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Your data matches 7 different statistics following compositions of up to 3 maps.
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Matching statistic: St000721
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000721: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000721: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [(1,2)]
=> 1
[1,0,1,0]
=> [(1,2),(3,4)]
=> 2
[1,1,0,0]
=> [(1,4),(2,3)]
=> 4
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 3
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 5
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 5
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 7
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 9
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 4
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 6
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 6
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 8
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 10
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 6
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 8
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 8
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 10
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> 12
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 10
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> 12
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> 14
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 16
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> 7
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> 7
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> 9
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> 11
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> 7
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 9
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> 9
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> 11
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 13
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> 11
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> 13
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 15
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 17
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> 7
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> 9
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 9
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> 11
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 13
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> 9
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> 11
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> 11
[1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> 13
[1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> 15
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> 13
[1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> 15
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> 17
[1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> 19
Description
The sum of the partition sizes in the oscillating tableau corresponding to a perfect matching.
Sundaram's map sends a perfect matching on $1,\dots,2n$ to a oscillating tableau, a sequence of $n$ partitions, starting and ending with the empty partition and where two consecutive partitions differ by precisely one cell.
This statistic is the sum of the sizes of these partitions, called the weight of the perfect matching in [1].
Matching statistic: St000290
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 58%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 58%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 10 => 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1100 => 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1010 => 4
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 111000 => 3
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 101100 => 5
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 101100 => 5
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 110010 => 7
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 101010 => 9
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 4
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 6
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 6
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 8
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 10
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 6
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 8
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 8
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 10
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 12
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 10
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 12
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 14
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 16
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 7
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 7
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 9
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 11
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 7
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 9
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 9
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 11
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 13
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 11
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 13
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 15
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 17
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 7
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 9
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 9
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 11
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 13
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 9
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 11
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 11
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => 13
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 15
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 13
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 15
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 17
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 19
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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]
=> 11111110000000 => ? = 7
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 10101111100000 => ? = 13
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 11100011110000 => ? = 13
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 10101111100000 => ? = 13
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 10110011110000 => ? = 15
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> 11001011110000 => ? = 17
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10101011110000 => ? = 19
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 10110011110000 => ? = 15
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 11110000111000 => ? = 15
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 10101111100000 => ? = 13
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 10110011110000 => ? = 15
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> 10111000111000 => ? = 17
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> 11001100111000 => ? = 19
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> 11100010111000 => ? = 21
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> 10110010111000 => ? = 23
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10101011110000 => ? = 19
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> 10101100111000 => ? = 21
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> 11001010111000 => ? = 25
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 10101010111000 => ? = 27
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 10110011110000 => ? = 15
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 11100011110000 => ? = 13
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 10110011110000 => ? = 15
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> 10101100111000 => ? = 21
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 10101011001100 => ? = 29
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 10101111100000 => ? = 13
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 10110011110000 => ? = 15
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 10110011110000 => ? = 15
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> 11001100111000 => ? = 19
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> 10111100001100 => ? = 19
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> 11001110001100 => ? = 21
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> 11100011001100 => ? = 23
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 10110011001100 => ? = 25
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> 11001100101100 => ? = 29
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> 10101100101100 => ? = 31
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10101011110000 => ? = 19
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> 10101100111000 => ? = 21
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> 10101110001100 => ? = 23
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> 11001100101100 => ? = 29
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> 11001011001100 => ? = 27
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> 11001100101100 => ? = 29
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> 11100010101100 => ? = 31
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> 10110010101100 => ? = 33
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 10101010111000 => ? = 27
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 10101011001100 => ? = 29
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> 10101100101100 => ? = 31
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> 10110010101100 => ? = 33
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> 11001010101100 => ? = 35
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => ? = 37
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 10110011110000 => ? = 15
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 11100011110000 => ? = 13
Description
The major index of a binary word.
This is the sum of the positions of descents, i.e., a one followed by a zero.
For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
Matching statistic: St000330
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 58%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 58%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [[1],[2]]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 4
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 3
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 5
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 5
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 7
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 9
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 6
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 6
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 8
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 10
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 6
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 8
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 8
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 10
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 12
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 10
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 12
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 14
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 16
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> 7
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> 7
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> 9
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> 11
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> 7
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> 9
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> 9
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> 11
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> 13
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> 11
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> 13
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,7,8],[3,4,6,9,10]]
=> 15
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> 17
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> 7
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> 9
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> 9
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> 11
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> 13
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> 9
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> 11
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> 11
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> 13
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> 15
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> 13
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> 15
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[1,2,5,6,9],[3,4,7,8,10]]
=> 17
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> 19
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 7
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,5,6,7,8,9],[2,4,10,11,12,13,14]]
=> ? = 13
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [[1,2,3,7,8,9,10],[4,5,6,11,12,13,14]]
=> ? = 13
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,5,6,7,8,9],[2,4,10,11,12,13,14]]
=> ? = 13
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,7,8,9,10],[2,5,6,11,12,13,14]]
=> ? = 15
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,2,5,7,8,9,10],[3,4,6,11,12,13,14]]
=> ? = 17
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,3,5,7,8,9,10],[2,4,6,11,12,13,14]]
=> ? = 19
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,7,8,9,10],[2,5,6,11,12,13,14]]
=> ? = 15
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [[1,2,3,4,9,10,11],[5,6,7,8,12,13,14]]
=> ? = 15
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,5,6,7,8,9],[2,4,10,11,12,13,14]]
=> ? = 13
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,7,8,9,10],[2,5,6,11,12,13,14]]
=> ? = 15
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [[1,3,4,5,9,10,11],[2,6,7,8,12,13,14]]
=> ? = 17
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,9,10,11],[3,4,7,8,12,13,14]]
=> ? = 19
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [[1,2,3,7,9,10,11],[4,5,6,8,12,13,14]]
=> ? = 21
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [[1,3,4,7,9,10,11],[2,5,6,8,12,13,14]]
=> ? = 23
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,3,5,7,8,9,10],[2,4,6,11,12,13,14]]
=> ? = 19
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [[1,3,5,6,9,10,11],[2,4,7,8,12,13,14]]
=> ? = 21
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,2,5,7,9,10,11],[3,4,6,8,12,13,14]]
=> ? = 25
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,7,9,10,11],[2,4,6,8,12,13,14]]
=> ? = 27
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,7,8,9,10],[2,5,6,11,12,13,14]]
=> ? = 15
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [[1,2,3,7,8,9,10],[4,5,6,11,12,13,14]]
=> ? = 13
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,7,8,9,10],[2,5,6,11,12,13,14]]
=> ? = 15
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [[1,3,5,6,9,10,11],[2,4,7,8,12,13,14]]
=> ? = 21
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,3,5,7,8,11,12],[2,4,6,9,10,13,14]]
=> ? = 29
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,5,6,7,8,9],[2,4,10,11,12,13,14]]
=> ? = 13
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,7,8,9,10],[2,5,6,11,12,13,14]]
=> ? = 15
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,7,8,9,10],[2,5,6,11,12,13,14]]
=> ? = 15
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,9,10,11],[3,4,7,8,12,13,14]]
=> ? = 19
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [[1,3,4,5,6,11,12],[2,7,8,9,10,13,14]]
=> ? = 19
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [[1,2,5,6,7,11,12],[3,4,8,9,10,13,14]]
=> ? = 21
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [[1,2,3,7,8,11,12],[4,5,6,9,10,13,14]]
=> ? = 23
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8,11,12],[2,5,6,9,10,13,14]]
=> ? = 25
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,6,9,11,12],[3,4,7,8,10,13,14]]
=> ? = 29
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,3,5,6,9,11,12],[2,4,7,8,10,13,14]]
=> ? = 31
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,3,5,7,8,9,10],[2,4,6,11,12,13,14]]
=> ? = 19
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [[1,3,5,6,9,10,11],[2,4,7,8,12,13,14]]
=> ? = 21
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [[1,3,5,6,7,11,12],[2,4,8,9,10,13,14]]
=> ? = 23
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,6,9,11,12],[3,4,7,8,10,13,14]]
=> ? = 29
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,2,5,7,8,11,12],[3,4,6,9,10,13,14]]
=> ? = 27
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,6,9,11,12],[3,4,7,8,10,13,14]]
=> ? = 29
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [[1,2,3,7,9,11,12],[4,5,6,8,10,13,14]]
=> ? = 31
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [[1,3,4,7,9,11,12],[2,5,6,8,10,13,14]]
=> ? = 33
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,7,9,10,11],[2,4,6,8,12,13,14]]
=> ? = 27
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,3,5,7,8,11,12],[2,4,6,9,10,13,14]]
=> ? = 29
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,3,5,6,9,11,12],[2,4,7,8,10,13,14]]
=> ? = 31
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [[1,3,4,7,9,11,12],[2,5,6,8,10,13,14]]
=> ? = 33
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,2,5,7,9,11,12],[3,4,6,8,10,13,14]]
=> ? = 35
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,11,12],[2,4,6,8,10,13,14]]
=> ? = 37
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,7,8,9,10],[2,5,6,11,12,13,14]]
=> ? = 15
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [[1,2,3,7,8,9,10],[4,5,6,11,12,13,14]]
=> ? = 13
Description
The (standard) major index of a standard tableau.
A descent of a standard tableau $T$ is an index $i$ such that $i+1$ appears in a row strictly below the row of $i$. The (standard) major index is the the sum of the descents.
Matching statistic: St000161
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
St000161: Binary trees ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 58%
Mp00008: Binary trees —to complete tree⟶ Ordered trees
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
St000161: Binary trees ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 58%
Values
[1,0]
=> [.,.]
=> [[],[]]
=> [.,[.,.]]
=> 1
[1,0,1,0]
=> [[.,.],.]
=> [[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> 2
[1,1,0,0]
=> [.,[.,.]]
=> [[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> 4
[1,0,1,0,1,0]
=> [[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> [[[.,[.,.]],[.,.]],[.,.]]
=> 3
[1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> [[.,[.,.]],[[.,[.,.]],.]]
=> 5
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> [[[],[[],[]]],[]]
=> [[.,[[.,[.,.]],.]],[.,.]]
=> 5
[1,1,0,1,0,0]
=> [.,[[.,.],.]]
=> [[],[[[],[]],[]]]
=> [.,[[[.,[.,.]],[.,.]],.]]
=> 7
[1,1,1,0,0,0]
=> [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> [.,[[.,[[.,[.,.]],.]],.]]
=> 9
[1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> [[[[[],[]],[]],[]],[]]
=> [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> 4
[1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> [[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> 6
[1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> [[[[],[]],[[],[]]],[]]
=> [[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
=> 6
[1,0,1,1,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> [[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> 8
[1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> [[.,[.,.]],[[.,[[.,[.,.]],.]],.]]
=> 10
[1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> [[[[],[[],[]]],[]],[]]
=> [[[.,[[.,[.,.]],.]],[.,.]],[.,.]]
=> 6
[1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> [[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
=> 8
[1,1,0,1,0,0,1,0]
=> [[.,[[.,.],.]],.]
=> [[[],[[[],[]],[]]],[]]
=> [[.,[[[.,[.,.]],[.,.]],.]],[.,.]]
=> 8
[1,1,0,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> [[],[[[[],[]],[]],[]]]
=> [.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> 10
[1,1,0,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> [.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> 12
[1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> [[[],[[],[[],[]]]],[]]
=> [[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
=> 10
[1,1,1,0,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> [[],[[[],[[],[]]],[]]]
=> [.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> 12
[1,1,1,0,1,0,0,0]
=> [.,[.,[[.,.],.]]]
=> [[],[[],[[[],[]],[]]]]
=> [.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> 14
[1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> 16
[1,0,1,0,1,0,1,0,1,0]
=> [[[[[.,.],.],.],.],.]
