- FindStat

Your data matches 92 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000216
(load all 2 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000216: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The absolute length of a permutation. The absolute length of a permutation

$\sigma$ of length

$n$ is the shortest

$k$ such that

$\sigma = t_1 \dots t_k$ for transpositions

$t_i$ . Also, this is equal to

$n$ minus the number of cycles of

$\sigma$ .

Matching statistic: St001179

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
St001179: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra.

Matching statistic: St001583
(load all 4 compositions to match this statistic)

Mapping

Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%

Values

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

Description

The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.

Matching statistic: St000209

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000209: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 80%

Values

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

Description

Maximum difference of elements in cycles. Given a cycle

$C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is

$\max \{ a_i-a_j | a_i, a_j \in C \}$ . The statistic is then the maximum of this value over all cycles in the permutation.

Matching statistic: St001232
(load all 3 compositions to match this statistic)

Mapping

Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 100%

Values

[1] => 1 => [1,1] => [1,0,1,0]
 => 1
[1,1] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => ? = 2
[2] => 10 => [1,2] => [1,0,1,1,0,0]
 => 2
[1,1,1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => ? = 3
[1,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => ? = 3
[2,1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
[3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3
[1,1,1,1] => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 3
[1,1,2] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => ? = 4
[1,2,1] => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => ? = 4
[1,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[2,1,1] => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => ? = 4
[2,2] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
[3,1] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 4
[4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4
[1,1,1,1,1] => 11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 4
[1,1,1,2] => 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 4
[1,1,2,1] => 11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 4
[1,1,3] => 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 5
[1,2,1,1] => 11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 5
[1,2,2] => 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 5
[1,3,1] => 11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 5
[1,4] => 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 5
[2,1,1,1] => 10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4
[2,1,2] => 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 5
[2,2,1] => 10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
[2,3] => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 5
[3,1,1] => 10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 5
[3,2] => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 5
[4,1] => 10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5
[5] => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5

Description

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

Matching statistic: St000080
(load all 3 compositions to match this statistic)

Mapping

Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000080: Posets ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 1 => ([(0,1)],2)
 => 1
[1,1] => [2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2] => [1,1] => 11 => ([(0,2),(2,1)],3)
 => 2
[1,1,1] => [3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[1,2] => [1,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[2,1] => [2,1] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 3
[3] => [1,1,1] => 111 => ([(0,3),(2,1),(3,2)],4)
 => 3
[1,1,1,1] => [4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 3
[1,1,2] => [1,3] => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 4
[1,2,1] => [2,2] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 4
[1,3] => [1,1,2] => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4
[2,1,1] => [3,1] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 4
[2,2] => [1,2,1] => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4
[3,1] => [2,1,1] => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4
[4] => [1,1,1,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,1,1,1,1] => [5] => 10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 4
[1,1,1,2] => [1,4] => 11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 4
[1,1,2,1] => [2,3] => 10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => ? = 4
[1,1,3] => [1,1,3] => 11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 5
[1,2,1,1] => [3,2] => 10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => ? = 5
[1,2,2] => [1,2,2] => 11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => ? = 5
[1,3,1] => [2,1,2] => 10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => ? = 5
[1,4] => [1,1,1,2] => 11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 5
[2,1,1,1] => [4,1] => 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
 => ? = 4
[2,1,2] => [1,3,1] => 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 5
[2,2,1] => [2,2,1] => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
 => ? = 5
[2,3] => [1,1,2,1] => 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 5
[3,1,1] => [3,1,1] => 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 5
[3,2] => [1,2,1,1] => 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
 => ? = 5
[4,1] => [2,1,1,1] => 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 5
[5] => [1,1,1,1,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5

Description

The rank of the poset.

