- FindStat

Your data matches 111 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000031
(load all 32 compositions to match this statistic)

Mapping

St000031: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of cycles in the cycle decomposition of a permutation.

Matching statistic: St000010
(load all 15 compositions to match this statistic)

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%

Values

[1] => [1]
 => 1
[1,2] => [1,1]
 => 2
[2,1] => [2]
 => 1
[1,2,3] => [1,1,1]
 => 3
[1,3,2] => [2,1]
 => 2
[2,1,3] => [2,1]
 => 2
[2,3,1] => [3]
 => 1
[3,1,2] => [3]
 => 1
[3,2,1] => [2,1]
 => 2
[1,2,3,4] => [1,1,1,1]
 => 4
[1,2,4,3] => [2,1,1]
 => 3
[1,3,2,4] => [2,1,1]
 => 3
[1,3,4,2] => [3,1]
 => 2
[1,4,2,3] => [3,1]
 => 2
[1,4,3,2] => [2,1,1]
 => 3
[2,1,3,4] => [2,1,1]
 => 3
[2,1,4,3] => [2,2]
 => 2
[2,3,1,4] => [3,1]
 => 2
[2,3,4,1] => [4]
 => 1
[2,4,1,3] => [4]
 => 1
[2,4,3,1] => [3,1]
 => 2
[3,1,2,4] => [3,1]
 => 2
[3,1,4,2] => [4]
 => 1
[3,2,1,4] => [2,1,1]
 => 3
[3,2,4,1] => [3,1]
 => 2
[3,4,1,2] => [2,2]
 => 2
[3,4,2,1] => [4]
 => 1
[4,1,2,3] => [4]
 => 1
[4,1,3,2] => [3,1]
 => 2
[4,2,1,3] => [3,1]
 => 2
[4,2,3,1] => [2,1,1]
 => 3
[4,3,1,2] => [4]
 => 1
[4,3,2,1] => [2,2]
 => 2
[1,2,3,4,5] => [1,1,1,1,1]
 => 5
[1,2,3,5,4] => [2,1,1,1]
 => 4
[1,2,4,3,5] => [2,1,1,1]
 => 4
[1,2,4,5,3] => [3,1,1]
 => 3
[1,2,5,3,4] => [3,1,1]
 => 3
[1,2,5,4,3] => [2,1,1,1]
 => 4
[1,3,2,4,5] => [2,1,1,1]
 => 4
[1,3,2,5,4] => [2,2,1]
 => 3
[1,3,4,2,5] => [3,1,1]
 => 3
[1,3,4,5,2] => [4,1]
 => 2
[1,3,5,2,4] => [4,1]
 => 2
[1,3,5,4,2] => [3,1,1]
 => 3
[1,4,2,3,5] => [3,1,1]
 => 3
[1,4,2,5,3] => [4,1]
 => 2
[1,4,3,2,5] => [2,1,1,1]
 => 4
[1,4,3,5,2] => [3,1,1]
 => 3
[1,4,5,2,3] => [2,2,1]
 => 3
[6,3,2,5,4,1,8,7,10,9] => ?
 => ? = 5
[8,3,2,5,4,7,6,1,10,9] => ?
 => ? = 5
[10,3,2,5,4,7,6,9,8,1] => ?
 => ? = 5
[2,1,6,5,4,3,8,7,10,9] => ?
 => ? = 5
[8,5,4,3,2,7,6,1,10,9] => ?
 => ? = 5
[10,5,4,3,2,7,6,9,8,1] => ?
 => ? = 5
[2,1,8,5,4,7,6,3,10,9] => ?
 => ? = 5
[8,7,4,3,6,5,2,1,10,9] => ?
 => ? = 5
[10,7,4,3,6,5,2,9,8,1] => ?
 => ? = 5
[2,1,10,5,4,7,6,9,8,3] => ?
 => ? = 5
[10,9,4,3,6,5,8,7,2,1] => ?
 => ? = 5
[2,1,4,3,8,7,6,5,10,9] => ?
 => ? = 5
[8,3,2,7,6,5,4,1,10,9] => ?
 => ? = 5
[10,3,2,7,6,5,4,9,8,1] => ?
 => ? = 5
[2,1,8,7,6,5,4,3,10,9] => ?
 => ? = 5
[10,7,6,5,4,3,2,9,8,1] => ?
 => ? = 5
[2,1,10,7,6,5,4,9,8,3] => ?
 => ? = 5
[10,9,6,5,4,3,8,7,2,1] => ?
 => ? = 5
[2,1,4,3,10,7,6,9,8,5] => ?
 => ? = 5
[4,3,2,1,10,7,6,9,8,5] => ?
 => ? = 5
[10,3,2,9,6,5,8,7,4,1] => ?
 => ? = 5
[2,1,10,9,6,5,8,7,4,3] => ?
 => ? = 5
[10,9,8,5,4,7,6,3,2,1] => ?
 => ? = 5
[2,1,4,3,6,5,10,9,8,7] => ?
 => ? = 5
[4,3,2,1,6,5,10,9,8,7] => ?
 => ? = 5
[6,3,2,5,4,1,10,9,8,7] => ?
 => ? = 5
[10,3,2,5,4,9,8,7,6,1] => ?
 => ? = 5
[2,1,6,5,4,3,10,9,8,7] => ?
 => ? = 5
[10,5,4,3,2,9,8,7,6,1] => ?
 => ? = 5
[2,1,10,5,4,9,8,7,6,3] => ?
 => ? = 5
[10,9,4,3,8,7,6,5,2,1] => ?
 => ? = 5
[2,1,4,3,10,9,8,7,6,5] => ?
 => ? = 5
[10,3,2,9,8,7,6,5,4,1] => ?
 => ? = 5
[2,1,4,3,6,5,8,7,12,11,10,9] => ?
 => ? = 6
[2,1,4,3,6,5,10,9,8,7,12,11] => ?
 => ? = 6
[2,1,4,3,6,5,12,9,8,11,10,7] => ?
 => ? = 6
[2,1,4,3,6,5,12,11,10,9,8,7] => ?
 => ? = 6
[2,1,4,3,8,7,6,5,10,9,12,11] => ?
 => ? = 6
[2,1,4,3,8,7,6,5,12,11,10,9] => ?
 => ? = 6
[2,1,4,3,10,7,6,9,8,5,12,11] => ?
 => ? = 6
[2,1,4,3,12,7,6,9,8,11,10,5] => ?
 => ? = 6
[2,1,4,3,12,7,6,11,10,9,8,5] => ?
 => ? = 6
[2,1,4,3,10,9,8,7,6,5,12,11] => ?
 => ? = 6
[2,1,4,3,12,9,8,7,6,11,10,5] => ?
 => ? = 6
[2,1,4,3,12,11,8,7,10,9,6,5] => ?
 => ? = 6
[2,1,4,3,12,11,10,9,8,7,6,5] => ?
 => ? = 6
[2,1,6,5,4,3,8,7,10,9,12,11] => ?
 => ? = 6
[2,1,6,5,4,3,8,7,12,11,10,9] => ?
 => ? = 6
[2,1,6,5,4,3,10,9,8,7,12,11] => ?
 => ? = 6
[2,1,6,5,4,3,12,9,8,11,10,7] => ?
 => ? = 6

Description

The length of the partition.

Matching statistic: St000147
(load all 2 compositions to match this statistic)

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%

Values

[1] => [1]
 => [1]
 => 1
[1,2] => [1,1]
 => [2]
 => 2
[2,1] => [2]
 => [1,1]
 => 1
[1,2,3] => [1,1,1]
 => [3]
 => 3
[1,3,2] => [2,1]
 => [2,1]
 => 2
[2,1,3] => [2,1]
 => [2,1]
 => 2
[2,3,1] => [3]
 => [1,1,1]
 => 1
[3,1,2] => [3]
 => [1,1,1]
 => 1
[3,2,1] => [2,1]
 => [2,1]
 => 2
[1,2,3,4] => [1,1,1,1]
 => [4]
 => 4
[1,2,4,3] => [2,1,1]
 => [3,1]
 => 3
[1,3,2,4] => [2,1,1]
 => [3,1]
 => 3
[1,3,4,2] => [3,1]
 => [2,1,1]
 => 2
[1,4,2,3] => [3,1]
 => [2,1,1]
 => 2
[1,4,3,2] => [2,1,1]
 => [3,1]
 => 3
[2,1,3,4] => [2,1,1]
 => [3,1]
 => 3
[2,1,4,3] => [2,2]
 => [2,2]
 => 2
[2,3,1,4] => [3,1]
 => [2,1,1]
 => 2
[2,3,4,1] => [4]
 => [1,1,1,1]
 => 1
[2,4,1,3] => [4]
 => [1,1,1,1]
 => 1
[2,4,3,1] => [3,1]
 => [2,1,1]
 => 2
[3,1,2,4] => [3,1]
 => [2,1,1]
 => 2
[3,1,4,2] => [4]
 => [1,1,1,1]
 => 1
[3,2,1,4] => [2,1,1]
 => [3,1]
 => 3
[3,2,4,1] => [3,1]
 => [2,1,1]
 => 2
[3,4,1,2] => [2,2]
 => [2,2]
 => 2
[3,4,2,1] => [4]
 => [1,1,1,1]
 => 1
[4,1,2,3] => [4]
 => [1,1,1,1]
 => 1
[4,1,3,2] => [3,1]
 => [2,1,1]
 => 2
[4,2,1,3] => [3,1]
 => [2,1,1]
 => 2
[4,2,3,1] => [2,1,1]
 => [3,1]
 => 3
[4,3,1,2] => [4]
 => [1,1,1,1]
 => 1
[4,3,2,1] => [2,2]
 => [2,2]
 => 2
[1,2,3,4,5] => [1,1,1,1,1]
 => [5]
 => 5
[1,2,3,5,4] => [2,1,1,1]
 => [4,1]
 => 4
[1,2,4,3,5] => [2,1,1,1]
 => [4,1]
 => 4
[1,2,4,5,3] => [3,1,1]
 => [3,1,1]
 => 3
[1,2,5,3,4] => [3,1,1]
 => [3,1,1]
 => 3
[1,2,5,4,3] => [2,1,1,1]
 => [4,1]
 => 4
[1,3,2,4,5] => [2,1,1,1]
 => [4,1]
 => 4
[1,3,2,5,4] => [2,2,1]
 => [3,2]
 => 3
[1,3,4,2,5] => [3,1,1]
 => [3,1,1]
 => 3
[1,3,4,5,2] => [4,1]
 => [2,1,1,1]
 => 2
[1,3,5,2,4] => [4,1]
 => [2,1,1,1]
 => 2
[1,3,5,4,2] => [3,1,1]
 => [3,1,1]
 => 3
[1,4,2,3,5] => [3,1,1]
 => [3,1,1]
 => 3
[1,4,2,5,3] => [4,1]
 => [2,1,1,1]
 => 2
[1,4,3,2,5] => [2,1,1,1]
 => [4,1]
 => 4
[1,4,3,5,2] => [3,1,1]
 => [3,1,1]
 => 3
[1,4,5,2,3] => [2,2,1]
 => [3,2]
 => 3
[6,3,2,5,4,1,8,7,10,9] => ?
 => ?
 => ? = 5
[8,3,2,5,4,7,6,1,10,9] => ?
 => ?
 => ? = 5
[10,3,2,5,4,7,6,9,8,1] => ?
 => ?
 => ? = 5
[2,1,6,5,4,3,8,7,10,9] => ?
 => ?
 => ? = 5
[8,5,4,3,2,7,6,1,10,9] => ?
 => ?
 => ? = 5
[10,5,4,3,2,7,6,9,8,1] => ?
 => ?
 => ? = 5
[2,1,8,5,4,7,6,3,10,9] => ?
 => ?
 => ? = 5
[8,7,4,3,6,5,2,1,10,9] => ?
 => ?
 => ? = 5
[10,7,4,3,6,5,2,9,8,1] => ?
 => ?
 => ? = 5
[2,1,10,5,4,7,6,9,8,3] => ?
 => ?
 => ? = 5
[10,9,4,3,6,5,8,7,2,1] => ?
 => ?
 => ? = 5
[2,1,4,3,8,7,6,5,10,9] => ?
 => ?
 => ? = 5
[8,3,2,7,6,5,4,1,10,9] => ?
 => ?
 => ? = 5
[10,3,2,7,6,5,4,9,8,1] => ?
 => ?
 => ? = 5
[2,1,8,7,6,5,4,3,10,9] => ?
 => ?
 => ? = 5
[10,7,6,5,4,3,2,9,8,1] => ?
 => ?
 => ? = 5
[2,1,10,7,6,5,4,9,8,3] => ?
 => ?
 => ? = 5
[10,9,6,5,4,3,8,7,2,1] => ?
 => ?
 => ? = 5
[2,1,4,3,10,7,6,9,8,5] => ?
 => ?
 => ? = 5
[4,3,2,1,10,7,6,9,8,5] => ?
 => ?
 => ? = 5
[10,3,2,9,6,5,8,7,4,1] => ?
 => ?
 => ? = 5
[2,1,10,9,6,5,8,7,4,3] => ?
 => ?
 => ? = 5
[10,9,8,5,4,7,6,3,2,1] => ?
 => ?
 => ? = 5
[2,1,4,3,6,5,10,9,8,7] => ?
 => ?
 => ? = 5
[4,3,2,1,6,5,10,9,8,7] => ?
 => ?
 => ? = 5
[6,3,2,5,4,1,10,9,8,7] => ?
 => ?
 => ? = 5
[10,3,2,5,4,9,8,7,6,1] => ?
 => ?
 => ? = 5
[2,1,6,5,4,3,10,9,8,7] => ?
 => ?
 => ? = 5
[10,5,4,3,2,9,8,7,6,1] => ?
 => ?
 => ? = 5
[2,1,10,5,4,9,8,7,6,3] => ?
 => ?
 => ? = 5
[10,9,4,3,8,7,6,5,2,1] => ?
 => ?
 => ? = 5
[2,1,4,3,10,9,8,7,6,5] => ?
 => ?
 => ? = 5
[10,3,2,9,8,7,6,5,4,1] => ?
 => ?
 => ? = 5
[2,1,4,3,6,5,8,7,12,11,10,9] => ?
 => ?
 => ? = 6
[2,1,4,3,6,5,10,9,8,7,12,11] => ?
 => ?
 => ? = 6
[2,1,4,3,6,5,12,9,8,11,10,7] => ?
 => ?
 => ? = 6
[2,1,4,3,6,5,12,11,10,9,8,7] => ?
 => ?
 => ? = 6
[2,1,4,3,8,7,6,5,10,9,12,11] => ?
 => ?
 => ? = 6
[2,1,4,3,8,7,6,5,12,11,10,9] => ?
 => ?
 => ? = 6
[2,1,4,3,10,7,6,9,8,5,12,11] => ?
 => ?
 => ? = 6
[2,1,4,3,12,7,6,9,8,11,10,5] => ?
 => ?
 => ? = 6
[2,1,4,3,12,7,6,11,10,9,8,5] => ?
 => ?
 => ? = 6
[2,1,4,3,10,9,8,7,6,5,12,11] => ?
 => ?
 => ? = 6
[2,1,4,3,12,9,8,7,6,11,10,5] => ?
 => ?
 => ? = 6
[2,1,4,3,12,11,8,7,10,9,6,5] => ?
 => ?
 => ? = 6
[2,1,4,3,12,11,10,9,8,7,6,5] => ?
 => ?
 => ? = 6
[2,1,6,5,4,3,8,7,10,9,12,11] => ?
 => ?
 => ? = 6
[2,1,6,5,4,3,8,7,12,11,10,9] => ?
 => ?
 => ? = 6
[2,1,6,5,4,3,10,9,8,7,12,11] => ?
 => ?
 => ? = 6
[2,1,6,5,4,3,12,9,8,11,10,7] => ?
 => ?
 => ? = 6

Description

The largest part of an integer partition.

Matching statistic: St000378
(load all 2 compositions to match this statistic)

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%

Values

[1] => [1]
 => [1]
 => 1
[1,2] => [1,1]
 => [2]
 => 2
[2,1] => [2]
 => [1,1]
 => 1
[1,2,3] => [1,1,1]
 => [2,1]
 => 3
[1,3,2] => [2,1]
 => [3]
 => 2
[2,1,3] => [2,1]
 => [3]
 => 2
[2,3,1] => [3]
 => [1,1,1]
 => 1
[3,1,2] => [3]
 => [1,1,1]
 => 1
[3,2,1] => [2,1]
 => [3]
 => 2
[1,2,3,4] => [1,1,1,1]
 => [3,1]
 => 4
[1,2,4,3] => [2,1,1]
 => [2,2]
 => 3
[1,3,2,4] => [2,1,1]
 => [2,2]
 => 3
[1,3,4,2] => [3,1]
 => [2,1,1]
 => 2
[1,4,2,3] => [3,1]
 => [2,1,1]
 => 2
[1,4,3,2] => [2,1,1]
 => [2,2]
 => 3
[2,1,3,4] => [2,1,1]
 => [2,2]
 => 3
[2,1,4,3] => [2,2]
 => [4]
 => 2
[2,3,1,4] => [3,1]
 => [2,1,1]
 => 2
[2,3,4,1] => [4]
 => [1,1,1,1]
 => 1
[2,4,1,3] => [4]
 => [1,1,1,1]
 => 1
[2,4,3,1] => [3,1]
 => [2,1,1]
 => 2
[3,1,2,4] => [3,1]
 => [2,1,1]
 => 2
[3,1,4,2] => [4]
 => [1,1,1,1]
 => 1
[3,2,1,4] => [2,1,1]
 => [2,2]
 => 3
[3,2,4,1] => [3,1]
 => [2,1,1]
 => 2
[3,4,1,2] => [2,2]
 => [4]
 => 2
[3,4,2,1] => [4]
 => [1,1,1,1]
 => 1
[4,1,2,3] => [4]
 => [1,1,1,1]
 => 1
[4,1,3,2] => [3,1]
 => [2,1,1]
 => 2
[4,2,1,3] => [3,1]
 => [2,1,1]
 => 2
[4,2,3,1] => [2,1,1]
 => [2,2]
 => 3
[4,3,1,2] => [4]
 => [1,1,1,1]
 => 1
[4,3,2,1] => [2,2]
 => [4]
 => 2
[1,2,3,4,5] => [1,1,1,1,1]
 => [3,2]
 => 5
[1,2,3,5,4] => [2,1,1,1]
 => [3,1,1]
 => 4
[1,2,4,3,5] => [2,1,1,1]
 => [3,1,1]
 => 4
[1,2,4,5,3] => [3,1,1]
 => [4,1]
 => 3
[1,2,5,3,4] => [3,1,1]
 => [4,1]
 => 3
[1,2,5,4,3] => [2,1,1,1]
 => [3,1,1]
 => 4
[1,3,2,4,5] => [2,1,1,1]
 => [3,1,1]
 => 4
[1,3,2,5,4] => [2,2,1]
 => [2,2,1]
 => 3
[1,3,4,2,5] => [3,1,1]
 => [4,1]
 => 3
[1,3,4,5,2] => [4,1]
 => [2,1,1,1]
 => 2
[1,3,5,2,4] => [4,1]
 => [2,1,1,1]
 => 2
[1,3,5,4,2] => [3,1,1]
 => [4,1]
 => 3
[1,4,2,3,5] => [3,1,1]
 => [4,1]
 => 3
[1,4,2,5,3] => [4,1]
 => [2,1,1,1]
 => 2
[1,4,3,2,5] => [2,1,1,1]
 => [3,1,1]
 => 4
[1,4,3,5,2] => [3,1,1]
 => [4,1]
 => 3
[1,4,5,2,3] => [2,2,1]
 => [2,2,1]
 => 3
[6,3,2,5,4,1,8,7,10,9] => ?
 => ?
 => ? = 5
[8,3,2,5,4,7,6,1,10,9] => ?
 => ?
 => ? = 5
[10,3,2,5,4,7,6,9,8,1] => ?
 => ?
 => ? = 5
[2,1,6,5,4,3,8,7,10,9] => ?
 => ?
 => ? = 5
[8,5,4,3,2,7,6,1,10,9] => ?
 => ?
 => ? = 5
[10,5,4,3,2,7,6,9,8,1] => ?
 => ?
 => ? = 5
[2,1,8,5,4,7,6,3,10,9] => ?
 => ?
 => ? = 5
[8,7,4,3,6,5,2,1,10,9] => ?
 => ?
 => ? = 5
[10,7,4,3,6,5,2,9,8,1] => ?
 => ?
 => ? = 5
[2,1,10,5,4,7,6,9,8,3] => ?
 => ?
 => ? = 5
[10,9,4,3,6,5,8,7,2,1] => ?
 => ?
 => ? = 5
[2,1,4,3,8,7,6,5,10,9] => ?
 => ?
 => ? = 5
[8,3,2,7,6,5,4,1,10,9] => ?
 => ?
 => ? = 5
[10,3,2,7,6,5,4,9,8,1] => ?
 => ?
 => ? = 5
[2,1,8,7,6,5,4,3,10,9] => ?
 => ?
 => ? = 5
[10,7,6,5,4,3,2,9,8,1] => ?
 => ?
 => ? = 5
[2,1,10,7,6,5,4,9,8,3] => ?
 => ?
 => ? = 5
[10,9,6,5,4,3,8,7,2,1] => ?
 => ?
 => ? = 5
[2,1,4,3,10,7,6,9,8,5] => ?
 => ?
 => ? = 5
[4,3,2,1,10,7,6,9,8,5] => ?
 => ?
 => ? = 5
[10,3,2,9,6,5,8,7,4,1] => ?
 => ?
 => ? = 5
[2,1,10,9,6,5,8,7,4,3] => ?
 => ?
 => ? = 5
[10,9,8,5,4,7,6,3,2,1] => ?
 => ?
 => ? = 5
[2,1,4,3,6,5,10,9,8,7] => ?
 => ?
 => ? = 5
[4,3,2,1,6,5,10,9,8,7] => ?
 => ?
 => ? = 5
[6,3,2,5,4,1,10,9,8,7] => ?
 => ?
 => ? = 5
[10,3,2,5,4,9,8,7,6,1] => ?
 => ?
 => ? = 5
[2,1,6,5,4,3,10,9,8,7] => ?
 => ?
 => ? = 5
[10,5,4,3,2,9,8,7,6,1] => ?
 => ?
 => ? = 5
[2,1,10,5,4,9,8,7,6,3] => ?
 => ?
 => ? = 5
[10,9,4,3,8,7,6,5,2,1] => ?
 => ?
 => ? = 5
[2,1,4,3,10,9,8,7,6,5] => ?
 => ?
 => ? = 5
[10,3,2,9,8,7,6,5,4,1] => ?
 => ?
 => ? = 5
[2,1,4,3,6,5,8,7,12,11,10,9] => ?
 => ?
 => ? = 6
[2,1,4,3,6,5,10,9,8,7,12,11] => ?
 => ?
 => ? = 6
[2,1,4,3,6,5,12,9,8,11,10,7] => ?
 => ?
 => ? = 6
[2,1,4,3,6,5,12,11,10,9,8,7] => ?
 => ?
 => ? = 6
[2,1,4,3,8,7,6,5,10,9,12,11] => ?
 => ?
 => ? = 6
[2,1,4,3,8,7,6,5,12,11,10,9] => ?
 => ?
 => ? = 6
[2,1,4,3,10,7,6,9,8,5,12,11] => ?
 => ?
 => ? = 6
[2,1,4,3,12,7,6,9,8,11,10,5] => ?
 => ?
 => ? = 6
[2,1,4,3,12,7,6,11,10,9,8,5] => ?
 => ?
 => ? = 6
[2,1,4,3,10,9,8,7,6,5,12,11] => ?
 => ?
 => ? = 6
[2,1,4,3,12,9,8,7,6,11,10,5] => ?
 => ?
 => ? = 6
[2,1,4,3,12,11,8,7,10,9,6,5] => ?
 => ?
 => ? = 6
[2,1,4,3,12,11,10,9,8,7,6,5] => ?
 => ?
 => ? = 6
[2,1,6,5,4,3,8,7,10,9,12,11] => ?
 => ?
 => ? = 6
[2,1,6,5,4,3,8,7,12,11,10,9] => ?
 => ?
 => ? = 6
[2,1,6,5,4,3,10,9,8,7,12,11] => ?
 => ?
 => ? = 6
[2,1,6,5,4,3,12,9,8,11,10,7] => ?
 => ?
 => ? = 6

