- FindStat

Your data matches 65 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000208

Mapping

St000208: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1]
 => 1
[2]
 => 2
[1,1]
 => 1
[3]
 => 3
[2,1]
 => 2
[1,1,1]
 => 1
[4]
 => 5
[3,1]
 => 4
[2,2]
 => 3
[2,1,1]
 => 2
[1,1,1,1]
 => 1
[5]
 => 7
[4,1]
 => 6
[3,2]
 => 4
[3,1,1]
 => 4
[2,2,1]
 => 2
[2,1,1,1]
 => 2
[1,1,1,1,1]
 => 1
[6]
 => 11
[5,1]
 => 10
[4,2]
 => 8
[4,1,1]
 => 7
[3,3]
 => 6
[3,2,1]
 => 4
[3,1,1,1]
 => 4
[2,2,2]
 => 3
[2,2,1,1]
 => 2
[2,1,1,1,1]
 => 2
[1,1,1,1,1,1]
 => 1
[7]
 => 15
[6,1]
 => 14
[5,2]
 => 12
[5,1,1]
 => 11
[4,3]
 => 8
[4,2,1]
 => 6
[4,1,1,1]
 => 7
[3,3,1]
 => 5
[3,2,2]
 => 5
[3,2,1,1]
 => 4
[3,1,1,1,1]
 => 4
[2,2,2,1]
 => 2
[2,2,1,1,1]
 => 2
[2,1,1,1,1,1]
 => 2
[1,1,1,1,1,1,1]
 => 1
[8]
 => 22
[7,1]
 => 21
[6,2]
 => 19
[6,1,1]
 => 17
[5,3]
 => 15
[5,2,1]
 => 9

Description

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

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

$w$ (weight) there are, such that

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

$w$ such that the Gelfand-Tsetlin polytope

$P_{\lambda,w}$ has only integer lattice points as vertices. See also [[St000205]], [[St000206]] and [[St000207]].

Matching statistic: St001232
(load all 10 compositions to match this statistic)

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 33%

Values

[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ? = 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 3
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {1,5}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 4
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ? ∊ {1,5}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {2,2,4,4,6,7}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ? ∊ {2,2,4,4,6,7}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {2,2,4,4,6,7}
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => ? ∊ {2,2,4,4,6,7}
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ? ∊ {2,2,4,4,6,7}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => ? ∊ {2,2,4,4,6,7}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {1,4,6,7,8,10,11}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? ∊ {1,4,6,7,8,10,11}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 4
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {1,4,6,7,8,10,11}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? ∊ {1,4,6,7,8,10,11}
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? ∊ {1,4,6,7,8,10,11}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => ? ∊ {1,4,6,7,8,10,11}
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,4,6,7,8,10,11}
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {2,2,2,5,7,8,11,12,14,15}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? ∊ {2,2,2,5,7,8,11,12,14,15}
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ? ∊ {2,2,2,5,7,8,11,12,14,15}
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {2,2,2,5,7,8,11,12,14,15}
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ? ∊ {2,2,2,5,7,8,11,12,14,15}
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {2,2,2,5,7,8,11,12,14,15}
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? ∊ {2,2,2,5,7,8,11,12,14,15}
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 4
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 6
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 5
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => ? ∊ {2,2,2,5,7,8,11,12,14,15}
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 4
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => ? ∊ {2,2,2,5,7,8,11,12,14,15}
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {2,2,2,5,7,8,11,12,14,15}
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {1,2,2,3,4,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? ∊ {1,2,2,3,4,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? ∊ {1,2,2,3,4,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {1,2,2,3,4,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => ? ∊ {1,2,2,3,4,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {1,2,2,3,4,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? ∊ {1,2,2,3,4,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? ∊ {1,2,2,3,4,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? ∊ {1,2,2,3,4,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? ∊ {1,2,2,3,4,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ? ∊ {1,2,2,3,4,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => ? ∊ {1,2,2,3,4,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => ? ∊ {1,2,2,3,4,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,1,0,0]
 => ? ∊ {1,2,2,3,4,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => ? ∊ {1,2,2,3,4,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => ? ∊ {1,2,2,3,4,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? ∊ {1,2,2,3,4,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? ∊ {1,2,2,3,4,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,2,2,3,4,4,5,7,7,9,9,9,12,12,15,17,19,21,22}

Description

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

Matching statistic: St001645

Mapping

Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 22% ●values known / values provided: 29%●distinct values known / distinct values provided: 22%