Matching statistic: St000528
(load all 3 compositions to match this statistic)

Mapping

Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000528: Posets ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 1 => ([(0,1)],2)
 => 2 = 1 + 1
[1,1] => [2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2] => [1,1] => 11 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[1,1,1] => [3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 4 = 3 + 1
[1,2] => [1,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 4 = 3 + 1
[2,1] => [2,1] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 4 = 3 + 1
[3] => [1,1,1] => 111 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[1,1,1,1] => [4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 3 + 1
[1,1,2] => [1,3] => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[1,2,1] => [2,2] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 4 + 1
[1,3] => [1,1,2] => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 + 1
[2,1,1] => [3,1] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 4 + 1
[2,2] => [1,2,1] => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 + 1
[3,1] => [2,1,1] => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 + 1
[4] => [1,1,1,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[1,1,1,1,1] => [5] => 10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 4 + 1
[1,1,1,2] => [1,4] => 11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 4 + 1
[1,1,2,1] => [2,3] => 10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => ? = 4 + 1
[1,1,3] => [1,1,3] => 11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 5 + 1
[1,2,1,1] => [3,2] => 10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => ? = 5 + 1
[1,2,2] => [1,2,2] => 11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => ? = 5 + 1
[1,3,1] => [2,1,2] => 10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => ? = 5 + 1
[1,4] => [1,1,1,2] => 11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 5 + 1
[2,1,1,1] => [4,1] => 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
 => ? = 4 + 1
[2,1,2] => [1,3,1] => 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 5 + 1
[2,2,1] => [2,2,1] => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
 => ? = 5 + 1
[2,3] => [1,1,2,1] => 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 5 + 1
[3,1,1] => [3,1,1] => 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 5 + 1
[3,2] => [1,2,1,1] => 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
 => ? = 5 + 1
[4,1] => [2,1,1,1] => 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 5 + 1
[5] => [1,1,1,1,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1

Description

The height of a poset. This equals the rank of the poset [[St000080]] plus one.

Matching statistic: St000906
(load all 3 compositions to match this statistic)

Mapping

Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000906: Posets ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 1 => ([(0,1)],2)
 => 2 = 1 + 1
[1,1] => [2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2] => [1,1] => 11 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[1,1,1] => [3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 4 = 3 + 1
[1,2] => [1,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 4 = 3 + 1
[2,1] => [2,1] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 4 = 3 + 1
[3] => [1,1,1] => 111 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[1,1,1,1] => [4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 3 + 1
[1,1,2] => [1,3] => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[1,2,1] => [2,2] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 4 + 1
[1,3] => [1,1,2] => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 + 1
[2,1,1] => [3,1] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 4 + 1
[2,2] => [1,2,1] => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 + 1
[3,1] => [2,1,1] => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 + 1
[4] => [1,1,1,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[1,1,1,1,1] => [5] => 10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 4 + 1
[1,1,1,2] => [1,4] => 11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 4 + 1
[1,1,2,1] => [2,3] => 10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => ? = 4 + 1
[1,1,3] => [1,1,3] => 11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 5 + 1
[1,2,1,1] => [3,2] => 10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => ? = 5 + 1
[1,2,2] => [1,2,2] => 11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => ? = 5 + 1
[1,3,1] => [2,1,2] => 10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => ? = 5 + 1
[1,4] => [1,1,1,2] => 11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 5 + 1
[2,1,1,1] => [4,1] => 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
 => ? = 4 + 1
[2,1,2] => [1,3,1] => 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 5 + 1
[2,2,1] => [2,2,1] => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
 => ? = 5 + 1
[2,3] => [1,1,2,1] => 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 5 + 1
[3,1,1] => [3,1,1] => 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 5 + 1
[3,2] => [1,2,1,1] => 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
 => ? = 5 + 1
[4,1] => [2,1,1,1] => 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 5 + 1
[5] => [1,1,1,1,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1

Description

The length of the shortest maximal chain in a poset.