Description

The diagonal inversion number of an integer partition. The dinv of a partition is the number of cells

$c$ in the diagram of an integer partition

$\lambda$ for which

$\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$ . See also exercise 3.19 of [2]. This statistic is equidistributed with the length of the partition, see [3].

Matching statistic: St000733
(load all 3 compositions to match this statistic)

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 91%

Values

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

Description

The row containing the largest entry of a standard tableau.

Matching statistic: St000288
(load all 15 compositions to match this statistic)

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 91%

Values

[1] => [1]
 => 10 => 1
[1,2] => [1,1]
 => 110 => 2
[2,1] => [2]
 => 100 => 1
[1,2,3] => [1,1,1]
 => 1110 => 3
[1,3,2] => [2,1]
 => 1010 => 2
[2,1,3] => [2,1]
 => 1010 => 2
[2,3,1] => [3]
 => 1000 => 1
[3,1,2] => [3]
 => 1000 => 1
[3,2,1] => [2,1]
 => 1010 => 2
[1,2,3,4] => [1,1,1,1]
 => 11110 => 4
[1,2,4,3] => [2,1,1]
 => 10110 => 3
[1,3,2,4] => [2,1,1]
 => 10110 => 3
[1,3,4,2] => [3,1]
 => 10010 => 2
[1,4,2,3] => [3,1]
 => 10010 => 2
[1,4,3,2] => [2,1,1]
 => 10110 => 3
[2,1,3,4] => [2,1,1]
 => 10110 => 3
[2,1,4,3] => [2,2]
 => 1100 => 2
[2,3,1,4] => [3,1]
 => 10010 => 2
[2,3,4,1] => [4]
 => 10000 => 1
[2,4,1,3] => [4]
 => 10000 => 1
[2,4,3,1] => [3,1]
 => 10010 => 2
[3,1,2,4] => [3,1]
 => 10010 => 2
[3,1,4,2] => [4]
 => 10000 => 1
[3,2,1,4] => [2,1,1]
 => 10110 => 3
[3,2,4,1] => [3,1]
 => 10010 => 2
[3,4,1,2] => [2,2]
 => 1100 => 2
[3,4,2,1] => [4]
 => 10000 => 1
[4,1,2,3] => [4]
 => 10000 => 1
[4,1,3,2] => [3,1]
 => 10010 => 2
[4,2,1,3] => [3,1]
 => 10010 => 2
[4,2,3,1] => [2,1,1]
 => 10110 => 3
[4,3,1,2] => [4]
 => 10000 => 1
[4,3,2,1] => [2,2]
 => 1100 => 2
[1,2,3,4,5] => [1,1,1,1,1]
 => 111110 => 5
[1,2,3,5,4] => [2,1,1,1]
 => 101110 => 4
[1,2,4,3,5] => [2,1,1,1]
 => 101110 => 4
[1,2,4,5,3] => [3,1,1]
 => 100110 => 3
[1,2,5,3,4] => [3,1,1]
 => 100110 => 3
[1,2,5,4,3] => [2,1,1,1]
 => 101110 => 4
[1,3,2,4,5] => [2,1,1,1]
 => 101110 => 4
[1,3,2,5,4] => [2,2,1]
 => 11010 => 3
[1,3,4,2,5] => [3,1,1]
 => 100110 => 3
[1,3,4,5,2] => [4,1]
 => 100010 => 2
[1,3,5,2,4] => [4,1]
 => 100010 => 2
[1,3,5,4,2] => [3,1,1]
 => 100110 => 3
[1,4,2,3,5] => [3,1,1]
 => 100110 => 3
[1,4,2,5,3] => [4,1]
 => 100010 => 2
[1,4,3,2,5] => [2,1,1,1]
 => 101110 => 4
[1,4,3,5,2] => [3,1,1]
 => 100110 => 3
[1,4,5,2,3] => [2,2,1]
 => 11010 => 3
[6,3,2,5,4,1,8,7,10,9] => ?
 => ? => ? = 5
[8,3,2,5,4,7,6,1,10,9] => ?
 => ? => ? = 5
[10,3,2,5,4,7,6,9,8,1] => ?
 => ? => ? = 5
[2,1,6,5,4,3,8,7,10,9] => ?
 => ? => ? = 5
[8,5,4,3,2,7,6,1,10,9] => ?
 => ? => ? = 5
[10,5,4,3,2,7,6,9,8,1] => ?
 => ? => ? = 5
[2,1,8,5,4,7,6,3,10,9] => ?
 => ? => ? = 5
[8,7,4,3,6,5,2,1,10,9] => ?
 => ? => ? = 5
[10,7,4,3,6,5,2,9,8,1] => ?
 => ? => ? = 5
[2,1,10,5,4,7,6,9,8,3] => ?
 => ? => ? = 5
[10,9,4,3,6,5,8,7,2,1] => ?
 => ? => ? = 5
[2,1,4,3,8,7,6,5,10,9] => ?
 => ? => ? = 5
[8,3,2,7,6,5,4,1,10,9] => ?
 => ? => ? = 5
[10,3,2,7,6,5,4,9,8,1] => ?
 => ? => ? = 5
[2,1,8,7,6,5,4,3,10,9] => ?
 => ? => ? = 5
[10,7,6,5,4,3,2,9,8,1] => ?
 => ? => ? = 5
[2,1,10,7,6,5,4,9,8,3] => ?
 => ? => ? = 5
[10,9,6,5,4,3,8,7,2,1] => ?
 => ? => ? = 5
[2,1,4,3,10,7,6,9,8,5] => ?
 => ? => ? = 5
[4,3,2,1,10,7,6,9,8,5] => ?
 => ? => ? = 5
[10,3,2,9,6,5,8,7,4,1] => ?
 => ? => ? = 5
[2,1,10,9,6,5,8,7,4,3] => ?
 => ? => ? = 5
[10,9,8,5,4,7,6,3,2,1] => ?
 => ? => ? = 5
[2,1,4,3,6,5,10,9,8,7] => ?
 => ? => ? = 5
[4,3,2,1,6,5,10,9,8,7] => ?
 => ? => ? = 5
[6,3,2,5,4,1,10,9,8,7] => ?
 => ? => ? = 5
[10,3,2,5,4,9,8,7,6,1] => ?
 => ? => ? = 5
[2,1,6,5,4,3,10,9,8,7] => ?
 => ? => ? = 5
[10,5,4,3,2,9,8,7,6,1] => ?
 => ? => ? = 5
[2,1,10,5,4,9,8,7,6,3] => ?
 => ? => ? = 5
[10,9,4,3,8,7,6,5,2,1] => ?
 => ? => ? = 5
[2,1,4,3,10,9,8,7,6,5] => ?
 => ? => ? = 5
[10,3,2,9,8,7,6,5,4,1] => ?
 => ? => ? = 5
[] => []
 =>  => ? = 0
[2,1,4,3,6,5,8,7,12,11,10,9] => ?
 => ? => ? = 6
[2,1,4,3,6,5,10,9,8,7,12,11] => ?
 => ? => ? = 6
[2,1,4,3,6,5,12,9,8,11,10,7] => ?
 => ? => ? = 6
[2,1,4,3,6,5,12,11,10,9,8,7] => ?
 => ? => ? = 6
[2,1,4,3,8,7,6,5,10,9,12,11] => ?
 => ? => ? = 6
[2,1,4,3,8,7,6,5,12,11,10,9] => ?
 => ? => ? = 6
[2,1,4,3,10,7,6,9,8,5,12,11] => ?
 => ? => ? = 6
[2,1,4,3,12,7,6,9,8,11,10,5] => ?
 => ? => ? = 6
[2,1,4,3,12,7,6,11,10,9,8,5] => ?
 => ? => ? = 6
[2,1,4,3,10,9,8,7,6,5,12,11] => ?
 => ? => ? = 6
[2,1,4,3,12,9,8,7,6,11,10,5] => ?
 => ? => ? = 6
[2,1,4,3,12,11,8,7,10,9,6,5] => ?
 => ? => ? = 6
[2,1,4,3,12,11,10,9,8,7,6,5] => ?
 => ? => ? = 6
[2,1,6,5,4,3,8,7,10,9,12,11] => ?
 => ? => ? = 6
[2,1,6,5,4,3,8,7,12,11,10,9] => ?
 => ? => ? = 6
[2,1,6,5,4,3,10,9,8,7,12,11] => ?
 => ? => ? = 6

