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Your data matches 63 different statistics following compositions of up to 3 maps.
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Matching statistic: St001280
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> 0
[1,0,1,0]
=> [1,1] => [1,1]
=> 0
[1,1,0,0]
=> [2] => [2]
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> 0
[1,0,1,1,0,0]
=> [1,2] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [3] => [3]
=> 1
[1,1,1,0,0,0]
=> [3] => [3]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [3,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1] => [3,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [4] => [4]
=> 1
[1,1,0,1,1,0,0,0]
=> [4] => [4]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [4] => [4]
=> 1
[1,1,1,0,1,0,0,0]
=> [4] => [4]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [3,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [4,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [3,2]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [5]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [5]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [5]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [5]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [5]
=> 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000292
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1 => 1 => 0
[1,0,1,0]
=> [1,1] => 11 => 11 => 0
[1,1,0,0]
=> [2] => 10 => 01 => 1
[1,0,1,0,1,0]
=> [1,1,1] => 111 => 111 => 0
[1,0,1,1,0,0]
=> [1,2] => 110 => 011 => 1
[1,1,0,0,1,0]
=> [2,1] => 101 => 101 => 1
[1,1,0,1,0,0]
=> [3] => 100 => 001 => 1
[1,1,1,0,0,0]
=> [3] => 100 => 001 => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 1111 => 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 0111 => 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 1011 => 1
[1,0,1,1,0,1,0,0]
=> [1,3] => 1100 => 0011 => 1
[1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 0011 => 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 1101 => 1
[1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 0101 => 2
[1,1,0,1,0,0,1,0]
=> [3,1] => 1001 => 1001 => 1
[1,1,0,1,0,1,0,0]
=> [4] => 1000 => 0001 => 1
[1,1,0,1,1,0,0,0]
=> [4] => 1000 => 0001 => 1
[1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 1001 => 1
[1,1,1,0,0,1,0,0]
=> [4] => 1000 => 0001 => 1
[1,1,1,0,1,0,0,0]
=> [4] => 1000 => 0001 => 1
[1,1,1,1,0,0,0,0]
=> [4] => 1000 => 0001 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 11111 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 01111 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 10111 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 11100 => 00111 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 00111 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 11011 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 01011 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 11001 => 10011 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 11000 => 00011 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 11000 => 00011 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 10011 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 11000 => 00011 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 11000 => 00011 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 00011 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 11101 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 01101 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 10101 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 10100 => 00101 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 00101 => 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 11001 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 10010 => 01001 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 10001 => 10001 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => 10000 => 00001 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => 10000 => 00001 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 10001 => 10001 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => 10000 => 00001 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => 10000 => 00001 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => 10000 => 00001 => 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => 10000000001 => 10000000001 => ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00000000011 => ? = 1
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1] => 100000000001 => 100000000001 => ? = 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => 10000000001 => 10000000001 => ? = 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00000000011 => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1] => 100000000001 => 100000000001 => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => 10000000001 => 10000000001 => ? = 1
[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => 10000000001 => 10000000001 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00000000011 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00000000011 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11] => 110000000000 => 000000000011 => ? = 1
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => 10000000001 => 10000000001 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00000000011 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00000000011 => ? = 1
[1,0,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00000000011 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,0,1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00000000011 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,0,1,1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00000000011 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,0,1,1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00000000011 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,1,1] => 1000000011 => 1100000001 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,8,1] => 1100000001 => 1000000011 => ? = 1
[1,0,1,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00000000011 => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,3] => 1111111100 => 0011111111 => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1] => 11111111111 => 11111111111 => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,3] => 1111111100 => 0011111111 => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,3] => 11111111100 => 00111111111 => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,6] => 11111100000 => 00000111111 => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,9] => 11100000000 => 00000000111 => ? = 1
Description
The number of ascents of a binary word.
Matching statistic: St000390
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1 => 0 => 0
[1,0,1,0]
=> [1,1] => 11 => 00 => 0
[1,1,0,0]
=> [2] => 10 => 01 => 1
[1,0,1,0,1,0]
=> [1,1,1] => 111 => 000 => 0
[1,0,1,1,0,0]
=> [1,2] => 110 => 001 => 1
[1,1,0,0,1,0]
=> [2,1] => 101 => 010 => 1
[1,1,0,1,0,0]
=> [3] => 100 => 011 => 1
[1,1,1,0,0,0]
=> [3] => 100 => 011 => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 0000 => 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 0001 => 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 0010 => 1
[1,0,1,1,0,1,0,0]
=> [1,3] => 1100 => 0011 => 1
[1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 0011 => 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 0100 => 1
[1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 0101 => 2
[1,1,0,1,0,0,1,0]
=> [3,1] => 1001 => 0110 => 1
[1,1,0,1,0,1,0,0]
=> [4] => 1000 => 0111 => 1
[1,1,0,1,1,0,0,0]
=> [4] => 1000 => 0111 => 1
[1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 0110 => 1
[1,1,1,0,0,1,0,0]
=> [4] => 1000 => 0111 => 1
[1,1,1,0,1,0,0,0]
=> [4] => 1000 => 0111 => 1
[1,1,1,1,0,0,0,0]
=> [4] => 1000 => 0111 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 00000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 00001 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 00010 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 11100 => 00011 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 00011 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 00100 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 00101 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 11001 => 00110 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 11000 => 00111 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 11000 => 00111 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 00110 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 11000 => 00111 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 11000 => 00111 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 00111 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 01000 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 01001 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 01010 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 10100 => 01011 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 01011 => 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 01100 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 10010 => 01101 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 10001 => 01110 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => 10000 => 01111 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => 10000 => 01111 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 10001 => 01110 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => 10000 => 01111 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => 10000 => 01111 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => 10000 => 01111 => 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 0111111110 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => 0011111111 => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => 10000000001 => 01111111110 => ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 0111111110 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => 0011111111 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00111111111 => ? = 1
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1] => 100000000001 => 011111111110 => ? = 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => 10000000001 => 01111111110 => ? = 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 0111111110 => ? = 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 0111111110 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => 0011111111 => ? = 1
[1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => 0011111111 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00111111111 => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1] => 100000000001 => 011111111110 => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => 10000000001 => 01111111110 => ? = 1
[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => 10000000001 => 01111111110 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 0111111110 => ? = 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 0111111110 => ? = 1
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 0111111110 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [1,9] => 1100000000 => 0011111111 => ? = 1
[1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => 0011111111 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00111111111 => ? = 1
[1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => 0011111111 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00111111111 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11] => 110000000000 => 001111111111 => ? = 1
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => 10000000001 => 01111111110 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 0111111110 => ? = 1
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 0111111110 => ? = 1
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 0111111110 => ? = 1
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 0111111110 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [1,9] => 1100000000 => 0011111111 => ? = 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0]
=> [1,9] => 1100000000 => 0011111111 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00111111111 => ? = 1
[1,0,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => 0011111111 => ? = 1
[1,0,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => 0011111111 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00111111111 => ? = 1
[1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => 0011111111 => ? = 1
[1,0,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00111111111 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 0111111110 => ? = 1
[1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 0111111110 => ? = 1
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 0111111110 => ? = 1
[1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 0111111110 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
=> [1,9] => 1100000000 => 0011111111 => ? = 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
=> [1,9] => 1100000000 => 0011111111 => ? = 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0]
=> [1,9] => 1100000000 => 0011111111 => ? = 1
[1,0,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => 0011111111 => ? = 1
[1,0,1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => 0011111111 => ? = 1
[1,0,1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00111111111 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,1,0]
=> [9,1] => 1000000001 => 0111111110 => ? = 1
[1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 0111111110 => ? = 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000291
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1 => 0
[1,0,1,0]
=> [1,1] => 11 => 0
[1,1,0,0]
=> [2] => 10 => 1
[1,0,1,0,1,0]
=> [1,1,1] => 111 => 0
[1,0,1,1,0,0]
=> [1,2] => 110 => 1
[1,1,0,0,1,0]
=> [2,1] => 101 => 1
[1,1,0,1,0,0]
=> [3] => 100 => 1
[1,1,1,0,0,0]
=> [3] => 100 => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 1
[1,0,1,1,0,1,0,0]
=> [1,3] => 1100 => 1
[1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 1
[1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 2
[1,1,0,1,0,0,1,0]
=> [3,1] => 1001 => 1
[1,1,0,1,0,1,0,0]
=> [4] => 1000 => 1
[1,1,0,1,1,0,0,0]
=> [4] => 1000 => 1
[1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 1
[1,1,1,0,0,1,0,0]
=> [4] => 1000 => 1
[1,1,1,0,1,0,0,0]
=> [4] => 1000 => 1
[1,1,1,1,0,0,0,0]
=> [4] => 1000 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 11100 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 11001 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 11000 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 11000 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 11000 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 11000 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 10100 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 10010 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 10001 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => 10000 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => 10000 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 10001 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => 10000 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => 10000 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => 10000 => 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => 10000000001 => ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => ? = 1
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1] => 100000000001 => ? = 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => 10000000001 => ? = 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => ? = 1
[1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1] => 100000000001 => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => 10000000001 => ? = 1
[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => 10000000001 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [1,9] => 1100000000 => ? = 1
[1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => ? = 1
[1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11] => 110000000000 => ? = 1
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => 10000000001 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [1,9] => 1100000000 => ? = 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0]
=> [1,9] => 1100000000 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => ? = 1
[1,0,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => ? = 1
[1,0,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => ? = 1
[1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => ? = 1
[1,0,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1
[1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1
[1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
=> [1,9] => 1100000000 => ? = 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
=> [1,9] => 1100000000 => ? = 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0]
=> [1,9] => 1100000000 => ? = 1
[1,0,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => ? = 1
[1,0,1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => ? = 1
[1,0,1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1
[1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1
Description
The number of descents of a binary word.
Matching statistic: St000374
(load all 72 compositions to match this statistic)
(load all 72 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => [3,2,1] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => [4,3,2,1] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,3,2,1] => [4,3,2,1] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,2,3,1] => [4,3,2,1] => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [4,3,2,1,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [5,4,3,2,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,4,3,1] => [5,4,3,2,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,3,2,4,1] => [5,4,3,2,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,3,4,5,2,6,7] => [1,5,3,4,2,6,7] => [1,5,4,3,2,6,7] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,3,4,5,2,7,6] => [1,5,3,4,2,7,6] => [1,5,4,3,2,7,6] => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,6,2,7] => [1,6,3,4,5,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,3,4,6,5,2,7] => [1,6,3,5,4,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,3,5,4,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,3,5,4,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,3,5,4,6,2,7] => [1,6,4,3,5,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,3,6,4,5,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,3,6,5,4,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,4,3,5,2,6,7] => [1,5,3,4,2,6,7] => [1,5,4,3,2,6,7] => ? = 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,4,3,5,2,7,6] => [1,5,3,4,2,7,6] => [1,5,4,3,2,7,6] => ? = 2
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,4,3,5,6,2,7] => [1,6,3,4,5,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,4,3,6,5,2,7] => [1,6,3,5,4,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,5,3,4,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 1
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,5,3,4,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,5,3,4,6,2,7] => [1,6,4,3,5,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,6,3,4,5,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,6,3,5,4,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 2
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,5,4,3,6,2,7] => [1,6,4,3,5,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,6,4,3,5,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,6,4,5,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 2
