- FindStat

Your data matches 90 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000327
(load all 9 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000327: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of cover relations in a poset. Equivalently, this is also the number of edges in the Hasse diagram [1].

Matching statistic: St001074
(load all 14 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001074: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,2] => [1,2] => 2 = 1 + 1
[2,1] => [1,2] => [1,2] => 2 = 1 + 1
[1,2,3] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[1,3,2] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[2,1,3] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[2,3,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[3,1,2] => [1,3,2] => [1,3,2] => 5 = 4 + 1
[3,2,1] => [1,3,2] => [1,3,2] => 5 = 4 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 6 = 5 + 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 6 = 5 + 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 6 = 5 + 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 6 = 5 + 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 6 = 5 + 1
[3,1,4,2] => [1,3,4,2] => [1,2,4,3] => 6 = 5 + 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 6 = 5 + 1
[3,2,4,1] => [1,3,4,2] => [1,2,4,3] => 6 = 5 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1

Description

The number of inversions of the cyclic embedding of a permutation. The cyclic embedding of a permutation

$\pi$ of length

$n$ is given by the permutation of length

$n+1$ represented in cycle notation by

$(\pi_1,\ldots,\pi_n,n+1)$ . This reflects in particular the fact that the number of long cycles of length

$n+1$ equals

$n!$ . This statistic counts the number of inversions of this embedding, see [1]. As shown in [2], the sum of this statistic on all permutations of length

$n$ equals

$n!\cdot(3n-1)/12$ .

Matching statistic: St001127

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001127: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,2] => [1,1]
 => 2 = 1 + 1
[2,1] => [1,2] => [1,1]
 => 2 = 1 + 1
[1,2,3] => [1,2,3] => [1,1,1]
 => 3 = 2 + 1
[1,3,2] => [1,2,3] => [1,1,1]
 => 3 = 2 + 1
[2,1,3] => [1,2,3] => [1,1,1]
 => 3 = 2 + 1
[2,3,1] => [1,2,3] => [1,1,1]
 => 3 = 2 + 1
[3,1,2] => [1,3,2] => [2,1]
 => 5 = 4 + 1
[3,2,1] => [1,3,2] => [2,1]
 => 5 = 4 + 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => 4 = 3 + 1
[1,2,4,3] => [1,2,3,4] => [1,1,1,1]
 => 4 = 3 + 1
[1,3,2,4] => [1,2,3,4] => [1,1,1,1]
 => 4 = 3 + 1
[1,3,4,2] => [1,2,3,4] => [1,1,1,1]
 => 4 = 3 + 1
[1,4,2,3] => [1,2,4,3] => [2,1,1]
 => 6 = 5 + 1
[1,4,3,2] => [1,2,4,3] => [2,1,1]
 => 6 = 5 + 1
[2,1,3,4] => [1,2,3,4] => [1,1,1,1]
 => 4 = 3 + 1
[2,1,4,3] => [1,2,3,4] => [1,1,1,1]
 => 4 = 3 + 1
[2,3,1,4] => [1,2,3,4] => [1,1,1,1]
 => 4 = 3 + 1
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
 => 4 = 3 + 1
[2,4,1,3] => [1,2,4,3] => [2,1,1]
 => 6 = 5 + 1
[2,4,3,1] => [1,2,4,3] => [2,1,1]
 => 6 = 5 + 1
[3,1,2,4] => [1,3,2,4] => [2,1,1]
 => 6 = 5 + 1
[3,1,4,2] => [1,3,4,2] => [2,1,1]
 => 6 = 5 + 1
[3,2,1,4] => [1,3,2,4] => [2,1,1]
 => 6 = 5 + 1
[3,2,4,1] => [1,3,4,2] => [2,1,1]
 => 6 = 5 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 4 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 4 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 4 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 4 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 4 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 4 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 4 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 4 + 1
[2,1,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 4 + 1
[2,1,3,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 4 + 1
[2,1,4,3,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 4 + 1
[2,1,4,5,3] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 4 + 1
[2,3,1,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 4 + 1
[2,3,1,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 4 + 1
[2,3,4,1,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 4 + 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 4 + 1

Description

The sum of the squares of the parts of a partition.

Matching statistic: St000070
(load all 2 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000070: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of antichains in a poset. An antichain in a poset

$P$ is a subset of elements of

$P$ which are pairwise incomparable. An order ideal is a subset

$I$ of

$P$ such that

$a\in I$ and

$b \leq_P a$ implies

$b \in I$ . Since there is a one-to-one correspondence between antichains and order ideals, this statistic is also the number of order ideals in a poset.

Matching statistic: St000104
(load all 3 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000104: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St000151
(load all 3 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000151: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St001977
(load all 2 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St001977: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,0,1,0]
 => [1,1,0,0]
 => [[0,1],[1,0]]
 => 1
[2,1] => [1,1,0,0]
 => [1,0,1,0]
 => [[1,0],[0,1]]
 => 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => 2
[1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => 2
[2,1,3] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => 2
[2,3,1] => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => 2
[3,1,2] => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => 4
[3,2,1] => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => 4
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 3
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => 3
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => 5
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => 5
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => 3
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => 5
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => 5
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => 5
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => 5
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => 5
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => 5
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 4
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 4
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 4
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 4
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 4
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 4
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 4
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 4
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 4
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 4
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 4
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 4
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 4
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 4
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 4
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[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]]
 => 4

Description

The degree of an alternating sign matrix in the Hasse diagram of the corner sum lattice.

