1 = 0 + 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
 => 2 = 1 + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
 => 3 = 2 + 1
{{1,3,4},{2}}
 => [3,2,4,1] => [4,2,3,1] => [[1,3],[2],[4]]
 => 1 = 0 + 1
{{1,3},{2,4}}
 => [3,4,1,2] => [4,3,2,1] => [[1],[2],[3],[4]]
 => 1 = 0 + 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
 => 2 = 1 + 1
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
 => 1 = 0 + 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,3,2] => [[1,2],[3],[4]]
 => 2 = 1 + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
 => 3 = 2 + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [4,3,2,1] => [[1],[2],[3],[4]]
 => 1 = 0 + 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
 => 2 = 1 + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
 => 3 = 2 + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
 => 4 = 3 + 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,2,3,4,1] => [[1,3,4],[2],[5]]
 => 1 = 0 + 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,2,3,1,5] => [[1,3,5],[2],[4]]
 => 2 = 1 + 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [5,2,4,3,1] => [[1,3],[2],[4],[5]]
 => 1 = 0 + 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,2,1,5,4] => [[1,4],[2,5],[3]]
 => 2 = 1 + 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,2,1,4,5] => [[1,4,5],[2],[3]]
 => 3 = 2 + 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [5,3,2,4,1] => [[1,4],[2],[3],[5]]
 => 1 = 0 + 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,5,3,1,2] => [[1,2],[3,5],[4]]
 => 1 = 0 + 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [4,3,2,1,5] => [[1,5],[2],[3],[4]]
 => 2 = 1 + 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
 => 1 = 0 + 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
 => 2 = 1 + 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => [[1,3,5],[2,4]]
 => 3 = 2 + 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
 => 1 = 0 + 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
 => 2 = 1 + 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => [[1,3,4],[2,5]]
 => 3 = 2 + 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => 4 = 3 + 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [5,2,3,4,1] => [[1,3,4],[2],[5]]
 => 1 = 0 + 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
 => 1 = 0 + 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [4,2,3,1,5] => [[1,3,5],[2],[4]]
 => 2 = 1 + 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
 => 1 = 0 + 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [5,3,2,4,1] => [[1,4],[2],[3],[5]]
 => 1 = 0 + 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [4,3,2,1,5] => [[1,5],[2],[3],[4]]
 => 2 = 1 + 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [5,2,4,3,1] => [[1,3],[2],[4],[5]]
 => 1 = 0 + 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
 => 1 = 0 + 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [3,2,1,5,4] => [[1,4],[2,5],[3]]
 => 2 = 1 + 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,2,1,4,5] => [[1,4,5],[2],[3]]
 => 3 = 2 + 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [5,3,2,4,1] => [[1,4],[2],[3],[5]]
 => 1 = 0 + 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
 => 1 = 0 + 1

Description

The number of factors of a standard tableaux under concatenation. The concatenation of two standard Young tableaux

$T_1$ and

$T_2$ is obtained by adding the largest entry of

$T_1$ to each entry of

$T_2$ , and then appending the rows of the result to

$T_1$ , see [1, dfn 2.10]. This statistic returns the maximal number of standard tableaux such that their concatenation is the given tableau.

Matching statistic: St000439

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => [1,0]
 => [1,0]
 => 2 = 0 + 2
{{1,2}}
 => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 0 + 2
{{1},{2}}
 => [1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
{{1,2,3}}
 => [2,3,1] => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 0 + 2
{{1,2},{3}}
 => [2,1,3] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 3 = 1 + 2
{{1,3},{2}}
 => [3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2 = 0 + 2
{{1},{2,3}}
 => [1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 3 = 1 + 2
{{1},{2},{3}}
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 2 + 2
{{1,2,3,4}}
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
{{1,2,3},{4}}
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 3 = 1 + 2
{{1,2,4},{3}}
 => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 0 + 2
{{1,2},{3,4}}
 => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3 = 1 + 2
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 4 = 2 + 2
{{1,3,4},{2}}
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 0 + 2
{{1,3},{2,4}}
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 1 + 2
{{1,4},{2,3}}
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
{{1},{2,3,4}}
 => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 3 = 1 + 2
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 4 = 2 + 2
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 2 + 2
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 5 = 3 + 2
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 1 + 2
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 0 + 2
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3 = 1 + 2
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 2 + 2
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 0 + 2
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 0 + 2
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3 = 1 + 2
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2 = 0 + 2
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3 = 1 + 2
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 4 = 2 + 2
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2 = 0 + 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3 = 1 + 2
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 4 = 2 + 2
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 3 + 2
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 0 + 2
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 1 + 2
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 0 + 2
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 1 + 2
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2 = 0 + 2
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 0 + 2
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3 = 1 + 2
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 2 + 2
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 0 + 2
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 0 + 2

Description

The position of the first down step of a Dyck path.

