- FindStat

Your data matches 301 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001879
(load all 156 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 2
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 6
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => 7
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => 7
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 7
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => 12
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
 => 8
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => 8
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 8
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => ([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
 => 13
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
 => 8
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
 => 11
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 8
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
 => 13
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => 13
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
 => ([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
 => 12
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
 => 14
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
 => 20

Description

The 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.

Matching statistic: St000076
(load all 2 compositions to match this statistic)

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000076: Alternating sign matrices ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 58%

Values

[1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => 2
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => 3
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 4
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => 6
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
 => 7
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 7
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
 => 7
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]]
 => 12
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => ? = 6
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,0,1,0]]
 => ? = 8
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => ? = 8
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,0,1,0]]
 => ? = 8
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,-1,1],[0,0,0,0,0,1,0]]
 => ? = 13
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => ? = 8
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,-1,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,0,1,0]]
 => ? = 11
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => ? = 8
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,-1,1],[0,0,0,0,0,1,0]]
 => ? = 13
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => ? = 13
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,-1,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,0,1,0]]
 => ? = 12
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,-1,1],[0,0,0,0,0,1,0]]
 => ? = 14
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
 => ? = 20

Description

The rank of the alternating sign matrix in the alternating sign matrix poset. This rank is the sum of the entries of the monotone triangle minus

$\binom{n+2}{3}$ , which is the smallest sum of the entries in the set of all monotone triangles with bottom row

$1\dots n$ . Alternatively,

$rank(A)=\frac{1}{2} \sum_{i,j=1}^n (i-j)^2 a_{ij}$ , see [3, thm.5.1].

Matching statistic: St000104
(load all 57 compositions to match this statistic)

Mapping

Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000104: Posets ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 50%

Values

[1,0,1,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 5 = 4 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 7 = 6 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6 = 5 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 8 = 7 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 8 = 7 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 8 = 7 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? = 12 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 7 = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ? = 8 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ? = 8 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ? = 8 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ? = 13 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => ? = 8 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ? = 11 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ? = 8 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 13 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 13 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 12 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 14 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? = 20 + 1

Description

The number of facets in the order polytope of this poset.

Matching statistic: St000151
(load all 57 compositions to match this statistic)

Mapping

Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000151: Posets ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 50%

Values

[1,0,1,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 5 = 4 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 7 = 6 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6 = 5 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 8 = 7 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 8 = 7 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 8 = 7 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? = 12 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 7 = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ? = 8 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ? = 8 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ? = 8 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ? = 13 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => ? = 8 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ? = 11 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ? = 8 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 13 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 13 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 12 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 14 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? = 20 + 1

Description

The number of facets in the chain polytope of the poset.

Matching statistic: St001411
(load all 3 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St001411: Permutations ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 58%

Values

[1,0,1,0]
 => [3,1,2] => [3,1,2] => [3,2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [4,1,2,3] => [4,2,3,1] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [5,1,2,3,4] => [5,2,3,4,1] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [5,1,3,2,4] => [5,2,4,3,1] => 5 = 6 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [6,1,2,3,4,5] => [6,2,3,4,5,1] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [6,1,2,4,3,5] => [6,2,3,5,4,1] => 6 = 7 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [6,1,3,2,4,5] => [6,2,4,3,5,1] => 6 = 7 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [4,1,6,3,2,5] => [5,2,6,4,1,3] => 6 = 7 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [6,1,3,4,2,5] => [6,2,5,4,3,1] => 11 = 12 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => [7,2,3,4,5,6,1] => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,1,2,3,7,4,6] => [7,1,2,3,5,4,6] => [7,2,3,4,6,5,1] => ? = 8 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => [7,1,2,4,3,5,6] => [7,2,3,5,4,6,1] => ? = 8 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [7,1,2,5,3,4,6] => [5,1,2,7,4,3,6] => [6,2,3,7,5,1,4] => ? = 8 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => [7,1,2,4,5,3,6] => [7,2,3,6,5,4,1] => ? = 13 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => [7,1,3,2,4,5,6] => [7,2,4,3,5,6,1] => ? = 8 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => [7,1,3,2,5,4,6] => [7,2,4,3,6,5,1] => ? = 11 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,1,4,2,3,5,6] => [4,1,7,3,2,5,6] => [5,2,7,4,1,6,3] => ? = 8 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,1,4,2,7,3,6] => [4,1,7,3,5,2,6] => [6,2,7,4,5,1,3] => ? = 13 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => [7,1,3,4,2,5,6] => [7,2,5,4,3,6,1] => ? = 13 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [3,1,7,5,2,4,6] => [5,1,3,7,4,2,6] => [6,2,5,7,3,1,4] => ? = 12 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [7,1,4,5,2,3,6] => [4,1,7,5,3,2,6] => [6,2,7,5,4,1,3] => ? = 14 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => [7,1,3,4,5,2,6] => [7,2,6,4,5,3,1] => ? = 20 - 1