Values

[1]
 => 1 => [1] => ([],1)
 => 1
[2]
 => 0 => [1] => ([],1)
 => 1
[1,1]
 => 11 => [2] => ([],2)
 => ? = 2
[3]
 => 1 => [1] => ([],1)
 => 1
[2,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 2
[1,1,1]
 => 111 => [3] => ([],3)
 => ? = 3
[4]
 => 0 => [1] => ([],1)
 => 1
[3,1]
 => 11 => [2] => ([],2)
 => ? ∊ {2,3,4,5}
[2,2]
 => 00 => [2] => ([],2)
 => ? ∊ {2,3,4,5}
[2,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => ? ∊ {2,3,4,5}
[1,1,1,1]
 => 1111 => [4] => ([],4)
 => ? ∊ {2,3,4,5}
[5]
 => 1 => [1] => ([],1)
 => 1
[4,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 2
[3,2]
 => 10 => [1,1] => ([(0,1)],2)
 => 2
[3,1,1]
 => 111 => [3] => ([],3)
 => ? ∊ {4,6,7}
[2,2,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 4
[2,1,1,1]
 => 0111 => [1,3] => ([(2,3)],4)
 => ? ∊ {4,6,7}
[1,1,1,1,1]
 => 11111 => [5] => ([],5)
 => ? ∊ {4,6,7}
[6]
 => 0 => [1] => ([],1)
 => 1
[5,1]
 => 11 => [2] => ([],2)
 => ? ∊ {2,2,4,4,6,7,8,10,11}
[4,2]
 => 00 => [2] => ([],2)
 => ? ∊ {2,2,4,4,6,7,8,10,11}
[4,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => ? ∊ {2,2,4,4,6,7,8,10,11}
[3,3]
 => 11 => [2] => ([],2)
 => ? ∊ {2,2,4,4,6,7,8,10,11}
[3,2,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,1,1,1]
 => 1111 => [4] => ([],4)
 => ? ∊ {2,2,4,4,6,7,8,10,11}
[2,2,2]
 => 000 => [3] => ([],3)
 => ? ∊ {2,2,4,4,6,7,8,10,11}
[2,2,1,1]
 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {2,2,4,4,6,7,8,10,11}
[2,1,1,1,1]
 => 01111 => [1,4] => ([(3,4)],5)
 => ? ∊ {2,2,4,4,6,7,8,10,11}
[1,1,1,1,1,1]
 => 111111 => [6] => ([],6)
 => ? ∊ {2,2,4,4,6,7,8,10,11}
[7]
 => 1 => [1] => ([],1)
 => 1
[6,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 2
[5,2]
 => 10 => [1,1] => ([(0,1)],2)
 => 2
[5,1,1]
 => 111 => [3] => ([],3)
 => ? ∊ {4,5,5,6,7,8,11,12,14,15}
[4,3]
 => 01 => [1,1] => ([(0,1)],2)
 => 2
[4,2,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 4
[4,1,1,1]
 => 0111 => [1,3] => ([(2,3)],4)
 => ? ∊ {4,5,5,6,7,8,11,12,14,15}
[3,3,1]
 => 111 => [3] => ([],3)
 => ? ∊ {4,5,5,6,7,8,11,12,14,15}
[3,2,2]
 => 100 => [1,2] => ([(1,2)],3)
 => ? ∊ {4,5,5,6,7,8,11,12,14,15}
[3,2,1,1]
 => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {4,5,5,6,7,8,11,12,14,15}
[3,1,1,1,1]
 => 11111 => [5] => ([],5)
 => ? ∊ {4,5,5,6,7,8,11,12,14,15}
[2,2,2,1]
 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {4,5,5,6,7,8,11,12,14,15}
[2,2,1,1,1]
 => 00111 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {4,5,5,6,7,8,11,12,14,15}
[2,1,1,1,1,1]
 => 011111 => [1,5] => ([(4,5)],6)
 => ? ∊ {4,5,5,6,7,8,11,12,14,15}
[1,1,1,1,1,1,1]
 => 1111111 => [7] => ([],7)
 => ? ∊ {4,5,5,6,7,8,11,12,14,15}
[8]
 => 0 => [1] => ([],1)
 => 1
[7,1]
 => 11 => [2] => ([],2)
 => ? ∊ {2,2,2,3,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[6,2]
 => 00 => [2] => ([],2)
 => ? ∊ {2,2,2,3,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[6,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => ? ∊ {2,2,2,3,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[5,3]
 => 11 => [2] => ([],2)
 => ? ∊ {2,2,2,3,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[5,2,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[5,1,1,1]
 => 1111 => [4] => ([],4)
 => ? ∊ {2,2,2,3,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[4,4]
 => 00 => [2] => ([],2)
 => ? ∊ {2,2,2,3,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[4,3,1]
 => 011 => [1,2] => ([(1,2)],3)
 => ? ∊ {2,2,2,3,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[4,2,2]
 => 000 => [3] => ([],3)
 => ? ∊ {2,2,2,3,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[4,2,1,1]
 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {2,2,2,3,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[4,1,1,1,1]
 => 01111 => [1,4] => ([(3,4)],5)
 => ? ∊ {2,2,2,3,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[3,3,2]
 => 110 => [2,1] => ([(0,2),(1,2)],3)
 => 4
[3,3,1,1]
 => 1111 => [4] => ([],4)
 => ? ∊ {2,2,2,3,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[3,2,2,1]
 => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {2,2,2,3,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[3,2,1,1,1]
 => 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[3,1,1,1,1,1]
 => 111111 => [6] => ([],6)
 => ? ∊ {2,2,2,3,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[2,2,2,2]
 => 0000 => [4] => ([],4)
 => ? ∊ {2,2,2,3,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[2,2,2,1,1]
 => 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[2,2,1,1,1,1]
 => 001111 => [2,4] => ([(3,5),(4,5)],6)
 => ? ∊ {2,2,2,3,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[2,1,1,1,1,1,1]
 => 0111111 => [1,6] => ([(5,6)],7)
 => ? ∊ {2,2,2,3,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[1,1,1,1,1,1,1,1]
 => 11111111 => [8] => ([],8)
 => ? ∊ {2,2,2,3,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}

Description

The pebbling number of a connected graph.

Matching statistic: St001861

Mapping

Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001861: Signed permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 28%

