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Your data matches 44 different statistics following compositions of up to 3 maps.
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Matching statistic: St001589
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(load all 4 compositions to match this statistic)
St001589: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> 1
[(1,2),(3,4)]
=> 1
[(1,3),(2,4)]
=> 1
[(1,4),(2,3)]
=> 2
[(1,2),(3,4),(5,6)]
=> 1
[(1,3),(2,4),(5,6)]
=> 1
[(1,4),(2,3),(5,6)]
=> 2
[(1,5),(2,3),(4,6)]
=> 2
[(1,6),(2,3),(4,5)]
=> 2
[(1,6),(2,4),(3,5)]
=> 2
[(1,5),(2,4),(3,6)]
=> 2
[(1,4),(2,5),(3,6)]
=> 1
[(1,3),(2,5),(4,6)]
=> 1
[(1,2),(3,5),(4,6)]
=> 1
[(1,2),(3,6),(4,5)]
=> 2
[(1,3),(2,6),(4,5)]
=> 2
[(1,4),(2,6),(3,5)]
=> 2
[(1,5),(2,6),(3,4)]
=> 2
[(1,6),(2,5),(3,4)]
=> 3
[(1,2),(3,4),(5,6),(7,8)]
=> 1
[(1,3),(2,4),(5,6),(7,8)]
=> 1
[(1,4),(2,3),(5,6),(7,8)]
=> 2
[(1,5),(2,3),(4,6),(7,8)]
=> 2
[(1,6),(2,3),(4,5),(7,8)]
=> 2
[(1,7),(2,3),(4,5),(6,8)]
=> 2
[(1,8),(2,3),(4,5),(6,7)]
=> 2
[(1,8),(2,4),(3,5),(6,7)]
=> 2
[(1,7),(2,4),(3,5),(6,8)]
=> 2
[(1,6),(2,4),(3,5),(7,8)]
=> 2
[(1,5),(2,4),(3,6),(7,8)]
=> 2
[(1,4),(2,5),(3,6),(7,8)]
=> 1
[(1,3),(2,5),(4,6),(7,8)]
=> 1
[(1,2),(3,5),(4,6),(7,8)]
=> 1
[(1,2),(3,6),(4,5),(7,8)]
=> 2
[(1,3),(2,6),(4,5),(7,8)]
=> 2
[(1,4),(2,6),(3,5),(7,8)]
=> 2
[(1,5),(2,6),(3,4),(7,8)]
=> 2
[(1,6),(2,5),(3,4),(7,8)]
=> 3
[(1,7),(2,5),(3,4),(6,8)]
=> 3
[(1,8),(2,5),(3,4),(6,7)]
=> 3
[(1,8),(2,6),(3,4),(5,7)]
=> 3
[(1,7),(2,6),(3,4),(5,8)]
=> 3
[(1,6),(2,7),(3,4),(5,8)]
=> 2
[(1,5),(2,7),(3,4),(6,8)]
=> 2
[(1,4),(2,7),(3,5),(6,8)]
=> 2
[(1,3),(2,7),(4,5),(6,8)]
=> 2
[(1,2),(3,7),(4,5),(6,8)]
=> 2
[(1,2),(3,8),(4,5),(6,7)]
=> 2
[(1,3),(2,8),(4,5),(6,7)]
=> 2
[(1,4),(2,8),(3,5),(6,7)]
=> 2
Description
The nesting number of a perfect matching.
This is the maximal number of chords in the standard representation of a perfect matching that mutually nest.
Matching statistic: St001590
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(load all 8 compositions to match this statistic)
Mp00165: Perfect matchings —Chen Deng Du Stanley Yan⟶ Perfect matchings
St001590: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001590: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [(1,2)]
=> 1
[(1,2),(3,4)]
=> [(1,2),(3,4)]
=> 1
[(1,3),(2,4)]
=> [(1,4),(2,3)]
=> 1
[(1,4),(2,3)]
=> [(1,3),(2,4)]
=> 2
[(1,2),(3,4),(5,6)]
=> [(1,2),(3,4),(5,6)]
=> 1
[(1,3),(2,4),(5,6)]
=> [(1,4),(2,3),(5,6)]
=> 1
[(1,4),(2,3),(5,6)]
=> [(1,3),(2,4),(5,6)]
=> 2
[(1,5),(2,3),(4,6)]
=> [(1,3),(2,6),(4,5)]
=> 2
[(1,6),(2,3),(4,5)]
=> [(1,3),(2,5),(4,6)]
=> 2
[(1,6),(2,4),(3,5)]
=> [(1,5),(2,6),(3,4)]
=> 2
[(1,5),(2,4),(3,6)]
=> [(1,4),(2,6),(3,5)]
=> 2
[(1,4),(2,5),(3,6)]
=> [(1,6),(2,5),(3,4)]
=> 1
[(1,3),(2,5),(4,6)]
=> [(1,6),(2,3),(4,5)]
=> 1
[(1,2),(3,5),(4,6)]
=> [(1,2),(3,6),(4,5)]
=> 1
[(1,2),(3,6),(4,5)]
=> [(1,2),(3,5),(4,6)]
=> 2
[(1,3),(2,6),(4,5)]
=> [(1,5),(2,3),(4,6)]
=> 2
[(1,4),(2,6),(3,5)]
=> [(1,5),(2,4),(3,6)]
=> 2
[(1,5),(2,6),(3,4)]
=> [(1,6),(2,4),(3,5)]
=> 2
[(1,6),(2,5),(3,4)]
=> [(1,4),(2,5),(3,6)]
=> 3
[(1,2),(3,4),(5,6),(7,8)]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 1
[(1,3),(2,4),(5,6),(7,8)]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 1
[(1,4),(2,3),(5,6),(7,8)]
=> [(1,3),(2,4),(5,6),(7,8)]
=> 2
[(1,5),(2,3),(4,6),(7,8)]
=> [(1,3),(2,6),(4,5),(7,8)]
=> 2
[(1,6),(2,3),(4,5),(7,8)]
=> [(1,3),(2,5),(4,6),(7,8)]
=> 2
[(1,7),(2,3),(4,5),(6,8)]
=> [(1,3),(2,5),(4,8),(6,7)]
=> 2
[(1,8),(2,3),(4,5),(6,7)]
=> [(1,3),(2,5),(4,7),(6,8)]
=> 2
[(1,8),(2,4),(3,5),(6,7)]
=> [(1,5),(2,7),(3,4),(6,8)]
=> 2
[(1,7),(2,4),(3,5),(6,8)]
=> [(1,5),(2,8),(3,4),(6,7)]
=> 2
[(1,6),(2,4),(3,5),(7,8)]
=> [(1,5),(2,6),(3,4),(7,8)]
=> 2
[(1,5),(2,4),(3,6),(7,8)]
=> [(1,4),(2,6),(3,5),(7,8)]
=> 2
[(1,4),(2,5),(3,6),(7,8)]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 1
[(1,3),(2,5),(4,6),(7,8)]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 1
[(1,2),(3,5),(4,6),(7,8)]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 1
[(1,2),(3,6),(4,5),(7,8)]
=> [(1,2),(3,5),(4,6),(7,8)]
=> 2
[(1,3),(2,6),(4,5),(7,8)]
=> [(1,5),(2,3),(4,6),(7,8)]
=> 2
[(1,4),(2,6),(3,5),(7,8)]
=> [(1,5),(2,4),(3,6),(7,8)]
=> 2
[(1,5),(2,6),(3,4),(7,8)]
=> [(1,6),(2,4),(3,5),(7,8)]
=> 2
[(1,6),(2,5),(3,4),(7,8)]
=> [(1,4),(2,5),(3,6),(7,8)]
=> 3
[(1,7),(2,5),(3,4),(6,8)]
=> [(1,4),(2,5),(3,8),(6,7)]
=> 3
[(1,8),(2,5),(3,4),(6,7)]
=> [(1,4),(2,5),(3,7),(6,8)]
=> 3
[(1,8),(2,6),(3,4),(5,7)]
=> [(1,4),(2,7),(3,8),(5,6)]
=> 3
[(1,7),(2,6),(3,4),(5,8)]
=> [(1,4),(2,6),(3,8),(5,7)]
=> 3
[(1,6),(2,7),(3,4),(5,8)]
=> [(1,8),(2,4),(3,7),(5,6)]
=> 2
[(1,5),(2,7),(3,4),(6,8)]
=> [(1,8),(2,4),(3,5),(6,7)]
=> 2
[(1,4),(2,7),(3,5),(6,8)]
=> [(1,5),(2,4),(3,8),(6,7)]
=> 2
[(1,3),(2,7),(4,5),(6,8)]
=> [(1,5),(2,3),(4,8),(6,7)]
=> 2
[(1,2),(3,7),(4,5),(6,8)]
=> [(1,2),(3,5),(4,8),(6,7)]
=> 2
[(1,2),(3,8),(4,5),(6,7)]
=> [(1,2),(3,5),(4,7),(6,8)]
=> 2
[(1,3),(2,8),(4,5),(6,7)]
=> [(1,5),(2,3),(4,7),(6,8)]
=> 2
[(1,4),(2,8),(3,5),(6,7)]
=> [(1,5),(2,4),(3,7),(6,8)]
=> 2
Description
The crossing number of a perfect matching.
This is the maximal number of chords in the standard representation of a perfect matching that mutually cross.
