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Your data matches 176 different statistics following compositions of up to 3 maps.
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Matching statistic: St000291
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,2] => 0 => 0
[1,1,0,0]
=> [1,1,0,0]
=> [2,1] => 1 => 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 00 => 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 01 => 0
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 10 => 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 01 => 0
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 11 => 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 001 => 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 010 => 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 001 => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 011 => 0
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 0
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 011 => 0
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 101 => 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 110 => 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 011 => 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0001 => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0010 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0001 => 0
[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] => 0011 => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 0100 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1001 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0010 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0001 => 0
[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] => 0011 => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => 0101 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 0110 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 0011 => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 0111 => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1000 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 0101 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1010 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0101 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 1011 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0100 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1001 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0010 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0001 => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 0011 => 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 0101 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 0110 => 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] => 0011 => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => 0111 => 0
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1001 => 1
Description
The number of descents of a binary word.
Matching statistic: St000288
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 86%●distinct values known / distinct values provided: 83%
Mp00130: Permutations —descent tops⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 86%●distinct values known / distinct values provided: 83%
Values
[1,0,1,0]
=> [1,2] => 0 => 1 => 1 = 0 + 1
[1,1,0,0]
=> [2,1] => 1 => 1 => 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => 00 => 01 => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => 01 => 10 => 1 = 0 + 1
[1,1,0,0,1,0]
=> [2,1,3] => 10 => 11 => 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => 01 => 10 => 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,1,2] => 01 => 10 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 001 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 010 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 101 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 001 => 010 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 001 => 010 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 101 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 110 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 010 => 101 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 001 => 010 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 001 => 010 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 010 => 101 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 011 => 101 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 001 => 010 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 001 => 010 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0000 => 0001 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0001 => 0010 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0010 => 0101 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0001 => 0010 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 0001 => 0010 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0100 => 1001 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0101 => 1010 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0010 => 0101 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0001 => 0010 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0001 => 0010 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 0010 => 0101 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 0011 => 0101 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 0001 => 0010 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 0001 => 0010 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1000 => 1001 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1001 => 1010 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1010 => 1101 => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1001 => 1010 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1001 => 1010 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 0100 => 1001 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 0101 => 1010 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0010 => 0101 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0001 => 0010 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 0001 => 0010 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 0010 => 0101 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 0011 => 0101 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 0001 => 0010 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 0001 => 0010 => 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 0100 => 1001 => 2 = 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] => ? => ? => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6,8,7] => ? => ? => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,2,5,3,6,8,4,7] => ? => ? => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,2,5,6,7,3,4,8] => ? => ? => ? = 1 + 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] => ? => ? => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,3,2,4,7,5,6,8] => ? => ? => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,2,5,6,8,4,7] => ? => ? => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2,6,4,5,8,7] => ? => ? => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,3,2,7,4,5,8,6] => ? => ? => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,3,2,7,8,4,5,6] => ? => ? => ? = 1 + 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,4,6,2,7,8,5] => ? => ? => ? = 1 + 1
[1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,4,6,2,8,5,7] => ? => ? => ? = 1 + 1
[1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,6,2,8,4,5,7] => ? => ? => ? = 1 + 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,3,6,7,2,4,5,8] => ? => ? => ? = 1 + 1
[1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,6,7,2,8,4,5] => ? => ? => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,4,2,3,6,8,5,7] => ? => ? => ? = 1 + 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,4,2,5,6,3,7,8] => ? => ? => ? = 2 + 1
[1,0,1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,7,8,5] => ? => ? => ? = 2 + 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,4,2,6,7,3,8,5] => ? => ? => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [1,4,5,2,3,6,8,7] => ? => ? => ? = 1 + 1
[1,0,1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,4,5,7,2,8,3,6] => ? => ? => ? = 1 + 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,5,2,6,7,3,8,4] => ? => ? => ? = 2 + 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,5,6,2,3,7,4,8] => ? => ? => ? = 2 + 1
[1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,1,3,4,6,8,5,7] => ? => ? => ? = 1 + 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] => ? => ? => ? = 2 + 1
[1,1,0,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,3,6,4,7,5,8] => ? => ? => ? = 3 + 1
[1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,3,6,4,8,5,7] => ? => ? => ? = 2 + 1
[1,1,0,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,1,3,6,8,4,5,7] => ? => ? => ? = 1 + 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,7,4,5,6,8] => ? => ? => ? = 2 + 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,3,7,4,5,8,6] => ? => ? => ? = 2 + 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,3,5,7,6,8] => ? => ? => ? = 3 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,7,5,6,8] => ? => ? => ? = 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] => ? => ? => ? = 2 + 1
[1,1,0,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,1,4,5,3,8,6,7] => ? => ? => ? = 2 + 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,1,4,5,6,3,7,8] => ? => ? => ? = 2 + 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,1,4,5,7,3,6,8] => ? => ? => ? = 2 + 1
[1,1,0,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,1,4,7,3,5,6,8] => ? => ? => ? = 2 + 1
[1,1,0,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,1,4,7,3,8,5,6] => ? => ? => ? = 2 + 1
[1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,5,3,4,7,6,8] => ? => ? => ? = 3 + 1
[1,1,0,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,1,5,3,4,7,8,6] => ? => ? => ? = 2 + 1
[1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4,8,6,7] => ? => ? => ? = 2 + 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,1,5,3,7,4,6,8] => ? => ? => ? = 3 + 1
[1,1,0,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [2,1,5,6,3,8,4,7] => ? => ? => ? = 2 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,6,3,4,5,7,8] => ? => ? => ? = 2 + 1
[1,1,0,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,1,6,3,4,7,5,8] => ? => ? => ? = 3 + 1
[1,1,0,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,1,4,6,8,5,7] => ? => ? => ? = 1 + 1
[1,1,0,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,7,5,8,6] => ? => ? => ? = 2 + 1
[1,1,0,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,3,1,5,4,8,6,7] => ? => ? => ? = 2 + 1
[1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,6,4,8,7] => ? => ? => ? = 2 + 1
[1,1,0,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [2,3,1,5,7,4,6,8] => ? => ? => ? = 2 + 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the 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: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [1,2] => 0 => 0
[1,1,0,0]
=> [1,2] => [2,1] => 1 => 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => 00 => 0
[1,0,1,1,0,0]
=> [2,3,1] => [2,1,3] => 10 => 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 01 => 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,3,1] => 10 => 0
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => 11 => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => 000 => 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [2,1,3,4] => 100 => 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,3,2,4] => 010 => 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,3,1,4] => 100 => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,2,1,4] => 110 => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,4,3] => 001 => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => 101 => 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,4,2] => 010 => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,4,1] => 100 => 0
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,4,1] => 110 => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => 011 => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,4,3,1] => 101 => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [3,4,2,1] => 110 => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => 111 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [2,1,3,4,5] => 1000 => 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,3,2,4,5] => 0100 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [2,3,1,4,5] => 1000 => 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,2,1,4,5] => 1100 => 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,4,3,5] => 0010 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [2,1,4,3,5] => 1010 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,3,4,2,5] => 0100 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [2,3,4,1,5] => 1000 => 0
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2,4,1,5] => 1100 => 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,4,3,2,5] => 0110 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,4,3,1,5] => 1010 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,4,2,1,5] => 1100 => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => 1110 => 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,5,4] => 0001 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [2,1,3,5,4] => 1001 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,3,2,5,4] => 0101 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [2,3,1,5,4] => 1001 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,2,1,5,4] => 1101 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,2,4,5,3] => 0010 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [2,1,4,5,3] => 1010 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,3,4,5,2] => 0100 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [2,3,4,5,1] => 1000 => 0
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,2,4,5,1] => 1100 => 0
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,3,5,2] => 0110 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,4,3,5,1] => 1010 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,4,2,5,1] => 1100 => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,3,2,5,1] => 1110 => 0
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => 0011 => 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [8,7,5,4,6,3,2,1] => [1,2,4,5,3,6,7,8] => ? => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [8,5,6,4,7,3,2,1] => [1,4,3,5,2,6,7,8] => ? => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [7,6,8,5,3,4,2,1] => [2,3,1,4,6,5,7,8] => ? => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [8,6,5,7,3,4,2,1] => [1,3,4,2,6,5,7,8] => ? => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [8,5,6,7,3,4,2,1] => [1,4,3,2,6,5,7,8] => ? => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [6,5,7,8,3,4,2,1] => [3,4,2,1,6,5,7,8] => ? => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,3,5,2,1] => [2,1,3,5,6,4,7,8] => ? => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,3,6,2,1] => [2,1,4,5,6,3,7,8] => ? => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,5,6,4,3,8,2,1] => [2,4,3,5,6,1,7,8] => ? => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [8,7,4,5,3,6,2,1] => [1,2,5,4,6,3,7,8] => ? => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [6,7,4,5,3,8,2,1] => [3,2,5,4,6,1,7,8] => ? => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [7,5,4,6,3,8,2,1] => [2,4,5,3,6,1,7,8] => ? => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [7,4,5,6,3,8,2,1] => [2,5,4,3,6,1,7,8] => ? => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [7,8,6,3,4,5,2,1] => [2,1,3,6,5,4,7,8] => ? => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [8,6,5,3,4,7,2,1] => [1,3,4,6,5,2,7,8] => ? => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [6,7,5,3,4,8,2,1] => [3,2,4,6,5,1,7,8] => ? => ? = 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] => ? => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,7,4,3,5,8,2,1] => [3,2,5,6,4,1,7,8] => ? => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,7,8,2,1] => [4,3,5,6,2,1,7,8] => ? => ? = 0
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,4,5,3,6,8,2,1] => [2,5,4,6,3,1,7,8] => ? => ? = 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] => ? => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [5,4,6,3,7,8,2,1] => [4,5,3,6,2,1,7,8] => ? => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [7,8,3,4,5,6,2,1] => [2,1,6,5,4,3,7,8] => ? => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [8,6,3,4,5,7,2,1] => [1,3,6,5,4,2,7,8] => ? => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,5,8,2,1] => [3,2,6,5,4,1,7,8] => ? => ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [6,5,3,4,7,8,2,1] => [3,4,6,5,2,1,7,8] => ? => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [6,4,3,5,7,8,2,1] => [3,5,6,4,2,1,7,8] => ? => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [7,3,4,5,6,8,2,1] => [2,6,5,4,3,1,7,8] => ? => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,3,4,6,7,8,2,1] => [4,6,5,3,2,1,7,8] => ? => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [7,5,6,8,4,2,3,1] => [2,4,3,1,5,7,6,8] => ? => ? = 2
[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] => ? => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,6,2,3,1] => [2,1,4,5,3,7,6,8] => ? => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [6,7,5,4,8,2,3,1] => [3,2,4,5,1,7,6,8] => ? => ? = 1
[1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [5,6,7,4,8,2,3,1] => [4,3,2,5,1,7,6,8] => ? => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [8,7,4,5,6,2,3,1] => [1,2,5,4,3,7,6,8] => ? => ? = 2
[1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [5,6,4,7,8,2,3,1] => [4,3,5,2,1,7,6,8] => ? => ? = 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] => ? => ? = 2
[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] => ? => ? = 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] => ? => ? = 2
[1,0,1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [6,5,7,8,3,2,4,1] => [3,4,2,1,6,7,5,8] => ? => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,3,2,5,1] => [2,1,3,5,6,7,4,8] => ? => ? = 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [8,5,6,4,3,2,7,1] => [1,4,3,5,6,7,2,8] => ? => ? = 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [7,5,6,4,3,2,8,1] => [2,4,3,5,6,7,1,8] => ? => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [6,7,4,5,3,2,8,1] => [3,2,5,4,6,7,1,8] => ? => ? = 1
[1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [5,6,4,7,3,2,8,1] => [4,3,5,2,6,7,1,8] => ? => ? = 0
[1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [5,4,6,7,3,2,8,1] => [4,5,3,2,6,7,1,8] => ? => ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [7,8,6,3,4,2,5,1] => [2,1,3,6,5,7,4,8] => ? => ? = 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] => ? => ? = 2
[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] => ? => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [6,7,5,3,4,2,8,1] => [3,2,4,6,5,7,1,8] => ? => ? = 1
Description
The number of ascents of a binary word.
Matching statistic: St001712
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
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: 76%●distinct values known / distinct values provided: 67%
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: 76%●distinct values known / distinct values provided: 67%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,2] => [[1,2]]
=> 0
[1,1,0,0]
=> [1,1,0,0]
=> [2,1] => [[1],[2]]
=> 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [[1,2,3]]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [[1,2],[3]]
=> 0
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> 0
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => [[1],[2],[3]]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[1,2,4],[3]]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [[1,2,3],[4]]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [[1,2],[3],[4]]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [[1,2],[3],[4]]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [[1,3],[2],[4]]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [[1,2],[3],[4]]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [[1,2,3,4],[5]]
=> 0
[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] => [[1,2,3],[4],[5]]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[1,2,4,5],[3]]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[1,3,4],[2,5]]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [[1,2,3,5],[4]]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 0
[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] => [[1,2,3],[4],[5]]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => [[1,2,4],[3],[5]]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => [[1,2,5],[3],[4]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [[1,2,3],[4],[5]]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => [[1,2],[3],[4],[5]]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[1,2,4],[3,5]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [[1,3],[2,4],[5]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [[1,2,3],[4],[5]]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [[1,2,4],[3],[5]]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [[1,2,5],[3],[4]]
=> 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] => [[1,2,3],[4],[5]]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => [[1,2],[3],[4],[5]]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [[1,3,4],[2],[5]]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,1,0,0]
=> [3,4,5,2,1,7,8,6] => ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,4,5,6,3,7,2,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
=> [4,5,6,3,7,8,2,1] => ?
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,3,5,4,7,8,6,2] => ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,4,3,6,7,5,8,2] => ?
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,4,6,7,5,8] => ?
=> ? = 2
[1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,4,5,7,8,6,3,2] => ?
=> ? = 0
[1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [2,4,5,3,6,7,1,8] => ?
=> ? = 2
[1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,1,4,6,7,5,3,8] => ?
=> ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,3,5,6,4,7,8,2] => ?
=> ? = 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0,1,0]
=> [3,2,1,6,7,5,4,8] => ?
=> ? = 2
[1,0,1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,1,0,0,0,0]
=> [3,2,4,7,8,6,5,1] => ?
=> ? = 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> [3,6,7,5,8,4,2,1] => ?
=> ? = 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,1,0,0,0]
=> [4,5,3,2,6,8,7,1] => ?
=> ? = 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [2,6,7,5,4,8,3,1] => ?
=> ? = 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> [4,5,3,7,6,2,8,1] => ?
=> ? = 2
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
=> [3,2,6,5,8,7,4,1] => ?
=> ? = 2
[1,1,0,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,5,4,7,6,3,2,8] => ?
=> ? = 2
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6,8,7] => ?
=> ? = 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,3,4,6,5,7,2,8] => ?
=> ? = 2
[1,1,0,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,7,6,8,4] => ?
=> ? = 2
[1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,3,5,4,6,7,8,2] => ?
=> ? = 1
[1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,3,5,4,6,7,2,8] => ?
=> ? = 2
[1,1,0,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,3,5,4,6,2,7,8] => ?
=> ? = 2
[1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,3,5,4,2,6,7,8] => ?
=> ? = 1
[1,1,0,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,1,3,5,7,6,8,4] => ?
=> ? = 2
[1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,2,4,6,5,7,8,3] => ?
=> ? = 1
[1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,3,5,7,6,8,4,2] => ?
=> ? = 1
[1,1,0,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,3,5,7,6,4,8,2] => ?
=> ? = 1
[1,1,0,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,1,0,0,0]
=> [3,5,4,6,2,8,7,1] => ?
=> ? = 2
[1,1,1,0,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
=> [3,2,1,5,6,7,4,8] => ?
=> ? = 2
[1,1,1,0,0,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,4,3,5,2,8,7,6] => ?
=> ? = 2
[1,1,1,0,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [2,4,5,3,8,7,6,1] => ?
=> ? = 1
[1,1,1,0,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,4,3,5,2,7,8,6] => ?
=> ? = 2
[1,1,1,0,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,8,7,6] => ?
=> ? = 1
[1,1,1,0,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [2,4,3,5,8,7,6,1] => ?
=> ? = 1
[1,1,1,0,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [2,5,4,6,3,7,8,1] => ?
=> ? = 2
[1,1,1,0,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,1,4,7,6,8,5,3] => ?
=> ? = 2
[1,1,1,0,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,2,4,7,6,8,5,3] => ?
=> ? = 1
[1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [2,6,5,7,8,4,3,1] => ?
=> ? = 1
[1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,3,7,6,5,4,2,8] => ?
=> ? = 1
[1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,6,5,4,8,7,3,2] => ?
=> ? = 1
[1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,6,5,4,3,7,8,2] => ?
=> ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [6,5,4,8,7,3,2,1] => ?
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [8,7,9,6,5,4,3,2,1] => [[1,3],[2],[4],[5],[6],[7],[8],[9]]
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [8,9,7,6,5,4,3,2,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [9,8,10,7,6,5,4,3,2,1] => [[1,3],[2],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,7,9,6,5,4,3,2] => [[1,2,4],[3],[5],[6],[7],[8],[9]]
=> ? = 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,7,6,5,4,3,2] => [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [9,10,8,7,6,5,4,3,2,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 0
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: 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: 65% ●values known / values provided: 65%●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: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 67%
Values
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[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
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[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
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[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
[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
[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
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[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
[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
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> 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
[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
[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
[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
[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
[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
[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
[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
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> 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
[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
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,0,0,1,0,1,0]
=> 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] => ?
=> ?
=> ? = 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] => ?
=> ?
=> ? = 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] => ?
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [8,5,4,6,7,3,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [8,6,5,7,3,4,2,1] => ?
=> ?
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [7,5,6,8,3,4,2,1] => ?
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [8,5,6,4,3,7,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [5,6,4,7,3,8,2,1] => ?
=> ?
=> ? = 0
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,3,7,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [8,6,5,3,4,7,2,1] => ?
=> ?
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [6,7,5,3,4,8,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [8,5,6,3,4,7,2,1] => ?
=> ?
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [7,5,6,3,4,8,2,1] => ?
=> ?
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [6,5,7,3,4,8,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,7,4,3,5,8,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,7,8,2,1] => ?
=> ?
=> ? = 0
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [8,4,5,3,6,7,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,4,5,3,6,8,2,1] => ?
=> ?
=> ? = 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] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [5,4,6,3,7,8,2,1] => ?
=> ?
=> ? = 0
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [8,6,3,4,5,7,2,1] => ?
=> ?
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,5,8,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [5,6,3,4,7,8,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [6,4,3,5,7,8,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [4,5,3,6,7,8,2,1] => ?
=> ?
=> ? = 0
[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [6,7,8,5,4,2,3,1] => ?
=> ?
=> ? = 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] => ?
=> ?
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,6,2,3,1] => ?
=> ?
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [6,7,5,4,8,2,3,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [5,6,7,4,8,2,3,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [8,5,4,6,7,2,3,1] => ?
=> ?
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [5,6,4,7,8,2,3,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [7,4,5,6,8,2,3,1] => ?
=> ?
=> ? = 2
[1,0,1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [5,4,6,7,8,2,3,1] => ?
=> ?
=> ? = 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] => ?
=> ?
=> ? = 2
[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
[1,0,1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [6,5,7,8,3,2,4,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [8,7,4,5,3,2,6,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,3,2,6,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [7,5,4,6,3,2,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [5,6,4,7,3,2,8,1] => ?
=> ?
=> ? = 0
[1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,3,2,7,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [5,4,6,7,3,2,8,1] => ?
=> ?
=> ? = 0
[1,0,1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [6,7,5,3,4,2,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,4,2,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,7,2,8,1] => ?
=> ?
=> ? = 0
[1,0,1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [8,7,3,4,5,2,6,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [7,8,3,4,5,2,6,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,5,2,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [8,5,3,4,6,2,7,1] => ?
=> ?
=> ? = 2
Description
The number of factors DDU in a Dyck path.
Matching statistic: St001037
(load all 43 compositions to match this statistic)
(load all 43 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 67%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 67%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,1] => [1,0,1,0]
=> 0
[1,1,0,0]
=> [1,1,0,0]
=> [2] => [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> 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] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0,1,0]
=> [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0,1,0]
=> [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0,1,0]
=> [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,0,1,0]
=> [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0,1,0]
=> [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0,1,0]
=> [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,0,1,0]
=> [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0,1,0]
=> [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0,1,0]
=> [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0,1,0]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0,1,0]
=> [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0,1,0]
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0,1,0]
=> [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0,1,0]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,0,1,0]
=> [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0]
=> [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000201
(load all 44 compositions to match this statistic)
(load all 44 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 67%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 67%
Values
[1,0,1,0]
=> [1]
=> [1,0,1,0]
=> [.,[.,.]]
=> 1 = 0 + 1
[1,1,0,0]
=> []
=> []
=> .
=> ? = 0 + 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> [.,[.,.]]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> []
=> []
=> .
=> ? = 0 + 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> [.,[.,.]]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> .
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[.,.]]]]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [.,[[.,[.,.]],[.,.]]]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[.,[.,[.,.]]],[.,.]]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],.]],[.,.]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[[.,.],[.,.]],[.,.]]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> .
=> ? = 0 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> .
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> .
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,5,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [.,[.,[.,[.,[[.,[[.,.],.]],.]]]]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,3,2,1]
=> ?
=> ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,6,5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,2,1]
=> ?
=> ?
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,5,3,3,2,1]
=> ?
=> ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,2,1]
=> ?
=> ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,3,2,1]
=> ?
=> ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,6,4,3,3,2,1]
=> ?
=> ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,3,2,1]
=> ?
=> ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,5,4,3,3,2,1]
=> ?
=> ?
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,4,3,3,2,1]
=> ?
=> ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,4,3,3,2,1]
=> ?
=> ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,3,2,1]
=> ?
=> ?
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,4,4,3,3,2,1]
=> ?
=> ?
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,3,3,2,1]
=> ?
=> ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,6,3,3,3,2,1]
=> ?
=> ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [7,5,3,3,3,2,1]
=> ?
=> ?
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,5,3,3,3,2,1]
=> ?
=> ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,4,3,3,3,2,1]
=> ?
=> ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,3,2,1]
=> ?
=> ?
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,3,2,1]
=> ?
=> ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,3,3,3,2,1]
=> ?
=> ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,3,3,3,2,1]
=> ?
=> ?
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,2,1]
=> ?
=> ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> ? = 1 + 1
Description
The number of leaf nodes in a binary tree.
Equivalently, the number of cherries [1] in the complete binary tree.
The number of binary trees of size $n$, at least $1$, with exactly one leaf node for is $2^{n-1}$, see [2].
The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].
Matching statistic: St000568
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000568: Binary trees ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 67%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000568: Binary trees ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 67%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [[.,.],.]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [.,[.,.]]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => [[[.,.],.],.]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [[.,.],[.,.]]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [[[.,.],.],[.,.]]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [[[.,.],.],[.,.]]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 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] => [[[[[.,.],.],.],.],.]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> 2 = 1 + 1
[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] => [[[.,.],.],[[.,.],.]]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => [[[.,.],.],[.,[.,.]]]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,5,4,3,2,1,8,7] => [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [7,6,5,4,3,2,8,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [6,5,4,3,2,1,7,8] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,4,3,2,1,8,7,6] => [[[[[.,.],.],.],.],[[[.,.],.],.]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [5,4,3,2,1,7,6,8] => [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [6,5,4,3,2,8,7,1] => [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [7,6,5,4,3,8,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [6,5,4,3,2,7,1,8] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [5,4,3,2,1,6,8,7] => [[[[[.,.],.],.],.],[.,[[.,.],.]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [5,4,3,2,1,7,8,6] => [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [6,5,4,3,2,7,8,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [5,4,3,2,1,6,7,8] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [4,3,2,1,8,7,6,5] => [[[[.,.],.],.],[[[[.,.],.],.],.]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [4,3,2,1,7,6,5,8] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [4,3,2,1,6,5,8,7] => [[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
=> [4,3,2,1,7,6,8,5] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [4,3,2,1,6,5,7,8] => [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [5,4,3,2,8,7,6,1] => [[[[[.,.],.],.],.],[[[.,.],.],.]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> [5,4,3,2,7,6,1,8] => [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [6,5,4,3,8,7,2,1] => [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [7,6,5,4,8,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [6,5,4,3,7,2,1,8] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [5,4,3,2,6,1,8,7] => [[[[[.,.],.],.],.],[.,[[.,.],.]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> [5,4,3,2,7,6,8,1] => [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [6,5,4,3,7,2,8,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [5,4,3,2,6,1,7,8] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [4,3,2,1,5,8,7,6] => [[[[.,.],.],.],[.,[[[.,.],.],.]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [4,3,2,1,5,7,6,8] => [[[[.,.],.],.],[.,[[.,.],[.,.]]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> [4,3,2,1,6,8,7,5] => [[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> [4,3,2,1,7,8,6,5] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0,1,0]
=> [4,3,2,1,6,7,5,8] => [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0]
=> [5,4,3,2,6,8,7,1] => [[[[[.,.],.],.],.],[.,[[.,.],.]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,1,0,0,0]
=> [5,4,3,2,7,8,6,1] => [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [6,5,4,3,7,8,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [5,4,3,2,6,7,1,8] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [4,3,2,1,5,6,8,7] => [[[[.,.],.],.],[.,[.,[[.,.],.]]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,1,0,0]
=> [4,3,2,1,5,7,8,6] => [[[[.,.],.],.],[.,[[.,.],[.,.]]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,1,6,7,8,5] => [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,7,8,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,2,1,8,7,6,5,4] => [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [3,2,1,7,6,5,4,8] => [[[.,.],.],[[[[.,.],.],.],[.,.]]]
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,6,5,4,8,7] => [[[.,.],.],[[[.,.],.],[[.,.],.]]]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> [3,2,1,7,6,5,8,4] => [[[.,.],.],[[[[.,.],.],.],[.,.]]]
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [3,2,1,5,4,7,6,8] => [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [3,2,1,6,5,8,7,4] => [[[.,.],.],[[[.,.],.],[[.,.],.]]]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,1,0,0,0]
=> [3,2,1,7,6,8,5,4] => [[[.,.],.],[[[[.,.],.],.],[.,.]]]
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0,1,0]
=> [3,2,1,6,5,7,4,8] => [[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> ? = 1 + 1
Description
The hook number of a binary tree.
A hook of a binary tree is a vertex together with is left- and its right-most branch. Then there is a unique decomposition of the tree into hooks and the hook number is the number of hooks in this decomposition.
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: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 83%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 83%
Values
[1,0,1,0]
=> [2,1] => [2] => [2]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,2] => [2] => [2]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [2,3,1] => [3] => [3]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [2,1,3] => [3] => [3]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,2] => [2,1]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [3,1,2] => [3] => [3]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [3] => [3]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4] => [4]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [4] => [4]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2] => [2,2]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4] => [4]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [4] => [4]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,3] => [3,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3] => [3,1]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2] => [2,2]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4] => [4]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [4] => [4]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,2] => [2,2]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3] => [3,1]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4] => [4]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4] => [4]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5] => [5]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [5] => [5]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2] => [3,2]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5] => [5]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [5] => [5]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,3] => [3,2]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,3] => [3,2]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,2] => [3,2]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5] => [5]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [5] => [5]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [3,2] => [3,2]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,3] => [3,2]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5] => [5]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [5] => [5]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,4] => [4,1]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4] => [4,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,2,2] => [2,2,1]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,4] => [4,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,4] => [4,1]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,3] => [3,2]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [2,3] => [3,2]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,2] => [3,2]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5] => [5]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5] => [5]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2] => [3,2]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [2,3] => [3,2]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5] => [5]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [5] => [5]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [2,3] => [3,2]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,5,1,7,8,6] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,4,5,1,7,6,8] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,4,5,7,1,6,8] => ? => ?
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,4,5,1,6,8,7] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,3,4,6,1,7,8,5] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,6,7,1,8,5] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,7,1,5,8] => ? => ?
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,3,4,6,1,5,8,7] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,4,6,1,8,5,7] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,4,6,8,1,5,7] => ? => ?
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,6,1,5,7,8] => ? => ?
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,3,4,7,1,5,8,6] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,4,7,8,1,5,6] => ? => ?
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,7,1,5,6,8] => ? => ?
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,8,4] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,6,4,8,7] => ? => ?
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,3,1,5,6,8,4,7] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1,5,7,4,8,6] => ? => ?
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,7,8,4,6] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,3,1,5,7,4,6,8] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1,5,4,6,8,7] => ? => ?
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,3,1,5,4,8,6,7] => ? => ?
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,3,1,5,8,4,6,7] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,5,4,6,7,8] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,3,5,1,6,7,4,8] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,3,5,1,6,8,4,7] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,5,6,1,7,4,8] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,5,6,7,1,8,4] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,5,6,7,1,4,8] => ? => ?
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,5,6,1,4,8,7] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,6,1,8,4,7] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,6,8,1,4,7] => ? => ?
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4,7,8] => ? => ?
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,3,5,1,4,7,8,6] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,3,5,1,7,8,4,6] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,3,5,1,7,4,6,8] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,3,5,7,1,4,8,6] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,3,5,7,1,8,4,6] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,5,7,8,1,4,6] => ? => ?
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,5,7,1,4,6,8] => ? => ?
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,3,5,1,4,6,8,7] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,3,5,1,4,8,6,7] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,5,1,8,4,6,7] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,5,8,1,4,6,7] => ? => ?
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,5,1,4,6,7,8] => ? => ?
=> ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,1,4,6,7,5,8] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,1,4,6,5,8,7] => ? => ?
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,1,4,6,8,5,7] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,1,4,6,5,7,8] => ? => ?
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,3,1,6,4,7,8,5] => ? => ?
=> ? = 2 + 1
Description
The length of the partition.
Matching statistic: St000196
(load all 44 compositions to match this statistic)
(load all 44 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 67%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 67%
Values
[1,0,1,0]
=> [1]
=> [1,0,1,0]
=> [.,[.,.]]
=> 0
[1,1,0,0]
=> []
=> []
=> .
=> ? = 0
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 0
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 0
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 1
[1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> [.,[.,.]]
=> 0
[1,1,1,0,0,0]
=> []
=> []
=> .