Values

[1]
 => [[1]]
 => [1] => [1] => 0 = 1 - 1
[2]
 => [[1,2]]
 => [1,2] => [1,2] => 0 = 1 - 1
[1,1]
 => [[1],[2]]
 => [2,1] => [2,1] => 1 = 2 - 1
[3]
 => [[1,2,3]]
 => [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,1]
 => [[1,3],[2]]
 => [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => [3,4,1,2] => 4 = 5 - 1
[2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => 2 = 3 - 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {2,2,4,4,6,7} - 1
[3,2]
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => [3,4,1,2,5] => ? ∊ {2,2,4,4,6,7} - 1
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [3,2,1,4,5] => ? ∊ {2,2,4,4,6,7} - 1
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [4,2,5,1,3] => ? ∊ {2,2,4,4,6,7} - 1
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [4,3,2,1,5] => ? ∊ {2,2,4,4,6,7} - 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5,4,3,2,1] => ? ∊ {2,2,4,4,6,7} - 1
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,2,2,3,4,4,6,7,8,10,11} - 1
[5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [2,1,3,4,5,6] => ? ∊ {1,2,2,3,4,4,6,7,8,10,11} - 1
[4,2]
 => [[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? ∊ {1,2,2,3,4,4,6,7,8,10,11} - 1
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => [3,2,1,4,5,6] => ? ∊ {1,2,2,3,4,4,6,7,8,10,11} - 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? ∊ {1,2,2,3,4,4,6,7,8,10,11} - 1
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [4,2,5,1,3,6] => [4,2,5,1,3,6] => ? ∊ {1,2,2,3,4,4,6,7,8,10,11} - 1
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => [4,3,2,1,5,6] => ? ∊ {1,2,2,3,4,4,6,7,8,10,11} - 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => [5,6,3,4,1,2] => ? ∊ {1,2,2,3,4,4,6,7,8,10,11} - 1
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [5,3,2,6,1,4] => [5,3,2,6,1,4] => ? ∊ {1,2,2,3,4,4,6,7,8,10,11} - 1
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [5,4,3,2,1,6] => [5,4,3,2,1,6] => ? ∊ {1,2,2,3,4,4,6,7,8,10,11} - 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] => ? ∊ {1,2,2,3,4,4,6,7,8,10,11} - 1
[7]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => [3,4,1,2,5,6,7] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => [4,5,6,1,2,3,7] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [4,2,5,1,3,6,7] => [4,2,5,1,3,6,7] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [5,2,6,7,1,3,4] => [5,2,6,7,1,3,4] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [5,6,3,4,1,2,7] => [5,6,3,4,1,2,7] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => [5,3,2,6,1,4,7] => [5,3,2,6,1,4,7] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 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] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 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] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 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] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 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] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[7,1]
 => [[1,3,4,5,6,7,8],[2]]
 => [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [3,4,1,2,5,6,7,8] => [3,4,1,2,5,6,7,8] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[6,1,1]
 => [[1,4,5,6,7,8],[2],[3]]
 => [3,2,1,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => [4,5,6,1,2,3,7,8] => [4,5,6,1,2,3,7,8] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[5,2,1]
 => [[1,3,6,7,8],[2,5],[4]]
 => [4,2,5,1,3,6,7,8] => [4,2,5,1,3,6,7,8] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[5,1,1,1]
 => [[1,5,6,7,8],[2],[3],[4]]
 => [4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => [5,6,7,8,1,2,3,4] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[4,3,1]
 => [[1,3,4,8],[2,6,7],[5]]
 => [5,2,6,7,1,3,4,8] => [5,2,6,7,1,3,4,8] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => [5,6,3,4,1,2,7,8] => [5,6,3,4,1,2,7,8] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[4,2,1,1]
 => [[1,4,7,8],[2,6],[3],[5]]
 => [5,3,2,6,1,4,7,8] => [5,3,2,6,1,4,7,8] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[4,1,1,1,1]
 => [[1,6,7,8],[2],[3],[4],[5]]
 => [5,4,3,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[3,3,2]
 => [[1,2,5],[3,4,8],[6,7]]
 => [6,7,3,4,8,1,2,5] => [6,7,3,4,8,1,2,5] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => [6,3,2,7,8,1,4,5] => [6,3,2,7,8,1,4,5] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3,8] => [6,4,7,2,5,1,3,8] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[3,2,1,1,1]
 => [[1,5,8],[2,7],[3],[4],[6]]
 => [6,4,3,2,7,1,5,8] => [6,4,3,2,7,1,5,8] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[3,1,1,1,1,1]
 => [[1,7,8],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,7,8] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 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] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1

Description

The number of Bruhat lower covers of a permutation. This is, for a signed permutation

$\pi$ , the number of signed permutations

$\tau$ having a reduced word which is obtained by deleting a letter from a reduced word from

$\pi$ .