Matching statistic: St000288
(load all 8 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] =>  => ? = 0
{{1,2}}
 => [2,1] => 0 => 0
{{1},{2}}
 => [1,2] => 1 => 1
{{1,2,3}}
 => [2,3,1] => 00 => 0
{{1,2},{3}}
 => [2,1,3] => 01 => 1
{{1,3},{2}}
 => [3,2,1] => 00 => 0
{{1},{2,3}}
 => [1,3,2] => 10 => 1
{{1},{2},{3}}
 => [1,2,3] => 11 => 2
{{1,2,3,4}}
 => [2,3,4,1] => 000 => 0
{{1,2,3},{4}}
 => [2,3,1,4] => 001 => 1
{{1,2,4},{3}}
 => [2,4,3,1] => 000 => 0
{{1,2},{3,4}}
 => [2,1,4,3] => 010 => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => 011 => 2
{{1,3,4},{2}}
 => [3,2,4,1] => 000 => 0
{{1,3},{2,4}}
 => [3,4,1,2] => 000 => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => 001 => 1
{{1,4},{2,3}}
 => [4,3,2,1] => 000 => 0
{{1},{2,3,4}}
 => [1,3,4,2] => 100 => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => 101 => 2
{{1,4},{2},{3}}
 => [4,2,3,1] => 000 => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => 100 => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => 110 => 2
{{1},{2},{3},{4}}
 => [1,2,3,4] => 111 => 3
{{1,2,3,4,5}}
 => [2,3,4,5,1] => 0000 => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => 0001 => 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => 0000 => 0
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => 0010 => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => 0011 => 2
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => 0000 => 0
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => 0000 => 0
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => 0001 => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => 0000 => 0
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => 0100 => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => 0101 => 2
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => 0000 => 0
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => 0100 => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => 0110 => 2
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => 0111 => 3
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => 0000 => 0
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => 0000 => 0
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => 0001 => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => 0000 => 0
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => 0000 => 0
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => 0001 => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => 0000 => 0
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => 0000 => 0
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => 0010 => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => 0011 => 2
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => 0000 => 0
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => 0000 => 0
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => 0001 => 1

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

Matching statistic: St000675

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000675: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => [1,0]
 => [1,0]
 => ? = 0 + 1
{{1,2}}
 => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
{{1},{2}}
 => [1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
{{1,2,3}}
 => [2,3,1] => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
{{1,2},{3}}
 => [2,1,3] => [1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => 2 = 1 + 1
{{1,3},{2}}
 => [3,2,1] => [1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
{{1},{2,3}}
 => [1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
{{1},{2},{3}}
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
{{1,2,3,4}}
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
{{1,2,3},{4}}
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
{{1,2,4},{3}}
 => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
{{1,3,4},{2}}
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
{{1,3},{2,4}}
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
{{1,4},{2,3}}
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 3 = 2 + 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 0 + 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1 = 0 + 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1 = 0 + 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 0 + 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2 = 1 + 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3 = 2 + 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 0 + 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1 = 0 + 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 1 + 1

Description

The number of centered multitunnels of a Dyck path. This is the number of factorisations

$D = A B C$ of a Dyck path, such that

$B$ is a Dyck path and

$A$ and

$B$ have the same length.

Matching statistic: St000678
(load all 2 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => [1,0]
 => [1,0]
 => ? = 0 + 1
{{1,2}}
 => [2,1] => [1,1,0,0]
 => [1,1,0,0]
 => 1 = 0 + 1
{{1},{2}}
 => [1,2] => [1,0,1,0]
 => [1,0,1,0]
 => 2 = 1 + 1
{{1,2,3}}
 => [2,3,1] => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
{{1,2},{3}}
 => [2,1,3] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,3},{2}}
 => [3,2,1] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
{{1},{2,3}}
 => [1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
{{1},{2},{3}}
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
{{1,2,3,4}}
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
{{1,2,3},{4}}
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,2,4},{3}}
 => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,3,4},{2}}
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
{{1,3},{2,4}}
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
{{1,4},{2,3}}
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3 = 2 + 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 0 + 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1 = 0 + 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 0 + 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1 = 0 + 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1

Description

The number of up steps after the last double rise of a Dyck path.

The following 24 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000717The number of ordinal summands of a poset. St000553The number of blocks of a graph. St000546The number of global descents of a permutation. St000007The number of saliances of the permutation. St000237The number of small exceedances. St001461The number of topologically connected components of the chord diagram of a permutation. St000843The decomposition number of a perfect matching. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000989The number of final rises of a permutation. St000056The decomposition (or block) number of a permutation. St000287The number of connected components of a graph. St000084The number of subtrees. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000061The number of nodes on the left branch of a binary tree. St000181The number of connected components of the Hasse diagram for the poset. St001889The size of the connectivity set of a signed permutation. St000942The number of critical left to right maxima of the parking functions. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path.