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ? = 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,3,4,6,7,5] => [2,1,3,4,7,6,5] => [2,1,3,4,7,6,5] => ? = 2
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [2,1,3,4,7,6,5] => [2,1,3,4,7,6,5] => [2,1,3,4,7,6,5] => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => ? = 3
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,6,4,7] => [2,1,3,6,5,4,7] => [2,1,3,6,5,4,7] => ? = 2
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [2,1,3,5,6,7,4] => [2,1,3,7,5,6,4] => [2,1,3,7,6,5,4] => ? = 2
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [2,1,3,5,7,6,4] => [2,1,3,7,6,5,4] => [2,1,3,7,6,5,4] => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,6,5,4,7] => [2,1,3,6,5,4,7] => [2,1,3,6,5,4,7] => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [2,1,3,6,5,7,4] => [2,1,3,7,5,6,4] => [2,1,3,7,6,5,4] => ? = 2
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [2,1,3,7,5,6,4] => [2,1,3,7,6,5,4] => [2,1,3,7,6,5,4] => ? = 2
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,7,6,5,4] => [2,1,3,7,6,5,4] => [2,1,3,7,6,5,4] => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => ? = 3
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 3
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,3,6,7,5] => [2,1,4,3,7,6,5] => [2,1,4,3,7,6,5] => ? = 3
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,6,5] => [2,1,4,3,7,6,5] => [2,1,4,3,7,6,5] => ? = 3
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,5,3,6,7] => [2,1,5,4,3,6,7] => [2,1,5,4,3,6,7] => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [2,1,4,5,3,7,6] => [2,1,5,4,3,7,6] => [2,1,5,4,3,7,6] => ? = 3
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,6,3,7] => [2,1,6,4,5,3,7] => [2,1,6,5,4,3,7] => ? = 2
[1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,6,5,3,7] => [2,1,6,5,4,3,7] => [2,1,6,5,4,3,7] => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,5,4,3,6,7] => [2,1,5,4,3,6,7] => [2,1,5,4,3,6,7] => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,4,3,7,6] => [2,1,5,4,3,7,6] => [2,1,5,4,3,7,6] => ? = 3
[1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,5,4,6,3,7] => [2,1,6,4,5,3,7] => [2,1,6,5,4,3,7] => ? = 2
[1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,6,4,5,3,7] => [2,1,6,5,4,3,7] => [2,1,6,5,4,3,7] => ? = 2
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000996
(load all 58 compositions to match this statistic)
(load all 58 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => [3,2,1] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => [4,3,2,1] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,3,2,1] => [4,3,2,1] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,2,3,1] => [4,3,2,1] => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [4,3,2,1,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [5,4,3,2,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,4,3,1] => [5,4,3,2,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,3,2,4,1] => [5,4,3,2,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,3,4,5,2,6,7] => [1,5,3,4,2,6,7] => [1,5,4,3,2,6,7] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,3,4,5,2,7,6] => [1,5,3,4,2,7,6] => [1,5,4,3,2,7,6] => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,6,2,7] => [1,6,3,4,5,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,3,4,6,5,2,7] => [1,6,3,5,4,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,3,5,4,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,3,5,4,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,3,5,4,6,2,7] => [1,6,4,3,5,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,3,6,4,5,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,3,6,5,4,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,4,3,5,2,6,7] => [1,5,3,4,2,6,7] => [1,5,4,3,2,6,7] => ? = 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,4,3,5,2,7,6] => [1,5,3,4,2,7,6] => [1,5,4,3,2,7,6] => ? = 2
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,4,3,5,6,2,7] => [1,6,3,4,5,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,4,3,6,5,2,7] => [1,6,3,5,4,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,5,3,4,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 1
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,5,3,4,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,5,3,4,6,2,7] => [1,6,4,3,5,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,6,3,4,5,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,6,3,5,4,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 2
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,5,4,3,6,2,7] => [1,6,4,3,5,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,6,4,3,5,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,6,4,5,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 2
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ? = 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,3,4,6,7,5] => [2,1,3,4,7,6,5] => [2,1,3,4,7,6,5] => ? = 2
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [2,1,3,4,7,6,5] => [2,1,3,4,7,6,5] => [2,1,3,4,7,6,5] => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => ? = 3
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,6,4,7] => [2,1,3,6,5,4,7] => [2,1,3,6,5,4,7] => ? = 2
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [2,1,3,5,6,7,4] => [2,1,3,7,5,6,4] => [2,1,3,7,6,5,4] => ? = 2
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [2,1,3,5,7,6,4] => [2,1,3,7,6,5,4] => [2,1,3,7,6,5,4] => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,6,5,4,7] => [2,1,3,6,5,4,7] => [2,1,3,6,5,4,7] => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [2,1,3,6,5,7,4] => [2,1,3,7,5,6,4] => [2,1,3,7,6,5,4] => ? = 2
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [2,1,3,7,5,6,4] => [2,1,3,7,6,5,4] => [2,1,3,7,6,5,4] => ? = 2
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,7,6,5,4] => [2,1,3,7,6,5,4] => [2,1,3,7,6,5,4] => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => ? = 3
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 3
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,3,6,7,5] => [2,1,4,3,7,6,5] => [2,1,4,3,7,6,5] => ? = 3
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,6,5] => [2,1,4,3,7,6,5] => [2,1,4,3,7,6,5] => ? = 3
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,5,3,6,7] => [2,1,5,4,3,6,7] => [2,1,5,4,3,6,7] => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [2,1,4,5,3,7,6] => [2,1,5,4,3,7,6] => [2,1,5,4,3,7,6] => ? = 3