Matching statistic: St000141

Mapping

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 17%

Values

[1]
 => [1,0]
 => [(1,2)]
 => [2,1] => 1
[2]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 1
[1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => [3,4,2,1] => 2
[3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 1
[2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => 2
[1,1,1]
 => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => 3
[4]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,7,8,6,5] => ? ∊ {2,4,5}
[2,2]
 => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => 3
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,5,7,4,8,6,3] => ? ∊ {2,4,5}
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? ∊ {2,4,5}
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [2,1,4,3,6,5,9,10,8,7] => ? ∊ {2,2,4,4,6,7}
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,6,7,8,5,4,3] => ? ∊ {2,2,4,4,6,7}
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => [2,1,4,3,7,9,6,10,8,5] => ? ∊ {2,2,4,4,6,7}
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => ? ∊ {2,2,4,4,6,7}
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9)]
 => [2,1,5,7,4,9,6,10,8,3] => ? ∊ {2,2,4,4,6,7}
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [3,5,2,7,4,9,6,10,8,1] => ? ∊ {2,2,4,4,6,7}
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => 1
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
 => [2,1,4,3,6,5,8,7,11,12,10,9] => ? ∊ {2,2,3,4,4,6,7,8,10,11}
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => [2,1,4,3,8,9,10,7,6,5] => ? ∊ {2,2,3,4,4,6,7,8,10,11}
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
 => [2,1,4,3,6,5,9,11,8,12,10,7] => ? ∊ {2,2,3,4,4,6,7,8,10,11}
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => ? ∊ {2,2,3,4,4,6,7,8,10,11}
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,6,7,9,5,4,10,8,3] => ? ∊ {2,2,3,4,4,6,7,8,10,11}
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
 => [2,1,4,3,7,9,6,11,8,12,10,5] => ? ∊ {2,2,3,4,4,6,7,8,10,11}
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => ? ∊ {2,2,3,4,4,6,7,8,10,11}
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [4,5,7,3,2,9,6,10,8,1] => ? ∊ {2,2,3,4,4,6,7,8,10,11}
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => [2,1,5,7,4,9,6,11,8,12,10,3] => ? ∊ {2,2,3,4,4,6,7,8,10,11}
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
 => [3,5,2,7,4,9,6,11,8,12,10,1] => ? ∊ {2,2,3,4,4,6,7,8,10,11}
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
 => [2,1,4,3,6,5,8,7,10,9,12,11,14,13] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15}
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,14),(12,13)]
 => [2,1,4,3,6,5,8,7,10,9,13,14,12,11] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15}
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
 => [2,1,4,3,6,5,10,11,12,9,8,7] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15}
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,14),(10,11),(12,13)]
 => [2,1,4,3,6,5,8,7,11,13,10,14,12,9] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15}
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => [2,1,6,8,9,5,10,7,4,3] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15}
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
 => [2,1,4,3,8,9,11,7,6,12,10,5] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15}
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,14),(8,9),(10,11),(12,13)]
 => [2,1,4,3,6,5,9,11,8,13,10,14,12,7] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15}
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => [4,6,7,3,9,5,2,10,8,1] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15}
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [2,1,7,8,9,10,6,5,4,3] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15}
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
 => [2,1,6,7,9,5,4,11,8,12,10,3] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15}
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,14),(6,7),(8,9),(10,11),(12,13)]
 => [2,1,4,3,7,9,6,11,8,13,10,14,12,5] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15}
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => [5,6,7,9,4,3,2,10,8,1] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15}
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
 => [4,5,7,3,2,9,6,11,8,12,10,1] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15}
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,14),(4,5),(6,7),(8,9),(10,11),(12,13)]
 => [2,1,5,7,4,9,6,11,8,13,10,14,12,3] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15}
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
 => [3,5,2,7,4,9,6,11,8,13,10,14,12,1] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15}
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)]
 => [2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15)]
 => [2,1,4,3,6,5,8,7,10,9,12,11,15,16,14,13] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,14),(10,13),(11,12)]
 => [2,1,4,3,6,5,8,7,12,13,14,11,10,9] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,16),(12,13),(14,15)]
 => [2,1,4,3,6,5,8,7,10,9,13,15,12,16,14,11] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
 => [2,1,4,3,8,10,11,7,12,9,6,5] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,14),(8,11),(9,10),(12,13)]
 => [2,1,4,3,6,5,10,11,13,9,8,14,12,7] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,16),(10,11),(12,13),(14,15)]
 => [2,1,4,3,6,5,8,7,11,13,10,15,12,16,14,9] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8)]
 => [4,6,8,3,9,5,10,7,2,1] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]
 => [2,1,6,8,9,5,11,7,4,12,10,3] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
 => [2,1,4,3,9,10,11,12,8,7,6,5] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,14),(6,9),(7,8),(10,11),(12,13)]
 => [2,1,4,3,8,9,11,7,6,13,10,14,12,5] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,16),(8,9),(10,11),(12,13),(14,15)]
 => [2,1,4,3,6,5,9,11,8,13,10,15,12,16,14,7] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => [4,7,8,3,9,10,6,5,2,1] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [(1,12),(2,7),(3,4),(5,6),(8,9),(10,11)]
 => [4,6,7,3,9,5,2,11,8,12,10,1] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [(1,2),(3,12),(4,9),(5,8),(6,7),(10,11)]
 => [2,1,7,8,9,11,6,5,4,12,10,3] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,2),(3,14),(4,7),(5,6),(8,9),(10,11),(12,13)]
 => [2,1,6,7,9,5,4,11,8,13,10,14,12,3] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22}

Description

The maximum drop size of a permutation. The maximum drop size of a permutation

$\pi$ of

$[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of

$i-\pi(i)$ .

Matching statistic: St001880
(load all 6 compositions to match this statistic)

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 28%

Values

[1]
 => [1,0,1,0]
 => [1,0,1,0]
 => ([(0,1)],2)
 => ? = 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => ? ∊ {1,2}
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => ([(1,2)],3)
 => ? ∊ {1,2}
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => ? ∊ {1,2}
[2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => ? ∊ {1,2}
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ? ∊ {1,2,3,5}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => ? ∊ {1,2,3,5}
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {1,2,3,5}
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => ([(3,4)],5)
 => ? ∊ {1,2,3,5}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ? ∊ {2,2,4,4,6,7}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => ? ∊ {2,2,4,4,6,7}
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => ? ∊ {2,2,4,4,6,7}
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ? ∊ {2,2,4,4,6,7}
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {2,2,4,4,6,7}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(4,5)],6)
 => ? ∊ {2,2,4,4,6,7}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6)],7)
 => ? ∊ {2,2,3,4,6,7,8,10,11}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {2,2,3,4,6,7,8,10,11}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,4,6,7,8,10,11}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => ([(1,2),(1,3),(1,4)],5)
 => ? ∊ {2,2,3,4,6,7,8,10,11}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,4,6,7,8,10,11}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => ([(2,3),(2,4)],5)
 => ? ∊ {2,2,3,4,6,7,8,10,11}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ? ∊ {2,2,3,4,6,7,8,10,11}
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {2,2,3,4,6,7,8,10,11}
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ([(5,6)],7)
 => ? ∊ {2,2,3,4,6,7,8,10,11}
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7)],8)
 => ? ∊ {2,2,2,4,4,5,6,7,8,11,12,14,15}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {2,2,2,4,4,5,6,7,8,11,12,14,15}
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,4,4,5,6,7,8,11,12,14,15}
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ? ∊ {2,2,2,4,4,5,6,7,8,11,12,14,15}
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 5
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ? ∊ {2,2,2,4,4,5,6,7,8,11,12,14,15}
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ? ∊ {2,2,2,4,4,5,6,7,8,11,12,14,15}
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ? ∊ {2,2,2,4,4,5,6,7,8,11,12,14,15}
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? ∊ {2,2,2,4,4,5,6,7,8,11,12,14,15}
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,4,4,5,6,7,8,11,12,14,15}
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ([(1,4),(2,3),(2,4)],5)
 => ? ∊ {2,2,2,4,4,5,6,7,8,11,12,14,15}
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
 => ? ∊ {2,2,2,4,4,5,6,7,8,11,12,14,15}
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? ∊ {2,2,2,4,4,5,6,7,8,11,12,14,15}
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ([(6,7)],8)
 => ? ∊ {2,2,2,4,4,5,6,7,8,11,12,14,15}
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8)],9)
 => ? ∊ {1,2,2,3,3,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? ∊ {1,2,2,3,3,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? ∊ {1,2,2,3,3,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,2,3,3,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? ∊ {1,2,2,3,3,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => 2
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,2,2,3,3,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => ([(1,2),(1,3),(1,4),(1,5)],6)
 => ? ∊ {1,2,2,3,3,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 5
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ? ∊ {1,2,2,3,3,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => ([(0,5),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,2,2,3,3,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ? ∊ {1,2,2,3,3,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ? ∊ {1,2,2,3,3,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {1,2,2,3,3,4,5,7,7,9,9,9,12,12,15,17,19,21,22}
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
 => ? ∊ {1,2,2,3,3,4,5,7,7,9,9,9,12,12,15,17,19,21,22}