Matching statistic: St000643
(load all 3 compositions to match this statistic)

Mapping

Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000643: Posets ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 1 => ([(0,1)],2)
 => 3 = 1 + 2
[1,1] => [2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[2] => [1,1] => 11 => ([(0,2),(2,1)],3)
 => 4 = 2 + 2
[1,1,1] => [3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 5 = 3 + 2
[1,2] => [1,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 5 = 3 + 2
[2,1] => [2,1] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 5 = 3 + 2
[3] => [1,1,1] => 111 => ([(0,3),(2,1),(3,2)],4)
 => 5 = 3 + 2
[1,1,1,1] => [4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 3 + 2
[1,1,2] => [1,3] => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 4 + 2
[1,2,1] => [2,2] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 4 + 2
[1,3] => [1,1,2] => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 + 2
[2,1,1] => [3,1] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 4 + 2
[2,2] => [1,2,1] => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 + 2
[3,1] => [2,1,1] => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 + 2
[4] => [1,1,1,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6 = 4 + 2
[1,1,1,1,1] => [5] => 10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 4 + 2
[1,1,1,2] => [1,4] => 11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 4 + 2
[1,1,2,1] => [2,3] => 10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => ? = 4 + 2
[1,1,3] => [1,1,3] => 11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 5 + 2
[1,2,1,1] => [3,2] => 10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => ? = 5 + 2
[1,2,2] => [1,2,2] => 11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => ? = 5 + 2
[1,3,1] => [2,1,2] => 10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => ? = 5 + 2
[1,4] => [1,1,1,2] => 11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 5 + 2
[2,1,1,1] => [4,1] => 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
 => ? = 4 + 2
[2,1,2] => [1,3,1] => 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 5 + 2
[2,2,1] => [2,2,1] => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
 => ? = 5 + 2
[2,3] => [1,1,2,1] => 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 5 + 2
[3,1,1] => [3,1,1] => 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 5 + 2
[3,2] => [1,2,1,1] => 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
 => ? = 5 + 2
[4,1] => [2,1,1,1] => 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 5 + 2
[5] => [1,1,1,1,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 7 = 5 + 2

Description

The size of the largest orbit of antichains under Panyushev complementation.

Matching statistic: St000004

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000004: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 60%

Values

[1] => [1,0]
 => [[1],[2]]
 => [2,1] => 1
[1,1] => [1,0,1,0]
 => [[1,3],[2,4]]
 => [2,4,1,3] => 2
[2] => [1,1,0,0]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
[1,1,1] => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => [2,4,6,1,3,5] => 3
[1,2] => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => [2,5,6,1,3,4] => 3
[2,1] => [1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => [3,4,6,1,2,5] => 3
[3] => [1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => [2,4,6,8,1,3,5,7] => ? = 3
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => [2,4,7,8,1,3,5,6] => ? = 4
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => [2,5,6,8,1,3,4,7] => ? = 4
[1,3] => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => [2,6,7,8,1,3,4,5] => ? = 4
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => [3,4,6,8,1,2,5,7] => ? = 4
[2,2] => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => [3,4,7,8,1,2,5,6] => ? = 4
[3,1] => [1,1,1,0,0,0,1,0]
 => [[1,2,3,7],[4,5,6,8]]
 => [4,5,6,8,1,2,3,7] => ? = 4
[4] => [1,1,1,1,0,0,0,0]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => ? = 4
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9],[2,4,6,8,10]]
 => [2,4,6,8,10,1,3,5,7,9] => ? = 4
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,8],[2,4,6,9,10]]
 => [2,4,6,9,10,1,3,5,7,8] => ? = 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [[1,3,5,6,9],[2,4,7,8,10]]
 => [2,4,7,8,10,1,3,5,6,9] => ? = 4
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,6,7],[2,4,8,9,10]]
 => [2,4,8,9,10,1,3,5,6,7] => ? = 5
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [[1,3,4,7,9],[2,5,6,8,10]]
 => [2,5,6,8,10,1,3,4,7,9] => ? = 5
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [[1,3,4,7,8],[2,5,6,9,10]]
 => [2,5,6,9,10,1,3,4,7,8] => ? = 5
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => [2,6,7,8,10,1,3,4,5,9] => ? = 5
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => [2,7,8,9,10,1,3,4,5,6] => ? = 5
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [[1,2,5,7,9],[3,4,6,8,10]]
 => [3,4,6,8,10,1,2,5,7,9] => ? = 4
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [[1,2,5,7,8],[3,4,6,9,10]]
 => [3,4,6,9,10,1,2,5,7,8] => ? = 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [[1,2,5,6,9],[3,4,7,8,10]]
 => [3,4,7,8,10,1,2,5,6,9] => ? = 5
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => [3,4,8,9,10,1,2,5,6,7] => ? = 5
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [[1,2,3,7,9],[4,5,6,8,10]]
 => [4,5,6,8,10,1,2,3,7,9] => ? = 5
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => [4,5,6,9,10,1,2,3,7,8] => ? = 5
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [[1,2,3,4,9],[5,6,7,8,10]]
 => [5,6,7,8,10,1,2,3,4,9] => ? = 5
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [[1,2,3,4,5],[6,7,8,9,10]]
 => [6,7,8,9,10,1,2,3,4,5] => ? = 5