Description

The number of patterns 321 or 3412 in a permutation. A permutation is '''boolean''' if its principal order ideal in the (strong) Bruhat order is boolean. It is shown in [1, Theorem 5.3] that a permutation is boolean if and only if it avoids the two patterns 321 and 3412.

Matching statistic: St001614
(load all 5 compositions to match this statistic)

Mapping

Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001614: Skew partitions ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 50%

Values

[1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => [[2],[]]
 => 2
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [[3],[]]
 => 3
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => 4
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => 6
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[2,2,2,2],[]]
 => ? = 7
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => 7
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => 7
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[4,4],[]]
 => ? = 12
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [[6],[]]
 => 6
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [[2,2,2,2,2],[]]
 => ? = 8
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [[3,2,2,2],[]]
 => ? = 8
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [[3,3,3,3],[1,1,1]]
 => ? = 8
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [[5,5],[]]
 => ? = 13
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [[4,2,2],[]]
 => ? = 8
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [[4,4,4],[2]]
 => ? = 11
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [[4,3,3],[1,1]]
 => ? = 8
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [[4,4,2],[]]
 => ? = 13
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [[5,4],[]]
 => ? = 13
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [[4,4,4],[2,1]]
 => ? = 12
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [[5,5],[1]]
 => ? = 14
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [[3,3,3,3],[]]
 => ? = 20

Description

The cyclic permutation representation number of a skew partition. This is the size of the largest cyclic group

$C$ such that the fake degree is the character of a permutation representation of

$C$ . See [[St001527]] for the restriction of this statistic to integer partitions.

Matching statistic: St000034
(load all 7 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000034: Permutations ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 50%

Values

[1,0,1,0]
 => [3,1,2] => [3,1,2] => [3,2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [4,1,2,3] => [4,2,3,1] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [5,1,2,3,4] => [5,2,3,4,1] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [5,3,1,2,4] => [5,4,3,2,1] => 5 = 6 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [6,1,2,3,4,5] => [6,2,3,4,5,1] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [6,1,4,2,3,5] => [6,2,5,4,3,1] => 6 = 7 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [6,3,1,2,4,5] => [6,4,3,2,5,1] => 6 = 7 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [4,6,1,2,3,5] => [5,6,3,4,1,2] => 6 = 7 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [6,3,1,4,2,5] => [6,5,3,4,2,1] => ? = 12 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => [7,2,3,4,5,6,1] => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,1,2,3,7,4,6] => [7,1,2,5,3,4,6] => [7,2,3,6,5,4,1] => ? = 8 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => [7,1,4,2,3,5,6] => [7,2,5,4,3,6,1] => ? = 8 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [7,1,2,5,3,4,6] => [5,1,7,2,3,4,6] => [6,2,7,4,5,1,3] => ? = 8 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => [7,1,4,2,5,3,6] => [7,2,6,4,5,3,1] => ? = 13 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => [7,3,1,2,4,5,6] => [7,4,3,2,5,6,1] => ? = 8 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => [7,3,1,5,2,4,6] => [7,5,3,6,2,4,1] => ? = 11 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,1,4,2,3,5,6] => [4,7,1,2,3,5,6] => [5,7,3,4,1,6,2] => ? = 8 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,1,4,2,7,3,6] => [7,4,5,1,2,3,6] => [7,6,5,4,3,2,1] => ? = 13 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => [7,3,1,4,2,5,6] => [7,5,3,4,2,6,1] => ? = 13 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [3,1,7,5,2,4,6] => [5,3,1,2,7,4,6] => [6,4,3,2,7,1,5] => ? = 12 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [7,1,4,5,2,3,6] => [5,7,1,2,4,3,6] => [6,7,3,4,5,1,2] => ? = 14 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => [7,3,1,4,5,2,6] => [7,6,3,4,5,2,1] => ? = 20 - 1

Description

The maximum defect over any reduced expression for a permutation and any subexpression.

