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Your data matches 60 different statistics following compositions of up to 3 maps.
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Matching statistic: St001036
(load all 88 compositions to match this statistic)
(load all 88 compositions to match this statistic)
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001036: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000288
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%
Mp00130: Permutations —descent tops⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => => ? => ? = 0 + 1
[1,0,1,0]
=> [2,1] => 1 => 1 => 1 = 0 + 1
[1,1,0,0]
=> [1,2] => 0 => 1 => 1 = 0 + 1
[1,0,1,0,1,0]
=> [3,2,1] => 11 => 11 => 2 = 1 + 1
[1,0,1,1,0,0]
=> [2,3,1] => 01 => 10 => 1 = 0 + 1
[1,1,0,0,1,0]
=> [3,1,2] => 01 => 10 => 1 = 0 + 1
[1,1,0,1,0,0]
=> [2,1,3] => 10 => 11 => 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => 00 => 01 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 111 => 111 => 3 = 2 + 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 101 => 110 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 011 => 101 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 011 => 101 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 001 => 010 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 011 => 101 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 001 => 010 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 101 => 110 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 110 => 111 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 010 => 101 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 001 => 010 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 010 => 101 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 100 => 101 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 000 => 001 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 1111 => 1111 => 4 = 3 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 1101 => 1110 => 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 1011 => 1101 => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 1011 => 1101 => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 1001 => 1010 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 0111 => 1011 => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 0101 => 1010 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 0111 => 1011 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 0111 => 1011 => 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 0011 => 0101 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 0011 => 0101 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 0011 => 0101 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 0101 => 1010 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0001 => 0010 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 0111 => 1011 => 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 0101 => 1010 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0011 => 0101 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 0011 => 0101 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0001 => 0010 => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 1011 => 1101 => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 1001 => 1010 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 1101 => 1110 => 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 1110 => 1111 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 1010 => 1101 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 0101 => 1010 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 0110 => 1011 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 0110 => 1011 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 0010 => 0101 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 0011 => 0101 => 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] => ? => ? => ? = 4 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [8,5,6,4,3,7,2,1] => ? => ? => ? = 4 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [8,5,4,6,3,7,2,1] => ? => ? => ? = 4 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,3,7,2,1] => ? => ? => ? = 3 + 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] => ? => ? => ? = 4 + 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] => ? => ? => ? = 3 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [8,4,5,3,6,7,2,1] => ? => ? => ? = 3 + 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] => ? => ? => ? = 3 + 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] => ? => ? => ? = 3 + 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] => ? => ? => ? = 4 + 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,3,2,7,1] => ? => ? => ? = 3 + 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [8,7,3,4,5,2,6,1] => ? => ? => ? = 3 + 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] => ? => ? => ? = 3 + 1
[1,0,1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [8,7,4,5,2,3,6,1] => ? => ? => ? = 3 + 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [8,5,4,6,2,3,7,1] => ? => ? => ? = 3 + 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,2,3,7,1] => ? => ? => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [8,5,6,3,2,4,7,1] => ? => ? => ? = 3 + 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [7,4,5,3,2,6,8,1] => ? => ? => ? = 3 + 1
[1,0,1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [8,7,3,4,2,5,6,1] => ? => ? => ? = 3 + 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [8,6,3,4,2,5,7,1] => ? => ? => ? = 3 + 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [7,5,3,4,2,6,8,1] => ? => ? => ? = 3 + 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [8,4,3,5,2,6,7,1] => ? => ? => ? = 3 + 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [7,3,4,5,2,6,8,1] => ? => ? => ? = 2 + 1
[1,0,1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [8,7,5,2,3,4,6,1] => ? => ? => ? = 3 + 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [8,6,5,2,3,4,7,1] => ? => ? => ? = 3 + 1
[1,0,1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [8,5,6,2,3,4,7,1] => ? => ? => ? = 2 + 1
[1,0,1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [8,7,4,2,3,5,6,1] => ? => ? => ? = 3 + 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [8,7,3,2,4,5,6,1] => ? => ? => ? = 3 + 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [8,6,3,2,4,5,7,1] => ? => ? => ? = 3 + 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [8,5,3,2,4,6,7,1] => ? => ? => ? = 3 + 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [7,3,4,2,5,6,8,1] => ? => ? => ? = 2 + 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [8,6,2,3,4,5,7,1] => ? => ? => ? = 2 + 1
[1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,3,2,1,8] => ? => ? => ? = 4 + 1
[1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [6,4,5,3,7,2,1,8] => ? => ? => ? = 4 + 1
[1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [4,5,6,3,7,2,1,8] => ? => ? => ? = 3 + 1
[1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,5,2,1,8] => ? => ? => ? = 3 + 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [7,5,3,4,6,2,1,8] => ? => ? => ? = 4 + 1
[1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [6,5,3,4,7,2,1,8] => ? => ? => ? = 4 + 1
[1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [5,6,3,4,7,2,1,8] => ? => ? => ? = 3 + 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [7,4,3,5,6,2,1,8] => ? => ? => ? = 4 + 1
[1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [6,4,3,5,7,2,1,8] => ? => ? => ? = 4 + 1
[1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [4,5,3,6,7,2,1,8] => ? => ? => ? = 3 + 1
[1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [6,3,4,5,7,2,1,8] => ? => ? => ? = 3 + 1
[1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [5,3,4,6,7,2,1,8] => ? => ? => ? = 3 + 1
[1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [4,3,5,6,7,2,1,8] => ? => ? => ? = 3 + 1
[1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [6,5,7,4,2,3,1,8] => ? => ? => ? = 4 + 1
[1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,2,3,1,8] => ? => ? => ? = 3 + 1
[1,1,0,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [6,7,5,3,2,4,1,8] => ? => ? => ? = 4 + 1
[1,1,0,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [7,5,6,3,2,4,1,8] => ? => ? => ? = 4 + 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000318
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 86%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1,0]
=> []
=> ?
=> ? = 0 + 1
[1,0,1,0]
=> [1,0,1,0]
=> [1]
=> []
=> 1 = 0 + 1
[1,1,0,0]
=> [1,1,0,0]
=> []
=> ?
=> ? = 0 + 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2]
=> []
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,1]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> 3 = 2 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? = 0 + 1
[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
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,1]
=> [6,4,3,2,1]
=> ? = 5 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,1,1]
=> ?
=> ? = 5 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,1]
=> [6,5,3,2,1]
=> ? = 5 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,1]
=> ?
=> ? = 4 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,1,1]
=> ?
=> ? = 5 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,2,1]
=> ?
=> ? = 5 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,1,1,1]
=> [4,3,2,1,1,1]
=> ? = 4 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2,1]
=> [6,3,2,1]
=> ? = 4 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,1,1]
=> ?
=> ? = 4 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,1]
=> [6,5,4,2,1]
=> ? = 5 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> [7,6,4,2,1]
=> [6,4,2,1]
=> ? = 4 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [7,5,4,2,1,1]
=> ?
=> ? = 4 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,1,1]
=> ?
=> ? = 5 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,1,1,1]
=> ?
=> ? = 4 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,2,1]
=> ?
=> ? = 5 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,3,2,1]
=> ?
=> ? = 5 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,2,2,1]
=> ?
=> ? = 4 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2,1,1,1]
=> [6,3,2,1,1,1]
=> ? = 4 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,2,1,1]
=> ?
=> ? = 4 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,2,1,1]
=> ?
=> ? = 4 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2,1]
=> [6,5,2,1]
=> ? = 4 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [7,6,4,2,1,1]
=> ?
=> ? = 4 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [7,5,4,2,2,1]
=> ?
=> ? = 4 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2,1,1]
=> [6,3,2,1,1]
=> ? = 4 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,2,1]
=> ?
=> ? = 4 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,2,1]
=> ?
=> ? = 4 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,1]
=> [6,5,4,3,1]
=> ? = 5 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,1]
=> ?
=> ? = 4 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,1,1]
=> ?
=> ? = 4 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,1]
=> ?
=> ? = 4 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,1,1]
=> ?
=> ? = 4 + 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,3,1]
=> ?
=> ? = 4 + 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,1,1]
=> ?
=> ? = 5 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [7,5,4,2,1,1,1]
=> [5,4,2,1,1,1]
=> ? = 4 + 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [7,6,4,2,1,1,1]
=> ?
=> ? = 4 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [7,5,4,2,2,1,1]
=> ?
=> ? = 4 + 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,2,1]
=> ?
=> ? = 5 + 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,2,2,1]
=> ?
=> ? = 4 + 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,3,2,1]
=> ?
=> ? = 5 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,4,3,2,1]
=> [5,4,4,3,2,1]
=> ? = 5 + 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,4,3,3,3,2,1]
=> ?
=> ? = 4 + 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2,2,2,1]
=> ?
=> ? = 4 + 1
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St000159
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 71%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [1,0]
=> []
=> ?
=> ? = 0
[1,0,1,0]
=> [1,0,1,0]
=> [1]
=> []
=> 0
[1,1,0,0]
=> [1,1,0,0]
=> []
=> ?
=> ? = 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> 0
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2]
=> []
=> 0
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> 0
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> 0
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[1,1,1,1,0,0,0,0]
=> [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]
=> [4,3,2,1]
=> [3,2,1]
=> 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? = 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
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> [5,4,3,2,1]
=> ? = 5
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> [4,3,3,2,1]
=> ? = 4
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1]
=> [5,4,3,1]
=> ? = 4
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,1]
=> [5,4,2,1,1]
=> ? = 4
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> [5,3,2,2,1]
=> ? = 4
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> [4,3,3,2,1]
=> ? = 4
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> [4,4,3,2,1]
=> ? = 4
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> [5,4,3,2]
=> ? = 4
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> [4,4,3,2]
=> ? = 3
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> [5,4,3,1,1]
=> ? = 4
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,1,1]
=> [4,4,3,1,1]
=> ? = 3
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> [5,4,2,2,1]
=> ? = 4
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> [4,4,2,2,1]
=> ? = 3
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> [5,3,3,2,1]
=> ? = 4
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> [4,4,3,2,1]
=> ? = 4
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> [5,4,3,2,1]
=> ? = 5
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> [4,4,3,2,1]
=> ? = 4
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> [5,3,3,2,1]
=> ? = 4
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2,1]
=> [4,3,3,2,1]
=> ? = 4
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> [4,4,2,2,1]
=> ? = 3
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> [5,4,2,2,1]
=> ? = 4
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2,1]
=> [4,4,2,2,1]
=> ? = 3
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> [5,3,2,2,1]
=> ? = 4
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> [5,3,3,1,1]
=> ? = 3
[1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> [4,4,3,1,1]
=> ? = 3
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1,1]
=> [5,4,3,1,1]
=> ? = 4
[1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,1,1]
=> [4,4,3,1,1]
=> ? = 3
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> [5,3,3,1,1]
=> ? = 3
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1,1]
=> [5,4,2,1,1]
=> ? = 4
[1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2]
=> [5,4,2,2]
=> ? = 3
[1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2]
=> [5,3,3,2]
=> ? = 3
[1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2]
=> [4,4,3,2]
=> ? = 3
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> [5,4,3,2]
=> ? = 4
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2]
=> [4,4,3,2]
=> ? = 3
[1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2]
=> [5,3,3,2]
=> ? = 3
[1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2]
=> [5,4,2,2]
=> ? = 3
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1]
=> [5,4,3,1]
=> ? = 4
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? = 6
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,1]
=> [5,4,3,2,1]
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,1]
=> [6,4,3,2,1]
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,1,1]
=> ?
=> ? = 5
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,1]
=> [6,5,3,2,1]
=> ? = 5
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,1]
=> ?
=> ? = 4
Description
The number of distinct parts of the integer partition.
This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Matching statistic: St001124
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 71%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [1,0]
=> []
=> ?
=> ? = 0 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [1]
=> []
=> ? = 0 - 1
[1,1,0,0]
=> [1,1,0,0]
=> []
=> ?
=> ? = 0 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1]
=> ? = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? = 0 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2]
=> []
=> ? = 0 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> ? = 1 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? = 0 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> ? = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? = 0 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? = 0 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? = 0 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> ? = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? = 0 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1]
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1]
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? = 0 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? = 0 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? = 0 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> 1 = 2 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? = 0 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [3]
=> 0 = 1 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1]
=> ? = 1 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? = 0 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [4,3,2,1]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [3,2,1]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [3,2,1]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [3,2,1,1]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [4,2,1]
=> 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [3,2,1,1]
=> 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [3,2,2,1]
=> 2 = 3 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1]
=> ? = 1 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [1]
=> ? = 1 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [1]
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [1]
=> ? = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> []
=> ? = 0 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> []
=> ? = 0 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> []
=> ? = 0 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> []
=> ? = 0 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> []
=> ? = 0 - 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1]
=> ? = 1 - 1
[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
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,1]
=> [1]
=> ? = 1 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [3,1]
=> [1]
=> ? = 1 - 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [4,1]
=> [1]
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [5,1]
=> [1]
=> ? = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1]
=> [1]
=> ? = 1 - 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> []
=> ? = 0 - 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2]
=> []
=> ? = 0 - 1
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3]
=> []
=> ? = 0 - 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [4]
=> []
=> ? = 0 - 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [5]
=> []
=> ? = 0 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6]
=> []
=> ? = 0 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1]
=> [1]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> ?
=> ? = 0 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,1]
=> [6,4,3,2,1]
=> ? = 5 - 1
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
Matching statistic: St000996
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [] => 0
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => [1] => 0
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => [1] => 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => [2,1] => 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => [1,2] => 0
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => [1,2] => 0
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => [2,1] => 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2] => 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,3,1] => 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,2] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,1,2] => 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,1,3] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,3,2] => 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,2,3] => 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,3] => 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,3,1] => 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,3] => 0
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2] => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,3,4,1] => 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,4,1,2] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,4,1,2] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,4,1,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [4,1,2,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,1,4,2] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [4,1,2,3] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,4,1,3] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,3,1,4] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [4,1,2,3] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,1,2,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,1,2,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,3,4,2] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [1,4,2,3] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [1,4,2,3] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,3,2,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [1,2,3,4] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,3,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,3,1,4] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,3,4,1] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,4,1,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4] => 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,5,6,1,7,8,2] => ? => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,4,5,6,7,1,8,3] => [2,4,5,6,7,1,3] => ? = 5
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [4,5,6,7,1,2,8,3] => ? => ? = 4
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [3,4,5,1,6,7,8,2] => ? => ? = 5
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [4,5,6,1,7,2,8,3] => ? => ? = 4
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,4,5,6,1,7,8,3] => [2,4,5,6,1,7,3] => ? = 5
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,5,6,7,1,8,4] => ? => ? = 5
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [2,5,6,7,1,3,8,4] => ? => ? = 4
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [4,5,6,1,2,7,8,3] => ? => ? = 4
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [3,5,6,1,7,2,8,4] => ? => ? = 4
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [3,5,6,7,1,2,8,4] => ? => ? = 4
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [5,6,7,1,2,3,8,4] => ? => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,4,1,5,6,7,8,2] => [3,4,1,5,6,7,2] => ? = 5
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]
=> [4,5,1,6,7,2,8,3] => ? => ? = 4
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> [4,5,1,6,2,7,8,3] => ? => ? = 4
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [3,5,1,6,7,2,8,4] => ? => ? = 4
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> [5,6,1,7,2,3,8,4] => ? => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [2,4,5,1,6,7,8,3] => ? => ? = 5
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [2,5,6,1,7,3,8,4] => ? => ? = 4
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,5,6,1,7,8,4] => [2,3,5,6,1,7,4] => ? = 5
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,6,7,1,8,5] => ? => ? = 5
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [2,3,6,7,1,4,8,5] => ? => ? = 4
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [2,5,6,1,3,7,8,4] => ? => ? = 4
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [2,4,6,1,7,3,8,5] => [2,4,6,1,7,3,5] => ? = 4
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [2,4,6,7,1,3,8,5] => ? => ? = 4
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [4,5,1,2,6,7,8,3] => ? => ? = 4
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> [5,6,1,2,7,3,8,4] => ? => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [3,5,1,6,2,7,8,4] => ? => ? = 4
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [3,4,1,6,7,2,8,5] => ? => ? = 4
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [3,6,1,7,2,4,8,5] => [3,6,1,7,2,4,5] => ? = 3
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [3,5,6,1,2,7,8,4] => ? => ? = 4
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [3,4,6,1,7,2,8,5] => [3,4,6,1,7,2,5] => ? = 4
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,6,7,1,2,8,5] => [3,4,6,7,1,2,5] => ? = 4
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [3,6,7,1,2,4,8,5] => ? => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> [4,6,1,2,7,3,8,5] => ? => ? = 3
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0,1,0]
=> [4,6,1,7,2,3,8,5] => ? => ? = 3
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0,1,0]
=> [4,6,7,1,2,3,8,5] => ? => ? = 3
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,4,5,6,7,8,2] => [3,1,4,5,6,7,2] => ? = 5
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [4,1,5,6,7,2,8,3] => ? => ? = 4
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> [4,1,5,6,2,7,8,3] => ? => ? = 4
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [3,1,5,6,7,2,8,4] => ? => ? = 4
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> [5,1,6,7,2,3,8,4] => ? => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [4,1,5,2,6,7,8,3] => ? => ? = 4
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [5,1,6,2,7,3,8,4] => [5,1,6,2,7,3,4] => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [3,1,5,6,2,7,8,4] => ? => ? = 4
[1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [3,1,4,6,7,2,8,5] => ? => ? = 4
[1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [3,1,6,7,2,4,8,5] => ? => ? = 3
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [5,1,6,2,3,7,8,4] => ? => ? = 3
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0,1,0]
=> [4,1,6,2,7,3,8,5] => [4,1,6,2,7,3,5] => ? = 3
[1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0,1,0]
=> [4,1,6,7,2,3,8,5] => ? => ? = 3
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St000245
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [] => 0
[1,0,1,0]
=> [1,0,1,0]
=> [1,2] => [1] => 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] => [1,2] => 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [2,1] => 0
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1] => 0
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,2] => 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => [2,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => 0
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => 2
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => 0
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => 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] => 3
[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] => [2,3,4,1] => 2
[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] => [2,3,4,1] => 2
[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] => [1,3,4,2] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => [3,4,2,1] => 1
[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] => [2,3,1,4] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => [3,4,2,1] => 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] => [1,3,4,2] => 2
[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] => [1,2,4,3] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,4,3,1] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => [3,4,2,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => 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] => [1,4,3,2] => 1
[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] => [4,3,2,1] => 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] => [2,1,3,4] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [3,2,4,1] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [3,2,4,1] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [4,3,2,1] => 0
[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] => [1,3,2,4] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,1] => 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] => [1,2,4,3] => 2
[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] => 3
[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] => [2,3,4,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => 2
[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] => [1,3,4,2] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,4,3,1] => 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,6,7,2,8] => ? => ? = 5
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,6,7,2,1,8] => ? => ? = 4
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [3,4,5,6,2,7,1,8] => ? => ? = 4
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,3,4,5,6,2,7,8] => ? => ? = 5
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,7,3,8] => ? => ? = 5
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [2,4,5,6,7,3,1,8] => ? => ? = 4
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [3,4,5,6,2,1,7,8] => ? => ? = 4
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,4,5,6,3,7,2,8] => ? => ? = 4
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,4,5,6,7,3,2,8] => ? => ? = 4
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [4,5,6,7,3,2,1,8] => ? => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]
=> [3,4,5,2,6,7,1,8] => ? => ? = 4
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> [3,4,5,2,6,1,7,8] => ? => ? = 4
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,4,5,3,6,7,2,8] => ? => ? = 4
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> [4,5,6,3,7,2,1,8] => ? => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2,6,7,8] => ? => ? = 5
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [2,4,5,6,3,7,1,8] => ? => ? = 4
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,6,3,7,8] => ? => ? = 5
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,3,5,6,7,4,8] => ? => ? = 5
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [2,3,5,6,7,4,1,8] => ? => ? = 4
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [2,4,5,6,3,1,7,8] => ? => ? = 4
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,3,5,6,4,7,2,8] => ? => ? = 4
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,3,5,6,7,4,2,8] => ? => ? = 4
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [2,5,6,7,4,3,1,8] => ? => ? = 3
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [3,4,5,2,1,6,7,8] => ? => ? = 4
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> [4,5,6,3,2,7,1,8] => ? => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,4,5,3,6,2,7,8] => ? => ? = 4
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,2,5,4,6,7,3,8] => ? => ? = 4
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0,1,0]
=> [3,5,6,4,7,2,1,8] => ? => ? = 3
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,4,5,6,3,2,7,8] => ? => ? = 4
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,2,5,6,4,7,3,8] => ? => ? = 4
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,2,5,6,7,4,3,8] => ? => ? = 4
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0,1,0]
=> [3,5,6,7,4,2,1,8] => ? => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [4,5,6,3,2,1,7,8] => ? => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,5,6,4,3,7,2,8] => ? => ? = 3
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,5,6,4,7,3,2,8] => ? => ? = 3
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,5,6,7,4,3,2,8] => ? => ? = 3
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [5,6,7,4,3,2,1,8] => [5,6,7,4,3,2,1] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7,8] => ? => ? = 5
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [3,4,2,5,6,7,1,8] => ? => ? = 4
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> [3,4,2,5,6,1,7,8] => ? => ? = 4
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,4,3,5,6,7,2,8] => ? => ? = 4
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> [4,5,3,6,7,2,1,8] => ? => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [3,4,2,5,1,6,7,8] => ? => ? = 4
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [4,5,3,6,2,7,1,8] => [4,5,3,6,2,7,1] => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,4,3,5,6,2,7,8] => ? => ? = 4
[1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,2,5,6,7,4,8] => ? => ? = 4
[1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [2,5,4,6,7,3,1,8] => ? => ? = 3
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [4,5,3,6,2,1,7,8] => ? => ? = 3
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,5,4,6,3,7,2,8] => ? => ? = 3
[1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,5,4,6,7,3,2,8] => ? => ? = 3
Description
The number of ascents of a permutation.
Matching statistic: St000672
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [] => 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => [1] => 0
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => [1] => 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2] => 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1] => 0
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => [2,1] => 0
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,2] => 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => [2,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3] => 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,1] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,3,1] => 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,2,1] => 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3] => 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1] => 0
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,1,2] => 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3] => 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => 0
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,1,3] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1,2] => 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4] => 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,3,4,1] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,4,1,2] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,4,2,1] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,4,2,1] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,4,1,2] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [4,1,2,3] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,2,3,1] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,4,2,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,2,1,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,3,1,2] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,3,2,1] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,2,4,1] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,2,1,4] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [4,3,1,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,1] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [4,2,1,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [4,1,2,3] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,4] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,4,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [4,2,3,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,1,4] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,4,1,2] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [3,4,2,1] => 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,7,1,2,8] => ? => ? = 5
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,7,2,8,1] => ? => ? = 4
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,2,1,7,8] => ? => ? = 4
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,1,2,7,8] => ? => ? = 5
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [4,5,6,7,1,2,3,8] => ? => ? = 5
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [4,5,6,7,2,3,8,1] => [4,5,6,7,2,3,1] => ? = 4
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,2,7,8,1] => ? => ? = 4
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [4,5,6,7,2,3,1,8] => ? => ? = 4
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,5,6,7,3,8,1,2] => ? => ? = 4
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,5,6,7,3,8,2,1] => ? => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [3,4,5,2,6,1,7,8] => ? => ? = 4
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [3,4,5,2,1,6,7,8] => ? => ? = 4
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> [4,5,6,3,1,2,7,8] => ? => ? = 4
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [4,5,6,3,2,7,8,1] => ? => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [3,4,5,1,2,6,7,8] => ? => ? = 5
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0]
=> [4,5,6,2,1,3,7,8] => ? => ? = 4
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [5,6,7,2,3,4,8,1] => ? => ? = 4
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [4,5,6,2,3,7,8,1] => ? => ? = 4
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [5,6,7,2,3,4,1,8] => [5,6,7,2,3,4,1] => ? = 4
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,4,8,1,2] => ? => ? = 4
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,4,8,2,1] => ? => ? = 3
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [4,5,6,3,2,7,1,8] => ? => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [4,5,6,2,3,7,1,8] => ? => ? = 4
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,1,1,0,0,0,0]
=> [5,6,7,2,3,1,4,8] => ? => ? = 4
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,4,2,1,8] => [5,6,7,3,4,2,1] => ? = 3
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [4,5,6,3,7,8,1,2] => ? => ? = 4
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,4,1,2,8] => ? => ? = 4
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> [5,6,7,4,8,1,2,3] => ? => ? = 4
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [5,6,7,4,8,2,3,1] => ? => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [4,5,6,3,7,8,2,1] => ? => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,4,2,8,1] => ? => ? = 3
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [5,6,7,4,8,2,1,3] => ? => ? = 3
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,6,7,4,8,3,1,2] => ? => ? = 3
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,6,7,4,8,3,2,1] => ? => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7,8] => ? => ? = 5
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,6,1,7,8] => ? => ? = 4
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,1,6,7,8] => ? => ? = 4
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
=> [4,5,3,6,1,2,7,8] => ? => ? = 4
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [4,5,3,6,2,7,8,1] => ? => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,1,5,6,7,8] => ? => ? = 4
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,3,2,1,6,7,8] => ? => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
=> [4,5,3,1,2,6,7,8] => ? => ? = 4
[1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> [5,6,4,1,2,3,7,8] => ? => ? = 4
[1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [5,6,4,2,3,7,8,1] => ? => ? = 3
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,3,2,6,7,8,1] => ? => ? = 3
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [5,6,4,2,3,7,1,8] => ? => ? = 3
[1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,7,8,1,2] => ? => ? = 3
[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,7,8,2,1] => ? => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> [4,5,2,3,1,6,7,8] => [4,5,2,3,1,6,7] => ? = 4
[1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> [4,5,2,1,3,6,7,8] => ? => ? = 4
Description
The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Matching statistic: St000662
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [] => 0
[1,0,1,0]
=> [2,1] => [2,1] => [1] => 0
[1,1,0,0]
=> [1,2] => [1,2] => [1] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => [2,1] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => [1,2] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [1,2] => 0
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [2,1] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => [3,2,1] => 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,1,2] => [3,1,2] => 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,1,3,2] => [1,3,2] => 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,2,1,3] => [2,1,3] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,3,2] => [1,3,2] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,4,2,3] => [1,2,3] => 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1,4,2] => [3,1,2] => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1] => 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,1,2,4] => [3,1,2] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => [1,2,3] => 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => [1,3,2] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [4,3,2,1] => 3
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,4,3,1,2] => [4,3,1,2] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [5,4,1,3,2] => [4,1,3,2] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,4,2,1,3] => [4,2,1,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,1,2,3] => [4,1,2,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [5,1,4,3,2] => [1,4,3,2] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,1,4,2,3] => [1,4,2,3] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,3,1,4,2] => [3,1,4,2] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,3,2,1,4] => [3,2,1,4] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,1,2,4] => [3,1,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,2,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,5,4,3,2] => [1,4,3,2] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,5,4,2,3] => [1,4,2,3] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,5,2,4,3] => [1,2,4,3] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,5,3,2,4] => [1,3,2,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,5,2,3,4] => [1,2,3,4] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,1,5,3,2] => [4,1,3,2] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,1,5,2,3] => [4,1,2,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,3,1,5,2] => [4,3,1,2] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,3,1,2,5] => [4,3,1,2] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [4,1,2,5,3] => [4,1,2,3] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,1,3,2,5] => [4,1,3,2] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,2,1,3,5] => [4,2,1,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3] => 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,4,3,2,1] => [8,7,6,5,4,1,3,2] => [7,6,5,4,1,3,2] => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [8,7,5,6,4,3,2,1] => [8,7,6,5,1,4,3,2] => [7,6,5,1,4,3,2] => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [8,6,5,7,4,3,2,1] => [8,7,6,5,3,1,4,2] => [7,6,5,3,1,4,2] => ? = 5
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [8,5,6,7,4,3,2,1] => [8,7,6,5,1,2,4,3] => [7,6,5,1,2,4,3] => ? = 4
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [8,7,6,4,5,3,2,1] => [8,7,6,1,5,4,3,2] => [7,6,1,5,4,3,2] => ? = 5
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [8,6,7,4,5,3,2,1] => [8,7,6,1,5,2,4,3] => ? => ? = 4
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [8,7,5,4,6,3,2,1] => [8,7,6,4,1,5,3,2] => ? => ? = 5
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [8,6,5,4,7,3,2,1] => [8,7,6,4,3,1,5,2] => ? => ? = 5
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [8,5,6,4,7,3,2,1] => [8,7,6,4,1,2,5,3] => ? => ? = 4
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [8,7,4,5,6,3,2,1] => [8,7,6,1,2,5,4,3] => ? => ? = 4
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [8,6,4,5,7,3,2,1] => [8,7,6,1,4,2,5,3] => ? => ? = 4
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [8,5,4,6,7,3,2,1] => [8,7,6,3,1,2,5,4] => ? => ? = 4
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,7,3,2,1] => [8,7,6,1,2,3,5,4] => ? => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,3,4,2,1] => [8,7,1,6,5,4,3,2] => ? => ? = 5
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,3,4,2,1] => [8,7,1,6,5,2,4,3] => ? => ? = 4
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [8,7,5,6,3,4,2,1] => [8,7,1,6,2,5,4,3] => [7,1,6,2,5,4,3] => ? = 4
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [8,6,5,7,3,4,2,1] => ? => ? => ? = 4
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [8,5,6,7,3,4,2,1] => [8,7,1,6,2,3,5,4] => [7,1,6,2,3,5,4] => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [8,7,6,4,3,5,2,1] => [8,7,5,1,6,4,3,2] => ? => ? = 5
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [8,6,7,4,3,5,2,1] => [8,7,5,1,6,2,4,3] => ? => ? = 4
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [8,7,5,4,3,6,2,1] => [8,7,5,4,1,6,3,2] => [7,5,4,1,6,3,2] => ? = 5
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [8,6,5,4,3,7,2,1] => [8,7,5,4,3,1,6,2] => ? => ? = 5
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [8,5,6,4,3,7,2,1] => ? => ? => ? = 4
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [8,7,4,5,3,6,2,1] => [8,7,5,1,2,6,4,3] => ? => ? = 4
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [8,6,4,5,3,7,2,1] => [8,7,5,1,4,2,6,3] => ? => ? = 4
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [8,5,4,6,3,7,2,1] => [8,7,5,3,1,2,6,4] => ? => ? = 4
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,3,7,2,1] => ? => ? => ? = 3
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [8,7,6,3,4,5,2,1] => [8,7,1,2,6,5,4,3] => ? => ? = 4
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [8,6,7,3,4,5,2,1] => [8,7,1,2,6,3,5,4] => ? => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [8,7,5,3,4,6,2,1] => [8,7,1,5,2,6,4,3] => ? => ? = 4
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [8,6,5,3,4,7,2,1] => ? => ? => ? = 4
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [8,5,6,3,4,7,2,1] => ? => ? => ? = 3
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [8,7,4,3,5,6,2,1] => [8,7,4,1,2,6,5,3] => ? => ? = 4
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [8,6,4,3,5,7,2,1] => [8,7,4,1,5,2,6,3] => ? => ? = 4
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [8,5,4,3,6,7,2,1] => [8,7,4,3,1,2,6,5] => ? => ? = 4
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [8,4,5,3,6,7,2,1] => ? => ? => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [8,7,3,4,5,6,2,1] => [8,7,1,2,3,6,5,4] => [7,1,2,3,6,5,4] => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [8,6,3,4,5,7,2,1] => ? => ? => ? = 3
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [8,5,3,4,6,7,2,1] => [8,7,1,4,2,3,6,5] => ? => ? = 3
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [8,4,3,5,6,7,2,1] => [8,7,3,1,2,4,6,5] => ? => ? = 3
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,7,2,1] => [8,7,1,2,3,4,6,5] => ? => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,2,3,1] => [8,1,7,6,5,4,3,2] => ? => ? = 5
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,4,2,3,1] => [8,1,7,6,5,2,4,3] => [1,7,6,5,2,4,3] => ? = 4
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [8,7,5,6,4,2,3,1] => [8,1,7,6,2,5,4,3] => ? => ? = 4
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [8,6,5,7,4,2,3,1] => [8,1,7,6,4,2,5,3] => ? => ? = 4
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [8,5,6,7,4,2,3,1] => [8,1,7,6,2,3,5,4] => ? => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [8,7,6,4,5,2,3,1] => [8,1,7,2,6,5,4,3] => ? => ? = 4
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [8,6,7,4,5,2,3,1] => [8,1,7,2,6,3,5,4] => [1,7,2,6,3,5,4] => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [8,7,5,4,6,2,3,1] => [8,1,7,5,2,6,4,3] => ? => ? = 4
[1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [8,6,5,4,7,2,3,1] => [8,1,7,5,4,2,6,3] => ? => ? = 4
Description
The staircase size of the code of a permutation.
The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$.
The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$.
This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Matching statistic: St000249
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000249: Set partitions ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000249: Set partitions ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => {{1}}
=> ? = 0 + 2
[1,0,1,0]
=> [1,0,1,0]
=> [1,2] => {{1},{2}}
=> 2 = 0 + 2
[1,1,0,0]
=> [1,1,0,0]
=> [2,1] => {{1,2}}
=> 2 = 0 + 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 3 = 1 + 2
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> 2 = 0 + 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> 2 = 0 + 2
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => {{1},{2},{3}}
=> 3 = 1 + 2
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => {{1,3},{2}}
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 4 = 2 + 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 3 = 1 + 2
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => {{1,4},{2},{3}}
=> 3 = 1 + 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => {{1,3,4},{2}}
=> 3 = 1 + 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => {{1,3},{2},{4}}
=> 2 = 0 + 2
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 3 = 1 + 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 2 = 0 + 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => {{1,2,4},{3}}
=> 3 = 1 + 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 4 = 2 + 2
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 3 = 1 + 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => {{1},{2,4},{3}}
=> 2 = 0 + 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 3 = 1 + 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 3 = 1 + 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => {{1,4},{2,3}}
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 5 = 3 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => {{1,2,3,5},{4}}
=> 4 = 2 + 2
[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] => {{1,2,3,4},{5}}
=> 4 = 2 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 4 = 2 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 3 = 1 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 4 = 2 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 3 = 1 + 2
[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},{2,3,4},{5}}
=> 4 = 2 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 4 = 2 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 3 = 1 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => {{1,5},{2,3,4}}
=> 3 = 1 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 3 = 1 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => {{1,4,5},{2,3}}
=> 3 = 1 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 2 = 0 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => {{1,3,4,5},{2}}
=> 4 = 2 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => {{1,3,5},{2},{4}}
=> 3 = 1 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => {{1,3,4},{2},{5}}
=> 3 = 1 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 3 = 1 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 2 = 0 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 4 = 2 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 3 = 1 + 2
[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},{3,4},{5}}
=> 4 = 2 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 5 = 3 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 4 = 2 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 3 = 1 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 4 = 2 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => {{1,4,5},{2},{3}}
=> 4 = 2 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 3 = 1 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 3 = 1 + 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 6 + 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,5,7,6,8] => {{1},{2},{3},{4},{5},{6,7},{8}}
=> ? = 5 + 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 5 + 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 5 + 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,7,6] => {{1},{2},{3},{4},{5},{6,8},{7}}
=> ? = 4 + 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [8,2,3,4,5,6,7,1] => {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 5 + 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [8,2,3,4,5,7,6,1] => {{1,8},{2},{3},{4},{5},{6,7}}
=> ? = 4 + 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [7,2,3,4,5,6,8,1] => {{1,7,8},{2},{3},{4},{5},{6}}
=> ? = 5 + 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [6,2,3,4,5,7,8,1] => {{1,6,7,8},{2},{3},{4},{5}}
=> ? = 5 + 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [6,2,3,4,5,8,7,1] => {{1,6,8},{2},{3},{4},{5},{7}}
=> ? = 4 + 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [7,2,3,4,5,6,1,8] => {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 4 + 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [6,2,3,4,5,7,1,8] => {{1,6,7},{2},{3},{4},{5},{8}}
=> ? = 4 + 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [6,2,3,4,5,1,7,8] => {{1,6},{2},{3},{4},{5},{7},{8}}
=> ? = 4 + 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> [6,2,3,4,5,1,8,7] => {{1,6},{2},{3},{4},{5},{7,8}}
=> ? = 3 + 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 5 + 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,5,7,6,8] => {{1,2},{3},{4},{5},{6,7},{8}}
=> ? = 4 + 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,6,8,7] => {{1,2},{3},{4},{5},{6},{7,8}}
=> ? = 4 + 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,3,4,5,7,8,6] => {{1,2},{3},{4},{5},{6,7,8}}
=> ? = 4 + 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,1,3,4,5,8,7,6] => {{1,2},{3},{4},{5},{6,8},{7}}
=> ? = 3 + 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,8,3,4,5,6,7,1] => {{1,2,8},{3},{4},{5},{6},{7}}
=> ? = 5 + 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [2,8,3,4,5,7,6,1] => {{1,2,8},{3},{4},{5},{6,7}}
=> ? = 4 + 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [2,7,3,4,5,6,8,1] => ?
=> ? = 5 + 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [2,6,3,4,5,7,8,1] => {{1,2,6,7,8},{3},{4},{5}}
=> ? = 5 + 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [2,6,3,4,5,8,7,1] => {{1,2,6,8},{3},{4},{5},{7}}
=> ? = 4 + 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [2,7,3,4,5,6,1,8] => {{1,2,7},{3},{4},{5},{6},{8}}
=> ? = 4 + 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [2,6,3,4,5,7,1,8] => ?
=> ? = 4 + 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [2,6,3,4,5,1,7,8] => {{1,2,6},{3},{4},{5},{7},{8}}
=> ? = 4 + 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [2,6,3,4,5,1,8,7] => ?
=> ? = 3 + 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,8,3,4,5,6,7,2] => {{1},{2,8},{3},{4},{5},{6},{7}}
=> ? = 4 + 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,8,3,4,5,7,6,2] => ?
=> ? = 3 + 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,7,3,4,5,6,8,2] => {{1},{2,7,8},{3},{4},{5},{6}}
=> ? = 4 + 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,6,3,4,5,7,8,2] => ?
=> ? = 4 + 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,6,3,4,5,8,7,2] => {{1},{2,6,8},{3},{4},{5},{7}}
=> ? = 3 + 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,7,3,4,5,6,2,8] => {{1},{2,7},{3},{4},{5},{6},{8}}
=> ? = 4 + 2
[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]
=> [1,6,3,4,5,7,2,8] => ?
=> ? = 4 + 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,6,3,4,5,2,7,8] => {{1},{2,6},{3},{4},{5},{7},{8}}
=> ? = 4 + 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [1,6,3,4,5,2,8,7] => ?
=> ? = 3 + 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [8,7,3,4,5,6,2,1] => {{1,8},{2,7},{3},{4},{5},{6}}
=> ? = 3 + 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [8,6,3,4,5,7,2,1] => ?
=> ? = 3 + 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
=> [8,6,3,4,5,2,7,1] => {{1,8},{2,6},{3},{4},{5},{7}}
=> ? = 3 + 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> [7,6,3,4,5,2,8,1] => {{1,7,8},{2,6},{3},{4},{5}}
=> ? = 3 + 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [7,6,3,4,5,2,1,8] => {{1,7},{2,6},{3},{4},{5},{8}}
=> ? = 2 + 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7,8] => {{1},{2,3},{4},{5},{6},{7},{8}}
=> ? = 5 + 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,5,7,6,8] => {{1},{2,3},{4},{5},{6,7},{8}}
=> ? = 4 + 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,6,8,7] => {{1},{2,3},{4},{5},{6},{7,8}}
=> ? = 4 + 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,5,7,8,6] => {{1},{2,3},{4},{5},{6,7,8}}
=> ? = 4 + 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,5,8,7,6] => {{1},{2,3},{4},{5},{6,8},{7}}
=> ? = 3 + 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [8,3,2,4,5,6,7,1] => {{1,8},{2,3},{4},{5},{6},{7}}
=> ? = 4 + 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> [8,3,2,4,5,7,6,1] => ?
=> ? = 3 + 2
Description
The number of singletons ([[St000247]]) plus the number of antisingletons ([[St000248]]) of a set partition.
The following 50 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000021The number of descents of a permutation. St000204The number of internal nodes of a binary tree. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000155The number of exceedances (also excedences) of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000312The number of leaves in a graph. St000711The number of big exceedences of a permutation. St000619The number of cyclic descents of a permutation. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. 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. St000871The number of very big ascents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000445The number of rises of length 1 of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001083The number of boxed occurrences of 132 in a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001427The number of descents of a signed permutation. St000015The number of peaks of a Dyck path. St000236The number of cyclical small weak excedances. St000443The number of long tunnels of a Dyck path. St000824The sum of the number of descents and the number of recoils of a permutation. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001368The number of vertices of maximal degree in a graph. St001960The number of descents of a permutation minus one if its first entry is not one. St001964The interval resolution global dimension of a poset. St000731The number of double exceedences of a permutation. St000834The number of right outer peaks of a permutation. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St000215The number of adjacencies of a permutation, zero appended. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001520The number of strict 3-descents. St001570The minimal number of edges to add to make a graph Hamiltonian.
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