Matching statistic: St000147
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(load all 4 compositions to match this statistic)
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => [2]
=> 2 = 1 + 1
[(1,2),(3,4)]
=> [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[(1,3),(2,4)]
=> [3,4,1,2] => [2,1,1]
=> 2 = 1 + 1
[(1,4),(2,3)]
=> [3,4,2,1] => [3,1]
=> 3 = 2 + 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,2,2]
=> 2 = 1 + 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [2,2,1,1]
=> 2 = 1 + 1
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [3,2,1]
=> 3 = 2 + 1
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [3,1,1,1]
=> 3 = 2 + 1
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [3,2,1]
=> 3 = 2 + 1
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [3,1,1,1]
=> 3 = 2 + 1
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [3,1,1,1]
=> 3 = 2 + 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [2,1,1,1,1]
=> 2 = 1 + 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [2,2,1,1]
=> 2 = 1 + 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,2,1,1]
=> 2 = 1 + 1
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [3,2,1]
=> 3 = 2 + 1
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [3,2,1]
=> 3 = 2 + 1
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [3,1,1,1]
=> 3 = 2 + 1
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [3,1,1,1]
=> 3 = 2 + 1
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [4,1,1]
=> 4 = 3 + 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,2,2,2]
=> 2 = 1 + 1
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [2,2,2,1,1]
=> 2 = 1 + 1
[(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [3,2,2,1]
=> 3 = 2 + 1
[(1,5),(2,3),(4,6),(7,8)]
=> [3,5,2,6,1,4,8,7] => [3,2,1,1,1]
=> 3 = 2 + 1
[(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [3,2,2,1]
=> 3 = 2 + 1
[(1,7),(2,3),(4,5),(6,8)]
=> [3,5,2,7,4,8,1,6] => [3,2,1,1,1]
=> 3 = 2 + 1
[(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [3,2,2,1]
=> 3 = 2 + 1
[(1,8),(2,4),(3,5),(6,7)]
=> [4,5,7,2,3,8,6,1] => [3,2,1,1,1]
=> 3 = 2 + 1
[(1,7),(2,4),(3,5),(6,8)]
=> [4,5,7,2,3,8,1,6] => [3,1,1,1,1,1]
=> 3 = 2 + 1
[(1,6),(2,4),(3,5),(7,8)]
=> [4,5,6,2,3,1,8,7] => [3,2,1,1,1]
=> 3 = 2 + 1
[(1,5),(2,4),(3,6),(7,8)]
=> [4,5,6,2,1,3,8,7] => [3,2,1,1,1]
=> 3 = 2 + 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [2,2,1,1,1,1]
=> 2 = 1 + 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [2,2,2,1,1]
=> 2 = 1 + 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [2,2,2,1,1]
=> 2 = 1 + 1
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [3,2,2,1]
=> 3 = 2 + 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,5,1,6,4,2,8,7] => [3,2,2,1]
=> 3 = 2 + 1
[(1,4),(2,6),(3,5),(7,8)]
=> [4,5,6,1,3,2,8,7] => [3,2,1,1,1]
=> 3 = 2 + 1
[(1,5),(2,6),(3,4),(7,8)]
=> [4,5,6,3,1,2,8,7] => [3,2,1,1,1]
=> 3 = 2 + 1
[(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [4,2,1,1]
=> 4 = 3 + 1
[(1,7),(2,5),(3,4),(6,8)]
=> [4,5,7,3,2,8,1,6] => [4,1,1,1,1]
=> 4 = 3 + 1
[(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [4,2,1,1]
=> 4 = 3 + 1
[(1,8),(2,6),(3,4),(5,7)]
=> [4,6,7,3,8,2,5,1] => [4,1,1,1,1]
=> 4 = 3 + 1
[(1,7),(2,6),(3,4),(5,8)]
=> [4,6,7,3,8,2,1,5] => [4,1,1,1,1]
=> 4 = 3 + 1
[(1,6),(2,7),(3,4),(5,8)]
=> [4,6,7,3,8,1,2,5] => [3,1,1,1,1,1]
=> 3 = 2 + 1
[(1,5),(2,7),(3,4),(6,8)]
=> [4,5,7,3,1,8,2,6] => [3,2,1,1,1]
=> 3 = 2 + 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,5,7,1,3,8,2,6] => [3,1,1,1,1,1]
=> 3 = 2 + 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,5,1,7,4,8,2,6] => [3,2,1,1,1]
=> 3 = 2 + 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,5,7,4,8,3,6] => [3,2,1,1,1]
=> 3 = 2 + 1
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [3,2,2,1]
=> 3 = 2 + 1
[(1,3),(2,8),(4,5),(6,7)]
=> [3,5,1,7,4,8,6,2] => [3,2,2,1]
=> 3 = 2 + 1
[(1,4),(2,8),(3,5),(6,7)]
=> [4,5,7,1,3,8,6,2] => [3,2,1,1,1]
=> 3 = 2 + 1
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [3,5,2,6,1,4,8,7,10,9] => ?
=> ? = 2 + 1
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [3,5,2,7,4,8,1,6,10,9] => ?
=> ? = 2 + 1
[(1,9),(2,3),(4,5),(6,7),(8,10)]
=> [3,5,2,7,4,9,6,10,1,8] => ?
=> ? = 2 + 1
[(1,10),(2,4),(3,5),(6,7),(8,9)]
=> [4,5,7,2,3,9,6,10,8,1] => ?
=> ? = 2 + 1
[(1,9),(2,4),(3,5),(6,7),(8,10)]
=> [4,5,7,2,3,9,6,10,1,8] => ?
=> ? = 2 + 1
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [4,5,7,2,3,8,6,1,10,9] => ?
=> ? = 2 + 1
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [4,5,7,2,3,8,1,6,10,9] => ?
=> ? = 2 + 1
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [4,5,6,2,3,1,8,7,10,9] => ?
=> ? = 2 + 1
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [4,5,6,2,1,3,8,7,10,9] => ?
=> ? = 2 + 1
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [3,5,1,6,4,2,8,7,10,9] => ?
=> ? = 2 + 1
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [4,5,6,1,3,2,8,7,10,9] => ?
=> ? = 2 + 1
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [4,5,6,3,1,2,8,7,10,9] => ?
=> ? = 2 + 1
[(1,7),(2,5),(3,4),(6,8),(9,10)]
=> [4,5,7,3,2,8,1,6,10,9] => ?
=> ? = 3 + 1
[(1,9),(2,5),(3,4),(6,7),(8,10)]
=> [4,5,7,3,2,9,6,10,1,8] => ?
=> ? = 3 + 1
[(1,10),(2,6),(3,4),(5,7),(8,9)]
=> [4,6,7,3,9,2,5,10,8,1] => ?
=> ? = 3 + 1
[(1,9),(2,6),(3,4),(5,7),(8,10)]
=> [4,6,7,3,9,2,5,10,1,8] => ?
=> ? = 3 + 1
[(1,8),(2,6),(3,4),(5,7),(9,10)]
=> [4,6,7,3,8,2,5,1,10,9] => ?
=> ? = 3 + 1
[(1,7),(2,6),(3,4),(5,8),(9,10)]
=> [4,6,7,3,8,2,1,5,10,9] => ?
=> ? = 3 + 1
[(1,6),(2,7),(3,4),(5,8),(9,10)]
=> [4,6,7,3,8,1,2,5,10,9] => ?
=> ? = 2 + 1
[(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [4,5,7,3,1,8,2,6,10,9] => ?
=> ? = 2 + 1
[(1,4),(2,7),(3,5),(6,8),(9,10)]
=> [4,5,7,1,3,8,2,6,10,9] => ?
=> ? = 2 + 1
[(1,3),(2,7),(4,5),(6,8),(9,10)]
=> [3,5,1,7,4,8,2,6,10,9] => ?
=> ? = 2 + 1
[(1,2),(3,7),(4,5),(6,8),(9,10)]
=> [2,1,5,7,4,8,3,6,10,9] => ?
=> ? = 2 + 1
[(1,3),(2,8),(4,5),(6,7),(9,10)]
=> [3,5,1,7,4,8,6,2,10,9] => ?
=> ? = 2 + 1
[(1,4),(2,8),(3,5),(6,7),(9,10)]
=> [4,5,7,1,3,8,6,2,10,9] => ?
=> ? = 2 + 1
[(1,5),(2,8),(3,4),(6,7),(9,10)]
=> [4,5,7,3,1,8,6,2,10,9] => ?
=> ? = 2 + 1
[(1,6),(2,8),(3,4),(5,7),(9,10)]
=> [4,6,7,3,8,1,5,2,10,9] => ?
=> ? = 2 + 1
[(1,7),(2,8),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,1,2,10,9] => ?
=> ? = 2 + 1
[(1,9),(2,7),(3,4),(5,6),(8,10)]
=> [4,6,7,3,9,5,2,10,1,8] => ?
=> ? = 3 + 1
[(1,10),(2,8),(3,4),(5,6),(7,9)]
=> [4,6,8,3,9,5,10,2,7,1] => ?
=> ? = 3 + 1
[(1,9),(2,8),(3,4),(5,6),(7,10)]
=> [4,6,8,3,9,5,10,2,1,7] => ?
=> ? = 3 + 1
[(1,8),(2,9),(3,4),(5,6),(7,10)]
=> [4,6,8,3,9,5,10,1,2,7] => ?