Matching statistic: St001643

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001643: Dyck paths ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 58%

Values

[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 4 = 2 + 2
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 5 = 3 + 2
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 6 = 4 + 2
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 8 = 6 + 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => 7 = 5 + 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => 9 = 7 + 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => ? = 7 + 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => ? = 7 + 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => ? = 12 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 8 = 6 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => 10 = 8 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? = 8 + 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => ? = 8 + 2
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => ? = 13 + 2
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => ? = 8 + 2
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => ? = 11 + 2
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => ? = 8 + 2
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => ? = 13 + 2
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => ? = 13 + 2
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => ? = 12 + 2
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
 => ? = 14 + 2
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => ? = 20 + 2

Description

The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St000067
(load all 7 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000067: Alternating sign matrices ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 50%

Values

[1,0,1,0]
 => [3,1,2] => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
 => 2
[1,0,1,0,1,0]
 => [4,1,2,3] => [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => 3
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [5,1,2,3,4] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
 => 4
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [5,3,1,2,4] => [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0]]
 => 6
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [6,1,2,3,4,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [6,1,4,2,3,5] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
 => ? = 7
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [6,3,1,2,4,5] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
 => 7
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [4,6,1,2,3,5] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
 => 7
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [6,3,1,4,2,5] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
 => ? = 12
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => [[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
 => ? = 6
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,1,2,3,7,4,6] => [7,1,2,5,3,4,6] => [[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
 => ? = 8
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => [7,1,4,2,3,5,6] => [[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
 => ? = 8
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [7,1,2,5,3,4,6] => [5,1,7,2,3,4,6] => [[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0]]
 => ? = 8
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => [7,1,4,2,5,3,6] => [[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
 => ? = 13
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => [7,3,1,2,4,5,6] => [[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
 => ? = 8
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => [7,3,1,5,2,4,6] => [[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
 => ? = 11
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,1,4,2,3,5,6] => [4,7,1,2,3,5,6] => [[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0]]
 => ? = 8
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,1,4,2,7,3,6] => [7,4,5,1,2,3,6] => [[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
 => ? = 13
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => [7,3,1,4,2,5,6] => [[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
 => ? = 13
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [3,1,7,5,2,4,6] => [5,3,1,2,7,4,6] => [[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0]]
 => ? = 12
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [7,1,4,5,2,3,6] => [5,7,1,2,4,3,6] => [[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0]]
 => ? = 14
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => [7,3,1,4,5,2,6] => [[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
 => ? = 20

Description

The inversion number of the alternating sign matrix. If we denote the entries of the alternating sign matrix as

$a_{i,j}$ , the inversion number is defined as

$\sum_{i > k}\sum_{j < \ell} a_{i,j}a_{k,\ell}.$ When restricted to permutation matrices, this gives the usual inversion number of the permutation.

Matching statistic: St000332
(load all 7 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000332: Alternating sign matrices ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 50%

Values

[1,0,1,0]
 => [3,1,2] => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
 => 2
[1,0,1,0,1,0]
 => [4,1,2,3] => [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => 3
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [5,1,2,3,4] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
 => 4
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [5,3,1,2,4] => [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0]]
 => 6
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [6,1,2,3,4,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [6,1,4,2,3,5] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
 => ? = 7
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [6,3,1,2,4,5] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
 => 7
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [4,6,1,2,3,5] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
 => 7
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [6,3,1,4,2,5] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
 => ? = 12
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => [[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
 => ? = 6
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,1,2,3,7,4,6] => [7,1,2,5,3,4,6] => [[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
 => ? = 8
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => [7,1,4,2,3,5,6] => [[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
 => ? = 8
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [7,1,2,5,3,4,6] => [5,1,7,2,3,4,6] => [[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0]]
 => ? = 8
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => [7,1,4,2,5,3,6] => [[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
 => ? = 13
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => [7,3,1,2,4,5,6] => [[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
 => ? = 8
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => [7,3,1,5,2,4,6] => [[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
 => ? = 11
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,1,4,2,3,5,6] => [4,7,1,2,3,5,6] => [[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0]]
 => ? = 8
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,1,4,2,7,3,6] => [7,4,5,1,2,3,6] => [[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
 => ? = 13
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => [7,3,1,4,2,5,6] => [[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
 => ? = 13
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [3,1,7,5,2,4,6] => [5,3,1,2,7,4,6] => [[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0]]
 => ? = 12
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [7,1,4,5,2,3,6] => [5,7,1,2,4,3,6] => [[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0]]
 => ? = 14
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => [7,3,1,4,5,2,6] => [[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
 => ? = 20

