- FindStat

Your data matches 75 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000703
(load all 28 compositions to match this statistic)

Mapping

St000703: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of deficiencies of a permutation. This is defined as

$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$ The number of exceedances is [[St000155]].

Matching statistic: St000377

Mapping

Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00204: Permutations —LLPS⟶ 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] => [1,2] => [1,1]
 => [2]
 => 0
[2,1] => [2,1] => [2]
 => [1,1]
 => 1
[1,2,3] => [1,2,3] => [1,1,1]
 => [2,1]
 => 0
[1,3,2] => [1,3,2] => [2,1]
 => [3]
 => 1
[2,1,3] => [2,1,3] => [2,1]
 => [3]
 => 1
[2,3,1] => [3,1,2] => [2,1]
 => [3]
 => 1
[3,1,2] => [3,2,1] => [3]
 => [1,1,1]
 => 2
[3,2,1] => [2,3,1] => [2,1]
 => [3]
 => 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [3,1]
 => 0
[1,2,4,3] => [1,2,4,3] => [2,1,1]
 => [2,2]
 => 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
 => [2,2]
 => 1
[1,3,4,2] => [1,4,2,3] => [2,1,1]
 => [2,2]
 => 1
[1,4,2,3] => [1,4,3,2] => [3,1]
 => [2,1,1]
 => 2
[1,4,3,2] => [1,3,4,2] => [2,1,1]
 => [2,2]
 => 1
[2,1,3,4] => [2,1,3,4] => [2,1,1]
 => [2,2]
 => 1
[2,1,4,3] => [2,1,4,3] => [2,2]
 => [4]
 => 2
[2,3,1,4] => [3,1,2,4] => [2,1,1]
 => [2,2]
 => 1
[2,3,4,1] => [4,1,2,3] => [2,1,1]
 => [2,2]
 => 1
[2,4,1,3] => [4,3,1,2] => [3,1]
 => [2,1,1]
 => 2
[2,4,3,1] => [3,4,1,2] => [2,1,1]
 => [2,2]
 => 1
[3,1,2,4] => [3,2,1,4] => [3,1]
 => [2,1,1]
 => 2
[3,1,4,2] => [4,2,1,3] => [3,1]
 => [2,1,1]
 => 2
[3,2,1,4] => [2,3,1,4] => [2,1,1]
 => [2,2]
 => 1
[3,2,4,1] => [2,4,1,3] => [2,1,1]
 => [2,2]
 => 1
[3,4,1,2] => [3,1,4,2] => [2,2]
 => [4]
 => 2
[3,4,2,1] => [4,1,3,2] => [3,1]
 => [2,1,1]
 => 2
[4,1,2,3] => [4,3,2,1] => [4]
 => [1,1,1,1]
 => 3
[4,1,3,2] => [3,4,2,1] => [3,1]
 => [2,1,1]
 => 2
[4,2,1,3] => [2,4,3,1] => [3,1]
 => [2,1,1]
 => 2
[4,2,3,1] => [2,3,4,1] => [2,1,1]
 => [2,2]
 => 1
[4,3,1,2] => [4,2,3,1] => [3,1]
 => [2,1,1]
 => 2
[4,3,2,1] => [3,2,4,1] => [3,1]
 => [2,1,1]
 => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => [3,2]
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => [3,1,1]
 => 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
 => [3,1,1]
 => 1
[1,2,4,5,3] => [1,2,5,3,4] => [2,1,1,1]
 => [3,1,1]
 => 1
[1,2,5,3,4] => [1,2,5,4,3] => [3,1,1]
 => [4,1]
 => 2
[1,2,5,4,3] => [1,2,4,5,3] => [2,1,1,1]
 => [3,1,1]
 => 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
 => [3,1,1]
 => 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
 => [2,2,1]
 => 2
[1,3,4,2,5] => [1,4,2,3,5] => [2,1,1,1]
 => [3,1,1]
 => 1
[1,3,4,5,2] => [1,5,2,3,4] => [2,1,1,1]
 => [3,1,1]
 => 1
[1,3,5,2,4] => [1,5,4,2,3] => [3,1,1]
 => [4,1]
 => 2
[1,3,5,4,2] => [1,4,5,2,3] => [2,1,1,1]
 => [3,1,1]
 => 1
[1,4,2,3,5] => [1,4,3,2,5] => [3,1,1]
 => [4,1]
 => 2
[1,4,2,5,3] => [1,5,3,2,4] => [3,1,1]
 => [4,1]
 => 2
[1,4,3,2,5] => [1,3,4,2,5] => [2,1,1,1]
 => [3,1,1]
 => 1
[1,4,3,5,2] => [1,3,5,2,4] => [2,1,1,1]
 => [3,1,1]
 => 1
[1,4,5,2,3] => [1,4,2,5,3] => [2,2,1]
 => [2,2,1]
 => 2

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

Mapping

Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00204: Permutations —LLPS⟶ 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] => [1,2] => [1,1]
 => [2]
 => 0
[2,1] => [2,1] => [2]
 => [1,1]
 => 1
[1,2,3] => [1,2,3] => [1,1,1]
 => [3]
 => 0
[1,3,2] => [1,3,2] => [2,1]
 => [2,1]
 => 1
[2,1,3] => [2,1,3] => [2,1]
 => [2,1]
 => 1
[2,3,1] => [3,1,2] => [2,1]
 => [2,1]
 => 1
[3,1,2] => [3,2,1] => [3]
 => [1,1,1]
 => 2
[3,2,1] => [2,3,1] => [2,1]
 => [2,1]
 => 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 0
[1,2,4,3] => [1,2,4,3] => [2,1,1]
 => [3,1]
 => 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
 => [3,1]
 => 1
[1,3,4,2] => [1,4,2,3] => [2,1,1]
 => [3,1]
 => 1
[1,4,2,3] => [1,4,3,2] => [3,1]
 => [2,1,1]
 => 2
[1,4,3,2] => [1,3,4,2] => [2,1,1]
 => [3,1]
 => 1
[2,1,3,4] => [2,1,3,4] => [2,1,1]
 => [3,1]
 => 1
[2,1,4,3] => [2,1,4,3] => [2,2]
 => [2,2]
 => 2
[2,3,1,4] => [3,1,2,4] => [2,1,1]
 => [3,1]
 => 1
[2,3,4,1] => [4,1,2,3] => [2,1,1]
 => [3,1]
 => 1
[2,4,1,3] => [4,3,1,2] => [3,1]
 => [2,1,1]
 => 2
[2,4,3,1] => [3,4,1,2] => [2,1,1]
 => [3,1]
 => 1
[3,1,2,4] => [3,2,1,4] => [3,1]
 => [2,1,1]
 => 2
[3,1,4,2] => [4,2,1,3] => [3,1]
 => [2,1,1]
 => 2
[3,2,1,4] => [2,3,1,4] => [2,1,1]
 => [3,1]
 => 1
[3,2,4,1] => [2,4,1,3] => [2,1,1]
 => [3,1]
 => 1
[3,4,1,2] => [3,1,4,2] => [2,2]
 => [2,2]
 => 2
[3,4,2,1] => [4,1,3,2] => [3,1]
 => [2,1,1]
 => 2
[4,1,2,3] => [4,3,2,1] => [4]
 => [1,1,1,1]
 => 3
[4,1,3,2] => [3,4,2,1] => [3,1]
 => [2,1,1]
 => 2
[4,2,1,3] => [2,4,3,1] => [3,1]
 => [2,1,1]
 => 2
[4,2,3,1] => [2,3,4,1] => [2,1,1]
 => [3,1]
 => 1
[4,3,1,2] => [4,2,3,1] => [3,1]
 => [2,1,1]
 => 2
[4,3,2,1] => [3,2,4,1] => [3,1]
 => [2,1,1]
 => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => [5]
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => [4,1]
 => 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
 => [4,1]
 => 1
[1,2,4,5,3] => [1,2,5,3,4] => [2,1,1,1]
 => [4,1]
 => 1
[1,2,5,3,4] => [1,2,5,4,3] => [3,1,1]
 => [3,1,1]
 => 2
[1,2,5,4,3] => [1,2,4,5,3] => [2,1,1,1]
 => [4,1]
 => 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
 => [4,1]
 => 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
 => [3,2]
 => 2
[1,3,4,2,5] => [1,4,2,3,5] => [2,1,1,1]
 => [4,1]
 => 1
[1,3,4,5,2] => [1,5,2,3,4] => [2,1,1,1]
 => [4,1]
 => 1
[1,3,5,2,4] => [1,5,4,2,3] => [3,1,1]
 => [3,1,1]
 => 2
[1,3,5,4,2] => [1,4,5,2,3] => [2,1,1,1]
 => [4,1]
 => 1
[1,4,2,3,5] => [1,4,3,2,5] => [3,1,1]
 => [3,1,1]
 => 2
[1,4,2,5,3] => [1,5,3,2,4] => [3,1,1]
 => [3,1,1]
 => 2
[1,4,3,2,5] => [1,3,4,2,5] => [2,1,1,1]
 => [4,1]
 => 1
[1,4,3,5,2] => [1,3,5,2,4] => [2,1,1,1]
 => [4,1]
 => 1
[1,4,5,2,3] => [1,4,2,5,3] => [2,2,1]
 => [3,2]
 => 2
[] => [] => []
 => []
 => ? = 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 2 compositions to match this statistic)

Mapping

Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the partition.

Matching statistic: St000288
(load all 37 compositions to match this statistic)

Mapping

Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 91%

Values

[1] => [1] =>  =>  => ? = 0
[1,2] => [1,2] => 0 => 0 => 0
[2,1] => [2,1] => 1 => 1 => 1
[1,2,3] => [1,2,3] => 00 => 00 => 0
[1,3,2] => [1,3,2] => 01 => 01 => 1
[2,1,3] => [2,1,3] => 10 => 01 => 1
[2,3,1] => [3,1,2] => 10 => 01 => 1
[3,1,2] => [3,2,1] => 11 => 11 => 2
[3,2,1] => [2,3,1] => 10 => 01 => 1
[1,2,3,4] => [1,2,3,4] => 000 => 000 => 0
[1,2,4,3] => [1,2,4,3] => 001 => 001 => 1
[1,3,2,4] => [1,3,2,4] => 010 => 001 => 1
[1,3,4,2] => [1,4,2,3] => 010 => 001 => 1
[1,4,2,3] => [1,4,3,2] => 011 => 011 => 2
[1,4,3,2] => [1,3,4,2] => 010 => 001 => 1
[2,1,3,4] => [2,1,3,4] => 100 => 001 => 1
[2,1,4,3] => [2,1,4,3] => 101 => 011 => 2
[2,3,1,4] => [3,1,2,4] => 100 => 001 => 1
[2,3,4,1] => [4,1,2,3] => 100 => 001 => 1
[2,4,1,3] => [4,3,1,2] => 101 => 011 => 2
[2,4,3,1] => [3,4,1,2] => 100 => 001 => 1
[3,1,2,4] => [3,2,1,4] => 110 => 011 => 2
[3,1,4,2] => [4,2,1,3] => 110 => 011 => 2
[3,2,1,4] => [2,3,1,4] => 100 => 001 => 1
[3,2,4,1] => [2,4,1,3] => 100 => 001 => 1
[3,4,1,2] => [3,1,4,2] => 110 => 011 => 2
[3,4,2,1] => [4,1,3,2] => 110 => 011 => 2
[4,1,2,3] => [4,3,2,1] => 111 => 111 => 3
[4,1,3,2] => [3,4,2,1] => 110 => 011 => 2
[4,2,1,3] => [2,4,3,1] => 101 => 011 => 2
[4,2,3,1] => [2,3,4,1] => 100 => 001 => 1
[4,3,1,2] => [4,2,3,1] => 110 => 011 => 2
[4,3,2,1] => [3,2,4,1] => 110 => 011 => 2
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0000 => 0
[1,2,3,5,4] => [1,2,3,5,4] => 0001 => 0001 => 1
[1,2,4,3,5] => [1,2,4,3,5] => 0010 => 0001 => 1
[1,2,4,5,3] => [1,2,5,3,4] => 0010 => 0001 => 1
[1,2,5,3,4] => [1,2,5,4,3] => 0011 => 0011 => 2
[1,2,5,4,3] => [1,2,4,5,3] => 0010 => 0001 => 1
[1,3,2,4,5] => [1,3,2,4,5] => 0100 => 0001 => 1
[1,3,2,5,4] => [1,3,2,5,4] => 0101 => 0101 => 2
[1,3,4,2,5] => [1,4,2,3,5] => 0100 => 0001 => 1
[1,3,4,5,2] => [1,5,2,3,4] => 0100 => 0001 => 1
[1,3,5,2,4] => [1,5,4,2,3] => 0101 => 0101 => 2
[1,3,5,4,2] => [1,4,5,2,3] => 0100 => 0001 => 1
[1,4,2,3,5] => [1,4,3,2,5] => 0110 => 0011 => 2
[1,4,2,5,3] => [1,5,3,2,4] => 0110 => 0011 => 2
[1,4,3,2,5] => [1,3,4,2,5] => 0100 => 0001 => 1
[1,4,3,5,2] => [1,3,5,2,4] => 0100 => 0001 => 1
[1,4,5,2,3] => [1,4,2,5,3] => 0110 => 0011 => 2
[1,4,5,3,2] => [1,5,2,4,3] => 0110 => 0011 => 2
[4,7,1,2,3,5,6,8] => [7,6,5,3,1,4,2,8] => ? => ? => ? = 5
[5,7,1,2,3,4,6,8] => [5,3,1,7,6,4,2,8] => ? => ? => ? = 5
[1,5,8,2,3,4,6,7] => [1,8,7,6,4,2,5,3] => ? => ? => ? = 5
[6,1,2,7,3,4,5,8] => [7,5,3,2,1,6,4,8] => ? => ? => ? = 5
[1,7,2,3,8,4,5,6] => [1,8,6,4,3,2,7,5] => ? => ? => ? = 5
[4,3,2,1,10,7,6,9,8,5] => [3,2,4,1,7,6,9,8,10,5] => 110011010 => 000110111 => ? = 5
[6,3,2,5,4,1,10,9,8,7] => [3,2,5,4,6,1,9,8,10,7] => 110100110 => 000110111 => ? = 5
[2,1,6,5,4,3,10,9,8,7] => [2,1,5,4,6,3,9,8,10,7] => 101100110 => 000110111 => ? = 5
[2,1,10,5,4,9,8,7,6,3] => [2,1,5,4,8,7,9,6,10,3] => 101101100 => 000110111 => ? = 5
[] => [] =>  =>  => ? = 0
[2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,4,3,6,5,8,7,10,9,12,11] => 10101010101 => 01010101011 => ? = 6
[2,1,4,3,6,5,8,7,12,11,10,9] => [2,1,4,3,6,5,8,7,11,10,12,9] => 10101010110 => ? => ? = 6
[2,1,4,3,6,5,10,9,8,7,12,11] => [2,1,4,3,6,5,9,8,10,7,12,11] => 10101011001 => ? => ? = 6
[2,1,4,3,6,5,12,9,8,11,10,7] => [2,1,4,3,6,5,9,8,11,10,12,7] => 10101011010 => ? => ? = 6
[2,1,4,3,6,5,12,11,10,9,8,7] => [2,1,4,3,6,5,10,9,11,8,12,7] => 10101011100 => ? => ? = 6
[2,1,4,3,8,7,6,5,10,9,12,11] => [2,1,4,3,7,6,8,5,10,9,12,11] => 10101100101 => ? => ? = 6
[2,1,4,3,8,7,6,5,12,11,10,9] => [2,1,4,3,7,6,8,5,11,10,12,9] => 10101100110 => ? => ? = 6
[2,1,4,3,10,7,6,9,8,5,12,11] => [2,1,4,3,7,6,9,8,10,5,12,11] => 10101101001 => ? => ? = 6
[2,1,4,3,12,7,6,9,8,11,10,5] => [2,1,4,3,7,6,9,8,11,10,12,5] => 10101101010 => ? => ? = 6
[2,1,4,3,12,7,6,11,10,9,8,5] => [2,1,4,3,7,6,10,9,11,8,12,5] => 10101101100 => ? => ? = 6
[2,1,4,3,10,9,8,7,6,5,12,11] => [2,1,4,3,8,7,9,6,10,5,12,11] => 10101110001 => ? => ? = 6
[2,1,4,3,12,9,8,7,6,11,10,5] => [2,1,4,3,8,7,9,6,11,10,12,5] => 10101110010 => ? => ? = 6
[2,1,4,3,12,11,8,7,10,9,6,5] => [2,1,4,3,8,7,10,9,11,6,12,5] => 10101110100 => ? => ? = 6
[2,1,4,3,12,11,10,9,8,7,6,5] => [2,1,4,3,9,8,10,7,11,6,12,5] => 10101111000 => ? => ? = 6
[2,1,6,5,4,3,8,7,10,9,12,11] => [2,1,5,4,6,3,8,7,10,9,12,11] => 10110010101 => ? => ? = 6
[2,1,6,5,4,3,8,7,12,11,10,9] => [2,1,5,4,6,3,8,7,11,10,12,9] => 10110010110 => ? => ? = 6
[2,1,6,5,4,3,10,9,8,7,12,11] => [2,1,5,4,6,3,9,8,10,7,12,11] => 10110011001 => ? => ? = 6
[2,1,6,5,4,3,12,9,8,11,10,7] => [2,1,5,4,6,3,9,8,11,10,12,7] => 10110011010 => ? => ? = 6
[2,1,6,5,4,3,12,11,10,9,8,7] => [2,1,5,4,6,3,10,9,11,8,12,7] => 10110011100 => ? => ? = 6
[2,1,8,5,4,7,6,3,10,9,12,11] => [2,1,5,4,7,6,8,3,10,9,12,11] => 10110100101 => ? => ? = 6
[2,1,8,5,4,7,6,3,12,11,10,9] => [2,1,5,4,7,6,8,3,11,10,12,9] => 10110100110 => ? => ? = 6
[2,1,10,5,4,7,6,9,8,3,12,11] => [2,1,5,4,7,6,9,8,10,3,12,11] => 10110101001 => ? => ? = 6
[2,1,12,5,4,7,6,9,8,11,10,3] => [2,1,5,4,7,6,9,8,11,10,12,3] => 10110101010 => ? => ? = 6
[2,1,12,5,4,7,6,11,10,9,8,3] => [2,1,5,4,7,6,10,9,11,8,12,3] => 10110101100 => ? => ? = 6
[2,1,10,5,4,9,8,7,6,3,12,11] => [2,1,5,4,8,7,9,6,10,3,12,11] => 10110110001 => ? => ? = 6
[2,1,12,5,4,9,8,7,6,11,10,3] => [2,1,5,4,8,7,9,6,11,10,12,3] => 10110110010 => ? => ? = 6
[2,1,12,5,4,11,8,7,10,9,6,3] => [2,1,5,4,8,7,10,9,11,6,12,3] => 10110110100 => ? => ? = 6
[2,1,12,5,4,11,10,9,8,7,6,3] => [2,1,5,4,9,8,10,7,11,6,12,3] => 10110111000 => ? => ? = 6
[2,1,8,7,6,5,4,3,10,9,12,11] => [2,1,6,5,7,4,8,3,10,9,12,11] => 10111000101 => ? => ? = 6
[2,1,8,7,6,5,4,3,12,11,10,9] => [2,1,6,5,7,4,8,3,11,10,12,9] => 10111000110 => ? => ? = 6
[2,1,10,7,6,5,4,9,8,3,12,11] => [2,1,6,5,7,4,9,8,10,3,12,11] => 10111001001 => ? => ? = 6
[2,1,12,7,6,5,4,9,8,11,10,3] => [2,1,6,5,7,4,9,8,11,10,12,3] => 10111001010 => ? => ? = 6
[2,1,12,7,6,5,4,11,10,9,8,3] => [2,1,6,5,7,4,10,9,11,8,12,3] => 10111001100 => ? => ? = 6
[2,1,10,9,6,5,8,7,4,3,12,11] => [2,1,6,5,8,7,9,4,10,3,12,11] => 10111010001 => ? => ? = 6
[2,1,12,9,6,5,8,7,4,11,10,3] => [2,1,6,5,8,7,9,4,11,10,12,3] => 10111010010 => ? => ? = 6
[2,1,12,11,6,5,8,7,10,9,4,3] => [2,1,6,5,8,7,10,9,11,4,12,3] => 10111010100 => ? => ? = 6
[2,1,12,11,6,5,10,9,8,7,4,3] => [2,1,6,5,9,8,10,7,11,4,12,3] => 10111011000 => ? => ? = 6
[2,1,10,9,8,7,6,5,4,3,12,11] => [2,1,7,6,8,5,9,4,10,3,12,11] => 10111100001 => ? => ? = 6
[2,1,12,9,8,7,6,5,4,11,10,3] => [2,1,7,6,8,5,9,4,11,10,12,3] => 10111100010 => ? => ? = 6