=> ? = 2 + 1
[(1,7),(2,9),(3,4),(5,6),(8,10)]
=> [4,6,7,3,9,5,1,10,2,8] => ?
=> ? = 2 + 1
[(1,6),(2,9),(3,4),(5,7),(8,10)]
=> [4,6,7,3,9,1,5,10,2,8] => ?
=> ? = 2 + 1
[(1,5),(2,9),(3,4),(6,7),(8,10)]
=> [4,5,7,3,1,9,6,10,2,8] => ?
=> ? = 2 + 1
[(1,4),(2,9),(3,5),(6,7),(8,10)]
=> [4,5,7,1,3,9,6,10,2,8] => ?
=> ? = 2 + 1
[(1,3),(2,9),(4,5),(6,7),(8,10)]
=> [3,5,1,7,4,9,6,10,2,8] => ?
=> ? = 2 + 1
[(1,2),(3,9),(4,5),(6,7),(8,10)]
=> [2,1,5,7,4,9,6,10,3,8] => ?
=> ? = 2 + 1
[(1,3),(2,10),(4,5),(6,7),(8,9)]
=> [3,5,1,7,4,9,6,10,8,2] => ?
=> ? = 2 + 1
[(1,4),(2,10),(3,5),(6,7),(8,9)]
=> [4,5,7,1,3,9,6,10,8,2] => ?
=> ? = 2 + 1
[(1,5),(2,10),(3,4),(6,7),(8,9)]
=> [4,5,7,3,1,9,6,10,8,2] => ?
=> ? = 2 + 1
[(1,6),(2,10),(3,4),(5,7),(8,9)]
=> [4,6,7,3,9,1,5,10,8,2] => ?
=> ? = 2 + 1
[(1,7),(2,10),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,1,10,8,2] => ?
=> ? = 2 + 1
[(1,8),(2,10),(3,4),(5,6),(7,9)]
=> [4,6,8,3,9,5,10,1,7,2] => ?
=> ? = 2 + 1
[(1,9),(2,10),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,1,2] => ?
=> ? = 2 + 1
[(1,10),(2,9),(3,5),(4,6),(7,8)]
=> [5,6,8,9,3,4,10,7,2,1] => ?
=> ? = 3 + 1
[(1,9),(2,10),(3,5),(4,6),(7,8)]
=> [5,6,8,9,3,4,10,7,1,2] => ?
=> ? = 2 + 1
[(1,8),(2,10),(3,5),(4,6),(7,9)]
=> [5,6,8,9,3,4,10,1,7,2] => ?
=> ? = 2 + 1
[(1,7),(2,10),(3,5),(4,6),(8,9)]
=> [5,6,7,9,3,4,1,10,8,2] => ?
=> ? = 2 + 1
[(1,6),(2,10),(3,5),(4,7),(8,9)]
=> [5,6,7,9,3,1,4,10,8,2] => ?
=> ? = 2 + 1
Description
The largest part of an integer partition.
Matching statistic: St000010
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => [2]
=> [1,1]
=> 2 = 1 + 1
[(1,2),(3,4)]
=> [2,1,4,3] => [2,2]
=> [2,2]
=> 2 = 1 + 1
[(1,3),(2,4)]
=> [3,4,1,2] => [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[(1,4),(2,3)]
=> [3,4,2,1] => [3,1]
=> [2,1,1]
=> 3 = 2 + 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,2,2]
=> [3,3]
=> 2 = 1 + 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [3,2,1]
=> [3,2,1]
=> 3 = 2 + 1
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [3,1,1,1]
=> [4,1,1]
=> 3 = 2 + 1
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [3,2,1]
=> [3,2,1]
=> 3 = 2 + 1
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [3,1,1,1]
=> [4,1,1]
=> 3 = 2 + 1
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [3,1,1,1]
=> [4,1,1]
=> 3 = 2 + 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [2,1,1,1,1]
=> [5,1]
=> 2 = 1 + 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [3,2,1]
=> [3,2,1]
=> 3 = 2 + 1
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [3,2,1]
=> [3,2,1]
=> 3 = 2 + 1
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [3,1,1,1]
=> [4,1,1]
=> 3 = 2 + 1
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [3,1,1,1]
=> [4,1,1]
=> 3 = 2 + 1
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [4,1,1]
=> [3,1,1,1]
=> 4 = 3 + 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,2,2,2]
=> [4,4]
=> 2 = 1 + 1
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [2,2,2,1,1]
=> [5,3]
=> 2 = 1 + 1
[(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [3,2,2,1]
=> [4,3,1]
=> 3 = 2 + 1
[(1,5),(2,3),(4,6),(7,8)]
=> [3,5,2,6,1,4,8,7] => [3,2,1,1,1]
=> [5,2,1]
=> 3 = 2 + 1
[(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [3,2,2,1]
=> [4,3,1]
=> 3 = 2 + 1
[(1,7),(2,3),(4,5),(6,8)]
=> [3,5,2,7,4,8,1,6] => [3,2,1,1,1]
=> [5,2,1]
=> 3 = 2 + 1
[(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [3,2,2,1]
=> [4,3,1]
=> 3 = 2 + 1
[(1,8),(2,4),(3,5),(6,7)]
=> [4,5,7,2,3,8,6,1] => [3,2,1,1,1]
=> [5,2,1]
=> 3 = 2 + 1
[(1,7),(2,4),(3,5),(6,8)]
=> [4,5,7,2,3,8,1,6] => [3,1,1,1,1,1]
=> [6,1,1]
=> 3 = 2 + 1
[(1,6),(2,4),(3,5),(7,8)]
=> [4,5,6,2,3,1,8,7] => [3,2,1,1,1]
=> [5,2,1]
=> 3 = 2 + 1
[(1,5),(2,4),(3,6),(7,8)]
=> [4,5,6,2,1,3,8,7] => [3,2,1,1,1]
=> [5,2,1]
=> 3 = 2 + 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [2,2,1,1,1,1]
=> [6,2]
=> 2 = 1 + 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [2,2,2,1,1]
=> [5,3]
=> 2 = 1 + 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [2,2,2,1,1]
=> [5,3]
=> 2 = 1 + 1
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [3,2,2,1]
=> [4,3,1]
=> 3 = 2 + 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,5,1,6,4,2,8,7] => [3,2,2,1]
=> [4,3,1]
=> 3 = 2 + 1
[(1,4),(2,6),(3,5),(7,8)]
=> [4,5,6,1,3,2,8,7] => [3,2,1,1,1]
=> [5,2,1]
=> 3 = 2 + 1
[(1,5),(2,6),(3,4),(7,8)]
=> [4,5,6,3,1,2,8,7] => [3,2,1,1,1]
=> [5,2,1]
=> 3 = 2 + 1
[(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [4,2,1,1]
=> [4,2,1,1]
=> 4 = 3 + 1
[(1,7),(2,5),(3,4),(6,8)]
=> [4,5,7,3,2,8,1,6] => [4,1,1,1,1]
=> [5,1,1,1]
=> 4 = 3 + 1
[(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [4,2,1,1]
=> [4,2,1,1]
=> 4 = 3 + 1
[(1,8),(2,6),(3,4),(5,7)]
=> [4,6,7,3,8,2,5,1] => [4,1,1,1,1]
=> [5,1,1,1]
=> 4 = 3 + 1
[(1,7),(2,6),(3,4),(5,8)]
=> [4,6,7,3,8,2,1,5] => [4,1,1,1,1]
=> [5,1,1,1]
=> 4 = 3 + 1
[(1,6),(2,7),(3,4),(5,8)]
=> [4,6,7,3,8,1,2,5] => [3,1,1,1,1,1]
=> [6,1,1]
=> 3 = 2 + 1
[(1,5),(2,7),(3,4),(6,8)]
=> [4,5,7,3,1,8,2,6] => [3,2,1,1,1]
=> [5,2,1]
=> 3 = 2 + 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,5,7,1,3,8,2,6] => [3,1,1,1,1,1]
=> [6,1,1]
=> 3 = 2 + 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,5,1,7,4,8,2,6] => [3,2,1,1,1]
=> [5,2,1]
=> 3 = 2 + 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,5,7,4,8,3,6] => [3,2,1,1,1]
=> [5,2,1]
=> 3 = 2 + 1
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [3,2,2,1]
=> [4,3,1]
=> 3 = 2 + 1
[(1,3),(2,8),(4,5),(6,7)]
=> [3,5,1,7,4,8,6,2] => [3,2,2,1]
=> [4,3,1]
=> 3 = 2 + 1
[(1,4),(2,8),(3,5),(6,7)]
=> [4,5,7,1,3,8,6,2] => [3,2,1,1,1]
=> [5,2,1]
=> 3 = 2 + 1
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [3,5,2,6,1,4,8,7,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [3,5,2,7,4,8,1,6,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,9),(2,3),(4,5),(6,7),(8,10)]
=> [3,5,2,7,4,9,6,10,1,8] => ?
=> ?
=> ? = 2 + 1
[(1,10),(2,4),(3,5),(6,7),(8,9)]
=> [4,5,7,2,3,9,6,10,8,1] => ?
=> ?
=> ? = 2 + 1
[(1,9),(2,4),(3,5),(6,7),(8,10)]
=> [4,5,7,2,3,9,6,10,1,8] => ?
=> ?