Description

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

Matching statistic: St000734

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 91%

Values

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

Description

The last entry in the first row of a standard tableau.

Matching statistic: St000507
(load all 2 compositions to match this statistic)

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%

Values

[1] => [1]
 => [[1]]
 => [[1]]
 => 1
[1,2] => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2
[2,1] => [2]
 => [[1,2]]
 => [[1],[2]]
 => 1
[1,2,3] => [1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 3
[1,3,2] => [2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 2
[2,1,3] => [2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 2
[2,3,1] => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 1
[3,1,2] => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 1
[3,2,1] => [2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 2
[1,2,3,4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 4
[1,2,4,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 3
[1,3,2,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 3
[1,3,4,2] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 2
[1,4,2,3] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 2
[1,4,3,2] => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 3
[2,1,3,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 3
[2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[2,3,1,4] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 2
[2,3,4,1] => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 1
[2,4,1,3] => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 1
[2,4,3,1] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 2
[3,1,2,4] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 2
[3,1,4,2] => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 1
[3,2,1,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 3
[3,2,4,1] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 2
[3,4,1,2] => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[3,4,2,1] => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 1
[4,1,2,3] => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 1
[4,1,3,2] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 2
[4,2,1,3] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 2
[4,2,3,1] => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 3
[4,3,1,2] => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 1
[4,3,2,1] => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[1,2,3,4,5] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => 5
[1,2,3,5,4] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 4
[1,2,4,3,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 4
[1,2,4,5,3] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 3
[1,2,5,3,4] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 3
[1,2,5,4,3] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 4
[1,3,2,4,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 4
[1,3,2,5,4] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 3
[1,3,4,2,5] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 3
[1,3,4,5,2] => [4,1]
 => [[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => 2
[1,3,5,2,4] => [4,1]
 => [[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => 2
[1,3,5,4,2] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 3
[1,4,2,3,5] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 3
[1,4,2,5,3] => [4,1]
 => [[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => 2
[1,4,3,2,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 4
[1,4,3,5,2] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 3
[1,4,5,2,3] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 3
[6,3,2,5,4,1,8,7,10,9] => ?
 => ?
 => ?
 => ? = 5
[8,3,2,5,4,7,6,1,10,9] => ?
 => ?
 => ?
 => ? = 5
[10,3,2,5,4,7,6,9,8,1] => ?
 => ?
 => ?
 => ? = 5
[2,1,6,5,4,3,8,7,10,9] => ?
 => ?
 => ?
 => ? = 5
[8,5,4,3,2,7,6,1,10,9] => ?
 => ?
 => ?
 => ? = 5
[10,5,4,3,2,7,6,9,8,1] => ?
 => ?
 => ?
 => ? = 5
[2,1,8,5,4,7,6,3,10,9] => ?
 => ?
 => ?
 => ? = 5
[8,7,4,3,6,5,2,1,10,9] => ?
 => ?
 => ?
 => ? = 5
[10,7,4,3,6,5,2,9,8,1] => ?
 => ?
 => ?
 => ? = 5
[2,1,10,5,4,7,6,9,8,3] => ?
 => ?
 => ?
 => ? = 5
[10,9,4,3,6,5,8,7,2,1] => ?
 => ?
 => ?
 => ? = 5
[2,1,4,3,8,7,6,5,10,9] => ?
 => ?
 => ?
 => ? = 5
[8,3,2,7,6,5,4,1,10,9] => ?
 => ?
 => ?
 => ? = 5
[10,3,2,7,6,5,4,9,8,1] => ?
 => ?
 => ?
 => ? = 5
[2,1,8,7,6,5,4,3,10,9] => ?
 => ?
 => ?
 => ? = 5
[10,7,6,5,4,3,2,9,8,1] => ?
 => ?
 => ?
 => ? = 5
[2,1,10,7,6,5,4,9,8,3] => ?
 => ?
 => ?
 => ? = 5
[10,9,6,5,4,3,8,7,2,1] => ?
 => ?
 => ?
 => ? = 5
[2,1,4,3,10,7,6,9,8,5] => ?
 => ?
 => ?
 => ? = 5
[4,3,2,1,10,7,6,9,8,5] => ?
 => ?
 => ?
 => ? = 5
[10,3,2,9,6,5,8,7,4,1] => ?
 => ?
 => ?
 => ? = 5
[2,1,10,9,6,5,8,7,4,3] => ?
 => ?
 => ?
 => ? = 5
[10,9,8,5,4,7,6,3,2,1] => ?
 => ?
 => ?
 => ? = 5
[2,1,4,3,6,5,10,9,8,7] => ?
 => ?
 => ?
 => ? = 5
[4,3,2,1,6,5,10,9,8,7] => ?
 => ?
 => ?
 => ? = 5
[6,3,2,5,4,1,10,9,8,7] => ?
 => ?
 => ?
 => ? = 5
[10,3,2,5,4,9,8,7,6,1] => ?
 => ?
 => ?
 => ? = 5
[2,1,6,5,4,3,10,9,8,7] => ?
 => ?
 => ?
 => ? = 5
[10,5,4,3,2,9,8,7,6,1] => ?
 => ?
 => ?
 => ? = 5
[2,1,10,5,4,9,8,7,6,3] => ?
 => ?
 => ?
 => ? = 5
[10,9,4,3,8,7,6,5,2,1] => ?
 => ?
 => ?
 => ? = 5
[2,1,4,3,10,9,8,7,6,5] => ?
 => ?
 => ?
 => ? = 5
[10,3,2,9,8,7,6,5,4,1] => ?
 => ?
 => ?
 => ? = 5
[2,1,4,3,6,5,8,7,12,11,10,9] => ?
 => ?
 => ?
 => ? = 6
[2,1,4,3,6,5,10,9,8,7,12,11] => ?
 => ?
 => ?
 => ? = 6
[2,1,4,3,6,5,12,9,8,11,10,7] => ?
 => ?
 => ?
 => ? = 6
[2,1,4,3,6,5,12,11,10,9,8,7] => ?
 => ?
 => ?
 => ? = 6
[2,1,4,3,8,7,6,5,10,9,12,11] => ?
 => ?
 => ?
 => ? = 6
[2,1,4,3,8,7,6,5,12,11,10,9] => ?
 => ?
 => ?
 => ? = 6
[2,1,4,3,10,7,6,9,8,5,12,11] => ?
 => ?
 => ?
 => ? = 6
[2,1,4,3,12,7,6,9,8,11,10,5] => ?
 => ?
 => ?
 => ? = 6
[2,1,4,3,12,7,6,11,10,9,8,5] => ?
 => ?
 => ?
 => ? = 6
[2,1,4,3,10,9,8,7,6,5,12,11] => ?
 => ?
 => ?
 => ? = 6
[2,1,4,3,12,9,8,7,6,11,10,5] => ?
 => ?
 => ?
 => ? = 6
[2,1,4,3,12,11,8,7,10,9,6,5] => ?
 => ?
 => ?
 => ? = 6
[2,1,4,3,12,11,10,9,8,7,6,5] => ?
 => ?
 => ?
 => ? = 6
[2,1,6,5,4,3,8,7,10,9,12,11] => ?
 => ?
 => ?
 => ? = 6
[2,1,6,5,4,3,8,7,12,11,10,9] => ?
 => ?
 => ?
 => ? = 6
[2,1,6,5,4,3,10,9,8,7,12,11] => ?
 => ?
 => ?
 => ? = 6
[2,1,6,5,4,3,12,9,8,11,10,7] => ?
 => ?
 => ?
 => ? = 6

Description

The number of ascents of a standard tableau. Entry

$i$ of a standard Young tableau is an '''ascent''' if

$i+1$ appears to the right or above

$i$ in the tableau (with respect to the English notation for tableaux).

Matching statistic: St000007
(load all 12 compositions to match this statistic)

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%

Values

[1] => [1]
 => [[1]]
 => [1] => 1
[1,2] => [1,1]
 => [[1],[2]]
 => [2,1] => 2
[2,1] => [2]
 => [[1,2]]
 => [1,2] => 1
[1,2,3] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3
[1,3,2] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[2,1,3] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[2,3,1] => [3]
 => [[1,2,3]]
 => [1,2,3] => 1
[3,1,2] => [3]
 => [[1,2,3]]
 => [1,2,3] => 1
[3,2,1] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[1,2,3,4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 4
[1,2,4,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 3
[1,3,2,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 3
[1,3,4,2] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 2
[1,4,2,3] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 2
[1,4,3,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 3
[2,1,3,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 3
[2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
[2,3,1,4] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 2
[2,3,4,1] => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1
[2,4,1,3] => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1
[2,4,3,1] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 2
[3,1,2,4] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 2
[3,1,4,2] => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1
[3,2,1,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 3
[3,2,4,1] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 2
[3,4,1,2] => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
[3,4,2,1] => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1
[4,1,2,3] => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1
[4,1,3,2] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 2
[4,2,1,3] => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 2
[4,2,3,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 3
[4,3,1,2] => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1
[4,3,2,1] => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
[1,2,3,4,5] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 5
[1,2,3,5,4] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 4
[1,2,4,3,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 4
[1,2,4,5,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 3
[1,2,5,3,4] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 3
[1,2,5,4,3] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 4
[1,3,2,4,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 4
[1,3,2,5,4] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 3
[1,3,4,2,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 3
[1,3,4,5,2] => [4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 2
[1,3,5,2,4] => [4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 2
[1,3,5,4,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 3
[1,4,2,3,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 3
[1,4,2,5,3] => [4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 2
[1,4,3,2,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 4
[1,4,3,5,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 3
[1,4,5,2,3] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 3
[6,3,2,5,4,1,8,7,10,9] => ?
 => ?
 => ? => ? = 5
[8,3,2,5,4,7,6,1,10,9] => ?
 => ?
 => ? => ? = 5
[10,3,2,5,4,7,6,9,8,1] => ?
 => ?
 => ? => ? = 5
[2,1,6,5,4,3,8,7,10,9] => ?
 => ?
 => ? => ? = 5
[8,5,4,3,2,7,6,1,10,9] => ?
 => ?
 => ? => ? = 5
[10,5,4,3,2,7,6,9,8,1] => ?
 => ?
 => ? => ? = 5
[2,1,8,5,4,7,6,3,10,9] => ?
 => ?
 => ? => ? = 5
[8,7,4,3,6,5,2,1,10,9] => ?
 => ?
 => ? => ? = 5
[10,7,4,3,6,5,2,9,8,1] => ?
 => ?
 => ? => ? = 5
[2,1,10,5,4,7,6,9,8,3] => ?
 => ?
 => ? => ? = 5
[10,9,4,3,6,5,8,7,2,1] => ?
 => ?
 => ? => ? = 5
[2,1,4,3,8,7,6,5,10,9] => ?
 => ?
 => ? => ? = 5
[8,3,2,7,6,5,4,1,10,9] => ?
 => ?
 => ? => ? = 5
[10,3,2,7,6,5,4,9,8,1] => ?
 => ?
 => ? => ? = 5
[2,1,8,7,6,5,4,3,10,9] => ?
 => ?
 => ? => ? = 5
[10,7,6,5,4,3,2,9,8,1] => ?
 => ?
 => ? => ? = 5
[2,1,10,7,6,5,4,9,8,3] => ?
 => ?
 => ? => ? = 5
[10,9,6,5,4,3,8,7,2,1] => ?
 => ?
 => ? => ? = 5
[2,1,4,3,10,7,6,9,8,5] => ?
 => ?
 => ? => ? = 5
[4,3,2,1,10,7,6,9,8,5] => ?
 => ?
 => ? => ? = 5
[10,3,2,9,6,5,8,7,4,1] => ?
 => ?
 => ? => ? = 5
[2,1,10,9,6,5,8,7,4,3] => ?
 => ?
 => ? => ? = 5
[10,9,8,5,4,7,6,3,2,1] => ?
 => ?
 => ? => ? = 5
[2,1,4,3,6,5,10,9,8,7] => ?
 => ?
 => ? => ? = 5
[4,3,2,1,6,5,10,9,8,7] => ?
 => ?
 => ? => ? = 5
[6,3,2,5,4,1,10,9,8,7] => ?
 => ?
 => ? => ? = 5
[10,3,2,5,4,9,8,7,6,1] => ?
 => ?
 => ? => ? = 5
[2,1,6,5,4,3,10,9,8,7] => ?
 => ?
 => ? => ? = 5
[10,5,4,3,2,9,8,7,6,1] => ?
 => ?
 => ? => ? = 5
[2,1,10,5,4,9,8,7,6,3] => ?
 => ?
 => ? => ? = 5
[10,9,4,3,8,7,6,5,2,1] => ?
 => ?
 => ? => ? = 5
[2,1,4,3,10,9,8,7,6,5] => ?
 => ?
 => ? => ? = 5
[10,3,2,9,8,7,6,5,4,1] => ?
 => ?
 => ? => ? = 5
[2,1,4,3,6,5,8,7,10,9,12,11] => [2,2,2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
 => [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 6
[2,1,4,3,6,5,8,7,12,11,10,9] => ?
 => ?
 => ? => ? = 6
[2,1,4,3,6,5,10,9,8,7,12,11] => ?
 => ?
 => ? => ? = 6
[2,1,4,3,6,5,12,9,8,11,10,7] => ?
 => ?
 => ? => ? = 6
[2,1,4,3,6,5,12,11,10,9,8,7] => ?
 => ?
 => ? => ? = 6
[2,1,4,3,8,7,6,5,10,9,12,11] => ?
 => ?
 => ? => ? = 6
[2,1,4,3,8,7,6,5,12,11,10,9] => ?
 => ?
 => ? => ? = 6
[2,1,4,3,10,7,6,9,8,5,12,11] => ?
 => ?
 => ? => ? = 6
[2,1,4,3,12,7,6,9,8,11,10,5] => ?
 => ?
 => ? => ? = 6
[2,1,4,3,12,7,6,11,10,9,8,5] => ?
 => ?
 => ? => ? = 6
[2,1,4,3,10,9,8,7,6,5,12,11] => ?
 => ?
 => ? => ? = 6
[2,1,4,3,12,9,8,7,6,11,10,5] => ?
 => ?
 => ? => ? = 6
[2,1,4,3,12,11,8,7,10,9,6,5] => ?
 => ?
 => ? => ? = 6
[2,1,4,3,12,11,10,9,8,7,6,5] => ?
 => ?
 => ? => ? = 6
[2,1,6,5,4,3,8,7,10,9,12,11] => ?
 => ?
 => ? => ? = 6
[2,1,6,5,4,3,8,7,12,11,10,9] => ?
 => ?
 => ? => ? = 6
[2,1,6,5,4,3,10,9,8,7,12,11] => ?
 => ?
 => ? => ? = 6