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,6,3,7] => [2,1,6,4,5,3,7] => [2,1,6,5,4,3,7] => ? = 2
[1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,6,5,3,7] => [2,1,6,5,4,3,7] => [2,1,6,5,4,3,7] => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,5,4,3,6,7] => [2,1,5,4,3,6,7] => [2,1,5,4,3,6,7] => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,4,3,7,6] => [2,1,5,4,3,7,6] => [2,1,5,4,3,7,6] => ? = 3
[1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,5,4,6,3,7] => [2,1,6,4,5,3,7] => [2,1,6,5,4,3,7] => ? = 2
[1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,6,4,5,3,7] => [2,1,6,5,4,3,7] => [2,1,6,5,4,3,7] => ? = 2
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St000659
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00314: Integer compositions —Foata bijection⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 71%
Mp00314: Integer compositions —Foata bijection⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [1] => [1] => [1,0]
=> ? = 0
[1,0,1,0]
=> [1,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,0]
=> [2] => [2] => [1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [3] => [3] => [1,1,1,0,0,0]
=> 1
[1,1,1,0,0,0]
=> [3] => [3] => [1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,3,1] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,3,1] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,3,2] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,4,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,4,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,3,2] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,4,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,4,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,4,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,3,2,1] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,4,2] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,4,2] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,3,2,1] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,4,2] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,4,2] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,4,2] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,3,2] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,3,2] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3
[1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,1,2,1] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,1,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,1,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,3,1] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,3,1] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,4,2,1] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,4,3] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [1,4,3] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,5,2] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,6,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,6,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,5,2] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,6,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,6,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,6,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,4,2,1] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,4,3] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St000245
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 57%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [2,1] => 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [1,2] => 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => [4,5,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [4,3,5,2,1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => [3,5,4,2,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => [3,5,4,2,1] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,5,1] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => [3,5,2,4,1] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [3,2,5,4,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [3,2,5,4,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => [3,5,2,1,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [3,2,5,1,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2,1,5,4] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => [6,7,5,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => [6,5,7,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [5,7,6,4,3,2,1] => [5,7,6,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,4,3,2,1] => [5,7,6,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,5,4,7,3,2,1] => [6,5,4,7,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,6,4,7,3,2,1] => [5,7,4,6,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [5,4,7,6,3,2,1] => [5,4,7,6,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,7,6,5,3,2,1] => [4,7,6,5,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,6,7,5,3,2,1] => [4,7,6,5,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [5,4,6,7,3,2,1] => [5,4,7,6,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [4,6,5,7,3,2,1] => [4,7,6,5,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,7,6,3,2,1] => [4,7,6,5,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [4,7,6,5,3,2,1] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [6,5,4,3,7,2,1] => [6,5,4,3,7,2,1] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,6,4,3,7,2,1] => [5,7,4,3,6,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => [5,4,7,3,6,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [4,6,5,3,7,2,1] => [4,7,6,3,5,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,5,6,3,7,2,1] => [4,7,6,3,5,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [5,4,3,7,6,2,1] => [5,4,3,7,6,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [4,5,3,7,6,2,1] => [4,7,3,6,5,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [4,3,7,6,5,2,1] => [4,3,7,6,5,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,7,6,5,4,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,6,7,5,4,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [4,3,6,7,5,2,1] => [4,3,7,6,5,2,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [3,6,5,7,4,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,5,7,6,4,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,5,6,7,4,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => [5,4,3,7,6,2,1] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [4,5,3,6,7,2,1] => [4,7,3,6,5,2,1] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,3,6,5,7,2,1] => [4