=> ? = 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 0
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 0
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> 0
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> [.,[.,.]]
=> 0
[1,1,1,1,0,0,0,0]
=> []
=> []
=> .
=> ? = 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[.,.]]]]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [.,[[.,[.,.]],[.,.]]]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[.,[.,[.,.]]],[.,.]]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],.]],[.,.]]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[[.,.],[.,.]],[.,.]]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> .
=> ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> .
=> ? = 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> .
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,5,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [.,[.,[.,[.,[[.,[[.,.],.]],.]]]]]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,3,2,1]
=> ?
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,6,5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,2,1]
=> ?
=> ?
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,5,3,3,2,1]
=> ?
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,2,1]
=> ?
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,3,2,1]
=> ?
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,6,4,3,3,2,1]
=> ?
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,3,2,1]
=> ?
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,5,4,3,3,2,1]
=> ?
=> ?
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,4,3,3,2,1]
=> ?
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,4,3,3,2,1]
=> ?
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,3,2,1]
=> ?
=> ?
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,4,4,3,3,2,1]
=> ?
=> ?
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,3,3,2,1]
=> ?
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,6,3,3,3,2,1]
=> ?
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [7,5,3,3,3,2,1]
=> ?
=> ?
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,5,3,3,3,2,1]
=> ?
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,4,3,3,3,2,1]
=> ?
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,3,2,1]
=> ?
=> ?
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,3,2,1]
=> ?
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,3,3,3,2,1]
=> ?
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,3,3,3,2,1]
=> ?
=> ?
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,2,1]
=> ?
=> ?
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> ? = 1
Description
The number of occurrences of the contiguous pattern {{{[[.,.],[.,.]]}}} in a binary tree.
Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by $2^{n-2}$.
The following 166 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000069The number of maximal elements of a poset. St000071The number of maximal chains in a poset. St000527The width of the poset. St000257The number of distinct parts of a partition that occur at least twice. St000097The order of the largest clique of the graph. St001581The achromatic number of a graph. St000068The number of minimal elements in a poset. St000052The number of valleys of a Dyck path not on the x-axis. St001732The number of peaks visible from the left. St000806The semiperimeter of the associated bargraph. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St001175The size of a partition minus the hook length of the base cell. 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. St001389The number of partitions of the same length below the given integer partition. St001092The number of distinct even parts of a partition. 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. St000142The number of even parts of a partition. St001214The aft of an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000035The number of left outer peaks of a permutation. St001083The number of boxed occurrences of 132 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. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000353The number of inner valleys of a permutation. St000157The number of descents of a standard tableau. St000632The jump number of the poset. St001840The number of descents of a set partition. 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. 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. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St001746The coalition number of a graph. St000021The number of descents of 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. St000325The width of the tree associated to a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000647The number of big descents of a permutation. St000092The number of outer peaks of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. 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. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000919The number of maximal left branches of a binary tree. St000159The number of distinct parts of the integer partition. St000659The number of rises of length at least 2 of a Dyck path. St000455The second largest eigenvalue of a graph if it is integral. St000312The number of leaves in a graph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000318The number of addable cells of the Ferrers diagram of an integer 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. St000711The number of big exceedences of a permutation. St001728The number of invisible descents of a permutation. St000710The number of big deficiencies of a permutation. St000646The number of big ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. 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. St000374The number of exclusive right-to-left minima of a permutation. St000360The number of occurrences of the pattern 32-1. 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. St000099The number of valleys of a permutation, including the boundary. 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. St001307The number of induced stars on four vertices in a graph. St001812The biclique partition number of a graph. St000409The number of pitchforks in a binary tree. St000822The Hadwiger number of the graph. St000636The hull number of a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001883The mutual visibility number of a graph. St001597The Frobenius rank of a skew partition. St001330The hat guessing number of a graph. St000670The reversal length of a permutation. St001487The number of inner corners of a skew partition. St000100The number of linear extensions of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000662The staircase size of the code of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000307The number of rowmotion orbits of a poset. 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. St000252The number of nodes of degree 3 of a binary tree. 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. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St001964The interval resolution global dimension of a poset. St000314The number of left-to-right-maxima of a permutation. St000663The number of right floats 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. 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. 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. 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. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001896The number of right descents of a signed permutations. St000807The sum of the heights of the valleys of the associated bargraph. 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. St000805The number of peaks of the associated bargraph. St000864The number of circled entries of the shifted recording tableau of a permutation. St000872The number of very big descents 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 $K[x]/(x^n)$.
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