Description

The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern

$([1], {(1,1)})$ , i.e., the upper right quadrant is shaded, see [1].

Matching statistic: St001777

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St001777: Integer compositions ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 91%

Values

[1] => [1]
 => [[1]]
 => [1] => 0 = 1 - 1
[1,2] => [1,1]
 => [[1],[2]]
 => [1,1] => 1 = 2 - 1
[2,1] => [2]
 => [[1,2]]
 => [2] => 0 = 1 - 1
[1,2,3] => [1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 2 = 3 - 1
[1,3,2] => [2,1]
 => [[1,2],[3]]
 => [2,1] => 1 = 2 - 1
[2,1,3] => [2,1]
 => [[1,2],[3]]
 => [2,1] => 1 = 2 - 1
[2,3,1] => [3]
 => [[1,2,3]]
 => [3] => 0 = 1 - 1
[3,1,2] => [3]
 => [[1,2,3]]
 => [3] => 0 = 1 - 1
[3,2,1] => [2,1]
 => [[1,2],[3]]
 => [2,1] => 1 = 2 - 1
[1,2,3,4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 3 = 4 - 1
[1,2,4,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 2 = 3 - 1
[1,3,2,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 2 = 3 - 1
[1,3,4,2] => [3,1]
 => [[1,2,3],[4]]
 => [3,1] => 1 = 2 - 1
[1,4,2,3] => [3,1]
 => [[1,2,3],[4]]
 => [3,1] => 1 = 2 - 1
[1,4,3,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 2 = 3 - 1
[2,1,3,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 2 = 3 - 1
[2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => [2,2] => 1 = 2 - 1
[2,3,1,4] => [3,1]
 => [[1,2,3],[4]]
 => [3,1] => 1 = 2 - 1
[2,3,4,1] => [4]
 => [[1,2,3,4]]
 => [4] => 0 = 1 - 1
[2,4,1,3] => [4]
 => [[1,2,3,4]]
 => [4] => 0 = 1 - 1
[2,4,3,1] => [3,1]
 => [[1,2,3],[4]]
 => [3,1] => 1 = 2 - 1
[3,1,2,4] => [3,1]
 => [[1,2,3],[4]]
 => [3,1] => 1 = 2 - 1
[3,1,4,2] => [4]
 => [[1,2,3,4]]
 => [4] => 0 = 1 - 1
[3,2,1,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 2 = 3 - 1
[3,2,4,1] => [3,1]
 => [[1,2,3],[4]]
 => [3,1] => 1 = 2 - 1
[3,4,1,2] => [2,2]
 => [[1,2],[3,4]]
 => [2,2] => 1 = 2 - 1
[3,4,2,1] => [4]
 => [[1,2,3,4]]
 => [4] => 0 = 1 - 1
[4,1,2,3] => [4]
 => [[1,2,3,4]]
 => [4] => 0 = 1 - 1
[4,1,3,2] => [3,1]
 => [[1,2,3],[4]]
 => [3,1] => 1 = 2 - 1
[4,2,1,3] => [3,1]
 => [[1,2,3],[4]]
 => [3,1] => 1 = 2 - 1
[4,2,3,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 2 = 3 - 1
[4,3,1,2] => [4]
 => [[1,2,3,4]]
 => [4] => 0 = 1 - 1
[4,3,2,1] => [2,2]
 => [[1,2],[3,4]]
 => [2,2] => 1 = 2 - 1
[1,2,3,4,5] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => 4 = 5 - 1
[1,2,3,5,4] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => 3 = 4 - 1
[1,2,4,3,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => 3 = 4 - 1
[1,2,4,5,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [3,1,1] => 2 = 3 - 1
[1,2,5,3,4] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [3,1,1] => 2 = 3 - 1
[1,2,5,4,3] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => 3 = 4 - 1
[1,3,2,4,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => 3 = 4 - 1
[1,3,2,5,4] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [2,2,1] => 2 = 3 - 1
[1,3,4,2,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [3,1,1] => 2 = 3 - 1
[1,3,4,5,2] => [4,1]
 => [[1,2,3,4],[5]]
 => [4,1] => 1 = 2 - 1
[1,3,5,2,4] => [4,1]
 => [[1,2,3,4],[5]]
 => [4,1] => 1 = 2 - 1
[1,3,5,4,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [3,1,1] => 2 = 3 - 1
[1,4,2,3,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [3,1,1] => 2 = 3 - 1
[1,4,2,5,3] => [4,1]
 => [[1,2,3,4],[5]]
 => [4,1] => 1 = 2 - 1
[1,4,3,2,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => 3 = 4 - 1
[1,4,3,5,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [3,1,1] => 2 = 3 - 1
[1,4,5,2,3] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [2,2,1] => 2 = 3 - 1
[6,3,2,5,4,1,8,7,10,9] => ?
 => ?
 => ? => ? = 5 - 1
[8,3,2,5,4,7,6,1,10,9] => ?
 => ?
 => ? => ? = 5 - 1
[10,3,2,5,4,7,6,9,8,1] => ?
 => ?
 => ? => ? = 5 - 1
[2,1,6,5,4,3,8,7,10,9] => ?
 => ?
 => ? => ? = 5 - 1
[8,5,4,3,2,7,6,1,10,9] => ?
 => ?
 => ? => ? = 5 - 1
[10,5,4,3,2,7,6,9,8,1] => ?
 => ?
 => ? => ? = 5 - 1
[2,1,8,5,4,7,6,3,10,9] => ?
 => ?
 => ? => ? = 5 - 1
[8,7,4,3,6,5,2,1,10,9] => ?
 => ?
 => ? => ? = 5 - 1
[10,7,4,3,6,5,2,9,8,1] => ?
 => ?
 => ? => ? = 5 - 1
[2,1,10,5,4,7,6,9,8,3] => ?
 => ?
 => ? => ? = 5 - 1
[10,9,4,3,6,5,8,7,2,1] => ?
 => ?
 => ? => ? = 5 - 1
[2,1,4,3,8,7,6,5,10,9] => ?
 => ?
 => ? => ? = 5 - 1
[8,3,2,7,6,5,4,1,10,9] => ?
 => ?
 => ? => ? = 5 - 1
[10,3,2,7,6,5,4,9,8,1] => ?
 => ?
 => ? => ? = 5 - 1
[2,1,8,7,6,5,4,3,10,9] => ?
 => ?
 => ? => ? = 5 - 1
[10,7,6,5,4,3,2,9,8,1] => ?
 => ?
 => ? => ? = 5 - 1
[2,1,10,7,6,5,4,9,8,3] => ?
 => ?
 => ? => ? = 5 - 1
[10,9,6,5,4,3,8,7,2,1] => ?
 => ?
 => ? => ? = 5 - 1
[2,1,4,3,10,7,6,9,8,5] => ?
 => ?
 => ? => ? = 5 - 1
[4,3,2,1,10,7,6,9,8,5] => ?
 => ?
 => ? => ? = 5 - 1
[10,3,2,9,6,5,8,7,4,1] => ?
 => ?
 => ? => ? = 5 - 1
[2,1,10,9,6,5,8,7,4,3] => ?
 => ?
 => ? => ? = 5 - 1
[10,9,8,5,4,7,6,3,2,1] => ?
 => ?
 => ? => ? = 5 - 1