=> ? = 2 + 1
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [4,5,7,2,3,8,6,1,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [4,5,7,2,3,8,1,6,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [4,5,6,2,3,1,8,7,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [4,5,6,2,1,3,8,7,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [3,5,1,6,4,2,8,7,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [4,5,6,1,3,2,8,7,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [4,5,6,3,1,2,8,7,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,5),(3,4),(6,8),(9,10)]
=> [4,5,7,3,2,8,1,6,10,9] => ?
=> ?
=> ? = 3 + 1
[(1,9),(2,5),(3,4),(6,7),(8,10)]
=> [4,5,7,3,2,9,6,10,1,8] => ?
=> ?
=> ? = 3 + 1
[(1,10),(2,6),(3,4),(5,7),(8,9)]
=> [4,6,7,3,9,2,5,10,8,1] => ?
=> ?
=> ? = 3 + 1
[(1,9),(2,6),(3,4),(5,7),(8,10)]
=> [4,6,7,3,9,2,5,10,1,8] => ?
=> ?
=> ? = 3 + 1
[(1,8),(2,6),(3,4),(5,7),(9,10)]
=> [4,6,7,3,8,2,5,1,10,9] => ?
=> ?
=> ? = 3 + 1
[(1,7),(2,6),(3,4),(5,8),(9,10)]
=> [4,6,7,3,8,2,1,5,10,9] => ?
=> ?
=> ? = 3 + 1
[(1,6),(2,7),(3,4),(5,8),(9,10)]
=> [4,6,7,3,8,1,2,5,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [4,5,7,3,1,8,2,6,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,4),(2,7),(3,5),(6,8),(9,10)]
=> [4,5,7,1,3,8,2,6,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,3),(2,7),(4,5),(6,8),(9,10)]
=> [3,5,1,7,4,8,2,6,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,2),(3,7),(4,5),(6,8),(9,10)]
=> [2,1,5,7,4,8,3,6,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,3),(2,8),(4,5),(6,7),(9,10)]
=> [3,5,1,7,4,8,6,2,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,4),(2,8),(3,5),(6,7),(9,10)]
=> [4,5,7,1,3,8,6,2,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,5),(2,8),(3,4),(6,7),(9,10)]
=> [4,5,7,3,1,8,6,2,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,6),(2,8),(3,4),(5,7),(9,10)]
=> [4,6,7,3,8,1,5,2,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,8),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,1,2,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,9),(2,7),(3,4),(5,6),(8,10)]
=> [4,6,7,3,9,5,2,10,1,8] => ?
=> ?
=> ? = 3 + 1
[(1,10),(2,8),(3,4),(5,6),(7,9)]
=> [4,6,8,3,9,5,10,2,7,1] => ?
=> ?
=> ? = 3 + 1
[(1,9),(2,8),(3,4),(5,6),(7,10)]
=> [4,6,8,3,9,5,10,2,1,7] => ?
=> ?
=> ? = 3 + 1
[(1,8),(2,9),(3,4),(5,6),(7,10)]
=> [4,6,8,3,9,5,10,1,2,7] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,9),(3,4),(5,6),(8,10)]
=> [4,6,7,3,9,5,1,10,2,8] => ?
=> ?
=> ? = 2 + 1
[(1,6),(2,9),(3,4),(5,7),(8,10)]
=> [4,6,7,3,9,1,5,10,2,8] => ?
=> ?
=> ? = 2 + 1
[(1,5),(2,9),(3,4),(6,7),(8,10)]
=> [4,5,7,3,1,9,6,10,2,8] => ?
=> ?
=> ? = 2 + 1
[(1,4),(2,9),(3,5),(6,7),(8,10)]
=> [4,5,7,1,3,9,6,10,2,8] => ?
=> ?
=> ? = 2 + 1
[(1,3),(2,9),(4,5),(6,7),(8,10)]
=> [3,5,1,7,4,9,6,10,2,8] => ?
=> ?
=> ? = 2 + 1
[(1,2),(3,9),(4,5),(6,7),(8,10)]
=> [2,1,5,7,4,9,6,10,3,8] => ?
=> ?
=> ? = 2 + 1
[(1,3),(2,10),(4,5),(6,7),(8,9)]
=> [3,5,1,7,4,9,6,10,8,2] => ?
=> ?
=> ? = 2 + 1
[(1,4),(2,10),(3,5),(6,7),(8,9)]
=> [4,5,7,1,3,9,6,10,8,2] => ?
=> ?
=> ? = 2 + 1
[(1,5),(2,10),(3,4),(6,7),(8,9)]
=> [4,5,7,3,1,9,6,10,8,2] => ?
=> ?
=> ? = 2 + 1
[(1,6),(2,10),(3,4),(5,7),(8,9)]
=> [4,6,7,3,9,1,5,10,8,2] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,10),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,1,10,8,2] => ?
=> ?
=> ? = 2 + 1
[(1,8),(2,10),(3,4),(5,6),(7,9)]
=> [4,6,8,3,9,5,10,1,7,2] => ?
=> ?
=> ? = 2 + 1
[(1,9),(2,10),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,1,2] => ?
=> ?
=> ? = 2 + 1
[(1,10),(2,9),(3,5),(4,6),(7,8)]
=> [5,6,8,9,3,4,10,7,2,1] => ?
=> ?
=> ? = 3 + 1
[(1,9),(2,10),(3,5),(4,6),(7,8)]
=> [5,6,8,9,3,4,10,7,1,2] => ?
=> ?
=> ? = 2 + 1
[(1,8),(2,10),(3,5),(4,6),(7,9)]
=> [5,6,8,9,3,4,10,1,7,2] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,10),(3,5),(4,6),(8,9)]
=> [5,6,7,9,3,4,1,10,8,2] => ?
=> ?
=> ? = 2 + 1
[(1,6),(2,10),(3,5),(4,7),(8,9)]
=> [5,6,7,9,3,1,4,10,8,2] => ?
=> ?
=> ? = 2 + 1
Description
The length of the partition.
Matching statistic: St000734
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => [2]
=> [[1,2]]
=> 2 = 1 + 1
[(1,2),(3,4)]
=> [2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> 2 = 1 + 1
[(1,3),(2,4)]
=> [3,4,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[(1,4),(2,3)]
=> [3,4,2,1] => [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 2 = 1 + 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 2 = 1 + 1
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3 = 2 + 1
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 3 = 2 + 1
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3 = 2 + 1
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 3 = 2 + 1
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 3 = 2 + 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 2 = 1 + 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 2 = 1 + 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 2 = 1 + 1
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3 = 2 + 1
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3 = 2 + 1
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 3 = 2 + 1
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 3 = 2 + 1
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 4 = 3 + 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> 2 = 1 + 1
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> 2 = 1 + 1
[(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> 3 = 2 + 1
[(1,5),(2,3),(4,6),(7,8)]
=> [3,5,2,6,1,4,8,7] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> 3 = 2 + 1
[(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> 3 = 2 + 1
[(1,7),(2,3),(4,5),(6,8)]
=> [3,5,2,7,4,8,1,6] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> 3 = 2 + 1
[(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> 3 = 2 + 1
[(1,8),(2,4),(3,5),(6,7)]
=> [4,5,7,2,3,8,6,1] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> 3 = 2 + 1
[(1,7),(2,4),(3,5),(6,8)]
=> [4,5,7,2,3,8,1,6] => [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> 3 = 2 + 1
[(1,6),(2,4),(3,5),(7,8)]
=> [4,5,6,2,3,1,8,7] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> 3 = 2 + 1
[(1,5),(2,4),(3,6),(7,8)]
=> [4,5,6,2,1,3,8,7] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> 3 = 2 + 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> 2 = 1 + 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> 2 = 1 + 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> 2 = 1 + 1
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> 3 = 2 + 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,5,1,6,4,2,8,7] => [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> 3 = 2 + 1
[(1,4),(2,6),(3,5),(7,8)]
=> [4,5,6,1,3,2,8,7] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> 3 = 2 + 1
[(1,5),(2,6),(3,4),(7,8)]
=> [4,5,6,3,1,2,8,7] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> 3 = 2 + 1
[(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 4 = 3 + 1
[(1,7),(2,5),(3,4),(6,8)]
=> [4,5,7,3,2,8,1,6] => [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> 4 = 3 + 1
[(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 4 = 3 + 1
[(1,8),(2,6),(3,4),(5,7)]
=> [4,6,7,3,8,2,5,1] => [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> 4 = 3 + 1
[(1,7),(2,6),(3,4),(5,8)]
=> [4,6,7,3,8,2,1,5] => [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> 4 = 3 + 1
[(1,6),(2,7),(3,4),(5,8)]
=> [4,6,7,3,8,1,2,5] => [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> 3 = 2 + 1
[(1,5),(2,7),(3,4),(6,8)]
=> [4,5,7,3,1,8,2,6] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> 3 = 2 + 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,5,7,1,3,8,2,6] => [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> 3 = 2 + 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,5,1,7,4,8,2,6] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> 3 = 2 + 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,5,7,4,8,3,6] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> 3 = 2 + 1
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> 3 = 2 + 1
[(1,3),(2,8),(4,5),(6,7)]
=> [3,5,1,7,4,8,6,2] => [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> 3 = 2 + 1
[(1,4),(2,8),(3,5),(6,7)]
=> [4,5,7,1,3,8,6,2] => [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> 3 = 2 + 1
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [3,5,2,6,1,4,8,7,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [3,5,2,7,4,8,1,6,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,9),(2,3),(4,5),(6,7),(8,10)]
=> [3,5,2,7,4,9,6,10,1,8] => ?
=> ?
=> ? = 2 + 1
[(1,10),(2,4),(3,5),(6,7),(8,9)]
=> [4,5,7,2,3,9,6,10,8,1] => ?
=> ?
=> ? = 2 + 1
[(1,9),(2,4),(3,5),(6,7),(8,10)]
=> [4,5,7,2,3,9,6,10,1,8] => ?
=> ?
=> ? = 2 + 1
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [4,5,7,2,3,8,6,1,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [4,5,7,2,3,8,1,6,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [4,5,6,2,3,1,8,7,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [4,5,6,2,1,3,8,7,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [3,5,1,6,4,2,8,7,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [4,5,6,1,3,2,8,7,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [4,5,6,3,1,2,8,7,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,5),(3,4),(6,8),(9,10)]
=> [4,5,7,3,2,8,1,6,10,9] => ?
=> ?
=> ? = 3 + 1
[(1,9),(2,5),(3,4),(6,7),(8,10)]
=> [4,5,7,3,2,9,6,10,1,8] => ?
=> ?
=> ? = 3 + 1
[(1,10),(2,6),(3,4),(5,7),(8,9)]
=> [4,6,7,3,9,2,5,10,8,1] => ?
=> ?
=> ? = 3 + 1
[(1,9),(2,6),(3,4),(5,7),(8,10)]
=> [4,6,7,3,9,2,5,10,1,8] => ?
=> ?
=> ? = 3 + 1
[(1,8),(2,6),(3,4),(5,7),(9,10)]
=> [4,6,7,3,8,2,5,1,10,9] => ?
=> ?
=> ? = 3 + 1
[(1,7),(2,6),(3,4),(5,8),(9,10)]
=> [4,6,7,3,8,2,1,5,10,9] => ?
=> ?
=> ? = 3 + 1
[(1,6),(2,7),(3,4),(5,8),(9,10)]
=> [4,6,7,3,8,1,2,5,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [4,5,7,3,1,8,2,6,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,4),(2,7),(3,5),(6,8),(9,10)]
=> [4,5,7,1,3,8,2,6,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,3),(2,7),(4,5),(6,8),(9,10)]
=> [3,5,1,7,4,8,2,6,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,2),(3,7),(4,5),(6,8),(9,10)]
=> [2,1,5,7,4,8,3,6,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,3),(2,8),(4,5),(6,7),(9,10)]
=> [3,5,1,7,4,8,6,2,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,4),(2,8),(3,5),(6,7),(9,10)]
=> [4,5,7,1,3,8,6,2,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,5),(2,8),(3,4),(6,7),(9,10)]
=> [4,5,7,3,1,8,6,2,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,6),(2,8),(3,4),(5,7),(9,10)]
=> [4,6,7,3,8,1,5,2,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,8),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,1,2,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,9),(2,7),(3,4),(5,6),(8,10)]
=> [4,6,7,3,9,5,2,10,1,8] => ?
=> ?
=> ? = 3 + 1
[(1,10),(2,8),(3,4),(5,6),(7,9)]
=> [4,6,8,3,9,5,10,2,7,1] => ?
=> ?
=> ? = 3 + 1
[(1,9),(2,8),(3,4),(5,6),(7,10)]
=> [4,6,8,3,9,5,10,2,1,7] => ?
=> ?
=> ? = 3 + 1
[(1,8),(2,9),(3,4),(5,6),(7,10)]
=> [4,6,8,3,9,5,10,1,2,7] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,9),(3,4),(5,6),(8,10)]
=> [4,6,7,3,9,5,1,10,2,8] => ?
=> ?
=> ? = 2 + 1
[(1,6),(2,9),(3,4),(5,7),(8,10)]
=> [4,6,7,3,9,1,5,10,2,8] => ?
=> ?
=> ? = 2 + 1
[(1,5),(2,9),(3,4),(6,7),(8,10)]
=> [4,5,7,3,1,9,6,10,2,8] => ?
=> ?
=> ? = 2 + 1
[(1,4),(2,9),(3,5),(6,7),(8,10)]
=> [4,5,7,1,3,9,6,10,2,8] => ?
=> ?
=> ? = 2 + 1
[(1,3),(2,9),(4,5),(6,7),(8,10)]
=> [3,5,1,7,4,9,6,10,2,8] => ?
=> ?
=> ? = 2 + 1
[(1,2),(3,9),(4,5),(6,7),(8,10)]
=> [2,1,5,7,4,9,6,10,3,8] => ?
=> ?
=> ? = 2 + 1
[(1,3),(2,10),(4,5),(6,7),(8,9)]
=> [3,5,1,7,4,9,6,10,8,2] => ?
=> ?
=> ? = 2 + 1
[(1,4),(2,10),(3,5),(6,7),(8,9)]
=> [4,5,7,1,3,9,6,10,8,2] => ?
=> ?
=> ? = 2 + 1
[(1,5),(2,10),(3,4),(6,7),(8,9)]
=> [4,5,7,3,1,9,6,10,8,2] => ?
=> ?
=> ? = 2 + 1
[(1,6),(2,10),(3,4),(5,7),(8,9)]
=> [4,6,7,3,9,1,5,10,8,2] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,10),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,1,10,8,2] => ?
=> ?
=> ? = 2 + 1
[(1,8),(2,10),(3,4),(5,6),(7,9)]
=> [4,6,8,3,9,5,10,1,7,2] => ?
=> ?
=> ? = 2 + 1
[(1,9),(2,10),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,1,2] => ?
=> ?
=> ? = 2 + 1
[(1,10),(2,9),(3,5),(4,6),(7,8)]
=> [5,6,8,9,3,4,10,7,2,1] => ?
=> ?
=> ? = 3 + 1
[(1,9),(2,10),(3,5),(4,6),(7,8)]
=> [5,6,8,9,3,4,10,7,1,2] => ?
=> ?
=> ? = 2 + 1
[(1,8),(2,10),(3,5),(4,6),(7,9)]
=> [5,6,8,9,3,4,10,1,7,2] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,10),(3,5),(4,6),(8,9)]
=> [5,6,7,9,3,4,1,10,8,2] => ?
=> ?
=> ? = 2 + 1
[(1,6),(2,10),(3,5),(4,7),(8,9)]
=> [5,6,7,9,3,1,4,10,8,2] => ?
=> ?
=> ? = 2 + 1
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000157
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => [1,1]
=> [[1],[2]]
=> 1
[(1,2),(3,4)]
=> [2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> 1
[(1,3),(2,4)]
=> [3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> 1
[(1,4),(2,3)]
=> [3,4,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [3,3]
=> [[1,2,3],[4,5,6]]
=> 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [3,3]
=> [[1,2,3],[4,5,6]]
=> 1
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 2
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 2
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 2
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 2
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 2
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [3,3]
=> [[1,2,3],[4,5,6]]
=> 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [3,3]
=> [[1,2,3],[4,5,6]]
=> 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [3,3]
=> [[1,2,3],[4,5,6]]
=> 1
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 2
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 2
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 2
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 2
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 3
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 1
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 1
[(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 2
[(1,5),(2,3),(4,6),(7,8)]
=> [3,5,2,6,1,4,8,7] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 2
[(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 2
[(1,7),(2,3),(4,5),(6,8)]
=> [3,5,2,7,4,8,1,6] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 2
[(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 2
[(1,8),(2,4),(3,5),(6,7)]
=> [4,5,7,2,3,8,6,1] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 2
[(1,7),(2,4),(3,5),(6,8)]
=> [4,5,7,2,3,8,1,6] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 2
[(1,6),(2,4),(3,5),(7,8)]
=> [4,5,6,2,3,1,8,7] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 2
[(1,5),(2,4),(3,6),(7,8)]
=> [4,5,6,2,1,3,8,7] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 2
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 1
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 2
[(1,3),(2,6),(4,5),(7,8)]
=> [3,5,1,6,4,2,8,7] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 2
[(1,4),(2,6),(3,5),(7,8)]
=> [4,5,6,1,3,2,8,7] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 2
[(1,5),(2,6),(3,4),(7,8)]
=> [4,5,6,3,1,2,8,7] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 2
[(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 3
[(1,7),(2,5),(3,4),(6,8)]
=> [4,5,7,3,2,8,1,6] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 3
[(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 3
[(1,8),(2,6),(3,4),(5,7)]
=> [4,6,7,3,8,2,5,1] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 3
[(1,7),(2,6),(3,4),(5,8)]
=> [4,6,7,3,8,2,1,5] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 3
[(1,6),(2,7),(3,4),(5,8)]
=> [4,6,7,3,8,1,2,5] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 2
[(1,5),(2,7),(3,4),(6,8)]
=> [4,5,7,3,1,8,2,6] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 2
[(1,4),(2,7),(3,5),(6,8)]
=> [4,5,7,1,3,8,2,6] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 2
[(1,3),(2,7),(4,5),(6,8)]
=> [3,5,1,7,4,8,2,6] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 2
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,5,7,4,8,3,6] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 2
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 2
[(1,3),(2,8),(4,5),(6,7)]
=> [3,5,1,7,4,8,6,2] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 2
[(1,4),(2,8),(3,5),(6,7)]
=> [4,5,7,1,3,8,6,2] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 2
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [3,5,2,6,1,4,8,7,10,9] => ?
=> ?
=> ? = 2
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [3,5,2,7,4,8,1,6,10,9] => ?
=> ?
=> ? = 2
[(1,9),(2,3),(4,5),(6,7),(8,10)]
=> [3,5,2,7,4,9,6,10,1,8] => ?
=> ?
=> ? = 2
[(1,10),(2,4),(3,5),(6,7),(8,9)]
=> [4,5,7,2,3,9,6,10,8,1] => ?
=> ?
=> ? = 2
[(1,9),(2,4),(3,5),(6,7),(8,10)]
=> [4,5,7,2,3,9,6,10,1,8] => ?
=> ?
=> ? = 2
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [4,5,7,2,3,8,6,1,10,9] => ?
=> ?
=> ? = 2
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [4,5,7,2,3,8,1,6,10,9] => ?
=> ?
=> ? = 2
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [4,5,6,2,3,1,8,7,10,9] => ?
=> ?
=> ? = 2
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [4,5,6,2,1,3,8,7,10,9] => ?
=> ?
=> ? = 2
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [3,5,1,6,4,2,8,7,10,9] => ?
=> ?
=> ? = 2
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [4,5,6,1,3,2,8,7,10,9] => ?
=> ?
=> ? = 2
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [4,5,6,3,1,2,8,7,10,9] => ?
=> ?
=> ? = 2
[(1,7),(2,5),(3,4),(6,8),(9,10)]
=> [4,5,7,3,2,8,1,6,10,9] => ?
=> ?
=> ? = 3
[(1,9),(2,5),(3,4),(6,7),(8,10)]
=> [4,5,7,3,2,9,6,10,1,8] => ?
=> ?
=> ? = 3
[(1,10),(2,6),(3,4),(5,7),(8,9)]
=> [4,6,7,3,9,2,5,10,8,1] => ?
=> ?
=> ? = 3
[(1,9),(2,6),(3,4),(5,7),(8,10)]
=> [4,6,7,3,9,2,5,10,1,8] => ?
=> ?
=> ? = 3
[(1,8),(2,6),(3,4),(5,7),(9,10)]
=> [4,6,7,3,8,2,5,1,10,9] => ?
=> ?
=> ? = 3
[(1,7),(2,6),(3,4),(5,8),(9,10)]
=> [4,6,7,3,8,2,1,5,10,9] => ?
=> ?
=> ? = 3
[(1,6),(2,7),(3,4),(5,8),(9,10)]
=> [4,6,7,3,8,1,2,5,10,9] => ?
=> ?
=> ? = 2
[(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [4,5,7,3,1,8,2,6,10,9] => ?
=> ?
=> ? = 2
[(1,4),(2,7),(3,5),(6,8),(9,10)]
=> [4,5,7,1,3,8,2,6,10,9] => ?
=> ?
=> ? = 2
[(1,3),(2,7),(4,5),(6,8),(9,10)]
=> [3,5,1,7,4,8,2,6,10,9] => ?
=> ?
=> ? = 2
[(1,2),(3,7),(4,5),(6,8),(9,10)]
=> [2,1,5,7,4,8,3,6,10,9] => ?
=> ?
=> ? = 2
[(1,3),(2,8),(4,5),(6,7),(9,10)]
=> [3,5,1,7,4,8,6,2,10,9] => ?
=> ?
=> ? = 2
[(1,4),(2,8),(3,5),(6,7),(9,10)]
=> [4,5,7,1,3,8,6,2,10,9] => ?
=> ?
=> ? = 2
[(1,5),(2,8),(3,4),(6,7),(9,10)]
=> [4,5,7,3,1,8,6,2,10,9] => ?
=> ?
=> ? = 2
[(1,6),(2,8),(3,4),(5,7),(9,10)]
=> [4,6,7,3,8,1,5,2,10,9] => ?
=> ?
=> ? = 2
[(1,7),(2,8),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,1,2,10,9] => ?
=> ?
=> ? = 2
[(1,9),(2,7),(3,4),(5,6),(8,10)]
=> [4,6,7,3,9,5,2,10,1,8] => ?
=> ?
=> ? = 3
[(1,10),(2,8),(3,4),(5,6),(7,9)]
=> [4,6,8,3,9,5,10,2,7,1] => ?
=> ?
=> ? = 3
[(1,9),(2,8),(3,4),(5,6),(7,10)]
=> [4,6,8,3,9,5,10,2,1,7] => ?
=> ?
=> ? = 3
[(1,8),(2,9),(3,4),(5,6),(7,10)]
=> [4,6,8,3,9,5,10,1,2,7] => ?
=> ?
=> ? = 2
[(1,7),(2,9),(3,4),(5,6),(8,10)]
=> [4,6,7,3,9,5,1,10,2,8] => ?
=> ?
=> ? = 2
[(1,6),(2,9),(3,4),(5,7),(8,10)]
=> [4,6,7,3,9,1,5,10,2,8] => ?
=> ?
=> ? = 2
[(1,5),(2,9),(3,4),(6,7),(8,10)]
=> [4,5,7,3,1,9,6,10,2,8] => ?
=> ?
=> ? = 2
[(1,4),(2,9),(3,5),(6,7),(8,10)]
=> [4,5,7,1,3,9,6,10,2,8] => ?
=> ?
=> ? = 2
[(1,3),(2,9),(4,5),(6,7),(8,10)]
=> [3,5,1,7,4,9,6,10,2,8] => ?
=> ?
=> ? = 2
[(1,2),(3,9),(4,5),(6,7),(8,10)]
=> [2,1,5,7,4,9,6,10,3,8] => ?
=> ?
=> ? = 2
[(1,3),(2,10),(4,5),(6,7),(8,9)]
=> [3,5,1,7,4,9,6,10,8,2] => ?
=> ?
=> ? = 2
[(1,4),(2,10),(3,5),(6,7),(8,9)]
=> [4,5,7,1,3,9,6,10,8,2] => ?
=> ?
=> ? = 2
[(1,5),(2,10),(3,4),(6,7),(8,9)]
=> [4,5,7,3,1,9,6,10,8,2] => ?
=> ?
=> ? = 2
[(1,6),(2,10),(3,4),(5,7),(8,9)]
=> [4,6,7,3,9,1,5,10,8,2] => ?
=> ?
=> ? = 2
[(1,7),(2,10),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,1,10,8,2] => ?
=> ?
=> ? = 2
[(1,8),(2,10),(3,4),(5,6),(7,9)]
=> [4,6,8,3,9,5,10,1,7,2] => ?
=> ?
=> ? = 2
[(1,9),(2,10),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,1,2] => ?
=> ?
=> ? = 2
[(1,10),(2,9),(3,5),(4,6),(7,8)]
=> [5,6,8,9,3,4,10,7,2,1] => ?
=> ?
=> ? = 3
[(1,9),(2,10),(3,5),(4,6),(7,8)]
=> [5,6,8,9,3,4,10,7,1,2] => ?
=> ?
=> ? = 2
[(1,8),(2,10),(3,5),(4,6),(7,9)]
=> [5,6,8,9,3,4,10,1,7,2] => ?
=> ?
=> ? = 2
[(1,7),(2,10),(3,5),(4,6),(8,9)]
=> [5,6,7,9,3,4,1,10,8,2] => ?
=> ?
=> ? = 2
[(1,6),(2,10),(3,5),(4,7),(8,9)]
=> [5,6,7,9,3,1,4,10,8,2] => ?
=> ?
=> ? = 2
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000288
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => [1,1]
=> 110 => 2 = 1 + 1
[(1,2),(3,4)]
=> [2,1,4,3] => [2,2]
=> 1100 => 2 = 1 + 1
[(1,3),(2,4)]
=> [3,4,1,2] => [2,2]
=> 1100 => 2 = 1 + 1
[(1,4),(2,3)]
=> [3,4,2,1] => [2,1,1]
=> 10110 => 3 = 2 + 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [3,3]
=> 11000 => 2 = 1 + 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [3,3]
=> 11000 => 2 = 1 + 1
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [3,2,1]
=> 101010 => 3 = 2 + 1
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [3,2,1]
=> 101010 => 3 = 2 + 1
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [3,2,1]
=> 101010 => 3 = 2 + 1
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [3,2,1]
=> 101010 => 3 = 2 + 1
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [3,2,1]
=> 101010 => 3 = 2 + 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [3,3]
=> 11000 => 2 = 1 + 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [3,3]
=> 11000 => 2 = 1 + 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [3,3]
=> 11000 => 2 = 1 + 1
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [3,2,1]
=> 101010 => 3 = 2 + 1
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [3,2,1]
=> 101010 => 3 = 2 + 1
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [3,2,1]
=> 101010 => 3 = 2 + 1
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [3,2,1]
=> 101010 => 3 = 2 + 1
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [3,1,1,1]
=> 1001110 => 4 = 3 + 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [4,4]
=> 110000 => 2 = 1 + 1
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [4,4]
=> 110000 => 2 = 1 + 1
[(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [4,3,1]
=> 1010010 => 3 = 2 + 1
[(1,5),(2,3),(4,6),(7,8)]
=> [3,5,2,6,1,4,8,7] => [4,3,1]
=> 1010010 => 3 = 2 + 1
[(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [4,3,1]
=> 1010010 => 3 = 2 + 1
[(1,7),(2,3),(4,5),(6,8)]
=> [3,5,2,7,4,8,1,6] => [4,3,1]
=> 1010010 => 3 = 2 + 1
[(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [4,3,1]
=> 1010010 => 3 = 2 + 1
[(1,8),(2,4),(3,5),(6,7)]
=> [4,5,7,2,3,8,6,1] => [4,3,1]
=> 1010010 => 3 = 2 + 1
[(1,7),(2,4),(3,5),(6,8)]
=> [4,5,7,2,3,8,1,6] => [4,3,1]
=> 1010010 => 3 = 2 + 1
[(1,6),(2,4),(3,5),(7,8)]
=> [4,5,6,2,3,1,8,7] => [4,3,1]
=> 1010010 => 3 = 2 + 1
[(1,5),(2,4),(3,6),(7,8)]
=> [4,5,6,2,1,3,8,7] => [4,3,1]
=> 1010010 => 3 = 2 + 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [4,4]
=> 110000 => 2 = 1 + 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [4,4]
=> 110000 => 2 = 1 + 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [4,4]
=> 110000 => 2 = 1 + 1
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [4,3,1]
=> 1010010 => 3 = 2 + 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,5,1,6,4,2,8,7] => [4,3,1]
=> 1010010 => 3 = 2 + 1
[(1,4),(2,6),(3,5),(7,8)]
=> [4,5,6,1,3,2,8,7] => [4,3,1]
=> 1010010 => 3 = 2 + 1
[(1,5),(2,6),(3,4),(7,8)]
=> [4,5,6,3,1,2,8,7] => [4,3,1]
=> 1010010 => 3 = 2 + 1
[(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [4,2,1,1]
=> 10010110 => 4 = 3 + 1
[(1,7),(2,5),(3,4),(6,8)]
=> [4,5,7,3,2,8,1,6] => [4,2,1,1]
=> 10010110 => 4 = 3 + 1
[(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [4,2,1,1]
=> 10010110 => 4 = 3 + 1
[(1,8),(2,6),(3,4),(5,7)]
=> [4,6,7,3,8,2,5,1] => [4,2,1,1]
=> 10010110 => 4 = 3 + 1
[(1,7),(2,6),(3,4),(5,8)]
=> [4,6,7,3,8,2,1,5] => [4,2,1,1]
=> 10010110 => 4 = 3 + 1
[(1,6),(2,7),(3,4),(5,8)]
=> [4,6,7,3,8,1,2,5] => [4,3,1]
=> 1010010 => 3 = 2 + 1
[(1,5),(2,7),(3,4),(6,8)]
=> [4,5,7,3,1,8,2,6] => [4,3,1]
=> 1010010 => 3 = 2 + 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,5,7,1,3,8,2,6] => [4,3,1]
=> 1010010 => 3 = 2 + 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,5,1,7,4,8,2,6] => [4,3,1]
=> 1010010 => 3 = 2 + 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,5,7,4,8,3,6] => [4,3,1]
=> 1010010 => 3 = 2 + 1
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [4,3,1]
=> 1010010 => 3 = 2 + 1
[(1,3),(2,8),(4,5),(6,7)]
=> [3,5,1,7,4,8,6,2] => [4,3,1]
=> 1010010 => 3 = 2 + 1
[(1,4),(2,8),(3,5),(6,7)]
=> [4,5,7,1,3,8,6,2] => [4,3,1]
=> 1010010 => 3 = 2 + 1
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [3,5,2,6,1,4,8,7,10,9] => ?
=> ? => ? = 2 + 1
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [3,5,2,7,4,8,1,6,10,9] => ?
=> ? => ? = 2 + 1
[(1,9),(2,3),(4,5),(6,7),(8,10)]
=> [3,5,2,7,4,9,6,10,1,8] => ?
=> ? => ? = 2 + 1
[(1,10),(2,4),(3,5),(6,7),(8,9)]
=> [4,5,7,2,3,9,6,10,8,1] => ?
=> ? => ? = 2 + 1
[(1,9),(2,4),(3,5),(6,7),(8,10)]
=> [4,5,7,2,3,9,6,10,1,8] => ?
=> ? => ? = 2 + 1
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [4,5,7,2,3,8,6,1,10,9] => ?
=> ? => ? = 2 + 1
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [4,5,7,2,3,8,1,6,10,9] => ?
=> ? => ? = 2 + 1
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [4,5,6,2,3,1,8,7,10,9] => ?
=> ? => ? = 2 + 1
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [4,5,6,2,1,3,8,7,10,9] => ?
=> ? => ? = 2 + 1
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [3,5,1,6,4,2,8,7,10,9] => ?
=> ? => ? = 2 + 1
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [4,5,6,1,3,2,8,7,10,9] => ?
=> ? => ? = 2 + 1
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [4,5,6,3,1,2,8,7,10,9] => ?
=> ? => ? = 2 + 1
[(1,7),(2,5),(3,4),(6,8),(9,10)]
=> [4,5,7,3,2,8,1,6,10,9] => ?
=> ? => ? = 3 + 1
[(1,9),(2,5),(3,4),(6,7),(8,10)]
=> [4,5,7,3,2,9,6,10,1,8] => ?
=> ? => ? = 3 + 1
[(1,10),(2,6),(3,4),(5,7),(8,9)]
=> [4,6,7,3,9,2,5,10,8,1] => ?
=> ? => ? = 3 + 1
[(1,9),(2,6),(3,4),(5,7),(8,10)]
=> [4,6,7,3,9,2,5,10,1,8] => ?
=> ? => ? = 3 + 1
[(1,8),(2,6),(3,4),(5,7),(9,10)]
=> [4,6,7,3,8,2,5,1,10,9] => ?
=> ? => ? = 3 + 1
[(1,7),(2,6),(3,4),(5,8),(9,10)]
=> [4,6,7,3,8,2,1,5,10,9] => ?
=> ? => ? = 3 + 1
[(1,6),(2,7),(3,4),(5,8),(9,10)]
=> [4,6,7,3,8,1,2,5,10,9] => ?
=> ? => ? = 2 + 1
[(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [4,5,7,3,1,8,2,6,10,9] => ?
=> ? => ? = 2 + 1
[(1,4),(2,7),(3,5),(6,8),(9,10)]
=> [4,5,7,1,3,8,2,6,10,9] => ?
=> ? => ? = 2 + 1
[(1,3),(2,7),(4,5),(6,8),(9,10)]
=> [3,5,1,7,4,8,2,6,10,9] => ?
=> ? => ? = 2 + 1
[(1,2),(3,7),(4,5),(6,8),(9,10)]
=> [2,1,5,7,4,8,3,6,10,9] => ?
=> ? => ? = 2 + 1
[(1,3),(2,8),(4,5),(6,7),(9,10)]
=> [3,5,1,7,4,8,6,2,10,9] => ?
=> ? => ? = 2 + 1
[(1,4),(2,8),(3,5),(6,7),(9,10)]
=> [4,5,7,1,3,8,6,2,10,9] => ?
=> ? => ? = 2 + 1
[(1,5),(2,8),(3,4),(6,7),(9,10)]
=> [4,5,7,3,1,8,6,2,10,9] => ?
=> ? => ? = 2 + 1
[(1,6),(2,8),(3,4),(5,7),(9,10)]
=> [4,6,7,3,8,1,5,2,10,9] => ?
=> ? => ? = 2 + 1
[(1,7),(2,8),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,1,2,10,9] => ?
=> ? => ? = 2 + 1
[(1,9),(2,7),(3,4),(5,6),(8,10)]
=> [4,6,7,3,9,5,2,10,1,8] => ?
=> ? => ? = 3 + 1
[(1,10),(2,8),(3,4),(5,6),(7,9)]
=> [4,6,8,3,9,5,10,2,7,1] => ?
=> ? => ? = 3 + 1
[(1,9),(2,8),(3,4),(5,6),(7,10)]
=> [4,6,8,3,9,5,10,2,1,7] => ?
=> ? => ? = 3 + 1
[(1,8),(2,9),(3,4),(5,6),(7,10)]
=> [4,6,8,3,9,5,10,1,2,7] => ?
=> ? => ? = 2 + 1
[(1,7),(2,9),(3,4),(5,6),(8,10)]
=> [4,6,7,3,9,5,1,10,2,8] => ?
=> ? => ? = 2 + 1
[(1,6),(2,9),(3,4),(5,7),(8,10)]
=> [4,6,7,3,9,1,5,10,2,8] => ?
=> ? => ? = 2 + 1
[(1,5),(2,9),(3,4),(6,7),(8,10)]
=> [4,5,7,3,1,9,6,10,2,8] => ?
=> ? => ? = 2 + 1
[(1,4),(2,9),(3,5),(6,7),(8,10)]
=> [4,5,7,1,3,9,6,10,2,8] => ?
=> ? => ? = 2 + 1
[(1,3),(2,9),(4,5),(6,7),(8,10)]
=> [3,5,1,7,4,9,6,10,2,8] => ?
=> ? => ? = 2 + 1
[(1,2),(3,9),(4,5),(6,7),(8,10)]
=> [2,1,5,7,4,9,6,10,3,8] => ?