,3,7,6,5,2,1] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [3,6,5,4,7,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [3,5,6,4,7,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [4,3,5,7,6,2,1] => [4,3,7,6,5,2,1] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,5,4,7,6,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,7,6,5,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [3,4,6,7,5,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [4,3,5,6,7,2,1] => [4,3,7,6,5,2,1] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [3,5,4,6,7,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [3,4,6,5,7,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,4,5,7,6,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,7,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,5,4,3,2,7,1] => [6,5,4,3,2,7,1] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [5,6,4,3,2,7,1] => [5,7,4,3,2,6,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [5,4,6,3,2,7,1] => [5,4,7,3,2,6,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [4,6,5,3,2,7,1] => [4,7,6,3,2,5,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [4,5,6,3,2,7,1] => [4,7,6,3,2,5,1] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [5,4,3,6,2,7,1] => [5,4,3,7,2,6,1] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [4,5,3,6,2,7,1] => [4,7,3,6,2,5,1] => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [4,3,6,5,2,7,1] => [4,3,7,6,2,5,1] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [3,6,5,4,2,7,1] => [3,7,6,5,2,4,1] => ? = 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)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 57%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [2,1] => 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [1,2] => 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => [4,5,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [4,3,5,2,1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => [3,5,4,2,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => [3,5,4,2,1] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,5,1] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => [3,5,2,4,1] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [3,2,5,4,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [3,2,5,4,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => [3,5,2,1,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [3,2,5,1,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2,1,5,4] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => [6,7,5,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => [6,5,7,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [5,7,6,4,3,2,1] => [5,7,6,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,4,3,2,1] => [5,7,6,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,5,4,7,3,2,1] => [6,5,4,7,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,6,4,7,3,2,1] => [5,7,4,6,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [5,4,7,6,3,2,1] => [5,4,7,6,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,7,6,5,3,2,1] => [4,7,6,5,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,6,7,5,3,2,1] => [4,7,6,5,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [5,4,6,7,3,2,1] => [5,4,7,6,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [4,6,5,7,3,2,1] => [4,7,6,5,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,7,6,3,2,1] => [4,7,6,5,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [4,7,6,5,3,2,1] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [6,5,4,3,7,2,1] => [6,5,4,3,7,2,1] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,6,4,3,7,2,1] => [5,7,4,3,6,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => [5,4,7,3,6,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [4,6,5,3,7,2,1] => [4,7,6,3,5,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,5,6,3,7,2,1] => [4,7,6,3,5,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [5,4,3,7,6,2,1] => [5,4,3,7,6,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [4,5,3,7,6,2,1] => [4,7,3,6,5,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [4,3,7,6,5,2,1] => [4,3,7,6,5,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,7,6,5,4,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,6,7,5,4,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [4,3,6,7,5,2,1] => [4,3,7,6,5,2,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [3,6,5,7,4,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,5,7,6,4,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,5,6,7,4,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => [5,4,3,7,6,2,1] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [4,5,3,6,7,2,1] => [4,7,3,6,5,2,1] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,3,6,5,7,2,1] => [4,3,7,6,5,2,1] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [3,6,5,4,7,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [3,5,6,4,7,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [4,3,5,7,6,2,1] => [4,3,7,6,5,2,1] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,5,4,7,6,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,7,6,5,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [3,4,6,7,5,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [4,3,5,6,7,2,1] => [4,3,7,6,5,2,1] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [3,5,4,6,7,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [3,4,6,5,7,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,4,5,7,6,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,7,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,5,4,3,2,7,1] => [6,5,4,3,2,7,1] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [5,6,4,3,2,7,1] => [5,7,4,3,2,6,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [5,4,6,3,2,7,1] => [5,4,7,3,2,6,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [4,6,5,3,2,7,1] => [4,7,6,3,2,5,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [4,5,6,3,2,7,1] => [4,7,6,3,2,5,1] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [5,4,3,6,2,7,1] => [5,4,3,7,2,6,1] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [4,5,3,6,2,7,1] => [4,7,3,6,2,5,1] => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [4,3,6,5,2,7,1] => [4,3,7,6,2,5,1] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [3,6,5,4,2,7,1] => [3,7,6,5,2,4,1] => ? = 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: St000834
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 57%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [2,1] => 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [1,2] => 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => [4,5,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [4,3,5,2,1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => [3,5,4,2,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => [3,5,4,2,1] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,5,1] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => [3,5,2,4,1] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [3,2,5,4,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [3,2,5,4,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => [3,5,2,1,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [3,2,5,1,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2,1,5,4] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => [6,7,5,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => [6,5,7,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [5,7,6,4,3,2,1] => [5,7,6,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,4,3,2,1] => [5,7,6,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,5,4,7,3,2,1] => [6,5,4,7,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,6,4,7,3,2,1] => [5,7,4,6,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [5,4,7,6,3,2,1] => [5,4,7,6,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,7,6,5,3,2,1] => [4,7,6,5,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,6,7,5,3,2,1] => [4,7,6,5,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [5,4,6,7,3,2,1] => [5,4,7,6,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [4,6,5,7,3,2,1] => [4,7,6,5,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,7,6,3,2,1] => [4,7,6,5,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [4,7,6,5,3,2,1] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [6,5,4,3,7,2,1] => [6,5,4,3,7,2,1] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,6,4,3,7,2,1] => [5,7,4,3,6,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => [5,4,7,3,6,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [4,6,5,3,7,2,1] => [4,7,6,3,5,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,5,6,3,7,2,1] => [4,7,6,3,5,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [5,4,3,7,6,2,1] => [5,4,3,7,6,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [4,5,3,7,6,2,1] => [4,7,3,6,5,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [4,3,7,6,5,2,1] => [4,3,7,6,5,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,7,6,5,4,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,6,7,5,4,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [4,3,6,7,5,2,1] => [4,3,7,6,5,2,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [3,6,5,7,4,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,5,7,6,4,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,5,6,7,4,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => [5,4,3,7,6,2,1] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [4,5,3,6,7,2,1] => [4,7,3,6,5,2,1] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,3,6,5,7,2,1] => [4,3,7,6,5,2,1] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [3,6,5,4,7,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [3,5,6,4,7,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [4,3,5,7,6,2,1] => [4,3,7,6,5,2,1] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,5,4,7,6,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,7,6,5,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [3,4,6,7,5,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [4,3,5,6,7,2,1] => [4,3,7,6,5,2,1] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [3,5,4,6,7,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [3,4,6,5,7,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,4,5,7,6,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,7,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,5,4,3,2,7,1] => [6,5,4,3,2,7,1] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [5,6,4,3,2,7,1] => [5,7,4,3,2,6,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [5,4,6,3,2,7,1] => [5,4,7,3,2,6,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [4,6,5,3,2,7,1] => [4,7,6,3,2,5,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [4,5,6,3,2,7,1] => [4,7,6,3,2,5,1] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [5,4,3,6,2,7,1] => [5,4,3,7,2,6,1] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [4,5,3,6,2,7,1] => [4,7,3,6,2,5,1] => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [4,3,6,5,2,7,1] => [4,3,7,6,2,5,1] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [3,6,5,4,2,7,1] => [3,7,6,5,2,4,1] => ? = 2
Description
The number of right outer peaks of a permutation.
A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
The following 53 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000919The number of maximal left branches of a binary tree. St001613The binary logarithm of the size of the center of a lattice. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001333The cardinality of a minimal edge-isolating set of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St000258The burning number of a graph. St000658The number of rises of length 2 of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001354The number of series nodes in the modular decomposition of a graph. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001060The distinguishing index of a graph. St000260The radius of a connected graph. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000456The monochromatic index of a connected graph. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001298The number of repeated entries in the Lehmer code of a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001545The second Elser number of a connected graph. St001330The hat guessing number of a graph. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000455The second largest eigenvalue of a graph if it is integral. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000640The rank of the largest boolean interval in a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000660The number of rises of length at least 3 of a Dyck path. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St000264The girth of a graph, which is not a tree. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001597The Frobenius rank of a skew partition. 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 $eA$. St001335The cardinality of a minimal cycle-isolating set of a graph. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001624The breadth of a lattice.
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