Description

The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.

Matching statistic: St000123

Mapping

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000123: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 22%

Values

[1]
 => [1,0]
 => [(1,2)]
 => [2,1] => 0 = 1 - 1
[2]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 0 = 1 - 1
[1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => [3,4,2,1] => 1 = 2 - 1
[3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 0 = 1 - 1
[2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => 1 = 2 - 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => 2 = 3 - 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => 0 = 1 - 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,7,8,6,5] => ? ∊ {2,3,5} - 1
[2,2]
 => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => 3 = 4 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,5,7,4,8,6,3] => ? ∊ {2,3,5} - 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? ∊ {2,3,5} - 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => 0 = 1 - 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [2,1,4,3,6,5,9,10,8,7] => ? ∊ {2,2,4,4,6,7} - 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,6,7,8,5,4,3] => ? ∊ {2,2,4,4,6,7} - 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => [2,1,4,3,7,9,6,10,8,5] => ? ∊ {2,2,4,4,6,7} - 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => ? ∊ {2,2,4,4,6,7} - 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9)]
 => [2,1,5,7,4,9,6,10,8,3] => ? ∊ {2,2,4,4,6,7} - 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [3,5,2,7,4,9,6,10,8,1] => ? ∊ {2,2,4,4,6,7} - 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 1 - 1
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
 => [2,1,4,3,6,5,8,7,11,12,10,9] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => [2,1,4,3,8,9,10,7,6,5] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
 => [2,1,4,3,6,5,9,11,8,12,10,7] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,6,7,9,5,4,10,8,3] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
 => [2,1,4,3,7,9,6,11,8,12,10,5] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [4,5,7,3,2,9,6,10,8,1] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => [2,1,5,7,4,9,6,11,8,12,10,3] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
 => [3,5,2,7,4,9,6,11,8,12,10,1] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
 => [2,1,4,3,6,5,8,7,10,9,12,11,14,13] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,14),(12,13)]
 => [2,1,4,3,6,5,8,7,10,9,13,14,12,11] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
 => [2,1,4,3,6,5,10,11,12,9,8,7] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,14),(10,11),(12,13)]
 => [2,1,4,3,6,5,8,7,11,13,10,14,12,9] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => [2,1,6,8,9,5,10,7,4,3] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
 => [2,1,4,3,8,9,11,7,6,12,10,5] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,14),(8,9),(10,11),(12,13)]
 => [2,1,4,3,6,5,9,11,8,13,10,14,12,7] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => [4,6,7,3,9,5,2,10,8,1] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [2,1,7,8,9,10,6,5,4,3] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
 => [2,1,6,7,9,5,4,11,8,12,10,3] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,14),(6,7),(8,9),(10,11),(12,13)]
 => [2,1,4,3,7,9,6,11,8,13,10,14,12,5] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => [5,6,7,9,4,3,2,10,8,1] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
 => [4,5,7,3,2,9,6,11,8,12,10,1] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,14),(4,5),(6,7),(8,9),(10,11),(12,13)]
 => [2,1,5,7,4,9,6,11,8,13,10,14,12,3] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
 => [3,5,2,7,4,9,6,11,8,13,10,14,12,1] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)]
 => [2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15)]
 => [2,1,4,3,6,5,8,7,10,9,12,11,15,16,14,13] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,14),(10,13),(11,12)]
 => [2,1,4,3,6,5,8,7,12,13,14,11,10,9] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,16),(12,13),(14,15)]
 => [2,1,4,3,6,5,8,7,10,9,13,15,12,16,14,11] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
 => [2,1,4,3,8,10,11,7,12,9,6,5] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,14),(8,11),(9,10),(12,13)]
 => [2,1,4,3,6,5,10,11,13,9,8,14,12,7] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,16),(10,11),(12,13),(14,15)]
 => [2,1,4,3,6,5,8,7,11,13,10,15,12,16,14,9] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8)]
 => [4,6,8,3,9,5,10,7,2,1] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]
 => [2,1,6,8,9,5,11,7,4,12,10,3] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
 => [2,1,4,3,9,10,11,12,8,7,6,5] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,14),(6,9),(7,8),(10,11),(12,13)]
 => [2,1,4,3,8,9,11,7,6,13,10,14,12,5] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,16),(8,9),(10,11),(12,13),(14,15)]
 => [2,1,4,3,6,5,9,11,8,13,10,15,12,16,14,7] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => [4,7,8,3,9,10,6,5,2,1] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [(1,12),(2,7),(3,4),(5,6),(8,9),(10,11)]
 => [4,6,7,3,9,5,2,11,8,12,10,1] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [(1,2),(3,12),(4,9),(5,8),(6,7),(10,11)]
 => [2,1,7,8,9,11,6,5,4,12,10,3] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,2),(3,14),(4,7),(5,6),(8,9),(10,11),(12,13)]
 => [2,1,6,7,9,5,4,11,8,13,10,14,12,3] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1

Description

The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. The Simion-Schmidt map takes a permutation and turns each occcurrence of [3,2,1] into an occurrence of [3,1,2], thus reducing the number of inversions of the permutation. This statistic records the difference in length of the permutation and its image. Apparently, this statistic can be described as the number of occurrences of the mesh pattern ([3,2,1], {(0,3),(0,2)}). Equivalent mesh patterns are ([3,2,1], {(0,2),(1,2)}), ([3,2,1], {(0,3),(1,3)}) and ([3,2,1], {(1,2),(1,3)}).

Matching statistic: St000223

Mapping

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 22%

Values

[1]
 => [1,0]
 => [(1,2)]
 => [2,1] => 0 = 1 - 1
[2]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 0 = 1 - 1
[1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => [3,4,2,1] => 1 = 2 - 1
[3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 0 = 1 - 1
[2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => 1 = 2 - 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => 2 = 3 - 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => 0 = 1 - 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,7,8,6,5] => ? ∊ {2,3,5} - 1
[2,2]
 => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => 3 = 4 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,5,7,4,8,6,3] => ? ∊ {2,3,5} - 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? ∊ {2,3,5} - 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => 0 = 1 - 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [2,1,4,3,6,5,9,10,8,7] => ? ∊ {2,2,4,4,6,7} - 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,6,7,8,5,4,3] => ? ∊ {2,2,4,4,6,7} - 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => [2,1,4,3,7,9,6,10,8,5] => ? ∊ {2,2,4,4,6,7} - 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => ? ∊ {2,2,4,4,6,7} - 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9)]
 => [2,1,5,7,4,9,6,10,8,3] => ? ∊ {2,2,4,4,6,7} - 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [3,5,2,7,4,9,6,10,8,1] => ? ∊ {2,2,4,4,6,7} - 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 1 - 1
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
 => [2,1,4,3,6,5,8,7,11,12,10,9] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => [2,1,4,3,8,9,10,7,6,5] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
 => [2,1,4,3,6,5,9,11,8,12,10,7] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,6,7,9,5,4,10,8,3] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
 => [2,1,4,3,7,9,6,11,8,12,10,5] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [4,5,7,3,2,9,6,10,8,1] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => [2,1,5,7,4,9,6,11,8,12,10,3] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
 => [3,5,2,7,4,9,6,11,8,12,10,1] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
 => [2,1,4,3,6,5,8,7,10,9,12,11,14,13] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,14),(12,13)]
 => [2,1,4,3,6,5,8,7,10,9,13,14,12,11] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
 => [2,1,4,3,6,5,10,11,12,9,8,7] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,14),(10,11),(12,13)]
 => [2,1,4,3,6,5,8,7,11,13,10,14,12,9] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => [2,1,6,8,9,5,10,7,4,3] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
 => [2,1,4,3,8,9,11,7,6,12,10,5] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,14),(8,9),(10,11),(12,13)]
 => [2,1,4,3,6,5,9,11,8,13,10,14,12,7] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => [4,6,7,3,9,5,2,10,8,1] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [2,1,7,8,9,10,6,5,4,3] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
 => [2,1,6,7,9,5,4,11,8,12,10,3] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,14),(6,7),(8,9),(10,11),(12,13)]
 => [2,1,4,3,7,9,6,11,8,13,10,14,12,5] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => [5,6,7,9,4,3,2,10,8,1] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
 => [4,5,7,3,2,9,6,11,8,12,10,1] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,14),(4,5),(6,7),(8,9),(10,11),(12,13)]
 => [2,1,5,7,4,9,6,11,8,13,10,14,12,3] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
 => [3,5,2,7,4,9,6,11,8,13,10,14,12,1] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)]
 => [2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15)]
 => [2,1,4,3,6,5,8,7,10,9,12,11,15,16,14,13] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,14),(10,13),(11,12)]
 => [2,1,4,3,6,5,8,7,12,13,14,11,10,9] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,16),(12,13),(14,15)]
 => [2,1,4,3,6,5,8,7,10,9,13,15,12,16,14,11] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
 => [2,1,4,3,8,10,11,7,12,9,6,5] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,14),(8,11),(9,10),(12,13)]
 => [2,1,4,3,6,5,10,11,13,9,8,14,12,7] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,16),(10,11),(12,13),(14,15)]
 => [2,1,4,3,6,5,8,7,11,13,10,15,12,16,14,9] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8)]
 => [4,6,8,3,9,5,10,7,2,1] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]
 => [2,1,6,8,9,5,11,7,4,12,10,3] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
 => [2,1,4,3,9,10,11,12,8,7,6,5] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,14),(6,9),(7,8),(10,11),(12,13)]
 => [2,1,4,3,8,9,11,7,6,13,10,14,12,5] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,16),(8,9),(10,11),(12,13),(14,15)]
 => [2,1,4,3,6,5,9,11,8,13,10,15,12,16,14,7] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => [4,7,8,3,9,10,6,5,2,1] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [(1,12),(2,7),(3,4),(5,6),(8,9),(10,11)]
 => [4,6,7,3,9,5,2,11,8,12,10,1] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [(1,2),(3,12),(4,9),(5,8),(6,7),(10,11)]
 => [2,1,7,8,9,11,6,5,4,12,10,3] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,2),(3,14),(4,7),(5,6),(8,9),(10,11),(12,13)]
 => [2,1,6,7,9,5,4,11,8,13,10,14,12,3] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1

Description

The number of nestings in the permutation.