=> [[[[[[],[]],[]],[]],[]],[]]
=> [[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,.]]
=> [[[[[],[]],[]],[]],[[],[]]]
=> [[[[.,[.,.]],[.,.]],[.,.]],[[.,[.,.]],.]]
=> 7
[1,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],.]
=> [[[[[],[]],[]],[[],[]]],[]]
=> [[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],[.,.]]
=> 7
[1,0,1,0,1,1,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> [[[[],[]],[]],[[[],[]],[]]]
=> [[[.,[.,.]],[.,.]],[[[.,[.,.]],[.,.]],.]]
=> 9
[1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> [[[[],[]],[]],[[],[[],[]]]]
=> [[[.,[.,.]],[.,.]],[[.,[[.,[.,.]],.]],.]]
=> 11
[1,0,1,1,0,0,1,0,1,0]
=> [[[[.,.],[.,.]],.],.]
=> [[[[[],[]],[[],[]]],[]],[]]
=> [[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],[.,.]]
=> 7
[1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> [[[[],[]],[[],[]]],[[],[]]]
=> [[[.,[.,.]],[[.,[.,.]],.]],[[.,[.,.]],.]]
=> 9
[1,0,1,1,0,1,0,0,1,0]
=> [[[.,.],[[.,.],.]],.]
=> [[[[],[]],[[[],[]],[]]],[]]
=> [[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],[.,.]]
=> 9
[1,0,1,1,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> [[[],[]],[[[[],[]],[]],[]]]
=> [[.,[.,.]],[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> 11
[1,0,1,1,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> [[[],[]],[[[],[]],[[],[]]]]
=> [[.,[.,.]],[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> 13
[1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> [[[[],[]],[[],[[],[]]]],[]]
=> [[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],[.,.]]
=> 11
[1,0,1,1,1,0,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> [[[],[]],[[[],[[],[]]],[]]]
=> [[.,[.,.]],[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> 13
[1,0,1,1,1,0,1,0,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> [[[],[]],[[],[[[],[]],[]]]]
=> [[.,[.,.]],[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> 15
[1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> [[.,[.,.]],[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> 17
[1,1,0,0,1,0,1,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> [[[[[],[[],[]]],[]],[]],[]]
=> [[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],[.,.]]
=> 7
[1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> [[[[],[[],[]]],[]],[[],[]]]
=> [[[.,[[.,[.,.]],.]],[.,.]],[[.,[.,.]],.]]
=> 9
[1,1,0,0,1,1,0,0,1,0]
=> [[[.,[.,.]],[.,.]],.]
=> [[[[],[[],[]]],[[],[]]],[]]
=> [[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],[.,.]]
=> 9
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> [[[],[[],[]]],[[[],[]],[]]]
=> [[.,[[.,[.,.]],.]],[[[.,[.,.]],[.,.]],.]]
=> 11
[1,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[.,[.,.]]]
=> [[[],[[],[]]],[[],[[],[]]]]
=> [[.,[[.,[.,.]],.]],[[.,[[.,[.,.]],.]],.]]
=> 13
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[[.,.],.]],.],.]
=> [[[[],[[[],[]],[]]],[]],[]]
=> [[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],[.,.]]
=> 9
[1,1,0,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> [[[],[[[],[]],[]]],[[],[]]]
=> [[.,[[[.,[.,.]],[.,.]],.]],[[.,[.,.]],.]]
=> 11
[1,1,0,1,0,1,0,0,1,0]
=> [[.,[[[.,.],.],.]],.]
=> [[[],[[[[],[]],[]],[]]],[]]
=> [[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],[.,.]]
=> 11
[1,1,0,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [[],[[[[[],[]],[]],[]],[]]]
=> [.,[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],.]]
=> 13
[1,1,0,1,0,1,1,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> [[],[[[[],[]],[]],[[],[]]]]
=> [.,[[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],.]]
=> 15
[1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],[.,.]]],.]
=> [[[],[[[],[]],[[],[]]]],[]]
=> [[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],[.,.]]
=> 13
[1,1,0,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [[],[[[[],[]],[[],[]]],[]]]
=> [.,[[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],.]]
=> 15
[1,1,0,1,1,0,1,0,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> [[],[[[],[]],[[[],[]],[]]]]
=> [.,[[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],.]]
=> 17
[1,1,0,1,1,1,0,0,0,0]
=> [.,[[.,.],[.,[.,.]]]]
=> [[],[[[],[]],[[],[[],[]]]]]
=> [.,[[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],.]]
=> 19
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> [[[[[[[[],[]],[]],[]],[]],[]],[]],[]]
=> ?
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[[[[[.,.],.],.],.],.],[.,.]]
=> [[[[[[[],[]],[]],[]],[]],[]],[[],[]]]
=> ?
=> ? = 9
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [[[[[[.,.],.],.],.],[.,.]],.]
=> [[[[[[[],[]],[]],[]],[]],[[],[]]],[]]
=> ?
=> ? = 9
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[[[[.,.],.],.],.],[[.,.],.]]
=> [[[[[[],[]],[]],[]],[]],[[[],[]],[]]]
=> ?
=> ? = 11
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[[[[.,.],.],.],.],[.,[.,.]]]
=> [[[[[[],[]],[]],[]],[]],[[],[[],[]]]]
=> ?
=> ? = 13
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [[[[[[.,.],.],.],[.,.]],.],.]
=> [[[[[[[],[]],[]],[]],[[],[]]],[]],[]]
=> ?
=> ? = 9
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[[[[.,.],.],.],[.,.]],[.,.]]
=> [[[[[[],[]],[]],[]],[[],[]]],[[],[]]]
=> ?
=> ? = 11
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],[[[.,.],.],.]]
=> [[[[[],[]],[]],[]],[[[[],[]],[]],[]]]
=> ?
=> ? = 13
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [[[[[.,.],.],.],[.,[.,.]]],.]
=> [[[[[[],[]],[]],[]],[[],[[],[]]]],[]]
=> ?
=> ? = 13
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [[[[.,.],.],.],[[.,[.,.]],.]]
=> [[[[[],[]],[]],[]],[[[],[[],[]]],[]]]
=> ?
=> ? = 15
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [[[[.,.],.],.],[.,[[.,.],.]]]
=> [[[[[],[]],[]],[]],[[],[[[],[]],[]]]]
=> ?
=> ? = 17
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [[[[[],[]],[]],[]],[[],[[],[[],[]]]]]
=> ?
=> ? = 19
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[[[[[.,.],.],[.,.]],.],.],.]
=> [[[[[[[],[]],[]],[[],[]]],[]],[]],[]]
=> ?
=> ? = 9
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [[[[[.,.],.],[.,.]],.],[.,.]]
=> [[[[[[],[]],[]],[[],[]]],[]],[[],[]]]
=> ?
=> ? = 11
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [[[[[.,.],.],[.,.]],[.,.]],.]
=> [[[[[[],[]],[]],[[],[]]],[[],[]]],[]]
=> ?
=> ? = 11
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [[[[[],[]],[]],[[],[]]],[[],[[],[]]]]
=> ?
=> ? = 15
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[[.,.],.],[[[[.,.],.],.],.]]
=> [[[[],[]],[]],[[[[[],[]],[]],[]],[]]]
=> ?
=> ? = 15
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [[[[[.,.],.],[.,[.,.]]],.],.]
=> [[[[[[],[]],[]],[[],[[],[]]]],[]],[]]
=> ?
=> ? = 13
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [[[[.,.],.],[.,[.,.]]],[.,.]]
=> [[[[[],[]],[]],[[],[[],[]]]],[[],[]]]
=> ?
=> ? = 15
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [[[.,.],.],[[[.,[.,.]],.],.]]
=> [[[[],[]],[]],[[[[],[[],[]]],[]],[]]]
=> ?
=> ? = 17
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [[[.,.],.],[[.,[[.,.],.]],.]]
=> [[[[],[]],[]],[[[],[[[],[]],[]]],[]]]
=> ?
=> ? = 19
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [[[.,.],.],[.,[[[.,.],.],.]]]
=> [[[[],[]],[]],[[],[[[[],[]],[]],[]]]]
=> ?
=> ? = 21
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [[[.,.],.],[.,[[.,.],[.,.]]]]
=> [[[[],[]],[]],[[],[[[],[]],[[],[]]]]]
=> ?
=> ? = 23
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],[.,[.,[.,.]]]],.]
=> [[[[[],[]],[]],[[],[[],[[],[]]]]],[]]
=> ?
=> ? = 19
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,[.,[.,.]]],.]]
=> [[[[],[]],[]],[[[],[[],[[],[]]]],[]]]
=> ?
=> ? = 21
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [[[.,.],.],[.,[.,[[.,.],.]]]]
=> [[[[],[]],[]],[[],[[],[[[],[]],[]]]]]
=> ?
=> ? = 25
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [[[[],[]],[]],[[],[[],[[],[[],[]]]]]]
=> ?
=> ? = 27
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [[[[[[.,.],[.,.]],.],.],.],.]
=> [[[[[[[],[]],[[],[]]],[]],[]],[]],[]]
=> ?
=> ? = 9
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [[[[[.,.],[.,.]],.],.],[.,.]]
=> [[[[[[],[]],[[],[]]],[]],[]],[[],[]]]
=> ?
=> ? = 11
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [[[[[.,.],[.,.]],.],[.,.]],.]
=> [[[[[[],[]],[[],[]]],[]],[[],[]]],[]]
=> ?
=> ? = 11
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],[.,[.,.]]]
=> [[[[[],[]],[[],[]]],[]],[[],[[],[]]]]
=> ?
=> ? = 15
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [[[[[.,.],[.,.]],[.,.]],.],.]
=> [[[[[[],[]],[[],[]]],[[],[]]],[]],[]]
=> ?
=> ? = 11
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [[[[.,.],[.,.]],[.,.]],[.,.]]
=> [[[[[],[]],[[],[]]],[[],[]]],[[],[]]]
=> ?
=> ? = 13
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [[[[[],[]],[[],[]]],[[],[[],[]]]],[]]
=> ?
=> ? = 15
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [[[[],[]],[[],[]]],[[],[[],[[],[]]]]]
=> ?
=> ? = 21
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[[.,.],.],.],.],.]]
=> [[[],[]],[[[[[[],[]],[]],[]],[]],[]]]
=> ?
=> ? = 17
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [[[],[]],[[[],[]],[[],[[],[[],[]]]]]]
=> ?
=> ? = 29
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [[[[[.,.],[.,[.,.]]],.],.],.]
=> [[[[[[],[]],[[],[[],[]]]],[]],[]],[]]
=> ?
=> ? = 13
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [[[[.,.],[.,[.,.]]],.],[.,.]]
=> [[[[[],[]],[[],[[],[]]]],[]],[[],[]]]
=> ?
=> ? = 15
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [[[[.,.],[.,[.,.]]],[.,.]],.]
=> [[[[[],[]],[[],[[],[]]]],[[],[]]],[]]
=> ?
=> ? = 15
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [[[.,.],[.,[.,.]]],[.,[.,.]]]
=> [[[[],[]],[[],[[],[]]]],[[],[[],[]]]]
=> ?
=> ? = 19
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[.,[.,.]],.],.],.]]
=> [[[],[]],[[[[[],[[],[]]],[]],[]],[]]]
=> ?
=> ? = 19
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,[[.,.],.]],.],.]]
=> [[[],[]],[[[[],[[[],[]],[]]],[]],[]]]
=> ?
=> ? = 21
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [[.,.],[[.,[[[.,.],.],.]],.]]
=> [[[],[]],[[[],[[[[],[]],[]],[]]],[]]]
=> ?
=> ? = 23
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [[.,.],[[.,[[.,.],[.,.]]],.]]
=> [[[],[]],[[[],[[[],[]],[[],[]]]],[]]]
=> ?
=> ? = 25
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [[[],[]],[[],[[[],[]],[[[],[]],[]]]]]
=> ?
=> ? = 29
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [[[],[]],[[],[[[],[]],[[],[[],[]]]]]]
=> ?
=> ? = 31
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [[[[[],[]],[[],[[],[[],[]]]]],[]],[]]
=> ?
=> ? = 19
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [[[.,.],[.,[.,[.,.]]]],[.,.]]
=> [[[[],[]],[[],[[],[[],[]]]]],[[],[]]]
=> ?
=> ? = 21
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [[.,.],[[[.,[.,[.,.]]],.],.]]
=> [[[],[]],[[[[],[[],[[],[]]]],[]],[]]]
=> ?
=> ? = 23
Description
The sum of the sizes of the right subtrees of a binary tree.
This statistic corresponds to [[St000012]] under the Tamari Dyck path-binary tree bijection, and to [[St000018]] of the $312$-avoiding permutation corresponding to the binary tree.
It is also the sum of all heights $j$ of the coordinates $(i,j)$ of the Dyck path corresponding to the binary tree.
Matching statistic: St000018
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 28%
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 28%
Values
[1,0]
=> [(1,2)]
=> [(1,2)]
=> [2,1] => 1
[1,0,1,0]
=> [(1,2),(3,4)]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 2
[1,1,0,0]
=> [(1,4),(2,3)]
=> [(1,3),(2,4)]
=> [3,4,1,2] => 4
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => 5
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => 5
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => 7
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => 9
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 4
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [(1,2),(3,4),(5,7),(6,8)]
=> [2,1,4,3,7,8,5,6] => 6
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => 6
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [(1,2),(3,5),(4,7),(6,8)]
=> [2,1,5,7,3,8,4,6] => 8
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [(1,2),(3,6),(4,7),(5,8)]
=> [2,1,6,7,8,3,4,5] => 10
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => 6
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [(1,3),(2,4),(5,7),(6,8)]
=> [3,4,1,2,7,8,5,6] => 8
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => 8
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [(1,3),(2,5),(4,7),(6,8)]
=> [3,5,1,7,2,8,4,6] => 10
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [(1,3),(2,6),(4,7),(5,8)]
=> [3,6,1,7,8,2,4,5] => 12
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => 10
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [(1,4),(2,5),(3,7),(6,8)]
=> [4,5,7,1,2,8,3,6] => 12
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [(1,4),(2,6),(3,7),(5,8)]
=> [4,6,7,1,8,2,3,5] => 14
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [(1,5),(2,6),(3,7),(4,8)]
=> [5,6,7,8,1,2,3,4] => 16
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [(1,2),(3,4),(5,6),(7,9),(8,10)]
=> [2,1,4,3,6,5,9,10,7,8] => ? = 7
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [(1,2),(3,4),(5,7),(6,8),(9,10)]
=> [2,1,4,3,7,8,5,6,10,9] => ? = 7
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [(1,2),(3,4),(5,7),(6,9),(8,10)]
=> [2,1,4,3,7,9,5,10,6,8] => ? = 9
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [(1,2),(3,4),(5,8),(6,9),(7,10)]
=> [2,1,4,3,8,9,10,5,6,7] => ? = 11
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [(1,2),(3,5),(4,6),(7,8),(9,10)]
=> [2,1,5,6,3,4,8,7,10,9] => ? = 7
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [(1,2),(3,5),(4,6),(7,9),(8,10)]
=> [2,1,5,6,3,4,9,10,7,8] => ? = 9
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [(1,2),(3,5),(4,7),(6,8),(9,10)]
=> [2,1,5,7,3,8,4,6,10,9] => ? = 9
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [(1,2),(3,5),(4,7),(6,9),(8,10)]
=> [2,1,5,7,3,9,4,10,6,8] => ? = 11
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [(1,2),(3,5),(4,8),(6,9),(7,10)]
=> [2,1,5,8,3,9,10,4,6,7] => ? = 13
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [(1,2),(3,6),(4,7),(5,8),(9,10)]
=> [2,1,6,7,8,3,4,5,10,9] => ? = 11
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [(1,2),(3,6),(4,7),(5,9),(8,10)]
=> [2,1,6,7,9,3,4,10,5,8] => ? = 13
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [(1,2),(3,6),(4,8),(5,9),(7,10)]
=> [2,1,6,8,9,3,10,4,5,7] => ? = 15
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [(1,2),(3,7),(4,8),(5,9),(6,10)]
=> [2,1,7,8,9,10,3,4,5,6] => ? = 17
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [3,4,1,2,6,5,8,7,10,9] => ? = 7
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [(1,3),(2,4),(5,6),(7,9),(8,10)]
=> [3,4,1,2,6,5,9,10,7,8] => ? = 9
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [(1,3),(2,4),(5,7),(6,8),(9,10)]
=> [3,4,1,2,7,8,5,6,10,9] => ? = 9
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [(1,3),(2,4),(5,7),(6,9),(8,10)]
=> [3,4,1,2,7,9,5,10,6,8] => ? = 11
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [(1,3),(2,4),(5,8),(6,9),(7,10)]
=> [3,4,1,2,8,9,10,5,6,7] => ? = 13
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [(1,3),(2,5),(4,6),(7,8),(9,10)]
=> [3,5,1,6,2,4,8,7,10,9] => ? = 9
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [(1,3),(2,5),(4,6),(7,9),(8,10)]
=> [3,5,1,6,2,4,9,10,7,8] => ? = 11
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [(1,3),(2,5),(4,7),(6,8),(9,10)]
=> [3,5,1,7,2,8,4,6,10,9] => ? = 11
[1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [(1,3),(2,5),(4,7),(6,9),(8,10)]
=> [3,5,1,7,2,9,4,10,6,8] => ? = 13
[1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [(1,3),(2,5),(4,8),(6,9),(7,10)]
=> [3,5,1,8,2,9,10,4,6,7] => ? = 15
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [(1,3),(2,6),(4,7),(5,8),(9,10)]
=> [3,6,1,7,8,2,4,5,10,9] => ? = 13
[1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [(1,3),(2,6),(4,7),(5,9),(8,10)]
=> [3,6,1,7,9,2,4,10,5,8] => ? = 15
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [(1,3),(2,6),(4,8),(5,9),(7,10)]
=> [3,6,1,8,9,2,10,4,5,7] => ? = 17
[1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> [(1,3),(2,7),(4,8),(5,9),(6,10)]
=> [3,7,1,8,9,10,2,4,5,6] => ? = 19
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [(1,4),(2,5),(3,6),(7,8),(9,10)]
=> [4,5,6,1,2,3,8,7,10,9] => ? = 11
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [(1,4),(2,5),(3,6),(7,9),(8,10)]
=> [4,5,6,1,2,3,9,10,7,8] => ? = 13
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [(1,4),(2,5),(3,7),(6,8),(9,10)]
=> [4,5,7,1,2,8,3,6,10,9] => ? = 13
[1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [(1,4),(2,5),(3,7),(6,9),(8,10)]
=> [4,5,7,1,2,9,3,10,6,8] => ? = 15
[1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> [(1,4),(2,5),(3,8),(6,9),(7,10)]
=> [4,5,8,1,2,9,10,3,6,7] => ? = 17
[1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [(1,4),(2,6),(3,7),(5,8),(9,10)]
=> [4,6,7,1,8,2,3,5,10,9] => ? = 15
[1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [(1,4),(2,6),(3,7),(5,9),(8,10)]
=> [4,6,7,1,9,2,3,10,5,8] => ? = 17
[1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [(1,4),(2,6),(3,8),(5,9),(7,10)]
=> [4,6,8,1,9,2,10,3,5,7] => ? = 19
[1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [(1,4),(2,7),(3,8),(5,9),(6,10)]
=> [4,7,8,1,9,10,2,3,5,6] => ? = 21
[1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [(1,5),(2,6),(3,7),(4,8),(9,10)]
=> [5,6,7,8,1,2,3,4,10,9] => ? = 17
[1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [(1,5),(2,6),(3,7),(4,9),(8,10)]
=> [5,6,7,9,1,2,3,10,4,8] => ? = 19
[1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [(1,5),(2,6),(3,8),(4,9),(7,10)]
=> [5,6,8,9,1,2,10,3,4,7] => ? = 21
[1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [(1,5),(2,7),(3,8),(4,9),(6,10)]
=> [5,7,8,9,1,10,2,3,4,6] => ? = 23
[1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [(1,6),(2,7),(3,8),(4,9),(5,10)]
=> [6,7,8,9,10,1,2,3,4,5] => 25
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 6
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]
=> [2,1,4,3,6,5,8,7,11,12,9,10] => ? = 8
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> [(1,2),(3,4),(5,6),(7,9),(8,10),(11,12)]
=> [2,1,4,3,6,5,9,10,7,8,12,11] => ? = 8
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> [(1,2),(3,4),(5,6),(7,9),(8,11),(10,12)]
=> [2,1,4,3,6,5,9,11,7,12,8,10] => ? = 10
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> [(1,2),(3,4),(5,6),(7,10),(8,11),(9,12)]
=> [2,1,4,3,6,5,10,11,12,7,8,9] => ? = 12
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> [(1,2),(3,4),(5,7),(6,8),(9,10),(11,12)]
=> [2,1,4,3,7,8,5,6,10,9,12,11] => ? = 8
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> [(1,2),(3,4),(5,7),(6,8),(9,11),(10,12)]
=> [2,1,4,3,7,8,5,6,11,12,9,10] => ? = 10
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]
=> [(1,2),(3,4),(5,7),(6,9),(8,10),(11,12)]
=> [2,1,4,3,7,9,5,10,6,8,12,11] => ? = 10
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> [(1,2),(3,4),(5,7),(6,9),(8,11),(10,12)]
=> [2,1,4,3,7,9,5,11,6,12,8,10] => ? = 12
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,11),(9,10)]
=> [(1,2),(3,4),(5,7),(6,10),(8,11),(9,12)]
=> [2,1,4,3,7,10,5,11,12,6,8,9] => ? = 14
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
=> [(1,2),(3,4),(5,8),(6,9),(7,10),(11,12)]
=> [2,1,4,3,8,9,10,5,6,7,12,11] => ? = 12
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
=> [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12)]
=> [7,8,9,10,11,12,1,2,3,4,5,6] => 36
Description
The number of inversions of a permutation.
This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.
Matching statistic: St000394
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 26%
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 26%
Values
[1,0]
=> [(1,2)]
=> [2,1] => [1,1,0,0]
=> 1
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 4
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [1,1,0,0,1,1,1,0,1,0,0,0]
=> 5
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [1,1,1,0,1,0,0,0,1,1,0,0]
=> 5
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> 7
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> 9
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> 6
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> 6
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> 8
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => [1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> 10
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> 6
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> 8
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> 8
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> 10
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> 12
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]
=> 10
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> 12
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> 14
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> 16
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 7
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,7,8,6,5,10,9] => [1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> ? = 7
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,7,9,6,10,8,5] => [1,1,0,0,1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 9
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 11
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,5,6,4,3,8,7,10,9] => [1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 7
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [2,1,5,6,4,3,9,10,8,7] => [1,1,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 9
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,5,7,4,8,6,3,10,9] => [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 9
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => [1,1,0,0,1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> ? = 11
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,5,8,4,9,10,7,6,3] => [1,1,0,0,1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 13
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,6,7,8,5,4,3,10,9] => [1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]
=> ? = 11
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => [1,1,0,0,1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 13
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => [1,1,0,0,1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 15
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => [1,1,0,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 17
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [3,4,2,1,6,5,8,7,10,9] => [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 7
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [3,4,2,1,6,5,9,10,8,7] => [1,1,1,0,1,0,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 9
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [3,4,2,1,7,8,6,5,10,9] => [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> ? = 9
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [3,4,2,1,7,9,6,10,8,5] => [1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 11
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [3,4,2,1,8,9,10,7,6,5] => [1,1,1,0,1,0,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 13
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [3,5,2,6,4,1,8,7,10,9] => [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 9
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [3,5,2,6,4,1,9,10,8,7] => [1,1,1,0,1,1,0,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 11
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [3,5,2,7,4,8,6,1,10,9] => [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 11
[1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [3,5,2,7,4,9,6,10,8,1] => [1,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> ? = 13
[1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [3,5,2,8,4,9,10,7,6,1] => [1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 15
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [3,6,2,7,8,5,4,1,10,9] => [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0,1,1,0,0]
=> ? = 13
[1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [3,6,2,7,9,5,4,10,8,1] => [1,1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 15
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [3,6,2,8,9,5,10,7,4,1] => [1,1,1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 17
[1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> [3,7,2,8,9,10,6,5,4,1] => [1,1,1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 19
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => [1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 11
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [4,5,6,3,2,1,9,10,8,7] => [1,1,1,1,0,1,0,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 13
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> ? = 13
[1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [4,5,7,3,2,9,6,10,8,1] => [1,1,1,1,0,1,0,1,1,0,0,0,1,1,0,0,1,0,0,0]
=> ? = 15
[1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> [4,5,8,3,2,9,10,7,6,1] => [1,1,1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> ? = 17
[1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,2,1,10,9] => [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0,1,1,0,0]
=> ? = 15
[1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,2,10,8,1] => [1,1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,1,0,0,0]
=> ? = 17
[1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,2,1] => [1,1,1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 19
[1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [4,7,8,3,9,10,6,5,2,1] => [1,1,1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0,0,0]
=> ? = 21
[1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [5,6,7,8,4,3,2,1,10,9] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,1,0,0]
=> ? = 17
[1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [5,6,7,9,4,3,2,10,8,1] => [1,1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0,0,0]
=> ? = 19
[1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [5,6,8,9,4,3,10,7,2,1] => [1,1,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0,0,0]
=> ? = 21
[1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [5,7,8,9,4,10,6,3,2,1] => [1,1,1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0,0,0]
=> ? = 23
[1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [6,7,8,9,10,5,4,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> 25
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 6
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [2,1,4,3,6,5,8,7,11,12,10,9] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 8
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> [2,1,4,3,6,5,9,10,8,7,12,11] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> ? = 8
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> [2,1,4,3,6,5,9,11,8,12,10,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 10
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> [2,1,4,3,6,5,10,11,12,9,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 12
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> [2,1,4,3,7,8,6,5,10,9,12,11] => [1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 8
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> [2,1,4,3,7,8,6,5,11,12,10,9] => [1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 10
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]
=> [2,1,4,3,7,9,6,10,8,5,12,11] => [1,1,0,0,1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 10
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> [2,1,4,3,7,9,6,11,8,12,10,5] => [1,1,0,0,1,1,0,0,1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> ? = 12
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,11),(9,10)]
=> [2,1,4,3,7,10,6,11,12,9,8,5] => [1,1,0,0,1,1,0,0,1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 14
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000463
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000463: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 25%
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000463: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 25%
Values
[1,0]
=> [(1,2)]
=> [2,1] => [2,1] => 0 = 1 - 1
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
[1,1,0,0]
=> [(1,4),(2,3)]
=> [4,3,2,1] => [3,2,4,1] => 3 = 4 - 1
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [2,1,5,4,6,3] => 4 = 5 - 1
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [3,2,4,1,6,5] => 4 = 5 - 1
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [3,2,5,4,6,1] => 6 = 7 - 1
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [4,3,5,2,6,1] => 8 = 9 - 1
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,8,7,6,5] => [2,1,4,3,7,6,8,5] => 5 = 6 - 1
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [2,1,5,4,6,3,8,7] => 5 = 6 - 1
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [2,1,5,4,7,6,8,3] => 7 = 8 - 1
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => 9 = 10 - 1
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [3,2,4,1,6,5,8,7] => 5 = 6 - 1
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => [3,2,4,1,7,6,8,5] => 7 = 8 - 1
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [3,2,5,4,6,1,8,7] => 7 = 8 - 1
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [3,2,5,4,7,6,8,1] => 9 = 10 - 1
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [8,3,2,7,6,5,4,1] => [3,2,6,5,7,4,8,1] => 11 = 12 - 1
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [4,3,5,2,6,1,8,7] => 9 = 10 - 1
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [4,3,5,2,7,6,8,1] => 11 = 12 - 1
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [4,3,6,5,7,2,8,1] => 13 = 14 - 1
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => [5,4,6,3,7,2,8,1] => 15 = 16 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,10,9,8,7] => [2,1,4,3,6,5,9,8,10,7] => ? = 7 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,8,7,6,5,10,9] => [2,1,4,3,7,6,8,5,10,9] => ? = 7 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,10,7,6,9,8,5] => [2,1,4,3,7,6,9,8,10,5] => ? = 9 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,10,9,8,7,6,5] => [2,1,4,3,8,7,9,6,10,5] => ? = 11 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,6,5,4,3,8,7,10,9] => [2,1,5,4,6,3,8,7,10,9] => ? = 7 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [2,1,6,5,4,3,10,9,8,7] => [2,1,5,4,6,3,9,8,10,7] => ? = 9 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,8,5,4,7,6,3,10,9] => [2,1,5,4,7,6,8,3,10,9] => ? = 9 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,10,5,4,7,6,9,8,3] => [2,1,5,4,7,6,9,8,10,3] => ? = 11 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,10,5,4,9,8,7,6,3] => [2,1,5,4,8,7,9,6,10,3] => ? = 13 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,8,7,6,5,4,3,10,9] => [2,1,6,5,7,4,8,3,10,9] => ? = 11 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,10,7,6,5,4,9,8,3] => [2,1,6,5,7,4,9,8,10,3] => ? = 13 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,10,9,6,5,8,7,4,3] => [2,1,6,5,8,7,9,4,10,3] => ? = 15 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,10,9,8,7,6,5,4,3] => [2,1,7,6,8,5,9,4,10,3] => ? = 17 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [4,3,2,1,6,5,8,7,10,9] => [3,2,4,1,6,5,8,7,10,9] => ? = 7 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [4,3,2,1,6,5,10,9,8,7] => [3,2,4,1,6,5,9,8,10,7] => ? = 9 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [4,3,2,1,8,7,6,5,10,9] => [3,2,4,1,7,6,8,5,10,9] => ? = 9 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [4,3,2,1,10,7,6,9,8,5] => [3,2,4,1,7,6,9,8,10,5] => ? = 11 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [4,3,2,1,10,9,8,7,6,5] => [3,2,4,1,8,7,9,6,10,5] => ? = 13 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [6,3,2,5,4,1,8,7,10,9] => [3,2,5,4,6,1,8,7,10,9] => ? = 9 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [6,3,2,5,4,1,10,9,8,7] => [3,2,5,4,6,1,9,8,10,7] => ? = 11 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [8,3,2,5,4,7,6,1,10,9] => [3,2,5,4,7,6,8,1,10,9] => ? = 11 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [10,3,2,5,4,7,6,9,8,1] => [3,2,5,4,7,6,9,8,10,1] => ? = 13 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [10,3,2,5,4,9,8,7,6,1] => [3,2,5,4,8,7,9,6,10,1] => ? = 15 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [8,3,2,7,6,5,4,1,10,9] => [3,2,6,5,7,4,8,1,10,9] => ? = 13 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [10,3,2,7,6,5,4,9,8,1] => [3,2,6,5,7,4,9,8,10,1] => ? = 15 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [10,3,2,9,6,5,8,7,4,1] => [3,2,6,5,8,7,9,4,10,1] => ? = 17 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> [10,3,2,9,8,7,6,5,4,1] => [3,2,7,6,8,5,9,4,10,1] => ? = 19 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [6,5,4,3,2,1,8,7,10,9] => [4,3,5,2,6,1,8,7,10,9] => ? = 11 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [6,5,4,3,2,1,10,9,8,7] => [4,3,5,2,6,1,9,8,10,7] => ? = 13 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [8,5,4,3,2,7,6,1,10,9] => [4,3,5,2,7,6,8,1,10,9] => ? = 13 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [10,5,4,3,2,7,6,9,8,1] => [4,3,5,2,7,6,9,8,10,1] => ? = 15 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> [10,5,4,3,2,9,8,7,6,1] => [4,3,5,2,8,7,9,6,10,1] => ? = 17 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [8,7,4,3,6,5,2,1,10,9] => [4,3,6,5,7,2,8,1,10,9] => ? = 15 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [10,7,4,3,6,5,2,9,8,1] => [4,3,6,5,7,2,9,8,10,1] => ? = 17 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [10,9,4,3,6,5,8,7,2,1] => [4,3,6,5,8,7,9,2,10,1] => ? = 19 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [10,9,4,3,8,7,6,5,2,1] => [4,3,7,6,8,5,9,2,10,1] => ? = 21 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [8,7,6,5,4,3,2,1,10,9] => [5,4,6,3,7,2,8,1,10,9] => ? = 17 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [10,7,6,5,4,3,2,9,8,1] => [5,4,6,3,7,2,9,8,10,1] => ? = 19 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [10,9,6,5,4,3,8,7,2,1] => [5,4,6,3,8,7,9,2,10,1] => ? = 21 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [10,9,8,5,4,7,6,3,2,1] => [5,4,7,6,8,3,9,2,10,1] => ? = 23 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [10,9,8,7,6,5,4,3,2,1] => [6,5,7,4,8,3,9,2,10,1] => ? = 25 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,4,3,6,5,8,7,10,9,12,11] => ? = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [2,1,4,3,6,5,8,7,12,11,10,9] => [2,1,4,3,6,5,8,7,11,10,12,9] => ? = 8 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> [2,1,4,3,6,5,10,9,8,7,12,11] => [2,1,4,3,6,5,9,8,10,7,12,11] => ? = 8 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> [2,1,4,3,6,5,12,9,8,11,10,7] => [2,1,4,3,6,5,9,8,11,10,12,7] => ? = 10 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> [2,1,4,3,6,5,12,11,10,9,8,7] => [2,1,4,3,6,5,10,9,11,8,12,7] => ? = 12 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> [2,1,4,3,8,7,6,5,10,9,12,11] => [2,1,4,3,7,6,8,5,10,9,12,11] => ? = 8 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> [2,1,4,3,8,7,6,5,12,11,10,9] => [2,1,4,3,7,6,8,5,11,10,12,9] => ? = 10 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]
=> [2,1,4,3,10,7,6,9,8,5,12,11] => [2,1,4,3,7,6,9,8,10,5,12,11] => ? = 10 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> [2,1,4,3,12,7,6,9,8,11,10,5] => [2,1,4,3,7,6,9,8,11,10,12,5] => ? = 12 - 1
Description
The number of admissible inversions of a permutation.
Let $w = w_1,w_2,\dots,w_k$ be a word of length $k$ with distinct letters from $[n]$.
An admissible inversion of $w$ is a pair $(w_i,w_j)$ such that $1\leq i < j\leq k$ and $w_i > w_j$ that satisfies either of the following conditions:
$1 < i$ and $w_{i−1} < w_i$ or there is some $l$ such that $i < l < j$ and $w_i < w_l$.
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