Matching statistic: St000018

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of inversions of a permutation. This equals the minimal number of simple transpositions

$(i,i+1)$ needed to write

$\pi$ . Thus, it is also the Coxeter length of

$\pi$ .

Matching statistic: St000795
(load all 2 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000795: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,3,2] => 1 = 2 - 1
[1,3,2] => [1,2,3] => [1,2,3] => [1,3,2] => 1 = 2 - 1
[2,1,3] => [1,2,3] => [1,2,3] => [1,3,2] => 1 = 2 - 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,3,2] => 1 = 2 - 1
[3,1,2] => [1,3,2] => [2,3,1] => [2,3,1] => 3 = 4 - 1
[3,2,1] => [1,3,2] => [2,3,1] => [2,3,1] => 3 = 4 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,4,3,2] => 2 = 3 - 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,4,3,2] => 2 = 3 - 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,4,3,2] => 2 = 3 - 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,4,3,2] => 2 = 3 - 1
[1,4,2,3] => [1,2,4,3] => [2,3,4,1] => [2,4,3,1] => 4 = 5 - 1
[1,4,3,2] => [1,2,4,3] => [2,3,4,1] => [2,4,3,1] => 4 = 5 - 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,4,3,2] => 2 = 3 - 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,4,3,2] => 2 = 3 - 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,4,3,2] => 2 = 3 - 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,4,3,2] => 2 = 3 - 1
[2,4,1,3] => [1,2,4,3] => [2,3,4,1] => [2,4,3,1] => 4 = 5 - 1
[2,4,3,1] => [1,2,4,3] => [2,3,4,1] => [2,4,3,1] => 4 = 5 - 1
[3,1,2,4] => [1,3,2,4] => [2,3,1,4] => [2,4,1,3] => 4 = 5 - 1
[3,1,4,2] => [1,3,4,2] => [2,4,1,3] => [2,4,1,3] => 4 = 5 - 1
[3,2,1,4] => [1,3,2,4] => [2,3,1,4] => [2,4,1,3] => 4 = 5 - 1
[3,2,4,1] => [1,3,4,2] => [2,4,1,3] => [2,4,1,3] => 4 = 5 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,5,4,3,2] => 3 = 4 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,5,4,3,2] => 3 = 4 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,5,4,3,2] => 3 = 4 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,5,4,3,2] => 3 = 4 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,5,4,3,2] => 3 = 4 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,5,4,3,2] => 3 = 4 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,5,4,3,2] => 3 = 4 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,5,4,3,2] => 3 = 4 - 1
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,5,4,3,2] => 3 = 4 - 1
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,5,4,3,2] => 3 = 4 - 1
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,5,4,3,2] => 3 = 4 - 1
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,5,4,3,2] => 3 = 4 - 1
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,5,4,3,2] => 3 = 4 - 1
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,5,4,3,2] => 3 = 4 - 1
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,5,4,3,2] => 3 = 4 - 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,5,4,3,2] => 3 = 4 - 1

Description

The mad of a permutation. According to [1], this is the sum of twice the number of occurrences of the vincular pattern of

$(2\underline{31})$ plus the number of occurrences of the vincular patterns

$(\underline{31}2)$ and

$(\underline{21})$ , where matches of the underlined letters must be adjacent.

Matching statistic: St000890
(load all 2 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000890: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,0,1,0]
 => [1,1,0,0]
 => [[0,1],[1,0]]
 => 2 = 1 + 1
[2,1] => [1,1,0,0]
 => [1,0,1,0]
 => [[1,0],[0,1]]
 => 2 = 1 + 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => 3 = 2 + 1
[1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => 3 = 2 + 1
[2,1,3] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => 3 = 2 + 1
[2,3,1] => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => 3 = 2 + 1
[3,1,2] => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => 5 = 4 + 1
[3,2,1] => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => 5 = 4 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 4 = 3 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => 4 = 3 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => 4 = 3 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => 4 = 3 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => 6 = 5 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => 6 = 5 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => 4 = 3 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => 4 = 3 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => 4 = 3 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => 4 = 3 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => 6 = 5 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => 6 = 5 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => 6 = 5 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => 6 = 5 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => 6 = 5 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => 6 = 5 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 5 = 4 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 5 = 4 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 5 = 4 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 5 = 4 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 5 = 4 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 5 = 4 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 5 = 4 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 5 = 4 + 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 5 = 4 + 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 5 = 4 + 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 5 = 4 + 1
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 5 = 4 + 1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 5 = 4 + 1
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 5 = 4 + 1
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 5 = 4 + 1
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[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]]
 => 5 = 4 + 1

Description

The number of nonzero entries in an alternating sign matrix.

The following 80 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001249Sum of the odd parts of a partition. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001688The sum of the squares of the heights of the peaks of a Dyck path. St000867The sum of the hook lengths in the first row of an integer partition. 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. St000454The largest eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001875The number of simple modules with projective dimension at most 1. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000264The girth of a graph, which is not a tree. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000219The number of occurrences of the pattern 231 in a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000422The energy of a graph, if it is integral. St001926Sparre Andersen's position of the maximum of a signed permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000550The number of modular elements of a lattice. St000080The rank of the poset. St000307The number of rowmotion orbits of a poset. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000477The weight of a partition according to Alladi. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000997The even-odd crank of an integer partition. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001060The distinguishing index of a graph. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001118The acyclic chromatic index of a graph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000455The second largest eigenvalue of a graph if it is integral. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition.