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Your data matches 86 different statistics following compositions of up to 3 maps.
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Matching statistic: St000903
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
St000903: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,1] => 1
[2] => 1
[1,1,1] => 1
[1,2] => 2
[2,1] => 2
[3] => 1
[1,1,1,1] => 1
[1,1,2] => 2
[1,2,1] => 2
[1,3] => 2
[2,1,1] => 2
[2,2] => 1
[3,1] => 2
[4] => 1
[1,1,1,1,1] => 1
[1,1,1,2] => 2
[1,1,2,1] => 2
[1,1,3] => 2
[1,2,1,1] => 2
[1,2,2] => 2
[1,3,1] => 2
[1,4] => 2
[2,1,1,1] => 2
[2,1,2] => 2
[2,2,1] => 2
[2,3] => 2
[3,1,1] => 2
[3,2] => 2
[4,1] => 2
[5] => 1
[1,1,1,1,1,1] => 1
[1,1,1,1,2] => 2
[1,1,1,2,1] => 2
[1,1,1,3] => 2
[1,1,2,1,1] => 2
[1,1,2,2] => 2
[1,1,3,1] => 2
[1,1,4] => 2
[1,2,1,1,1] => 2
[1,2,1,2] => 2
[1,2,2,1] => 2
[1,2,3] => 3
[1,3,1,1] => 2
[1,3,2] => 3
[1,4,1] => 2
[1,5] => 2
[2,1,1,1,1] => 2
[2,1,1,2] => 2
[2,1,2,1] => 2
Description
The number of different parts of an integer composition.
Matching statistic: St000159
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 1
[1,1] => [1,1]
=> 1
[2] => [2]
=> 1
[1,1,1] => [1,1,1]
=> 1
[1,2] => [2,1]
=> 2
[2,1] => [2,1]
=> 2
[3] => [3]
=> 1
[1,1,1,1] => [1,1,1,1]
=> 1
[1,1,2] => [2,1,1]
=> 2
[1,2,1] => [2,1,1]
=> 2
[1,3] => [3,1]
=> 2
[2,1,1] => [2,1,1]
=> 2
[2,2] => [2,2]
=> 1
[3,1] => [3,1]
=> 2
[4] => [4]
=> 1
[1,1,1,1,1] => [1,1,1,1,1]
=> 1
[1,1,1,2] => [2,1,1,1]
=> 2
[1,1,2,1] => [2,1,1,1]
=> 2
[1,1,3] => [3,1,1]
=> 2
[1,2,1,1] => [2,1,1,1]
=> 2
[1,2,2] => [2,2,1]
=> 2
[1,3,1] => [3,1,1]
=> 2
[1,4] => [4,1]
=> 2
[2,1,1,1] => [2,1,1,1]
=> 2
[2,1,2] => [2,2,1]
=> 2
[2,2,1] => [2,2,1]
=> 2
[2,3] => [3,2]
=> 2
[3,1,1] => [3,1,1]
=> 2
[3,2] => [3,2]
=> 2
[4,1] => [4,1]
=> 2
[5] => [5]
=> 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 1
[1,1,1,1,2] => [2,1,1,1,1]
=> 2
[1,1,1,2,1] => [2,1,1,1,1]
=> 2
[1,1,1,3] => [3,1,1,1]
=> 2
[1,1,2,1,1] => [2,1,1,1,1]
=> 2
[1,1,2,2] => [2,2,1,1]
=> 2
[1,1,3,1] => [3,1,1,1]
=> 2
[1,1,4] => [4,1,1]
=> 2
[1,2,1,1,1] => [2,1,1,1,1]
=> 2
[1,2,1,2] => [2,2,1,1]
=> 2
[1,2,2,1] => [2,2,1,1]
=> 2
[1,2,3] => [3,2,1]
=> 3
[1,3,1,1] => [3,1,1,1]
=> 2
[1,3,2] => [3,2,1]
=> 3
[1,4,1] => [4,1,1]
=> 2
[1,5] => [5,1]
=> 2
[2,1,1,1,1] => [2,1,1,1,1]
=> 2
[2,1,1,2] => [2,2,1,1]
=> 2
[2,1,2,1] => [2,2,1,1]
=> 2
Description
The number of distinct parts of the integer partition.
This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Matching statistic: St000318
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 2 = 1 + 1
[1,1] => [1,1]
=> 2 = 1 + 1
[2] => [2]
=> 2 = 1 + 1
[1,1,1] => [1,1,1]
=> 2 = 1 + 1
[1,2] => [2,1]
=> 3 = 2 + 1
[2,1] => [2,1]
=> 3 = 2 + 1
[3] => [3]
=> 2 = 1 + 1
[1,1,1,1] => [1,1,1,1]
=> 2 = 1 + 1
[1,1,2] => [2,1,1]
=> 3 = 2 + 1
[1,2,1] => [2,1,1]
=> 3 = 2 + 1
[1,3] => [3,1]
=> 3 = 2 + 1
[2,1,1] => [2,1,1]
=> 3 = 2 + 1
[2,2] => [2,2]
=> 2 = 1 + 1
[3,1] => [3,1]
=> 3 = 2 + 1
[4] => [4]
=> 2 = 1 + 1
[1,1,1,1,1] => [1,1,1,1,1]
=> 2 = 1 + 1
[1,1,1,2] => [2,1,1,1]
=> 3 = 2 + 1
[1,1,2,1] => [2,1,1,1]
=> 3 = 2 + 1
[1,1,3] => [3,1,1]
=> 3 = 2 + 1
[1,2,1,1] => [2,1,1,1]
=> 3 = 2 + 1
[1,2,2] => [2,2,1]
=> 3 = 2 + 1
[1,3,1] => [3,1,1]
=> 3 = 2 + 1
[1,4] => [4,1]
=> 3 = 2 + 1
[2,1,1,1] => [2,1,1,1]
=> 3 = 2 + 1
[2,1,2] => [2,2,1]
=> 3 = 2 + 1
[2,2,1] => [2,2,1]
=> 3 = 2 + 1
[2,3] => [3,2]
=> 3 = 2 + 1
[3,1,1] => [3,1,1]
=> 3 = 2 + 1
[3,2] => [3,2]
=> 3 = 2 + 1
[4,1] => [4,1]
=> 3 = 2 + 1
[5] => [5]
=> 2 = 1 + 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 2 = 1 + 1
[1,1,1,1,2] => [2,1,1,1,1]
=> 3 = 2 + 1
[1,1,1,2,1] => [2,1,1,1,1]
=> 3 = 2 + 1
[1,1,1,3] => [3,1,1,1]
=> 3 = 2 + 1
[1,1,2,1,1] => [2,1,1,1,1]
=> 3 = 2 + 1
[1,1,2,2] => [2,2,1,1]
=> 3 = 2 + 1
[1,1,3,1] => [3,1,1,1]
=> 3 = 2 + 1
[1,1,4] => [4,1,1]
=> 3 = 2 + 1
[1,2,1,1,1] => [2,1,1,1,1]
=> 3 = 2 + 1
[1,2,1,2] => [2,2,1,1]
=> 3 = 2 + 1
[1,2,2,1] => [2,2,1,1]
=> 3 = 2 + 1
[1,2,3] => [3,2,1]
=> 4 = 3 + 1
[1,3,1,1] => [3,1,1,1]
=> 3 = 2 + 1
[1,3,2] => [3,2,1]
=> 4 = 3 + 1
[1,4,1] => [4,1,1]
=> 3 = 2 + 1
[1,5] => [5,1]
=> 3 = 2 + 1
[2,1,1,1,1] => [2,1,1,1,1]
=> 3 = 2 + 1
[2,1,1,2] => [2,2,1,1]
=> 3 = 2 + 1
[2,1,2,1] => [2,2,1,1]
=> 3 = 2 + 1
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St000291
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 10 => 1
[1,1] => [1,1]
=> 110 => 1
[2] => [2]
=> 100 => 1
[1,1,1] => [1,1,1]
=> 1110 => 1
[1,2] => [2,1]
=> 1010 => 2
[2,1] => [2,1]
=> 1010 => 2
[3] => [3]
=> 1000 => 1
[1,1,1,1] => [1,1,1,1]
=> 11110 => 1
[1,1,2] => [2,1,1]
=> 10110 => 2
[1,2,1] => [2,1,1]
=> 10110 => 2
[1,3] => [3,1]
=> 10010 => 2
[2,1,1] => [2,1,1]
=> 10110 => 2
[2,2] => [2,2]
=> 1100 => 1
[3,1] => [3,1]
=> 10010 => 2
[4] => [4]
=> 10000 => 1
[1,1,1,1,1] => [1,1,1,1,1]
=> 111110 => 1
[1,1,1,2] => [2,1,1,1]
=> 101110 => 2
[1,1,2,1] => [2,1,1,1]
=> 101110 => 2
[1,1,3] => [3,1,1]
=> 100110 => 2
[1,2,1,1] => [2,1,1,1]
=> 101110 => 2
[1,2,2] => [2,2,1]
=> 11010 => 2
[1,3,1] => [3,1,1]
=> 100110 => 2
[1,4] => [4,1]
=> 100010 => 2
[2,1,1,1] => [2,1,1,1]
=> 101110 => 2
[2,1,2] => [2,2,1]
=> 11010 => 2
[2,2,1] => [2,2,1]
=> 11010 => 2
[2,3] => [3,2]
=> 10100 => 2
[3,1,1] => [3,1,1]
=> 100110 => 2
[3,2] => [3,2]
=> 10100 => 2
[4,1] => [4,1]
=> 100010 => 2
[5] => [5]
=> 100000 => 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 1111110 => 1
[1,1,1,1,2] => [2,1,1,1,1]
=> 1011110 => 2
[1,1,1,2,1] => [2,1,1,1,1]
=> 1011110 => 2
[1,1,1,3] => [3,1,1,1]
=> 1001110 => 2
[1,1,2,1,1] => [2,1,1,1,1]
=> 1011110 => 2
[1,1,2,2] => [2,2,1,1]
=> 110110 => 2
[1,1,3,1] => [3,1,1,1]
=> 1001110 => 2
[1,1,4] => [4,1,1]
=> 1000110 => 2
[1,2,1,1,1] => [2,1,1,1,1]
=> 1011110 => 2
[1,2,1,2] => [2,2,1,1]
=> 110110 => 2
[1,2,2,1] => [2,2,1,1]
=> 110110 => 2
[1,2,3] => [3,2,1]
=> 101010 => 3
[1,3,1,1] => [3,1,1,1]
=> 1001110 => 2
[1,3,2] => [3,2,1]
=> 101010 => 3
[1,4,1] => [4,1,1]
=> 1000110 => 2
[1,5] => [5,1]
=> 1000010 => 2
[2,1,1,1,1] => [2,1,1,1,1]
=> 1011110 => 2
[2,1,1,2] => [2,2,1,1]
=> 110110 => 2
[2,1,2,1] => [2,2,1,1]
=> 110110 => 2
Description
The number of descents of a binary word.
Matching statistic: St000390
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 10 => 1
[1,1] => [1,1]
=> 110 => 1
[2] => [2]
=> 100 => 1
[1,1,1] => [1,1,1]
=> 1110 => 1
[1,2] => [2,1]
=> 1010 => 2
[2,1] => [2,1]
=> 1010 => 2
[3] => [3]
=> 1000 => 1
[1,1,1,1] => [1,1,1,1]
=> 11110 => 1
[1,1,2] => [2,1,1]
=> 10110 => 2
[1,2,1] => [2,1,1]
=> 10110 => 2
[1,3] => [3,1]
=> 10010 => 2
[2,1,1] => [2,1,1]
=> 10110 => 2
[2,2] => [2,2]
=> 1100 => 1
[3,1] => [3,1]
=> 10010 => 2
[4] => [4]
=> 10000 => 1
[1,1,1,1,1] => [1,1,1,1,1]
=> 111110 => 1
[1,1,1,2] => [2,1,1,1]
=> 101110 => 2
[1,1,2,1] => [2,1,1,1]
=> 101110 => 2
[1,1,3] => [3,1,1]
=> 100110 => 2
[1,2,1,1] => [2,1,1,1]
=> 101110 => 2
[1,2,2] => [2,2,1]
=> 11010 => 2
[1,3,1] => [3,1,1]
=> 100110 => 2
[1,4] => [4,1]
=> 100010 => 2
[2,1,1,1] => [2,1,1,1]
=> 101110 => 2
[2,1,2] => [2,2,1]
=> 11010 => 2
[2,2,1] => [2,2,1]
=> 11010 => 2
[2,3] => [3,2]
=> 10100 => 2
[3,1,1] => [3,1,1]
=> 100110 => 2
[3,2] => [3,2]
=> 10100 => 2
[4,1] => [4,1]
=> 100010 => 2
[5] => [5]
=> 100000 => 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 1111110 => 1
[1,1,1,1,2] => [2,1,1,1,1]
=> 1011110 => 2
[1,1,1,2,1] => [2,1,1,1,1]
=> 1011110 => 2
[1,1,1,3] => [3,1,1,1]
=> 1001110 => 2
[1,1,2,1,1] => [2,1,1,1,1]
=> 1011110 => 2
[1,1,2,2] => [2,2,1,1]
=> 110110 => 2
[1,1,3,1] => [3,1,1,1]
=> 1001110 => 2
[1,1,4] => [4,1,1]
=> 1000110 => 2
[1,2,1,1,1] => [2,1,1,1,1]
=> 1011110 => 2
[1,2,1,2] => [2,2,1,1]
=> 110110 => 2
[1,2,2,1] => [2,2,1,1]
=> 110110 => 2
[1,2,3] => [3,2,1]
=> 101010 => 3
[1,3,1,1] => [3,1,1,1]
=> 1001110 => 2
[1,3,2] => [3,2,1]
=> 101010 => 3
[1,4,1] => [4,1,1]
=> 1000110 => 2
[1,5] => [5,1]
=> 1000010 => 2
[2,1,1,1,1] => [2,1,1,1,1]
=> 1011110 => 2
[2,1,1,2] => [2,2,1,1]
=> 110110 => 2
[2,1,2,1] => [2,2,1,1]
=> 110110 => 2
Description
The number of runs of ones in a binary word.
Matching statistic: St000292
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 10 => 0 = 1 - 1
[1,1] => [1,1]
=> 110 => 0 = 1 - 1
[2] => [2]
=> 100 => 0 = 1 - 1
[1,1,1] => [1,1,1]
=> 1110 => 0 = 1 - 1
[1,2] => [2,1]
=> 1010 => 1 = 2 - 1
[2,1] => [2,1]
=> 1010 => 1 = 2 - 1
[3] => [3]
=> 1000 => 0 = 1 - 1
[1,1,1,1] => [1,1,1,1]
=> 11110 => 0 = 1 - 1
[1,1,2] => [2,1,1]
=> 10110 => 1 = 2 - 1
[1,2,1] => [2,1,1]
=> 10110 => 1 = 2 - 1
[1,3] => [3,1]
=> 10010 => 1 = 2 - 1
[2,1,1] => [2,1,1]
=> 10110 => 1 = 2 - 1
[2,2] => [2,2]
=> 1100 => 0 = 1 - 1
[3,1] => [3,1]
=> 10010 => 1 = 2 - 1
[4] => [4]
=> 10000 => 0 = 1 - 1
[1,1,1,1,1] => [1,1,1,1,1]
=> 111110 => 0 = 1 - 1
[1,1,1,2] => [2,1,1,1]
=> 101110 => 1 = 2 - 1
[1,1,2,1] => [2,1,1,1]
=> 101110 => 1 = 2 - 1
[1,1,3] => [3,1,1]
=> 100110 => 1 = 2 - 1
[1,2,1,1] => [2,1,1,1]
=> 101110 => 1 = 2 - 1
[1,2,2] => [2,2,1]
=> 11010 => 1 = 2 - 1
[1,3,1] => [3,1,1]
=> 100110 => 1 = 2 - 1
[1,4] => [4,1]
=> 100010 => 1 = 2 - 1
[2,1,1,1] => [2,1,1,1]
=> 101110 => 1 = 2 - 1
[2,1,2] => [2,2,1]
=> 11010 => 1 = 2 - 1
[2,2,1] => [2,2,1]
=> 11010 => 1 = 2 - 1
[2,3] => [3,2]
=> 10100 => 1 = 2 - 1
[3,1,1] => [3,1,1]
=> 100110 => 1 = 2 - 1
[3,2] => [3,2]
=> 10100 => 1 = 2 - 1
[4,1] => [4,1]
=> 100010 => 1 = 2 - 1
[5] => [5]
=> 100000 => 0 = 1 - 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 1111110 => 0 = 1 - 1
[1,1,1,1,2] => [2,1,1,1,1]
=> 1011110 => 1 = 2 - 1
[1,1,1,2,1] => [2,1,1,1,1]
=> 1011110 => 1 = 2 - 1
[1,1,1,3] => [3,1,1,1]
=> 1001110 => 1 = 2 - 1
[1,1,2,1,1] => [2,1,1,1,1]
=> 1011110 => 1 = 2 - 1
[1,1,2,2] => [2,2,1,1]
=> 110110 => 1 = 2 - 1
[1,1,3,1] => [3,1,1,1]
=> 1001110 => 1 = 2 - 1
[1,1,4] => [4,1,1]
=> 1000110 => 1 = 2 - 1
[1,2,1,1,1] => [2,1,1,1,1]
=> 1011110 => 1 = 2 - 1
[1,2,1,2] => [2,2,1,1]
=> 110110 => 1 = 2 - 1
[1,2,2,1] => [2,2,1,1]
=> 110110 => 1 = 2 - 1
[1,2,3] => [3,2,1]
=> 101010 => 2 = 3 - 1
[1,3,1,1] => [3,1,1,1]
=> 1001110 => 1 = 2 - 1
[1,3,2] => [3,2,1]
=> 101010 => 2 = 3 - 1
[1,4,1] => [4,1,1]
=> 1000110 => 1 = 2 - 1
[1,5] => [5,1]
=> 1000010 => 1 = 2 - 1
[2,1,1,1,1] => [2,1,1,1,1]
=> 1011110 => 1 = 2 - 1
[2,1,1,2] => [2,2,1,1]
=> 110110 => 1 = 2 - 1
[2,1,2,1] => [2,2,1,1]
=> 110110 => 1 = 2 - 1
Description
The number of ascents of a binary word.
Matching statistic: St000069
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1],[]]
=> ([],1)
=> 1
[1,1] => [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 1
[2] => [2]
=> [[2],[]]
=> ([(0,1)],2)
=> 1
[1,1,1] => [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2] => [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 2
[2,1] => [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 2
[3] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1] => [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,2] => [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[1,2,1] => [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[1,3] => [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[2,1,1] => [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[2,2] => [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,1] => [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[4] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,1,2] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[1,1,2,1] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[1,1,3] => [3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[1,2,1,1] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[1,2,2] => [2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2
[1,3,1] => [3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[1,4] => [4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[2,1,1,1] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[2,1,2] => [2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2
[2,2,1] => [2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2
[2,3] => [3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2
[3,1,1] => [3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[3,2] => [3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2
[4,1] => [4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[5] => [5]
=> [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 2
[1,1,1,2,1] => [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 2
[1,1,1,3] => [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 2
[1,1,2,1,1] => [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 2
[1,1,2,2] => [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 2
[1,1,3,1] => [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 2
[1,1,4] => [4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 2
[1,2,1,1,1] => [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 2
[1,2,1,2] => [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 2
[1,2,2,1] => [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 2
[1,2,3] => [3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> 3
[1,3,1,1] => [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 2
[1,3,2] => [3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> 3
[1,4,1] => [4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 2
[1,5] => [5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 2
[2,1,1,1,1] => [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 2
[2,1,1,2] => [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 2
[2,1,2,1] => [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 2
Description
The number of maximal elements of a poset.
Matching statistic: St000245
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0,1,0]
=> [1,2] => 1
[1,1] => [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[2] => [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[1,2] => [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 2
[2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 2
[3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[1,1,1,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 1
[1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 2
[1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 2
[1,1,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 2
[1,2,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => 2
[2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 2
[2,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[2,2,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[3,1,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => 2
[5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => 2
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => 2
[1,1,1,3] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => 2
[1,1,2,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 2
[1,1,3,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2
[1,1,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 2
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => 2
[1,2,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 2
[1,2,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 2
[1,2,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 3
[1,3,1,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2
[1,3,2] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 3
[1,4,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 2
[1,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => 2
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => 2
[2,1,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 2
[2,1,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 2
Description
The number of ascents of a permutation.
Matching statistic: St000672
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0,1,0]
=> [1,2] => 1
[1,1] => [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[2] => [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[1,2] => [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 2
[2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 2
[3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[1,1,1,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 1
[1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 2
[1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 2
[1,1,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 2
[1,2,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => 2
[2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 2
[2,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[2,2,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[3,1,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => 2
[5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => 2
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => 2
[1,1,1,3] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => 2
[1,1,2,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 2
[1,1,3,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2
[1,1,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 2
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => 2
[1,2,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 2
[1,2,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 2
[1,2,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 3
[1,3,1,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2
[1,3,2] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 3
[1,4,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 2
[1,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => 2
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => 2
[2,1,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 2
[2,1,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 2
Description
The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Matching statistic: St000760
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000760: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000760: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> [1] => 1
[1,1] => [1,1]
=> [[1],[2]]
=> [1,1] => 1
[2] => [2]
=> [[1,2]]
=> [2] => 1
[1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1
[1,2] => [2,1]
=> [[1,2],[3]]
=> [2,1] => 2
[2,1] => [2,1]
=> [[1,2],[3]]
=> [2,1] => 2
[3] => [3]
=> [[1,2,3]]
=> [3] => 1
[1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1
[1,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 2
[1,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 2
[1,3] => [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 2
[2,1,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 2
[2,2] => [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 1
[3,1] => [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 2
[4] => [4]
=> [[1,2,3,4]]
=> [4] => 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 1
[1,1,1,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 2
[1,1,2,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 2
[1,1,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 2
[1,2,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 2
[1,2,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 2
[1,3,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 2
[1,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 2
[2,1,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 2
[2,1,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 2
[2,2,1] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 2
[2,3] => [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 2
[3,1,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 2
[3,2] => [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 2
[4,1] => [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 2
[5] => [5]
=> [[1,2,3,4,5]]
=> [5] => 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 2
[1,1,1,2,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 2
[1,1,1,3] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 2
[1,1,2,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 2
[1,1,2,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 2
[1,1,3,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 2
[1,1,4] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 2
[1,2,1,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 2
[1,2,1,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 2
[1,2,2,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 2
[1,2,3] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 3
[1,3,1,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 2
[1,3,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 3
[1,4,1] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 2
[1,5] => [5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => 2
[2,1,1,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 2
[2,1,1,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 2
[2,1,2,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 2
Description
The length of the longest strictly decreasing subsequence of parts of an integer composition.
By the Greene-Kleitman theorem, regarding the composition as a word, this is the length of the partition associated by the Robinson-Schensted-Knuth correspondence.
The following 76 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000764The number of strong records in an integer composition. St000767The number of runs in an integer composition. St000996The number of exclusive left-to-right maxima of a permutation. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000761The number of ascents in an integer composition. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000925The number of topologically connected components of a set partition. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000053The number of valleys of the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000024The number of double up and double down steps of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000105The number of blocks in the set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000522The number of 1-protected nodes of a rooted tree. St000386The number of factors DDU in a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000021The number of descents of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000619The number of cyclic descents of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. 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 $eA$. St001083The number of boxed occurrences of 132 in a permutation. St000646The number of big ascents of a permutation. St001732The number of peaks visible from the left. St000647The number of big descents of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000317The cycle descent number of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000702The number of weak deficiencies of a permutation. St000991The number of right-to-left minima of a permutation. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001960The number of descents of a permutation minus one if its first entry is not one. St000388The number of orbits of vertices of a graph under automorphisms. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000264The girth of a graph, which is not a tree. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000891The number of distinct diagonal sums of a permutation matrix. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001568The smallest positive integer that does not appear twice in the partition. St001487The number of inner corners of a skew partition. St000893The number of distinct diagonal sums of an alternating sign matrix. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001712The number of natural descents of a standard Young tableau. St001624The breadth of a lattice.
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