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Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St001688
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001688: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001688: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,0,1,0]
=> 2
[1,1,0,0]
=> [1,1,0,0]
=> 4
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 5
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 5
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 5
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 9
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 6
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 6
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 6
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 6
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 8
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 6
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 8
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 10
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 8
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 16
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 7
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 7
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 7
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 11
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 7
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 9
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 7
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 9
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 11
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 11
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 9
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 11
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 17
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 7
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 9
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 9
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 9
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 13
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 7
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 9
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 9
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 9
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 13
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 11
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 9
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 13
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 17
Description
The sum of the squares of the heights of the peaks of a Dyck path.
Matching statistic: St000721
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
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%
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%
Values
[1,0]
=> [1,0]
=> [(1,2)]
=> 1
[1,0,1,0]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 2
[1,1,0,0]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 4
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 3
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 5
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 5
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 5
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 9
[1,0,1,0,1,0,1,0]
=> [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]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 6
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 6
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 6
[1,0,1,1,1,0,0,0]
=> [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]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 6
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 8
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 6
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 8
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 10
[1,1,1,0,0,0,1,0]
=> [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]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 8
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 10
[1,1,1,1,0,0,0,0]
=> [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]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> 7
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> 7
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> 7
[1,0,1,0,1,1,1,0,0,0]
=> [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]
=> [1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> 7
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 9
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> 7
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 9
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> 11
[1,0,1,1,1,0,0,0,1,0]
=> [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]
=> [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 9
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> 11
[1,0,1,1,1,1,0,0,0,0]
=> [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]
=> [1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> 7
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> 9
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 9
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> 9
[1,1,0,0,1,1,1,0,0,0]
=> [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]
=> [1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> 7
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 9
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 9
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 9
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 13
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> 11
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 9
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 13
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 17
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: St001127
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001127: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001127: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1]
=> 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,2] => [1,1]
=> 2
[1,1,0,0]
=> [1,1,0,0]
=> [2,1] => [2]
=> 4
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 3
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 5
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 5
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 5
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => [3]
=> 9
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 6
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 6
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 6
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,1]
=> 10
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 6
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 8
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 6
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 8
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,1]
=> 10
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1]
=> 10
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 8
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,1]
=> 10
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4]
=> 16
[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] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 7
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 7
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 7
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> 11
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 7
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 9
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 7
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 9
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> 11
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> 11
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 9
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> 11
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,1]
=> 17
[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] => [2,1,1,1]
=> 7
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 9
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 9
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 9
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2]
=> 13
[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] => [2,1,1,1]
=> 7
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 9
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 9
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 9
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2]
=> 13
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> 11
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 9
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2]
=> 13
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,1]
=> 17
Description
The sum of the squares of the parts of a partition.
Matching statistic: St000059
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000059: Standard tableaux ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 82%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000059: Standard tableaux ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 82%
Values
[1,0]
=> [1,0]
=> [[1],[2]]
=> 1
[1,0,1,0]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[1,1,0,0]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 4
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3
[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]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 5
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 5
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 9
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 6
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 6
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 6
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 10
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 6
[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
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 6
[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
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 10
[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
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 8
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 10
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 16
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> 5
[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]]
=> 7
[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]]
=> 7
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> 7
[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]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> 7
[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]]
=> 9
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> 7
[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]]
=> 9
[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
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> 11
[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]]
=> 9
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> 11
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> 17
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> 7
[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]]
=> 9
[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]]
=> 9
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,7,8],[3,4,6,9,10]]
=> 9
[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]]
=> 13
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> 7
[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]]
=> 9
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[1,2,5,6,9],[3,4,7,8,10]]
=> 9
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> 9
[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]]
=> 13
[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]]
=> 11
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> 9
[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]]
=> 13
[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]]
=> 17
[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]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> ? ∊ {7,9,9,9,9,9,9,9,9,9,9,9,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,27,27,27,27,27,27,27,27,27,27,27,49}
[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]]
=> ? ∊ {7,9,9,9,9,9,9,9,9,9,9,9,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,27,27,27,27,27,27,27,27,27,27,27,49}
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,7,9,10,13],[2,4,6,8,11,12,14]]
=> ? ∊ {7,9,9,9,9,9,9,9,9,9,9,9,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,27,27,27,27,27,27,27,27,27,27,27,49}
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,11,12],[2,4,6,8,10,13,14]]
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=> ? ∊ {7,9,9,9,9,9,9,9,9,9,9,9,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,27,27,27,27,27,27,27,27,27,27,27,49}
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,8,11,13],[2,5,6,9,10,12,14]]
=> ? ∊ {7,9,9,9,9,9,9,9,9,9,9,9,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,27,27,27,27,27,27,27,27,27,27,27,49}
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8,11,12],[2,5,6,9,10,13,14]]
=> ? ∊ {7,9,9,9,9,9,9,9,9,9,9,9,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,27,27,27,27,27,27,27,27,27,27,27,49}
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [[1,3,4,7,9,10,13],[2,5,6,8,11,12,14]]
=> ? ∊ {7,9,9,9,9,9,9,9,9,9,9,9,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,27,27,27,27,27,27,27,27,27,27,27,49}
Description
The inversion number of a standard tableau as defined by Haglund and Stevens.
Their inversion number is the total number of inversion pairs for the tableau. An inversion pair is defined as a pair of cells (a,b), (x,y) such that the content of (x,y) is greater than the content of (a,b) and (x,y) is north of the inversion path of (a,b), where the inversion path is defined in detail in [1].
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