Description

The major index of a permutation. This is the sum of the positions of its descents,

$\operatorname{maj}(\sigma) = \sum_{\sigma(i) > \sigma(i+1)} i.$ Its generating function is

$[n]_q! = [1]_q \cdot [2]_q \dots [n]_q$ for

$[k]_q = 1 + q + q^2 + \dots q^{k-1}$ . A statistic equidistributed with the major index is called '''Mahonian statistic'''.

The following 82 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000021The number of descents of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000105The number of blocks in the set partition. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000211The rank of the set partition. St000213The number of weak exceedances (also weak excedences) of a permutation. St000251The number of nonsingleton blocks of a set partition. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000354The number of recoils of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000653The last descent of a permutation. St000654The first descent of a permutation. St000702The number of weak deficiencies of a permutation. St000794The mak of a permutation. St000798The makl of a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000833The comajor index of a permutation. St000925The number of topologically connected components of a set partition. St000956The maximal displacement of a permutation. St000991The number of right-to-left minima of a permutation. St001114The number of odd descents of a permutation. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001220The width of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001497The position of the largest weak excedence of a permutation. St001517The length of a longest pair of twins in a permutation. St001665The number of pure excedances of a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001729The number of visible descents of a permutation. St001769The reflection length of a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001861The number of Bruhat lower covers of a permutation. St001874Lusztig's a-function for the symmetric group. St001894The depth of a signed permutation. St001928The number of non-overlapping descents in a permutation. St000133The "bounce" of a permutation. St000168The number of internal nodes of an ordered tree. St000325The width of the tree associated to a permutation. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000470The number of runs in a permutation. St000624The normalized sum of the minimal distances to a greater element. St000744The length of the path to the largest entry in a standard Young tableau. St000863The length of the first row of the shifted shape of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000989The number of final rises of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001927Sparre Andersen's number of positives of a signed permutation. St000044The number of vertices of the unicellular map given by a perfect matching. St001926Sparre Andersen's position of the maximum of a signed permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001637The number of (upper) dissectors of a poset. St001645The pebbling number of a connected graph. St001668The number of points of the poset minus the width of the poset. St000782The indicator function of whether a given perfect matching is an L & P matching. 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000739The first entry in the last row of a semistandard tableau. St001401The number of distinct entries in a semistandard tableau. St001569The maximal modular displacement of a permutation. St001686The order of promotion on a Gelfand-Tsetlin pattern. St000101The cocharge of a semistandard tableau. St000454The largest eigenvalue of a graph if it is integral. St001556The number of inversions of the third entry of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one.