Description

The positive inversions of an alternating sign matrix. This is defined as

$\sum_{i > k,j < l} A_{ij}A_{kl} - \text{the number of negative ones in the matrix}.$ After counter-clockwise rotation, this is also the number of osculations in the corresponding fan of Dyck paths.

The following 291 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001428The number of B-inversions of a signed permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001343The dimension of the reduced incidence algebra of a poset. St000002The number of occurrences of the pattern 123 in a permutation. St000018The number of inversions of a permutation. St000189The number of elements in the poset. St000656The number of cuts of a poset. St001717The largest size of an interval in a poset. St001894The depth of a signed permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000019The cardinality of the support of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000035The number of left outer peaks of a permutation. St000153The number of adjacent cycles of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000237The number of small exceedances. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001096The size of the overlap set of a permutation. St001330The hat guessing number of a graph. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000080The rank of the poset. St000245The number of ascents of a permutation. St000327The number of cover relations in a poset. St000356The number of occurrences of the pattern 13-2. St000454The largest eigenvalue of a graph if it is integral. St000672The number of minimal elements in Bruhat order not less than the permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001631The number of simple modules

$S$ with

$dim Ext^1(S,A)=1$ in the incidence algebra

$A$ of the poset. St000028The number of stack-sorts needed to sort a permutation. St000141The maximum drop size of a permutation. St000201The number of leaf nodes in a binary tree. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000528The height of a poset. St000680The Grundy value for Hackendot on posets. St000692Babson and Steingrímsson's statistic of a permutation. St000717The number of ordinal summands of a poset. St000789The number of crossing-similar perfect matchings of a perfect matching. St000906The length of the shortest maximal chain in a poset. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St000961The shifted major index of a permutation. St001052The length of the exterior of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001644The dimension of a graph. St001645The pebbling number of a connected graph. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St000050The depth or height of a binary tree. St000203The number of external nodes of a binary tree. St000451The length of the longest pattern of the form k 1 2. St000463The number of admissible inversions of a permutation. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000883The number of longest increasing subsequences of a permutation. St000909The number of maximal chains of maximal size in a poset. St001083The number of boxed occurrences of 132 in a permutation. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001664The number of non-isomorphic subposets of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St001875The number of simple modules with projective dimension at most 1. St000070The number of antichains in a poset. St001861The number of Bruhat lower covers of a permutation. St001074The number of inversions of the cyclic embedding of a permutation. St000884The number of isolated descents of a permutation. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001485The modular major index of a binary word. St001684The reduced word complexity of a permutation. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000806The semiperimeter of the associated bargraph. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St000004The major index of a permutation. St000007The number of saliances of the permutation. St000029The depth of a permutation. St000074The number of special entries. St000078The number of alternating sign matrices whose left key is the permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000222The number of alignments in the permutation. St000246The number of non-inversions of a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000295The length of the border of a binary word. St000305The inverse major index of a permutation. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000393The number of strictly increasing runs in a binary word. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000441The number of successions of a permutation. St000446The disorder of a permutation. St000462The major index minus the number of excedences of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000519The largest length of a factor maximising the subword complexity. St000626The minimal period of a binary word. St000665The number of rafts of a permutation. St000682The Grundy value of Welter's game on a binary word. St000691The number of changes of a binary word. St000693The modular (standard) major index of a standard tableau. St000719The number of alignments in a perfect matching. St000738The first entry in the last row of a standard tableau. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000794The mak of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000922The minimal number such that all substrings of this length are unique. St000957The number of Bruhat lower covers of a permutation. St000963The 2-shifted major index of a permutation. St000984The number of boxes below precisely one peak. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001030Half the number of non-boundary horizontal edges in the fully packed loop corresponding to the alternating sign matrix. St001220The width of a permutation. St001285The number of primes in the column sums of the two line notation of a permutation. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001313The number of Dyck paths above the lattice path given by a binary word. St001406The number of nonzero entries in a Gelfand Tsetlin pattern. St001437The flex of a binary word. St001697The shifted natural comajor index of a standard Young tableau. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St001959The product of the heights of the peaks of a Dyck path. St000011The number of touch points (or returns) of a Dyck path. St000030The sum of the descent differences of a permutations. St000154The sum of the descent bottoms of a permutation. St000197The number of entries equal to positive one in the alternating sign matrix. St000223The number of nestings in the permutation. St000224The sorting index of a permutation. St000288The number of ones in a binary word. St000458The number of permutations obtained by switching adjacencies or successions. St000539The number of odd inversions of a permutation. St000546The number of global descents of a permutation. St000647The number of big descents of a permutation. St000795The mad of a permutation. St000796The stat' of a permutation. St000798The makl of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000833The comajor index of a permutation. St000841The largest opener of a perfect matching. St000890The number of nonzero entries in an alternating sign matrix. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000910The number of maximal chains of minimal length in a poset. St000921The number of internal inversions of a binary word. St000951The dimension of

$Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000956The maximal displacement of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001115The number of even descents of a permutation. St001267The length of the Lyndon factorization of the binary word. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001671Haglund's hag of a permutation. St001726The number of visible inversions of a permutation. St001817The number of flag weak exceedances of a signed permutation. St001821The sorting index of a signed permutation. St001856The number of edges in the reduced word graph of a permutation. St001892The flag excedance statistic of a signed permutation. St001911A descent variant minus the number of inversions. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St000005The bounce statistic of a Dyck path. St000120The number of left tunnels of a Dyck path. St000136The dinv of a parking function. St000156The Denert index of a permutation. St000176The total number of tiles in the Gelfand-Tsetlin pattern. St000178Number of free entries. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000209Maximum difference of elements in cycles. St000226The convexity of a permutation. St000238The number of indices that are not small weak excedances. St000299The number of nonisomorphic vertex-induced subtrees. St000376The bounce deficit of a Dyck path. St000472The sum of the ascent bottoms of a permutation. St000625The sum of the minimal distances to a greater element. St000677The standardized bi-alternating inversion number of a permutation. St000763The sum of the positions of the strong records of an integer composition. St000797The stat`` of a permutation. St000849The number of 1/3-balanced pairs in a poset. St000868The aid statistic in the sense of Shareshian-Wachs. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001171The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001209The pmaj statistic of a parking function. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001537The number of cyclic crossings of a permutation. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001377The major index minus the number of inversions of a permutation. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St000670The reversal length of a permutation. St000022The number of fixed points of a permutation. St000081The number of edges of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000522The number of 1-protected nodes of a rooted tree. St000632The jump number of the poset. St000662The staircase size of the code of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. 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

$eA$ . St001621The number of atoms of a lattice. St001863The number of weak excedances of a signed permutation. St001889The size of the connectivity set of a signed permutation. St000054The first entry of the permutation. St000071The number of maximal chains in a poset. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000307The number of rowmotion orbits of a poset. St000352The Elizalde-Pak rank of a permutation. St000359The number of occurrences of the pattern 23-1. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000485The length of the longest cycle of a permutation. St000521The number of distinct subtrees of an ordered tree. St000534The number of 2-rises of a permutation. St000648The number of 2-excedences of a permutation. St000651The maximal size of a rise in a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000730The maximal arc length of a set partition. St000731The number of double exceedences of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001510The number of self-evacuating linear extensions of a finite poset. St001555The order of a signed permutation. St001769The reflection length of a signed permutation. St001770The number of facets of a certain subword complex associated with the signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001896The number of right descents of a signed permutations. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000232The number of crossings of a set partition. St000497The lcb statistic of a set partition. St000562The number of internal points of a set partition. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000848The balance constant multiplied with the number of linear extensions of a poset. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001388The number of non-attacking neighbors of a permutation. St001394The genus of a permutation. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001877Number of indecomposable injective modules with projective dimension 2. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000455The second largest eigenvalue of a graph if it is integral. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.