Matching statistic: St000371

Mapping

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 17%

Values

[1]
 => [1,0]
 => [(1,2)]
 => [2,1] => 0 = 1 - 1
[2]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 0 = 1 - 1
[1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => [3,4,2,1] => 1 = 2 - 1
[3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 0 = 1 - 1
[2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => 1 = 2 - 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => 2 = 3 - 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => 0 = 1 - 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,7,8,6,5] => ? ∊ {2,4,5} - 1
[2,2]
 => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => 2 = 3 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,5,7,4,8,6,3] => ? ∊ {2,4,5} - 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? ∊ {2,4,5} - 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => 0 = 1 - 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [2,1,4,3,6,5,9,10,8,7] => ? ∊ {2,2,4,4,6,7} - 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,6,7,8,5,4,3] => ? ∊ {2,2,4,4,6,7} - 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => [2,1,4,3,7,9,6,10,8,5] => ? ∊ {2,2,4,4,6,7} - 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => ? ∊ {2,2,4,4,6,7} - 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9)]
 => [2,1,5,7,4,9,6,10,8,3] => ? ∊ {2,2,4,4,6,7} - 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [3,5,2,7,4,9,6,10,8,1] => ? ∊ {2,2,4,4,6,7} - 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 1 - 1
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
 => [2,1,4,3,6,5,8,7,11,12,10,9] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => [2,1,4,3,8,9,10,7,6,5] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
 => [2,1,4,3,6,5,9,11,8,12,10,7] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,6,7,9,5,4,10,8,3] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
 => [2,1,4,3,7,9,6,11,8,12,10,5] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [4,5,7,3,2,9,6,10,8,1] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => [2,1,5,7,4,9,6,11,8,12,10,3] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
 => [3,5,2,7,4,9,6,11,8,12,10,1] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
 => [2,1,4,3,6,5,8,7,10,9,12,11,14,13] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,14),(12,13)]
 => [2,1,4,3,6,5,8,7,10,9,13,14,12,11] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
 => [2,1,4,3,6,5,10,11,12,9,8,7] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,14),(10,11),(12,13)]
 => [2,1,4,3,6,5,8,7,11,13,10,14,12,9] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => [2,1,6,8,9,5,10,7,4,3] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
 => [2,1,4,3,8,9,11,7,6,12,10,5] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,14),(8,9),(10,11),(12,13)]
 => [2,1,4,3,6,5,9,11,8,13,10,14,12,7] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => [4,6,7,3,9,5,2,10,8,1] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [2,1,7,8,9,10,6,5,4,3] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
 => [2,1,6,7,9,5,4,11,8,12,10,3] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,14),(6,7),(8,9),(10,11),(12,13)]
 => [2,1,4,3,7,9,6,11,8,13,10,14,12,5] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => [5,6,7,9,4,3,2,10,8,1] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
 => [4,5,7,3,2,9,6,11,8,12,10,1] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,14),(4,5),(6,7),(8,9),(10,11),(12,13)]
 => [2,1,5,7,4,9,6,11,8,13,10,14,12,3] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
 => [3,5,2,7,4,9,6,11,8,13,10,14,12,1] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)]
 => [2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15)]
 => [2,1,4,3,6,5,8,7,10,9,12,11,15,16,14,13] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,14),(10,13),(11,12)]
 => [2,1,4,3,6,5,8,7,12,13,14,11,10,9] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,16),(12,13),(14,15)]
 => [2,1,4,3,6,5,8,7,10,9,13,15,12,16,14,11] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
 => [2,1,4,3,8,10,11,7,12,9,6,5] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,14),(8,11),(9,10),(12,13)]
 => [2,1,4,3,6,5,10,11,13,9,8,14,12,7] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,16),(10,11),(12,13),(14,15)]
 => [2,1,4,3,6,5,8,7,11,13,10,15,12,16,14,9] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8)]
 => [4,6,8,3,9,5,10,7,2,1] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]
 => [2,1,6,8,9,5,11,7,4,12,10,3] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
 => [2,1,4,3,9,10,11,12,8,7,6,5] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,14),(6,9),(7,8),(10,11),(12,13)]
 => [2,1,4,3,8,9,11,7,6,13,10,14,12,5] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,16),(8,9),(10,11),(12,13),(14,15)]
 => [2,1,4,3,6,5,9,11,8,13,10,15,12,16,14,7] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => [4,7,8,3,9,10,6,5,2,1] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [(1,12),(2,7),(3,4),(5,6),(8,9),(10,11)]
 => [4,6,7,3,9,5,2,11,8,12,10,1] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [(1,2),(3,12),(4,9),(5,8),(6,7),(10,11)]
 => [2,1,7,8,9,11,6,5,4,12,10,3] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,2),(3,14),(4,7),(5,6),(8,9),(10,11),(12,13)]
 => [2,1,6,7,9,5,4,11,8,13,10,14,12,3] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1

Description

The number of mid points of decreasing subsequences of length 3 in a permutation. For a permutation

$\pi$ of

$\{1,\ldots,n\}$ , this is the number of indices

$j$ such that there exist indices

$i,k$ with

$i < j < k$ and

$\pi(i) > \pi(j) > \pi(k)$ . In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima. This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence. See also [[St000119]].

Matching statistic: St001115

Mapping

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St001115: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 17%

Values

[1]
 => [1,0]
 => [(1,2)]
 => [2,1] => 0 = 1 - 1
[2]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 0 = 1 - 1
[1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => [3,4,2,1] => 1 = 2 - 1
[3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 0 = 1 - 1
[2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => 1 = 2 - 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => 2 = 3 - 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => 0 = 1 - 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,7,8,6,5] => ? ∊ {3,4,5} - 1
[2,2]
 => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => 1 = 2 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,5,7,4,8,6,3] => ? ∊ {3,4,5} - 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? ∊ {3,4,5} - 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => 0 = 1 - 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [2,1,4,3,6,5,9,10,8,7] => ? ∊ {2,2,4,4,6,7} - 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,6,7,8,5,4,3] => ? ∊ {2,2,4,4,6,7} - 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => [2,1,4,3,7,9,6,10,8,5] => ? ∊ {2,2,4,4,6,7} - 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => ? ∊ {2,2,4,4,6,7} - 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9)]
 => [2,1,5,7,4,9,6,10,8,3] => ? ∊ {2,2,4,4,6,7} - 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [3,5,2,7,4,9,6,10,8,1] => ? ∊ {2,2,4,4,6,7} - 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 1 - 1
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
 => [2,1,4,3,6,5,8,7,11,12,10,9] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => [2,1,4,3,8,9,10,7,6,5] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
 => [2,1,4,3,6,5,9,11,8,12,10,7] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,6,7,9,5,4,10,8,3] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
 => [2,1,4,3,7,9,6,11,8,12,10,5] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [4,5,7,3,2,9,6,10,8,1] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => [2,1,5,7,4,9,6,11,8,12,10,3] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
 => [3,5,2,7,4,9,6,11,8,12,10,1] => ? ∊ {2,2,3,4,4,6,7,8,10,11} - 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
 => [2,1,4,3,6,5,8,7,10,9,12,11,14,13] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,14),(12,13)]
 => [2,1,4,3,6,5,8,7,10,9,13,14,12,11] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
 => [2,1,4,3,6,5,10,11,12,9,8,7] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,14),(10,11),(12,13)]
 => [2,1,4,3,6,5,8,7,11,13,10,14,12,9] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => [2,1,6,8,9,5,10,7,4,3] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
 => [2,1,4,3,8,9,11,7,6,12,10,5] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,14),(8,9),(10,11),(12,13)]
 => [2,1,4,3,6,5,9,11,8,13,10,14,12,7] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => [4,6,7,3,9,5,2,10,8,1] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [2,1,7,8,9,10,6,5,4,3] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
 => [2,1,6,7,9,5,4,11,8,12,10,3] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,14),(6,7),(8,9),(10,11),(12,13)]
 => [2,1,4,3,7,9,6,11,8,13,10,14,12,5] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => [5,6,7,9,4,3,2,10,8,1] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
 => [4,5,7,3,2,9,6,11,8,12,10,1] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,14),(4,5),(6,7),(8,9),(10,11),(12,13)]
 => [2,1,5,7,4,9,6,11,8,13,10,14,12,3] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
 => [3,5,2,7,4,9,6,11,8,13,10,14,12,1] => ? ∊ {1,2,2,2,4,4,5,5,6,7,8,11,12,14,15} - 1
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)]
 => [2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15)]
 => [2,1,4,3,6,5,8,7,10,9,12,11,15,16,14,13] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,14),(10,13),(11,12)]
 => [2,1,4,3,6,5,8,7,12,13,14,11,10,9] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,16),(12,13),(14,15)]
 => [2,1,4,3,6,5,8,7,10,9,13,15,12,16,14,11] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
 => [2,1,4,3,8,10,11,7,12,9,6,5] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,14),(8,11),(9,10),(12,13)]
 => [2,1,4,3,6,5,10,11,13,9,8,14,12,7] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,16),(10,11),(12,13),(14,15)]
 => [2,1,4,3,6,5,8,7,11,13,10,15,12,16,14,9] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8)]
 => [4,6,8,3,9,5,10,7,2,1] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]
 => [2,1,6,8,9,5,11,7,4,12,10,3] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
 => [2,1,4,3,9,10,11,12,8,7,6,5] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,14),(6,9),(7,8),(10,11),(12,13)]
 => [2,1,4,3,8,9,11,7,6,13,10,14,12,5] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,16),(8,9),(10,11),(12,13),(14,15)]
 => [2,1,4,3,6,5,9,11,8,13,10,15,12,16,14,7] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => [4,7,8,3,9,10,6,5,2,1] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [(1,12),(2,7),(3,4),(5,6),(8,9),(10,11)]
 => [4,6,7,3,9,5,2,11,8,12,10,1] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [(1,2),(3,12),(4,9),(5,8),(6,7),(10,11)]
 => [2,1,7,8,9,11,6,5,4,12,10,3] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,2),(3,14),(4,7),(5,6),(8,9),(10,11),(12,13)]
 => [2,1,6,7,9,5,4,11,8,13,10,14,12,3] => ? ∊ {1,2,2,2,3,3,4,4,5,5,7,7,9,9,9,12,12,15,17,19,21,22} - 1

Description

The number of even descents of a permutation.

The following 55 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

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

$eA$ . St001394The genus of a permutation. St000662The staircase size of the code of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000056The decomposition (or block) number of a permutation. St000091The descent variation of a composition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000314The number of left-to-right-maxima of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000991The number of right-to-left minima of a permutation. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001461The number of topologically connected components of the chord diagram of a permutation. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000039The number of crossings of a permutation. St000043The number of crossings plus two-nestings of a perfect matching. St000173The segment statistic of a semistandard tableau. St000234The number of global ascents of a permutation. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000317The cycle descent number of a permutation. St000360The number of occurrences of the pattern 32-1. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000491The number of inversions of a set partition. St000565The major index of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000650The number of 3-rises of a permutation. St001403The number of vertical separators in a permutation. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001727The number of invisible inversions of a permutation. St001781The interlacing number of a set partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001843The Z-index of a set partition. St000352The Elizalde-Pak rank of a permutation. St000356The number of occurrences of the pattern 13-2. St000456The monochromatic index of a connected graph. St000834The number of right outer peaks 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. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000007The number of saliances of the permutation. St000054The first entry of the permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation.