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Your data matches 172 different statistics following compositions of up to 3 maps.
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Matching statistic: St000010
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 1
[1,0,1,0]
=> [2,1] => [2] => [2]
=> 1
[1,1,0,0]
=> [1,2] => [2] => [2]
=> 1
[1,0,1,0,1,0]
=> [2,3,1] => [3] => [3]
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => [3] => [3]
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,2] => [2,1]
=> 2
[1,1,0,1,0,0]
=> [3,1,2] => [3] => [3]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [3] => [3]
=> 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4] => [4]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [4] => [4]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2] => [2,2]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4] => [4]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [4] => [4]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,3] => [3,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3] => [3,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2] => [2,2]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4] => [4]
=> 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [4] => [4]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,2] => [2,2]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3] => [3,1]
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4] => [4]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4] => [4]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5] => [5]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [5] => [5]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2] => [3,2]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5] => [5]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [5] => [5]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,3] => [3,2]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,3] => [3,2]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,2] => [3,2]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5] => [5]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [5] => [5]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [3,2] => [3,2]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,3] => [3,2]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5] => [5]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [5] => [5]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,4] => [4,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4] => [4,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,2,2] => [2,2,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,4] => [4,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,4] => [4,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,3] => [3,2]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [2,3] => [3,2]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,2] => [3,2]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5] => [5]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5] => [5]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2] => [3,2]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [2,3] => [3,2]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5] => [5]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [5] => [5]
=> 1
Description
The length of the partition.
Matching statistic: St000288
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1 => 1
[1,0,1,0]
=> [2,1] => [2] => 10 => 1
[1,1,0,0]
=> [1,2] => [2] => 10 => 1
[1,0,1,0,1,0]
=> [2,3,1] => [3] => 100 => 1
[1,0,1,1,0,0]
=> [2,1,3] => [3] => 100 => 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,2] => 110 => 2
[1,1,0,1,0,0]
=> [3,1,2] => [3] => 100 => 1
[1,1,1,0,0,0]
=> [1,2,3] => [3] => 100 => 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4] => 1000 => 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [4] => 1000 => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2] => 1010 => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4] => 1000 => 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [4] => 1000 => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,3] => 1100 => 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3] => 1100 => 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2] => 1010 => 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4] => 1000 => 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [4] => 1000 => 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,2] => 1010 => 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3] => 1100 => 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4] => 1000 => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4] => 1000 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5] => 10000 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [5] => 10000 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2] => 10010 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5] => 10000 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [5] => 10000 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,3] => 10100 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,3] => 10100 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,2] => 10010 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5] => 10000 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [5] => 10000 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [3,2] => 10010 => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,3] => 10100 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5] => 10000 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [5] => 10000 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,4] => 11000 => 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4] => 11000 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,2,2] => 11010 => 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,4] => 11000 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,4] => 11000 => 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,3] => 10100 => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [2,3] => 10100 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,2] => 10010 => 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5] => 10000 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5] => 10000 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2] => 10010 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [2,3] => 10100 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5] => 10000 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [5] => 10000 => 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,8,2,3,4,5,6,9,7] => [1,6,2] => 110000010 => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,7,6,9,8,10] => [3,2,2,3] => 1001010100 => ? = 4
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,8,9,10,2] => [1,9] => 1100000000 => ? = 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,4,5,6,7,8,9,10,3] => [2,8] => 1010000000 => ? = 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,5,6,7,8,9,10,4] => [3,7] => 1001000000 => ? = 2
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,6,7,8,9,10,5] => [4,6] => 1000100000 => ? = 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,7,8,9,10,6] => [5,5] => 1000010000 => ? = 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,8,9,10,7] => [6,4] => 1000001000 => ? = 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,4,5,6,7,8,9,10,2] => [2,8] => 1010000000 => ? = 2
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,10,2,3,4,5,6,7,8,9] => [1,9] => 1100000000 => ? = 2
[1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0]
=> [1,9,10,2,3,4,5,6,7,8] => [1,9] => 1100000000 => ? = 2
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
=> [1,11,2,3,4,5,6,7,8,9,10] => [1,10] => 11000000000 => ? = 2
[1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0]
=> [1,2,3,6,4,7,8,9,5] => [3,2,4] => 100101000 => ? = 3
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [2,4,6,8,10,12,1,3,5,7,9,11] => [12] => 100000000000 => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,4,5,6,7,8,9,10,11,2] => [2,9] => 10100000000 => ? = 2
[1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0]
=> [9,1,10,2,3,4,5,6,7,8] => [2,8] => 1010000000 => ? = 2
[1,0,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,4,1,6,3,8,5,10,7,9] => [3,2,2,3] => 1001010100 => ? = 4
[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,5,3,7,6,9,8,10] => [3,2,2,3] => 1001010100 => ? = 4
[1,0,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [2,4,1,6,3,8,5,7,9] => [3,2,4] => 100101000 => ? = 3
[1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,3,6,5,8,7,10,9] => [4,2,2,2] => 1000101010 => ? = 4
[1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [2,8,9,10,11,12,1,3,4,5,6,7] => [12] => 100000000000 => ? = 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000291
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 92%●distinct values known / distinct values provided: 83%
Mp00109: Permutations —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 92%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1] => => ? = 1 - 1
[1,0,1,0]
=> [1,2] => 0 => 0 = 1 - 1
[1,1,0,0]
=> [2,1] => 1 => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => 00 => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => 01 => 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1,3] => 10 => 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => 01 => 0 = 1 - 1
[1,1,1,0,0,0]
=> [3,2,1] => 11 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 001 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 011 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 010 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 001 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 011 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 110 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 101 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 011 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0001 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0010 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0001 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0011 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0100 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0101 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0010 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0001 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 0011 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0110 => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 0101 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 0011 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0111 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1000 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1001 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1010 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1001 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1011 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 0100 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 0101 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0010 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0001 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 0011 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 0110 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 0101 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 0011 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 0111 => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1100 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,2,4,5,3,6,7,8] => ? => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,2,4,6,5,7,3,8] => ? => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,2,5,4,6,3,8,7] => ? => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,2,5,4,7,8,6,3] => ? => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,2,5,6,4,3,8,7] => ? => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,2,6,5,7,4,8,3] => ? => ? = 3 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,2,6,7,5,4,8,3] => ? => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,3,2,4,6,7,5,8] => ? => ? = 3 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,3,2,4,7,8,6,5] => ? => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,4,2,7,6,5,8] => ? => ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,3,5,4,6,2,8,7] => ? => ? = 3 - 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,3,6,5,4,7,8,2] => ? => ? = 2 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,3,6,7,5,4,8,2] => ? => ? = 2 - 1
[1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,3,7,6,5,8,4,2] => ? => ? = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,4,5,6,3,7,8,2] => ? => ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,4,6,5,3,7,8,2] => ? => ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [1,5,4,6,3,8,7,2] => ? => ? = 3 - 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,5,4,7,6,3,8,2] => ? => ? = 3 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,5,6,4,3,7,2,8] => ? => ? = 3 - 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,5,4,6,7,8] => ? => ? = 3 - 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,3,5,6,7,4,8] => ? => ? = 3 - 1
[1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,3,7,6,8,5,4] => ? => ? = 3 - 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3,7,8,6] => ? => ? = 3 - 1
[1,1,0,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,1,4,6,5,7,3,8] => ? => ? = 4 - 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [2,3,1,5,4,7,8,6] => ? => ? = 3 - 1
[1,1,0,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,5,4,1,7,8,6] => ? => ? = 2 - 1
[1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [2,3,6,7,5,8,4,1] => ? => ? = 2 - 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [2,4,3,6,7,5,1,8] => ? => ? = 3 - 1
[1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [2,4,3,7,6,8,5,1] => ? => ? = 3 - 1
[1,1,0,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [2,4,5,3,7,6,1,8] => ? => ? = 3 - 1
[1,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [2,4,5,6,3,7,1,8] => ? => ? = 3 - 1
[1,1,0,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [2,4,5,7,6,3,1,8] => ? => ? = 2 - 1
[1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [2,4,5,7,6,8,3,1] => ? => ? = 2 - 1
[1,1,0,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [2,5,4,6,3,8,7,1] => ? => ? = 3 - 1
[1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [2,5,4,6,8,7,3,1] => ? => ? = 2 - 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [2,5,6,4,7,3,8,1] => ? => ? = 3 - 1
[1,1,1,0,0,0,1,0,1,1,0,1,0,0,1,0]
=> [3,2,1,4,6,7,5,8] => ? => ? = 3 - 1
[1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [3,2,1,5,4,8,7,6] => ? => ? = 3 - 1
[1,1,1,0,0,1,0,1,0,0,1,1,0,0,1,0]
=> [3,2,4,5,1,7,6,8] => ? => ? = 4 - 1
[1,1,1,0,0,1,0,1,0,1,0,0,1,1,0,0]
=> [3,2,4,5,6,1,8,7] => ? => ? = 3 - 1
[1,1,1,0,0,1,0,1,1,0,0,1,0,0,1,0]
=> [3,2,4,6,5,7,1,8] => ? => ? = 4 - 1
[1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> [3,2,5,4,8,7,6,1] => ? => ? = 3 - 1
[1,1,1,0,0,1,1,0,1,0,0,1,0,0,1,0]
=> [3,2,5,6,4,7,1,8] => ? => ? = 4 - 1
[1,1,1,0,0,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2,6,5,7,8,4,1] => ? => ? = 3 - 1
[1,1,1,0,0,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,6,7,8,5,4,1] => ? => ? = 2 - 1
[1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0]
=> [3,2,7,6,5,8,4,1] => ? => ? = 3 - 1
[1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0]
=> [3,2,7,6,8,5,4,1] => ? => ? = 3 - 1
[1,1,1,0,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,4,2,1,5,7,6,8] => ? => ? = 3 - 1
[1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
=> [3,4,5,7,6,2,1,8] => ? => ? = 2 - 1
Description
The number of descents of a binary word.
Matching statistic: St000292
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 92%●distinct values known / distinct values provided: 83%
Mp00069: Permutations —complement⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 92%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1] => [1] => => ? = 1 - 1
[1,0,1,0]
=> [2,1] => [1,2] => 0 => 0 = 1 - 1
[1,1,0,0]
=> [1,2] => [2,1] => 1 => 0 = 1 - 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => 00 => 0 = 1 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1,3] => 10 => 0 = 1 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 01 => 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,3,1] => 10 => 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => 11 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => 000 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [2,1,3,4] => 100 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,3,2,4] => 010 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,3,1,4] => 100 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,2,1,4] => 110 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,4,3] => 001 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => 101 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,4,2] => 010 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,4,1] => 100 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,4,1] => 110 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => 011 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,4,3,1] => 101 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [3,4,2,1] => 110 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => 111 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [2,1,3,4,5] => 1000 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,3,2,4,5] => 0100 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [2,3,1,4,5] => 1000 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,2,1,4,5] => 1100 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,4,3,5] => 0010 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [2,1,4,3,5] => 1010 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,3,4,2,5] => 0100 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [2,3,4,1,5] => 1000 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2,4,1,5] => 1100 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,4,3,2,5] => 0110 => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,4,3,1,5] => 1010 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,4,2,1,5] => 1100 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => 1110 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,5,4] => 0001 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [2,1,3,5,4] => 1001 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,3,2,5,4] => 0101 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [2,3,1,5,4] => 1001 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,2,1,5,4] => 1101 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,2,4,5,3] => 0010 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [2,1,4,5,3] => 1010 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,3,4,5,2] => 0100 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [2,3,4,5,1] => 1000 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,2,4,5,1] => 1100 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,3,5,2] => 0110 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,4,3,5,1] => 1010 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,4,2,5,1] => 1100 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,3,2,5,1] => 1110 => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => 0011 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [6,5,7,3,4,8,2,1] => [3,4,2,6,5,1,7,8] => ? => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,4,5,3,7,8,2,1] => [3,5,4,6,2,1,7,8] => ? => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,4,3,5,6,8,2,1] => [2,5,6,4,3,1,7,8] => ? => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [6,5,7,8,4,2,3,1] => [3,4,2,1,5,7,6,8] => ? => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,3,2,4,1] => [1,3,2,4,6,7,5,8] => ? => ? = 3 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [6,7,8,5,3,2,4,1] => [3,2,1,4,6,7,5,8] => ? => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [7,5,6,8,3,2,4,1] => [2,4,3,1,6,7,5,8] => ? => ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [8,7,5,3,4,2,6,1] => [1,2,4,6,5,7,3,8] => ? => ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [7,8,5,3,4,2,6,1] => [2,1,4,6,5,7,3,8] => ? => ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [7,5,4,3,6,2,8,1] => [2,4,5,6,3,7,1,8] => ? => ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [7,6,3,4,5,2,8,1] => [2,3,6,5,4,7,1,8] => ? => ? = 2 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [8,4,3,5,6,2,7,1] => [1,5,6,4,3,7,2,8] => ? => ? = 2 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [7,4,3,5,6,2,8,1] => [2,5,6,4,3,7,1,8] => ? => ? = 2 - 1
[1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [5,4,3,6,7,2,8,1] => [4,5,6,3,2,7,1,8] => ? => ? = 1 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [7,5,4,6,2,3,8,1] => [2,4,5,3,7,6,1,8] => ? => ? = 3 - 1
[1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [5,4,6,3,2,7,8,1] => [4,5,3,6,7,2,1,8] => ? => ? = 1 - 1
[1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [5,4,3,6,2,7,8,1] => [4,5,6,3,7,2,1,8] => ? => ? = 1 - 1
[1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [6,5,4,2,3,7,8,1] => [3,4,5,7,6,2,1,8] => ? => ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [8,6,3,2,4,5,7,1] => ? => ? => ? = 3 - 1
[1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [6,2,3,4,5,7,8,1] => [3,7,6,5,4,2,1,8] => ? => ? = 2 - 1
[1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,8,3,1,2] => ? => ? => ? = 3 - 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,6,8,4,3,5,1,2] => [2,3,1,5,6,4,8,7] => ? => ? = 3 - 1
[1,1,0,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [8,6,4,5,3,7,1,2] => [1,3,5,4,6,2,8,7] => ? => ? = 4 - 1
[1,1,0,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [8,7,4,3,5,6,1,2] => [1,2,5,6,4,3,8,7] => ? => ? = 3 - 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [7,6,8,4,5,2,1,3] => [2,3,1,5,4,7,8,6] => ? => ? = 3 - 1
[1,1,0,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [7,6,8,3,4,2,1,5] => [2,3,1,6,5,7,8,4] => ? => ? = 2 - 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [8,5,4,3,6,2,1,7] => [1,4,5,6,3,7,8,2] => ? => ? = 2 - 1
[1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [5,4,6,3,7,2,1,8] => [4,5,3,6,2,7,8,1] => ? => ? = 1 - 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [8,5,4,6,2,3,1,7] => [1,4,5,3,7,6,8,2] => ? => ? = 3 - 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [7,5,4,6,2,3,1,8] => [2,4,5,3,7,6,8,1] => ? => ? = 3 - 1
[1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [5,6,4,2,3,7,1,8] => [4,3,5,7,6,2,8,1] => ? => ? = 2 - 1
[1,1,0,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [8,4,5,2,3,6,1,7] => [1,5,4,7,6,3,8,2] => ? => ? = 3 - 1
[1,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [5,4,2,3,6,7,1,8] => [4,5,7,6,3,2,8,1] => ? => ? = 2 - 1
[1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [6,2,3,4,5,7,1,8] => [3,7,6,5,4,2,8,1] => ? => ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [6,7,8,4,5,1,2,3] => [3,2,1,5,4,8,7,6] => ? => ? = 3 - 1
[1,1,1,0,0,1,0,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,3,1,2,4] => [2,1,3,4,6,8,7,5] => ? => ? = 3 - 1
[1,1,1,0,0,1,0,1,1,0,0,1,0,0,1,0]
=> [8,6,4,5,3,1,2,7] => [1,3,5,4,6,8,7,2] => ? => ? = 4 - 1
[1,1,1,0,0,1,1,0,1,0,0,1,0,0,1,0]
=> [8,6,4,3,5,1,2,7] => [1,3,5,6,4,8,7,2] => ? => ? = 4 - 1
[1,1,1,0,0,1,1,1,0,0,1,0,1,0,0,0]
=> [6,5,3,4,7,1,2,8] => [3,4,6,5,2,8,7,1] => ? => ? = 3 - 1
[1,1,1,0,1,0,0,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,2,1,3,4] => [1,3,2,4,7,8,6,5] => ? => ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [7,6,8,4,2,1,3,5] => [2,3,1,5,7,8,6,4] => ? => ? = 3 - 1
[1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
=> [5,4,6,3,2,1,7,8] => [4,5,3,6,7,8,2,1] => ? => ? = 1 - 1
[1,1,1,0,1,0,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,2,1,5,8] => [3,2,6,5,7,8,4,1] => ? => ? = 2 - 1
[1,1,1,0,1,0,1,1,0,0,1,0,0,0,1,0]
=> [8,5,3,4,2,1,6,7] => [1,4,6,5,7,8,3,2] => ? => ? = 3 - 1
[1,1,1,0,1,1,0,0,0,0,1,0,1,1,0,0]
=> [7,8,6,2,3,1,4,5] => [2,1,3,7,6,8,5,4] => ? => ? = 2 - 1
[1,1,1,0,1,1,0,0,0,1,1,0,0,1,0,0]
=> [7,5,6,2,3,1,4,8] => [2,4,3,7,6,8,5,1] => ? => ? = 3 - 1
[1,1,1,0,1,1,0,0,1,0,0,0,1,0,1,0]
=> [8,7,4,2,3,1,5,6] => ? => ? => ? = 3 - 1
[1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> [7,6,3,2,4,1,5,8] => [2,3,6,7,5,8,4,1] => ? => ? = 2 - 1
[1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
=> [5,6,3,2,4,1,7,8] => [4,3,6,7,5,8,2,1] => ? => ? = 2 - 1
Description
The number of ascents of a binary word.
Matching statistic: St000097
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 89%●distinct values known / distinct values provided: 67%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 89%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1
[1,0,1,0]
=> [2,1] => [2] => ([],2)
=> 1
[1,1,0,0]
=> [1,2] => [2] => ([],2)
=> 1
[1,0,1,0,1,0]
=> [2,3,1] => [3] => ([],3)
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => [3] => ([],3)
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,2] => ([(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [3,1,2] => [3] => ([],3)
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [3] => ([],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4] => ([],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [4] => ([],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4] => ([],4)
=> 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [4] => ([],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,3] => ([(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3] => ([(2,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4] => ([],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [4] => ([],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3] => ([(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4] => ([],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4] => ([],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5] => ([],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [5] => ([],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5] => ([],5)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [5] => ([],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5] => ([],5)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [5] => ([],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5] => ([],5)
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [5] => ([],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5] => ([],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5] => ([],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5] => ([],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [5] => ([],5)
=> 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,9,8] => [7,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 2
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9] => [9] => ([],9)
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,8,10,9] => [8,2] => ([(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,9,7] => [7,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 2
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,9,1,3,4,5,6,7,8] => [9] => ([],9)
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9,10] => [10] => ([],10)
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [9,2] => ([(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)
=> ? = 2
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,10,8] => [8,2] => ([(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,8,2,3,4,5,6,9,7] => [1,6,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,9,8] => [7,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 2
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [2,1,9,3,4,5,6,7,8] => [2,7] => ([(6,8),(7,8)],9)
=> ? = 2
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7,9] => [9] => ([],9)
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [2,10,1,3,4,5,6,7,8,9] => [10] => ([],10)
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9,10,11] => [11] => ([],11)
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,11,9] => [9,2] => ([(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)
=> ? = 2
[1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [2,9,10,1,3,4,5,6,7,8] => [10] => ([],10)
=> ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> [1,2,3,4,5,8,6,9,7] => [5,2,2] => ([(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,2,3,4,5,6,8,9,7] => [6,3] => ([(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,1,0]
=> [7,1,2,3,4,5,8,9,6] => [6,3] => ([(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [1,2,3,4,5,6,7,9,10,8] => [7,3] => ([(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,8,9,1,2] => [9] => ([],9)
=> ? = 1
[1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,4,1,2,5,6,7,8,9] => [9] => ([],9)
=> ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,4,5,6,7,8,9,2,3] => [1,8] => ([(7,8)],9)
=> ? = 2
[1,1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> [4,5,6,1,2,3,7,8,9] => [9] => ([],9)
=> ? = 1
[1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [5,6,7,8,1,2,3,4,9] => [9] => ([],9)
=> ? = 1
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [6,7,8,9,1,2,3,4,5] => [9] => ([],9)
=> ? = 1
[1,1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0]
=> [6,8,1,2,3,4,5,7,9] => [9] => ([],9)
=> ? = 1
[1,1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0,0]
=> [1,7,9,2,3,4,5,6,8] => [1,8] => ([(7,8)],9)
=> ? = 2
[1,1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0,0]
=> [7,1,8,2,3,4,5,6,9] => [2,7] => ([(6,8),(7,8)],9)
=> ? = 2
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [7,8,9,1,2,3,4,5,6] => [9] => ([],9)
=> ? = 1
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> [7,8,1,2,3,4,5,6,9] => [9] => ([],9)
=> ? = 1
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [7,1,2,3,4,5,6,8,9] => [9] => ([],9)
=> ? = 1
[1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,0]
=> [1,8,2,9,3,4,5,6,7] => [1,2,6] => ([(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
[1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
=> [1,8,9,2,3,4,5,6,7] => [1,8] => ([(7,8)],9)
=> ? = 2
[1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0]
=> [8,1,2,3,9,4,5,6,7] => [4,5] => ([(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 2
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [8,9,1,2,3,4,5,6,7] => [9] => ([],9)
=> ? = 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7,9] => [9] => ([],9)
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,2,3,4,5,6,9,7,8] => [6,3] => ([(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 2
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,9,2,3,4,5,6,7,8] => [1,8] => ([(7,8)],9)
=> ? = 2
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [9,1,2,3,4,5,6,7,8] => [9] => ([],9)
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8,9] => [9] => ([],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,9,1] => [9] => ([],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,7,8,1,9] => [9] => ([],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,6,7,9,1,8] => [9] => ([],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,5,6,8,9,1,7] => [9] => ([],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,7,8,9,1,6] => [9] => ([],9)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,6,7,8,9,1,5] => [9] => ([],9)
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,8,9,1,4] => [9] => ([],9)
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,7,8,9,1,3] => [9] => ([],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,9,10,1] => [10] => ([],10)
=> ? = 1
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St001581
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 89%●distinct values known / distinct values provided: 67%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 89%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1
[1,0,1,0]
=> [2,1] => [2] => ([],2)
=> 1
[1,1,0,0]
=> [1,2] => [2] => ([],2)
=> 1
[1,0,1,0,1,0]
=> [2,3,1] => [3] => ([],3)
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => [3] => ([],3)
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,2] => ([(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [3,1,2] => [3] => ([],3)
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [3] => ([],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4] => ([],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [4] => ([],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4] => ([],4)
=> 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [4] => ([],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,3] => ([(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3] => ([(2,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4] => ([],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [4] => ([],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3] => ([(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4] => ([],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4] => ([],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5] => ([],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [5] => ([],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5] => ([],5)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [5] => ([],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5] => ([],5)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [5] => ([],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5] => ([],5)
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [5] => ([],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5] => ([],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5] => ([],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5] => ([],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [5] => ([],5)
=> 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,9,8] => [7,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 2
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9] => [9] => ([],9)
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,8,10,9] => [8,2] => ([(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,9,7] => [7,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 2
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,9,1,3,4,5,6,7,8] => [9] => ([],9)
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9,10] => [10] => ([],10)
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [9,2] => ([(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)
=> ? = 2
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,10,8] => [8,2] => ([(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,8,2,3,4,5,6,9,7] => [1,6,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,9,8] => [7,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 2
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [2,1,9,3,4,5,6,7,8] => [2,7] => ([(6,8),(7,8)],9)
=> ? = 2
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7,9] => [9] => ([],9)
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [2,10,1,3,4,5,6,7,8,9] => [10] => ([],10)
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9,10,11] => [11] => ([],11)
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,11,9] => [9,2] => ([(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)
=> ? = 2
[1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [2,9,10,1,3,4,5,6,7,8] => [10] => ([],10)
=> ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> [1,2,3,4,5,8,6,9,7] => [5,2,2] => ([(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,2,3,4,5,6,8,9,7] => [6,3] => ([(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,1,0]
=> [7,1,2,3,4,5,8,9,6] => [6,3] => ([(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [1,2,3,4,5,6,7,9,10,8] => [7,3] => ([(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,8,9,1,2] => [9] => ([],9)
=> ? = 1
[1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,4,1,2,5,6,7,8,9] => [9] => ([],9)
=> ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,4,5,6,7,8,9,2,3] => [1,8] => ([(7,8)],9)
=> ? = 2
[1,1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> [4,5,6,1,2,3,7,8,9] => [9] => ([],9)
=> ? = 1
[1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [5,6,7,8,1,2,3,4,9] => [9] => ([],9)
=> ? = 1
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [6,7,8,9,1,2,3,4,5] => [9] => ([],9)
=> ? = 1
[1,1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0]
=> [6,8,1,2,3,4,5,7,9] => [9] => ([],9)
=> ? = 1
[1,1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0,0]
=> [1,7,9,2,3,4,5,6,8] => [1,8] => ([(7,8)],9)
=> ? = 2
[1,1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0,0]
=> [7,1,8,2,3,4,5,6,9] => [2,7] => ([(6,8),(7,8)],9)
=> ? = 2
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [7,8,9,1,2,3,4,5,6] => [9] => ([],9)
=> ? = 1
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> [7,8,1,2,3,4,5,6,9] => [9] => ([],9)
=> ? = 1
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [7,1,2,3,4,5,6,8,9] => [9] => ([],9)
=> ? = 1
[1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,0]
=> [1,8,2,9,3,4,5,6,7] => [1,2,6] => ([(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
[1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
=> [1,8,9,2,3,4,5,6,7] => [1,8] => ([(7,8)],9)
=> ? = 2
[1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0]
=> [8,1,2,3,9,4,5,6,7] => [4,5] => ([(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 2
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [8,9,1,2,3,4,5,6,7] => [9] => ([],9)
=> ? = 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7,9] => [9] => ([],9)
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,2,3,4,5,6,9,7,8] => [6,3] => ([(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 2
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,9,2,3,4,5,6,7,8] => [1,8] => ([(7,8)],9)
=> ? = 2
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [9,1,2,3,4,5,6,7,8] => [9] => ([],9)
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8,9] => [9] => ([],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,9,1] => [9] => ([],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,7,8,1,9] => [9] => ([],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,6,7,9,1,8] => [9] => ([],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,5,6,8,9,1,7] => [9] => ([],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,7,8,9,1,6] => [9] => ([],9)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,6,7,8,9,1,5] => [9] => ([],9)
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,8,9,1,4] => [9] => ([],9)
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,7,8,9,1,3] => [9] => ([],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,9,10,1] => [10] => ([],10)
=> ? = 1
Description
The achromatic number of a graph.
This is the maximal number of colours of a proper colouring, such that for any pair of colours there are two adjacent vertices with these colours.
Matching statistic: St000386
(load all 187 compositions to match this statistic)
(load all 187 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 83%●distinct values known / distinct values provided: 67%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 83%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [.,.]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [6,7,8,4,5,3,2,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,6,3,2,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,6,3,2,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [8,6,4,5,3,7,2,1] => ?
=> ?
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [7,5,6,3,4,8,2,1] => ?
=> ?
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [6,5,7,3,4,8,2,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [7,8,4,3,5,6,2,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [8,4,5,3,6,7,2,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,4,5,3,7,8,2,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,4,3,5,6,8,2,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [6,5,7,8,4,2,3,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,8,2,3,1] => ?
=> ?
=> ? = 3 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,3,2,4,1] => ?
=> ?
=> ? = 3 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [7,5,6,8,3,2,4,1] => ?
=> ?
=> ? = 3 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [7,4,3,5,6,2,8,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [8,6,7,4,2,3,5,1] => ?
=> ?
=> ? = 4 - 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [8,5,6,4,2,3,7,1] => ?
=> ?
=> ? = 3 - 1
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [8,6,4,5,2,3,7,1] => ?
=> ?
=> ? = 4 - 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [7,6,4,3,2,5,8,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [7,6,3,4,2,5,8,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [6,4,3,5,2,7,8,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [8,7,4,2,3,5,6,1] => ?
=> ?
=> ? = 3 - 1
[1,0,1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [6,7,4,2,3,5,8,1] => ?
=> ?
=> ? = 3 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [8,6,3,2,4,5,7,1] => ?
=> ?
=> ? = 3 - 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [8,6,5,4,7,3,1,2] => ?
=> ?
=> ? = 3 - 1
[1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,8,3,1,2] => ?
=> ?
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,3,5,1,2] => ?
=> ?
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,6,8,4,3,5,1,2] => ?
=> ?
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [8,6,4,5,3,7,1,2] => ?
=> ?
=> ? = 4 - 1
[1,1,0,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [5,4,6,7,3,8,1,2] => ?
=> ?
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [7,5,6,3,4,8,1,2] => ?
=> ?
=> ? = 4 - 1
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,4,8,1,2] => ?
=> ?
=> ? = 3 - 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [8,6,7,4,5,2,1,3] => ?
=> ?
=> ? = 4 - 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [7,6,8,4,5,2,1,3] => ?
=> ?
=> ? = 3 - 1
[1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [8,6,4,5,7,2,1,3] => ?
=> ?
=> ? = 4 - 1
[1,1,0,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,8,2,1,3] => ?
=> ?
=> ? = 3 - 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [8,5,4,6,3,2,1,7] => ?
=> ?
=> ? = 2 - 1
[1,1,0,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [8,6,7,3,4,2,1,5] => ?
=> ?
=> ? = 3 - 1
[1,1,0,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [7,6,8,3,4,2,1,5] => ?
=> ?
=> ? = 2 - 1
[1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [8,7,4,3,5,2,1,6] => ?
=> ?
=> ? = 2 - 1
[1,1,0,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [7,8,4,3,5,2,1,6] => ?
=> ?
=> ? = 2 - 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [8,6,4,3,5,2,1,7] => ?
=> ?
=> ? = 3 - 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [8,5,4,3,6,2,1,7] => ?
=> ?
=> ? = 2 - 1
[1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [6,4,3,5,7,2,1,8] => ?
=> ?
=> ? = 2 - 1
[1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [6,3,4,5,7,2,1,8] => ?
=> ?
=> ? = 2 - 1
[1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [8,6,7,4,2,3,1,5] => ?
=> ?
=> ? = 4 - 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [8,6,4,5,2,3,1,7] => ?
=> ?
=> ? = 4 - 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [8,5,4,6,2,3,1,7] => ?
=> ?
=> ? = 3 - 1
[1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,2,3,1,8] => ?
=> ?
=> ? = 3 - 1
[1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [7,6,8,3,2,4,1,5] => ?
=> ?
=> ? = 2 - 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St001712
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 67% ●values known / values provided: 81%●distinct values known / distinct values provided: 67%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 67% ●values known / values provided: 81%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [[1]]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [[1,2]]
=> 0 = 1 - 1
[1,1,0,0]
=> [2,1] => [[1],[2]]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [[1,2,3]]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [[1,2],[3]]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [[1],[2],[3]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [[1,2,3],[4]]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [[1,2],[3],[4]]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [[1,2],[3],[4]]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [[1,3],[2],[4]]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [[1,2],[3],[4]]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [[1,2,3,5],[4]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [[1,2,3,4],[5]]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [[1,2,3],[4],[5]]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [[1,2,4],[3],[5]]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [[1,2,3],[4],[5]]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[1,3,4],[2,5]]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[1,2,4,5],[3]]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[1,2,4],[3,5]]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [[1,2,3],[4],[5]]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [[1,2,5],[3],[4]]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [[1,2,4],[3],[5]]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [[1,2,3],[4],[5]]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,2,4,5,3,6,7,8] => ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,2,4,6,5,7,3,8] => ?
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,2,5,4,6,3,8,7] => ?
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,2,5,4,7,8,6,3] => ?
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,2,5,6,4,3,8,7] => ?
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,2,6,5,7,4,8,3] => ?
=> ? = 3 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,2,6,7,5,4,8,3] => ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,3,2,4,6,7,5,8] => ?
=> ? = 3 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,3,2,4,7,8,6,5] => ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,7,6,8,4] => ?
=> ? = 3 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,8,7,6] => ?
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,4,2,7,6,5,8] => ?
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,3,5,4,2,6,7,8] => ?
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,3,5,4,6,2,7,8] => ?
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,3,5,4,6,2,8,7] => ?
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,3,5,4,6,7,2,8] => ?
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,3,5,4,6,7,8,2] => ?
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,3,5,4,7,8,6,2] => ?
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,3,5,7,6,4,8,2] => ?
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,3,5,7,6,8,4,2] => ?
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,3,6,5,4,7,8,2] => ?
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,3,6,7,5,4,8,2] => ?
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,3,7,6,5,8,4,2] => ?
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,4,3,6,7,5,8,2] => ?
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,4,5,6,3,7,8,2] => ?
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,4,5,7,8,6,3,2] => ?
=> ? = 1 - 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,4,6,5,3,7,8,2] => ?
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [1,5,4,6,3,8,7,2] => ?
=> ? = 3 - 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,5,4,7,6,3,2,8] => ?
=> ? = 3 - 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,5,4,7,6,3,8,2] => ?
=> ? = 3 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,5,6,4,3,7,2,8] => ?
=> ? = 3 - 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,5,4,6,7,8] => ?
=> ? = 3 - 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,3,5,6,7,4,8] => ?
=> ? = 3 - 1
[1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,3,7,6,8,5,4] => ?
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3,7,8,6] => ?
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,1,4,6,5,7,3,8] => ?
=> ? = 4 - 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [2,3,1,5,4,7,8,6] => ?
=> ? = 3 - 1
[1,1,0,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,5,4,1,7,8,6] => ?
=> ? = 2 - 1
[1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [2,3,6,7,5,8,4,1] => ?
=> ? = 2 - 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [2,4,3,6,7,5,1,8] => ?
=> ? = 3 - 1
[1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [2,4,3,7,6,8,5,1] => ?
=> ? = 3 - 1
[1,1,0,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [2,4,5,3,7,6,1,8] => ?
=> ? = 3 - 1
[1,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [2,4,5,6,3,7,1,8] => ?
=> ? = 3 - 1
[1,1,0,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [2,4,5,7,6,3,1,8] => ?
=> ? = 2 - 1
[1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [2,4,5,7,6,8,3,1] => ?
=> ? = 2 - 1
[1,1,0,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [2,5,4,6,3,8,7,1] => ?
=> ? = 3 - 1
[1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [2,5,4,6,8,7,3,1] => ?
=> ? = 2 - 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [2,5,6,4,7,3,8,1] => ?
=> ? = 3 - 1
[1,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [2,6,5,7,8,4,3,1] => ?
=> ? = 2 - 1
[1,1,1,0,0,0,1,0,1,1,0,1,0,0,1,0]
=> [3,2,1,4,6,7,5,8] => ?
=> ? = 3 - 1
Description
The number of natural descents of a standard Young tableau.
A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
Matching statistic: St000806
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000806: Integer compositions ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 83%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000806: Integer compositions ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1] => [1] => [1] => ? = 1 + 1
[1,0,1,0]
=> [2,1] => [2] => [1] => ? = 1 + 1
[1,1,0,0]
=> [1,2] => [2] => [1] => ? = 1 + 1
[1,0,1,0,1,0]
=> [2,3,1] => [3] => [1] => ? = 1 + 1
[1,0,1,1,0,0]
=> [2,1,3] => [3] => [1] => ? = 1 + 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,2] => [1,1] => 3 = 2 + 1
[1,1,0,1,0,0]
=> [3,1,2] => [3] => [1] => ? = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [3] => [1] => ? = 1 + 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4] => [1] => ? = 1 + 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [4] => [1] => ? = 1 + 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2] => [2] => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4] => [1] => ? = 1 + 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [4] => [1] => ? = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,3] => [1,1] => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3] => [1,1] => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2] => [2] => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4] => [1] => ? = 1 + 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [4] => [1] => ? = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,2] => [2] => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3] => [1,1] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4] => [1] => ? = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4] => [1] => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5] => [1] => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [5] => [1] => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2] => [1,1] => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5] => [1] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [5] => [1] => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,3] => [1,1] => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,3] => [1,1] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,2] => [1,1] => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5] => [1] => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [5] => [1] => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [3,2] => [1,1] => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,3] => [1,1] => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5] => [1] => ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [5] => [1] => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,4] => [1,1] => 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4] => [1,1] => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,2,2] => [1,2] => 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,4] => [1,1] => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,4] => [1,1] => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,3] => [1,1] => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [2,3] => [1,1] => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,2] => [1,1] => 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5] => [1] => ? = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5] => [1] => ? = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2] => [1,1] => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [2,3] => [1,1] => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5] => [1] => ? = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [5] => [1] => ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [2,3] => [1,1] => 3 = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [2,3] => [1,1] => 3 = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [1,2,2] => [1,2] => 4 = 3 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [1,4] => [1,1] => 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [1,4] => [1,1] => 3 = 2 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [3,2] => [1,1] => 3 = 2 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [2,3] => [1,1] => 3 = 2 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [5] => [1] => ? = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [5] => [1] => ? = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [3,2] => [1,1] => 3 = 2 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [2,3] => [1,1] => 3 = 2 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [1,4] => [1,1] => 3 = 2 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [5] => [1] => ? = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [5] => [1] => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => [6] => [1] => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => [6] => [1] => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => [4,2] => [1,1] => 3 = 2 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => [6] => [1] => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => [6] => [1] => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => [3,3] => [2] => 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => [3,3] => [2] => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,5,1,6,4] => [4,2] => [1,1] => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,5,6,1,4] => [6] => [1] => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => [6] => [1] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => [4,2] => [1,1] => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,4,5] => [3,3] => [2] => 3 = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => [6] => [1] => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => [6] => [1] => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => [2,4] => [1,1] => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => [2,4] => [1,1] => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5] => [2,2,2] => [3] => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,6,3,5] => [2,4] => [1,1] => 3 = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,5,6] => [2,4] => [1,1] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,4,1,5,6,3] => [3,3] => [2] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,5,3,6] => [3,3] => [2] => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,4,5,1,6,3] => [4,2] => [1,1] => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => [6] => [1] => ? = 1 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [6] => [1] => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,6,5] => [4,2] => [1,1] => 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,6,3,5] => [3,3] => [2] => 3 = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,6,1,3,5] => [6] => [1] => ? = 1 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,1,3,5,6] => [6] => [1] => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,5,6,4] => [3,3] => [2] => 3 = 2 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,5,6,1,3,4] => [6] => [1] => ? = 1 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,5,1,3,4,6] => [6] => [1] => ? = 1 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,6,1,3,4,5] => [6] => [1] => ? = 1 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => [6] => [1] => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => [6] => [1] => ? = 1 + 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,4,5,1,2,6] => [6] => [1] => ? = 1 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,4,6,1,2,5] => [6] => [1] => ? = 1 + 1
Description
The semiperimeter of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the semiperimeter of the polygon determined by the axis and the bargraph. Put differently, it is the sum of the number of up steps and the number of horizontal steps when regarding the bargraph as a path with up, horizontal and down steps.
Matching statistic: St001037
(load all 43 compositions to match this statistic)
(load all 43 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 76%●distinct values known / distinct values provided: 67%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 76%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [.,.]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,1] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,7,1,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,5,6,1,8,7] => [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,6,8,1,7] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,5,6,1,7,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,4,5,7,1,8,6] => [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,5,7,8,1,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,4,5,1,8,6,7] => [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,5,8,1,6,7] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1,6,7,8] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,4,1,6,7,8,5] => [[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,4,1,6,7,5,8] => [[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5,8,7] => [[.,[.,[.,.]]],[[.,.],[[.,.],.]]]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,3,4,1,6,8,5,7] => [[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,3,4,1,6,5,7,8] => [[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5,8] => [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,6,7,8,1,5] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,3,4,1,5,7,8,6] => [[.,[.,[.,.]]],[.,[[.,[.,.]],.]]]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,3,4,1,5,7,6,8] => [[.,[.,[.,.]]],[.,[[.,.],[.,.]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,3,4,1,7,5,8,6] => [[.,[.,[.,.]]],[[.,.],[[.,.],.]]]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,4,1,7,8,5,6] => [[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,3,4,1,7,5,6,8] => [[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,4,7,1,8,5,6] => [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,1,5,6,8,7] => [[.,[.,[.,.]]],[.,[.,[[.,.],.]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,4,1,5,8,6,7] => [[.,[.,[.,.]]],[.,[[.,.],[.,.]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,4,1,8,5,6,7] => [[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,4,8,1,5,6,7] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,3,1,5,6,7,4,8] => [[.,[.,.]],[[.,[.,[.,.]]],[.,.]]]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,7,6,8] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,5,1,6,7,8,4] => [[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,4,7,8] => ?
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,5,6,1,7,8,4] => [[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,8,1,4] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,3,5,1,7,4,8,6] => [[.,[.,[.,.]]],[[.,.],[[.,.],.]]]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,3,1,6,4,7,5,8] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,3,1,6,7,8,4,5] => [[.,[.,.]],[[.,[.,[.,.]]],[.,.]]]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,3,1,6,4,8,5,7] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,3,1,6,8,4,5,7] => [[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [2,3,6,1,4,7,5,8] => ?
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,6,1,7,8,4,5] => [[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [2,3,6,1,4,5,8,7] => [[.,[.,[.,.]]],[.,[.,[[.,.],.]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [2,3,6,1,8,4,5,7] => [[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,3,1,7,4,8,5,6] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [2,3,7,1,4,8,5,6] => ?
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,3,7,8,1,4,5,6] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,4,1,5,6,7,8,3] => [[.,[.,.]],[[.,[.,[.,[.,.]]]],.]]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [2,4,1,5,6,3,8,7] => [[.,[.,.]],[[.,[.,.]],[[.,.],.]]]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 3 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3,7,8] => [[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,5,3,7,6,8] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
The following 162 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000201The number of leaf nodes in a binary tree. St000568The hook number of a binary tree. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000069The number of maximal elements of a poset. St000071The number of maximal chains in a poset. St000527The width of the poset. St000052The number of valleys of a Dyck path not on the x-axis. St000068The number of minimal elements in a poset. St001732The number of peaks visible from the left. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000035The number of left outer peaks of a permutation. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000993The multiplicity of the largest part of an integer partition. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001280The number of parts of an integer partition that are at least two. St001657The number of twos in an integer partition. St000345The number of refinements of a partition. St000935The number of ordered refinements of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001389The number of partitions of the same length below the given integer partition. St000142The number of even parts of a partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St001083The number of boxed occurrences of 132 in a permutation. St000470The number of runs in a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001665The number of pure excedances of a permutation. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000157The number of descents of a standard tableau. St000632The jump number of the poset. St001840The number of descents of a set partition. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001116The game chromatic number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000536The pathwidth of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001277The degeneracy of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000021The number of descents of a permutation. St000325The width of the tree associated to a permutation. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001874Lusztig's a-function for the symmetric group. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001092The number of distinct even parts of a partition. St000353The number of inner valleys of a permutation. St000647The number of big descents of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000092The number of outer peaks of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000919The number of maximal left branches of a binary tree. St000659The number of rises of length at least 2 of a Dyck path. St000159The number of distinct parts of the integer partition. St000455The second largest eigenvalue of a graph if it is integral. St000312The number of leaves in a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000779The tier of a permutation. St001728The number of invisible descents of a permutation. St000711The number of big exceedences of a permutation. St000710The number of big deficiencies of a permutation. St000646The number of big ascents of a permutation. St000834The number of right outer peaks of a permutation. St000251The number of nonsingleton blocks of a set partition. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000360The number of occurrences of the pattern 32-1. St000099The number of valleys of a permutation, including the boundary. St000023The number of inner peaks of a permutation. St000523The number of 2-protected nodes of a rooted tree. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000409The number of pitchforks in a binary tree. St000822The Hadwiger number of the graph. St000317The cycle descent number of a permutation. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. St000636The hull number of a graph. St001307The number of induced stars on four vertices in a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001812The biclique partition number of a graph. St001883The mutual visibility number of a graph. St001960The number of descents of a permutation minus one if its first entry is not one. St000374The number of exclusive right-to-left minima of a permutation. St001597The Frobenius rank of a skew partition. St001330The hat guessing number of a graph. St000670The reversal length of a permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000100The number of linear extensions of a poset. St001487The number of inner corners of a skew partition. St000307The number of rowmotion orbits of a poset. St000662The staircase size of the code of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000245The number of ascents of a permutation. St000703The number of deficiencies of a permutation. St001427The number of descents of a signed permutation. St000640The rank of the largest boolean interval in a poset. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St000252The number of nodes of degree 3 of a binary tree. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001964The interval resolution global dimension of a poset. St000633The size of the automorphism group of a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001779The order of promotion on the set of linear extensions of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset. St001896The number of right descents of a signed permutations. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000807The sum of the heights of the valleys of the associated bargraph. St000805The number of peaks of the associated bargraph. St000864The number of circled entries of the shifted recording tableau of a permutation. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001513The number of nested exceedences of a permutation.
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