Description

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

Matching statistic: St000157
(load all 7 compositions to match this statistic)

Mapping

Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 91%

Values

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

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

$i$ .

Matching statistic: St000097

Mapping

Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 91%

Values

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

Description

The order of the largest clique of the graph. A clique in a graph

$G$ is a subset

$U \subseteq V(G)$ such that any pair of vertices in

$U$ are adjacent. I.e. the subgraph induced by

$U$ is a complete graph.

Matching statistic: St000098

Mapping

Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 91%

Values

[1] => [1] => [1] => ([],1)
 => 1 = 0 + 1
[1,2] => [1,2] => [2] => ([],2)
 => 1 = 0 + 1
[2,1] => [2,1] => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,2,3] => [1,2,3] => [3] => ([],3)
 => 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[2,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[2,3,1] => [3,1,2] => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[3,1,2] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,2,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
 => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,3,4,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,4,2,3] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,4,3,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,1,3,4] => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,3,1,4] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[2,3,4,1] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[2,4,1,3] => [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,4,3,1] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,1,2,4] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1,4,2] => [4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,2,1,4] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,2,4,1] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,4,1,2] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,4,2,1] => [4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,1,2,3] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,1,3,2] => [3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,2,1,3] => [2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,2,3,1] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[4,3,1,2] => [4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,3,2,1] => [3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
 => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,4,5,3] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,5,4,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,4,2,5] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,4,5,2] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,5,2,4] => [1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,5,4,2] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,2,5,3] => [1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,3,2,5] => [1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,4,3,5,2] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,4,5,2,3] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[7,6,5,4,8,3,2,1] => [4,8,1,7,2,6,3,5] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[6,5,7,4,8,3,2,1] => [4,8,1,6,3,7,2,5] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[7,6,5,4,3,2,8,1] => [4,5,3,6,2,8,1,7] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[6,5,7,4,3,2,8,1] => [4,8,1,6,2,5,3,7] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[6,5,4,3,7,2,8,1] => [4,3,8,1,6,2,5,7] => [1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[5,4,6,3,7,2,8,1] => [6,2,4,3,8,1,5,7] => [1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[3,4,5,2,6,7,8,1] => [4,2,8,1,3,5,6,7] => [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[5,2,3,4,6,7,8,1] => [2,3,4,8,1,5,6,7] => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 1 + 1
[4,3,2,5,6,7,8,1] => [3,2,8,1,4,5,6,7] => [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[3,4,2,5,6,7,8,1] => [8,1,3,2,4,5,6,7] => [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[4,2,3,5,6,7,8,1] => [2,3,8,1,4,5,6,7] => [3,5] => ([(4,7),(5,7),(6,7)],8)
 => ? = 1 + 1
[3,2,4,5,6,7,8,1] => [2,8,1,3,4,5,6,7] => [2,6] => ([(5,7),(6,7)],8)
 => ? = 1 + 1
[2,3,4,5,6,7,8,1] => [8,1,2,3,4,5,6,7] => [1,7] => ([(6,7)],8)
 => ? = 1 + 1
[3,4,5,6,7,8,1,2] => [7,1,3,5,8,2,4,6] => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[6,7,4,5,8,1,2,3] => [6,1,7,2,8,3,4,5] => [1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[7,6,5,4,3,2,1,8] => [4,5,3,6,2,7,1,8] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[6,7,5,4,3,2,1,8] => [4,5,3,7,1,6,2,8] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[6,5,7,4,3,2,1,8] => [4,7,1,6,2,5,3,8] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[5,6,7,4,3,2,1,8] => [4,6,2,7,1,5,3,8] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[7,6,4,5,3,2,1,8] => [5,3,4,6,2,7,1,8] => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[6,5,4,7,3,2,1,8] => [7,1,6,2,5,3,4,8] => [1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[7,5,6,3,4,2,1,8] => [6,2,5,4,3,7,1,8] => [1,2,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[7,6,4,3,5,2,1,8] => [4,3,5,6,2,7,1,8] => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[7,6,3,4,5,2,1,8] => [3,4,5,6,2,7,1,8] => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[7,5,4,3,6,2,1,8] => [4,3,6,2,5,7,1,8] => [1,2,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[6,7,5,4,2,3,1,8] => [4,7,1,6,3,5,2,8] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[6,5,7,3,2,4,1,8] => [5,2,7,1,6,4,3,8] => [1,2,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[5,6,7,2,3,4,1,8] => [6,4,2,7,1,5,3,8] => [1,1,2,2,2] => ([(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[2,3,4,5,6,7,1,8] => [7,1,2,3,4,5,6,8] => [1,7] => ([(6,7)],8)
 => ? = 1 + 1
[5,6,7,1,2,3,4,8] => [7,4,1,5,2,6,3,8] => [1,1,2,2,2] => ([(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[6,7,1,2,3,4,5,8] => [7,5,3,1,6,4,2,8] => [1,1,1,2,1,2] => ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[7,1,2,3,4,5,6,8] => [7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[6,5,4,3,2,1,7,8] => [4,3,5,2,6,1,7,8] => [1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[2,3,4,5,6,1,7,8] => [6,1,2,3,4,5,7,8] => [1,7] => ([(6,7)],8)
 => ? = 1 + 1
[5,6,3,4,1,2,7,8] => [3,4,5,1,6,2,7,8] => [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[4,5,6,1,2,3,7,8] => [4,1,5,2,6,3,7,8] => [1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[6,1,2,3,4,5,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[5,4,3,2,1,6,7,8] => [3,4,2,5,1,6,7,8] => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[2,3,4,5,1,6,7,8] => [5,1,2,3,4,6,7,8] => [1,7] => ([(6,7)],8)
 => ? = 1 + 1
[5,1,2,3,4,6,7,8] => [5,4,3,2,1,6,7,8] => [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[4,3,2,1,5,6,7,8] => [3,2,4,1,5,6,7,8] => [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[3,4,1,2,5,6,7,8] => [3,1,4,2,5,6,7,8] => [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[3,2,1,4,5,6,7,8] => [2,3,1,4,5,6,7,8] => [2,6] => ([(5,7),(6,7)],8)
 => ? = 1 + 1
[2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [1,7] => ([(6,7)],8)
 => ? = 1 + 1
[1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => [8] => ([],8)
 => ? = 0 + 1
[3,5,8,7,6,4,2,1] => [7,2,5,6,4,8,1,3] => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[4,3,6,5,8,7,2,1] => [7,2,3,6,8,1,4,5] => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[2,3,5,6,8,7,4,1] => [7,4,6,8,1,2,3,5] => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[3,2,6,5,4,8,7,1] => [2,5,4,7,8,1,3,6] => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[4,3,6,5,7,2,8,1] => [6,2,3,8,1,4,5,7] => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1

Description

The chromatic number of a graph. The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.

Matching statistic: St000011

Mapping

Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 91%

Values

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

Description

The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.

Matching statistic: St001581

Mapping

Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 64%

Values

[1] => [1] => [1] => ([],1)
 => 1 = 0 + 1
[1,2] => [1,2] => [2] => ([],2)
 => 1 = 0 + 1
[2,1] => [2,1] => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,2,3] => [1,2,3] => [3] => ([],3)
 => 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[2,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[2,3,1] => [3,1,2] => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[3,1,2] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,2,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
 => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,3,4,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,4,2,3] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,4,3,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,1,3,4] => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,3,1,4] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[2,3,4,1] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[2,4,1,3] => [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,4,3,1] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,1,2,4] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1,4,2] => [4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,2,1,4] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,2,4,1] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,4,1,2] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,4,2,1] => [4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,1,2,3] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,1,3,2] => [3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,2,1,3] => [2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,2,3,1] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[4,3,1,2] => [4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,3,2,1] => [3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
 => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,4,5,3] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,5,4,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,4,2,5] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,4,5,2] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,5,2,4] => [1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,5,4,2] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,2,5,3] => [1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,3,2,5] => [1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,4,3,5,2] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,4,5,2,3] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[8,7,6,5,4,3,2,1] => [5,4,6,3,7,2,8,1] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[7,6,8,5,4,3,2,1] => [5,4,8,1,7,2,6,3] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[7,8,5,6,4,3,2,1] => [6,3,5,4,8,1,7,2] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[7,8,6,5,3,4,2,1] => [6,4,5,3,8,1,7,2] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[8,6,7,5,3,4,2,1] => [7,2,6,4,5,3,8,1] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[8,7,5,6,3,4,2,1] => [5,3,6,4,7,2,8,1] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[6,7,8,3,4,5,2,1] => [7,2,8,1,6,5,4,3] => [1,2,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[7,8,6,5,4,2,3,1] => [5,4,8,1,7,3,6,2] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[8,6,7,5,4,2,3,1] => [5,4,6,2,7,3,8,1] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[8,7,5,6,4,2,3,1] => [7,3,5,4,6,2,8,1] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[8,6,5,7,3,2,4,1] => [5,3,6,2,7,4,8,1] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[6,7,8,5,2,3,4,1] => [7,4,5,2,8,1,6,3] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[8,5,6,7,2,3,4,1] => [5,2,6,3,7,4,8,1] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[7,8,3,2,4,5,6,1] => [3,8,1,7,6,5,4,2] => [2,2,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[8,7,6,5,4,3,1,2] => [5,4,6,3,8,2,7,1] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[7,8,6,5,4,3,1,2] => [5,4,6,3,7,1,8,2] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[8,6,7,5,4,3,1,2] => [5,4,8,2,6,3,7,1] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[8,7,5,6,4,3,1,2] => [6,3,5,4,8,2,7,1] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[8,7,6,5,3,4,1,2] => [6,4,5,3,8,2,7,1] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[8,7,5,6,3,4,1,2] => [5,3,6,4,8,2,7,1] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[7,8,5,6,3,4,1,2] => [5,3,6,4,7,1,8,2] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[5,6,7,8,3,4,1,2] => [7,1,5,3,8,2,6,4] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[8,7,6,5,4,2,1,3] => [5,4,8,3,6,2,7,1] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[7,6,8,5,4,2,1,3] => [5,4,6,2,7,1,8,3] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[8,7,6,5,4,1,2,3] => [5,4,7,2,8,3,6,1] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[6,7,8,5,4,1,2,3] => [5,4,6,1,7,2,8,3] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[8,5,6,7,4,1,2,3] => [7,2,5,4,8,3,6,1] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[7,6,5,8,3,2,1,4] => [5,3,6,2,7,1,8,4] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[7,5,6,8,2,3,1,4] => [5,2,6,3,7,1,8,4] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[6,7,5,8,3,1,2,4] => [5,3,6,1,7,2,8,4] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[6,5,7,8,2,1,3,4] => [5,2,6,1,7,3,8,4] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[8,7,6,5,1,2,3,4] => [7,3,6,2,8,4,5,1] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[8,7,5,6,1,2,3,4] => [8,4,6,2,7,3,5,1] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[7,8,5,6,1,2,3,4] => [7,3,5,1,8,4,6,2] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[8,5,6,7,1,2,3,4] => [8,4,7,3,6,2,5,1] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,8,4] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[8,7,6,4,3,2,1,5] => [4,8,5,3,6,2,7,1] => [2,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[8,7,6,4,2,1,3,5] => [4,8,5,2,7,3,6,1] => [2,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[8,7,6,1,2,3,4,5] => [6,3,8,5,2,7,4,1] => [1,2,1,2,1,1] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[8,6,7,1,2,3,4,5] => [8,5,2,6,3,7,4,1] => [1,1,2,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[6,7,8,1,2,3,4,5] => [8,5,2,7,4,1,6,3] => [1,1,2,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[7,8,5,4,2,1,3,6] => [4,8,6,1,7,3,5,2] => [2,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[7,8,5,4,1,2,3,6] => [4,7,3,5,1,8,6,2] => [2,2,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[8,7,1,2,3,4,5,6] => [8,6,4,2,7,5,3,1] => [1,1,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[7,8,1,2,3,4,5,6] => [7,5,3,1,8,6,4,2] => [1,1,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[8,6,5,4,3,2,1,7] => [4,5,3,6,2,8,7,1] => [2,2,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[8,6,5,4,3,1,2,7] => [4,5,3,8,7,2,6,1] => [2,2,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[8,5,6,3,4,1,2,7] => [8,7,2,5,4,3,6,1] => [1,1,2,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[8,6,5,4,2,1,3,7] => [4,8,7,3,5,2,6,1] => [2,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[8,6,4,3,2,1,5,7] => [4,3,8,7,5,2,6,1] => [1,2,1,1,2,1] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1

Description

The achromatic number of a graph. This is the maximal number of colours of a proper colouring, such that for any pair of colours there are two adjacent vertices with these colours.

The following 65 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000167The number of leaves of an ordered tree. St000306The bounce count of a Dyck path. St000211The rank of the set partition. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of 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. St000105The number of blocks in the set partition. St000507The number of ascents of a standard tableau. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . 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. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000662The staircase size of the code of a permutation. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000702The number of weak deficiencies of a permutation. St000021The number of descents 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. St000155The number of exceedances (also excedences) of a permutation. St000325The width of the tree associated to a permutation. St000168The number of internal nodes of an ordered tree. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000314The number of left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St000822The Hadwiger number of the graph. 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. 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. St000216The absolute length of a permutation. St001812The biclique partition number of a graph. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001427The number of descents of a signed permutation. St001330The hat guessing number of a graph. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St001935The number of ascents in a parking function.