- FindStat

Your data matches 212 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000245
(load all 69 compositions to match this statistic)

Mapping

St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of ascents of a permutation.

Matching statistic: St000377
(load all 2 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The dinv defect of an integer partition. This is the number of cells

$c$ in the diagram of an integer partition

$\lambda$ for which

$\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$ .

Matching statistic: St001176
(load all 4 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.

Matching statistic: St000010
(load all 12 compositions to match this statistic)

Mapping

Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the partition.

Matching statistic: St000147
(load all 6 compositions to match this statistic)

Mapping

Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest part of an integer partition.

Matching statistic: St000378

Mapping

Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

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

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: St000288
(load all 142 compositions to match this statistic)

Mapping

Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 99%●distinct values known / distinct values provided: 91%

Values

[1] => [1]
 => []
 =>  => ? = 0
[1,2] => [1,1]
 => [1]
 => 10 => 1
[2,1] => [2]
 => []
 =>  => ? = 0
[1,2,3] => [1,1,1]
 => [1,1]
 => 110 => 2
[1,3,2] => [2,1]
 => [1]
 => 10 => 1
[2,1,3] => [2,1]
 => [1]
 => 10 => 1
[2,3,1] => [2,1]
 => [1]
 => 10 => 1
[3,1,2] => [2,1]
 => [1]
 => 10 => 1
[3,2,1] => [3]
 => []
 =>  => ? = 0
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1110 => 3
[1,2,4,3] => [2,1,1]
 => [1,1]
 => 110 => 2
[1,3,2,4] => [2,1,1]
 => [1,1]
 => 110 => 2
[1,3,4,2] => [2,1,1]
 => [1,1]
 => 110 => 2
[1,4,2,3] => [2,1,1]
 => [1,1]
 => 110 => 2
[1,4,3,2] => [3,1]
 => [1]
 => 10 => 1
[2,1,3,4] => [2,1,1]
 => [1,1]
 => 110 => 2
[2,1,4,3] => [2,2]
 => [2]
 => 100 => 1
[2,3,1,4] => [2,1,1]
 => [1,1]
 => 110 => 2
[2,3,4,1] => [2,1,1]
 => [1,1]
 => 110 => 2
[2,4,1,3] => [2,1,1]
 => [1,1]
 => 110 => 2
[2,4,3,1] => [3,1]
 => [1]
 => 10 => 1
[3,1,2,4] => [2,1,1]
 => [1,1]
 => 110 => 2
[3,1,4,2] => [2,2]
 => [2]
 => 100 => 1
[3,2,1,4] => [3,1]
 => [1]
 => 10 => 1
[3,2,4,1] => [3,1]
 => [1]
 => 10 => 1
[3,4,1,2] => [2,1,1]
 => [1,1]
 => 110 => 2
[3,4,2,1] => [3,1]
 => [1]
 => 10 => 1
[4,1,2,3] => [2,1,1]
 => [1,1]
 => 110 => 2
[4,1,3,2] => [3,1]
 => [1]
 => 10 => 1
[4,2,1,3] => [3,1]
 => [1]
 => 10 => 1
[4,2,3,1] => [3,1]
 => [1]
 => 10 => 1
[4,3,1,2] => [3,1]
 => [1]
 => 10 => 1
[4,3,2,1] => [4]
 => []
 =>  => ? = 0
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 4
[1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 1110 => 3
[1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => 1110 => 3
[1,2,4,5,3] => [2,1,1,1]
 => [1,1,1]
 => 1110 => 3
[1,2,5,3,4] => [2,1,1,1]
 => [1,1,1]
 => 1110 => 3
[1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 110 => 2
[1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => 1110 => 3
[1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 1010 => 2
[1,3,4,2,5] => [2,1,1,1]
 => [1,1,1]
 => 1110 => 3
[1,3,4,5,2] => [2,1,1,1]
 => [1,1,1]
 => 1110 => 3
[1,3,5,2,4] => [2,1,1,1]
 => [1,1,1]
 => 1110 => 3
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 110 => 2
[1,4,2,3,5] => [2,1,1,1]
 => [1,1,1]
 => 1110 => 3
[1,4,2,5,3] => [2,2,1]
 => [2,1]
 => 1010 => 2
[1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 110 => 2
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => 110 => 2
[1,4,5,2,3] => [2,1,1,1]
 => [1,1,1]
 => 1110 => 3
[1,4,5,3,2] => [3,1,1]
 => [1,1]
 => 110 => 2
[1,5,2,3,4] => [2,1,1,1]
 => [1,1,1]
 => 1110 => 3
[1,5,2,4,3] => [3,1,1]
 => [1,1]
 => 110 => 2
[1,5,3,2,4] => [3,1,1]
 => [1,1]
 => 110 => 2
[5,4,3,2,1] => [5]
 => []
 =>  => ? = 0
[6,5,4,3,2,1] => [6]
 => []
 =>  => ? = 0
[7,6,5,4,3,2,1] => [7]
 => []
 =>  => ? = 0
[8,7,6,5,4,3,2,1] => [8]
 => []
 =>  => ? = 0
[8,7,5,6,4,2,3,1] => ?
 => ?
 => ? => ? = 2
[10,9,8,7,6,5,4,3,2,1] => [10]
 => []
 =>  => ? = 0
[] => []
 => ?
 => ? => ? = 0
[12,11,10,9,8,7,6,5,4,3,2,1] => [12]
 => []
 =>  => ? = 0
[9,8,7,6,5,4,3,2,1] => [9]
 => []
 =>  => ? = 0
[1,8,9,7,6,5,4,3,2] => ?
 => ?
 => ? => ? = 2
[6,8,7,5,4,3,2,1,9] => ?
 => ?
 => ? => ? = 2
[1,8,7,9,6,5,4,3,2] => ?
 => ?
 => ? => ? = 2
[1,7,9,8,6,5,4,3,2] => ?
 => ?
 => ? => ? = 2
[1,9,10,8,7,6,5,4,3,2] => ?
 => ?
 => ? => ? = 2
[4,7,2,5,8,3,6,1] => ?
 => ?
 => ? => ? = 4
[5,2,6,8,4,3,7,1] => ?
 => ?
 => ? => ? = 3
[5,7,1,8,4,3,2,6] => ?
 => ?
 => ? => ? = 3
[8,5,6,7,1,3,2,4] => ?
 => ?
 => ? => ? = 4
[8,4,6,1,7,3,2,5] => ?
 => ?
 => ? => ? = 3
[5,7,6,8,2,3,4,1] => ?
 => ?
 => ? => ? = 4
[5,7,2,8,4,3,6,1] => ?
 => ?
 => ? => ? = 3
[8,2,6,5,1,3,7,4] => ?
 => ?
 => ? => ? = 3
[5,7,6,8,4,1,2,3] => ?
 => ?
 => ? => ? = 4