[2,1,4,3,6,5,10,9,8,7] => ?
 => ?
 => ? => ? = 5 - 1
[4,3,2,1,6,5,10,9,8,7] => ?
 => ?
 => ? => ? = 5 - 1
[6,3,2,5,4,1,10,9,8,7] => ?
 => ?
 => ? => ? = 5 - 1
[10,3,2,5,4,9,8,7,6,1] => ?
 => ?
 => ? => ? = 5 - 1
[2,1,6,5,4,3,10,9,8,7] => ?
 => ?
 => ? => ? = 5 - 1
[10,5,4,3,2,9,8,7,6,1] => ?
 => ?
 => ? => ? = 5 - 1
[2,1,10,5,4,9,8,7,6,3] => ?
 => ?
 => ? => ? = 5 - 1
[10,9,4,3,8,7,6,5,2,1] => ?
 => ?
 => ? => ? = 5 - 1
[2,1,4,3,10,9,8,7,6,5] => ?
 => ?
 => ? => ? = 5 - 1
[10,3,2,9,8,7,6,5,4,1] => ?
 => ?
 => ? => ? = 5 - 1
[] => []
 => []
 => [0] => ? = 0 - 1
[2,1,4,3,6,5,8,7,10,9,12,11] => [2,2,2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
 => [2,2,2,2,2,2] => ? = 6 - 1
[2,1,4,3,6,5,8,7,12,11,10,9] => ?
 => ?
 => ? => ? = 6 - 1
[2,1,4,3,6,5,10,9,8,7,12,11] => ?
 => ?
 => ? => ? = 6 - 1
[2,1,4,3,6,5,12,9,8,11,10,7] => ?
 => ?
 => ? => ? = 6 - 1
[2,1,4,3,6,5,12,11,10,9,8,7] => ?
 => ?
 => ? => ? = 6 - 1
[2,1,4,3,8,7,6,5,10,9,12,11] => ?
 => ?
 => ? => ? = 6 - 1
[2,1,4,3,8,7,6,5,12,11,10,9] => ?
 => ?
 => ? => ? = 6 - 1
[2,1,4,3,10,7,6,9,8,5,12,11] => ?
 => ?
 => ? => ? = 6 - 1
[2,1,4,3,12,7,6,9,8,11,10,5] => ?
 => ?
 => ? => ? = 6 - 1
[2,1,4,3,12,7,6,11,10,9,8,5] => ?
 => ?
 => ? => ? = 6 - 1
[2,1,4,3,10,9,8,7,6,5,12,11] => ?
 => ?
 => ? => ? = 6 - 1
[2,1,4,3,12,9,8,7,6,11,10,5] => ?
 => ?
 => ? => ? = 6 - 1
[2,1,4,3,12,11,8,7,10,9,6,5] => ?
 => ?
 => ? => ? = 6 - 1
[2,1,4,3,12,11,10,9,8,7,6,5] => ?
 => ?
 => ? => ? = 6 - 1
[2,1,6,5,4,3,8,7,10,9,12,11] => ?
 => ?
 => ? => ? = 6 - 1
[2,1,6,5,4,3,8,7,12,11,10,9] => ?
 => ?
 => ? => ? = 6 - 1

Description

The number of weak descents in an integer composition. A weak descent of an integer composition

$\alpha=(a_1, \dots, a_n)$ is an index

$1\leq i < n$ such that

$a_i \geq a_{i+1}$ .

The following 101 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001372The length of a longest cyclic run of ones of a binary word. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000676The number of odd rises of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. 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. St001462The number of factors of a standard tableaux under concatenation. St000097The order of the largest clique of the graph. St001581The achromatic number of a graph. St000678The number of up steps after the last double rise of a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St000011The number of touch points (or returns) of a Dyck path. St000809The reduced reflection length of the permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000306The bounce count of a Dyck path. St000653The last descent of a permutation. St001480The number of simple summands of the module J^2/J^3. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000015The number of peaks of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000331The number of upper interactions of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000216The absolute length of a permutation. St000098The chromatic number of a graph. St000925The number of topologically connected components of a set partition. St000006The dinv of a Dyck path. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000245The number of ascents of a permutation. St000912The number of maximal antichains in a poset. St000167The number of leaves of an ordered tree. St000546The number of global descents of a permutation. St000069The number of maximal elements of a poset. St000105The number of blocks in the set partition. St000172The Grundy number of a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St000024The number of double up and double down steps of a Dyck path. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000292The number of ascents of a binary word. St000068The number of minimal elements in a poset. St000702The number of weak deficiencies of a permutation. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000084The number of subtrees. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000822The Hadwiger number of the graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001225The vector space dimension of the first extension group between J and itself when J is 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000061The number of nodes on the left branch of a binary tree. St000083The number of left oriented leafs of a binary tree except the first one. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001812The biclique partition number of a graph. St001152The number of pairs with even minimum in a perfect matching. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001330The hat guessing number of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000942The number of critical left to right maxima of the parking functions. St001712The number of natural descents of a standard Young tableau. 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

$eA$ .