Description

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

Matching statistic: St000733
(load all 4 compositions to match this statistic)

Mapping

Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 91% ●values known / values provided: 99%●distinct values known / distinct values provided: 91%

Values

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

Description

The row containing the largest entry of a standard tableau.

Matching statistic: St000507
(load all 11 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%

Values

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

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: St000157
(load all 22 compositions to match this statistic)

Mapping

Mp00069: Permutations —complement⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of descents of a standard tableau. Entry

$i$ of a standard Young tableau is a descent if

$i+1$ appears in a row below the row of

$i$ .

The following 202 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001250The number of parts of a partition that are not congruent 0 modulo 3. St000806The semiperimeter of the associated bargraph. St000734The last entry in the first row of a standard tableau. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000546The number of global descents of a permutation. St001777The number of weak descents in an integer composition. 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. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000006The dinv of a Dyck path. St001372The length of a longest cyclic run of ones of a binary word. St000691The number of changes 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. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000097The order of the largest clique of the graph. St001462The number of factors of a standard tableaux under concatenation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001389The number of partitions of the same length below the given integer partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000093The cardinality of a maximal independent set of vertices of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000676The number of odd rises of a Dyck path. St000013The height of a Dyck path. St000678The number of up steps after the last double rise 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. St000053The number of valleys of the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001581The achromatic number of a graph. St000809The reduced reflection length of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000098The chromatic number of a graph. St001733The number of weak left to right maxima of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000439The position of the first down step of a Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000340The number of non-final maximal constant sub-paths of length greater than one. St000441The number of successions of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000306The bounce count of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000071The number of maximal chains in a poset. St000214The number of adjacencies of a permutation. St000502The number of successions of a set partitions. St000925The number of topologically connected components of a set partition. St000527The width of the poset. St001809The index of the step at the first peak of maximal height in a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000874The position of the last double rise in a Dyck path. St000444The length of the maximal rise of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St000105The number of blocks in the set partition. St000211The rank of the set partition. St000234The number of global ascents of a permutation. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. 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. St000025The number of initial rises of a Dyck path. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000636The hull number of a graph. St000808The number of up steps of the associated bargraph. 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: St001342The number of vertices in the center of a graph. St001494The Alon-Tarsi number of a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000068The number of minimal elements in a poset. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St000996The number of exclusive left-to-right maxima of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000153The number of adjacent cycles of a permutation. St000632The jump number of the poset. St000167The number of leaves of an ordered tree. St001461The number of topologically connected components of the chord diagram of a permutation. St000702The number of weak deficiencies of a permutation. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. 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. 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. St000015The number of peaks of a Dyck path. St000216The absolute length of a permutation. St000662The staircase size of the code of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000141The maximum drop size of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000067The inversion number of the alternating sign matrix. St000332The positive inversions of an alternating sign matrix. St000052The number of valleys of a Dyck path not on the x-axis. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000159The number of distinct parts of the integer partition. St000308The height of the tree associated to a permutation. St000292The number of ascents of a binary word. St000164The number of short pairs. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St001427The number of descents of a signed permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St000795The mad of a permutation. St000831The number of indices that are either descents or recoils. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St000021The number of descents of a permutation. St000542The number of left-to-right-minima of a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000213The number of weak exceedances (also weak excedences) of a permutation. St000325The width of the tree associated to a permutation. St000168The number of internal nodes of an ordered tree. St000316The number of non-left-to-right-maxima 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000120The number of left tunnels of a Dyck path. St000238The number of indices that are not small weak excedances. 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

$eA$ . St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000201The number of leaf nodes in a binary tree. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000314The number of left-to-right-maxima of a permutation. St000822The Hadwiger number of the graph. St000991The number of right-to-left minima of a permutation. 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. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000083The number of left oriented leafs of a binary tree except the first one. St000061The number of nodes on the left branch of a binary tree. St001812The biclique partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001323The independence gap of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001645The pebbling number of a connected graph. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001152The number of pairs with even minimum in a perfect matching. St000731The number of double exceedences of a permutation. St000746The number of pairs with odd minimum in a perfect matching. St001330The hat guessing number of a graph. St000039The number of crossings of a permutation. St001726The number of visible inversions of a permutation. St000299The number of nonisomorphic vertex-induced subtrees. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001896The number of right descents of a signed permutations. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001712The number of natural descents of a standard Young tableau. St001905The number of preferred parking spots in a parking function less than the index of the car. St001668The number of points of the poset minus the width of the poset. 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

$eA$ . St001626The number of maximal proper sublattices of a lattice.