=> ? => ? = 2 + 1
[(1,3),(2,10),(4,5),(6,7),(8,9)]
=> [3,5,1,7,4,9,6,10,8,2] => ?
=> ? => ? = 2 + 1
[(1,4),(2,10),(3,5),(6,7),(8,9)]
=> [4,5,7,1,3,9,6,10,8,2] => ?
=> ? => ? = 2 + 1
[(1,5),(2,10),(3,4),(6,7),(8,9)]
=> [4,5,7,3,1,9,6,10,8,2] => ?
=> ? => ? = 2 + 1
[(1,6),(2,10),(3,4),(5,7),(8,9)]
=> [4,6,7,3,9,1,5,10,8,2] => ?
=> ? => ? = 2 + 1
[(1,7),(2,10),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,1,10,8,2] => ?
=> ? => ? = 2 + 1
[(1,8),(2,10),(3,4),(5,6),(7,9)]
=> [4,6,8,3,9,5,10,1,7,2] => ?
=> ? => ? = 2 + 1
[(1,9),(2,10),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,1,2] => ?
=> ? => ? = 2 + 1
[(1,10),(2,9),(3,5),(4,6),(7,8)]
=> [5,6,8,9,3,4,10,7,2,1] => ?
=> ? => ? = 3 + 1
[(1,9),(2,10),(3,5),(4,6),(7,8)]
=> [5,6,8,9,3,4,10,7,1,2] => ?
=> ? => ? = 2 + 1
[(1,8),(2,10),(3,5),(4,6),(7,9)]
=> [5,6,8,9,3,4,10,1,7,2] => ?
=> ? => ? = 2 + 1
[(1,7),(2,10),(3,5),(4,6),(8,9)]
=> [5,6,7,9,3,4,1,10,8,2] => ?
=> ? => ? = 2 + 1
[(1,6),(2,10),(3,5),(4,7),(8,9)]
=> [5,6,7,9,3,1,4,10,8,2] => ?
=> ? => ? = 2 + 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000378
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => [1,1]
=> [2]
=> 2 = 1 + 1
[(1,2),(3,4)]
=> [2,1,4,3] => [2,2]
=> [4]
=> 2 = 1 + 1
[(1,3),(2,4)]
=> [3,4,1,2] => [2,2]
=> [4]
=> 2 = 1 + 1
[(1,4),(2,3)]
=> [3,4,2,1] => [2,1,1]
=> [2,2]
=> 3 = 2 + 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [3,3]
=> [6]
=> 2 = 1 + 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [3,3]
=> [6]
=> 2 = 1 + 1
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [3,2,1]
=> [5,1]
=> 3 = 2 + 1
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [3,2,1]
=> [5,1]
=> 3 = 2 + 1
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [3,2,1]
=> [5,1]
=> 3 = 2 + 1
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [3,2,1]
=> [5,1]
=> 3 = 2 + 1
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [3,2,1]
=> [5,1]
=> 3 = 2 + 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [3,3]
=> [6]
=> 2 = 1 + 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [3,3]
=> [6]
=> 2 = 1 + 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [3,3]
=> [6]
=> 2 = 1 + 1
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [3,2,1]
=> [5,1]
=> 3 = 2 + 1
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [3,2,1]
=> [5,1]
=> 3 = 2 + 1
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [3,2,1]
=> [5,1]
=> 3 = 2 + 1
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [3,2,1]
=> [5,1]
=> 3 = 2 + 1
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [3,1,1,1]
=> [3,3]
=> 4 = 3 + 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [4,4]
=> [8]
=> 2 = 1 + 1
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [4,4]
=> [8]
=> 2 = 1 + 1
[(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [4,3,1]
=> [7,1]
=> 3 = 2 + 1
[(1,5),(2,3),(4,6),(7,8)]
=> [3,5,2,6,1,4,8,7] => [4,3,1]
=> [7,1]
=> 3 = 2 + 1
[(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [4,3,1]
=> [7,1]
=> 3 = 2 + 1
[(1,7),(2,3),(4,5),(6,8)]
=> [3,5,2,7,4,8,1,6] => [4,3,1]
=> [7,1]
=> 3 = 2 + 1
[(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [4,3,1]
=> [7,1]
=> 3 = 2 + 1
[(1,8),(2,4),(3,5),(6,7)]
=> [4,5,7,2,3,8,6,1] => [4,3,1]
=> [7,1]
=> 3 = 2 + 1
[(1,7),(2,4),(3,5),(6,8)]
=> [4,5,7,2,3,8,1,6] => [4,3,1]
=> [7,1]
=> 3 = 2 + 1
[(1,6),(2,4),(3,5),(7,8)]
=> [4,5,6,2,3,1,8,7] => [4,3,1]
=> [7,1]
=> 3 = 2 + 1
[(1,5),(2,4),(3,6),(7,8)]
=> [4,5,6,2,1,3,8,7] => [4,3,1]
=> [7,1]
=> 3 = 2 + 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [4,4]
=> [8]
=> 2 = 1 + 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [4,4]
=> [8]
=> 2 = 1 + 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [4,4]
=> [8]
=> 2 = 1 + 1
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [4,3,1]
=> [7,1]
=> 3 = 2 + 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,5,1,6,4,2,8,7] => [4,3,1]
=> [7,1]
=> 3 = 2 + 1
[(1,4),(2,6),(3,5),(7,8)]
=> [4,5,6,1,3,2,8,7] => [4,3,1]
=> [7,1]
=> 3 = 2 + 1
[(1,5),(2,6),(3,4),(7,8)]
=> [4,5,6,3,1,2,8,7] => [4,3,1]
=> [7,1]
=> 3 = 2 + 1
[(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [4,2,1,1]
=> [4,4]
=> 4 = 3 + 1
[(1,7),(2,5),(3,4),(6,8)]
=> [4,5,7,3,2,8,1,6] => [4,2,1,1]
=> [4,4]
=> 4 = 3 + 1
[(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [4,2,1,1]
=> [4,4]
=> 4 = 3 + 1
[(1,8),(2,6),(3,4),(5,7)]
=> [4,6,7,3,8,2,5,1] => [4,2,1,1]
=> [4,4]
=> 4 = 3 + 1
[(1,7),(2,6),(3,4),(5,8)]
=> [4,6,7,3,8,2,1,5] => [4,2,1,1]
=> [4,4]
=> 4 = 3 + 1
[(1,6),(2,7),(3,4),(5,8)]
=> [4,6,7,3,8,1,2,5] => [4,3,1]
=> [7,1]
=> 3 = 2 + 1
[(1,5),(2,7),(3,4),(6,8)]
=> [4,5,7,3,1,8,2,6] => [4,3,1]
=> [7,1]
=> 3 = 2 + 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,5,7,1,3,8,2,6] => [4,3,1]
=> [7,1]
=> 3 = 2 + 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,5,1,7,4,8,2,6] => [4,3,1]
=> [7,1]
=> 3 = 2 + 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,5,7,4,8,3,6] => [4,3,1]
=> [7,1]
=> 3 = 2 + 1
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [4,3,1]
=> [7,1]
=> 3 = 2 + 1
[(1,3),(2,8),(4,5),(6,7)]
=> [3,5,1,7,4,8,6,2] => [4,3,1]
=> [7,1]
=> 3 = 2 + 1
[(1,4),(2,8),(3,5),(6,7)]
=> [4,5,7,1,3,8,6,2] => [4,3,1]
=> [7,1]
=> 3 = 2 + 1
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [3,5,2,6,1,4,8,7,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [3,5,2,7,4,8,1,6,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,9),(2,3),(4,5),(6,7),(8,10)]
=> [3,5,2,7,4,9,6,10,1,8] => ?
=> ?
=> ? = 2 + 1
[(1,10),(2,4),(3,5),(6,7),(8,9)]
=> [4,5,7,2,3,9,6,10,8,1] => ?
=> ?
=> ? = 2 + 1
[(1,9),(2,4),(3,5),(6,7),(8,10)]
=> [4,5,7,2,3,9,6,10,1,8] => ?
=> ?
=> ? = 2 + 1
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [4,5,7,2,3,8,6,1,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [4,5,7,2,3,8,1,6,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [4,5,6,2,3,1,8,7,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [4,5,6,2,1,3,8,7,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [3,5,1,6,4,2,8,7,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [4,5,6,1,3,2,8,7,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [4,5,6,3,1,2,8,7,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,5),(3,4),(6,8),(9,10)]
=> [4,5,7,3,2,8,1,6,10,9] => ?
=> ?
=> ? = 3 + 1
[(1,9),(2,5),(3,4),(6,7),(8,10)]
=> [4,5,7,3,2,9,6,10,1,8] => ?
=> ?
=> ? = 3 + 1
[(1,10),(2,6),(3,4),(5,7),(8,9)]
=> [4,6,7,3,9,2,5,10,8,1] => ?
=> ?
=> ? = 3 + 1
[(1,9),(2,6),(3,4),(5,7),(8,10)]
=> [4,6,7,3,9,2,5,10,1,8] => ?
=> ?
=> ? = 3 + 1
[(1,8),(2,6),(3,4),(5,7),(9,10)]
=> [4,6,7,3,8,2,5,1,10,9] => ?
=> ?
=> ? = 3 + 1
[(1,7),(2,6),(3,4),(5,8),(9,10)]
=> [4,6,7,3,8,2,1,5,10,9] => ?
=> ?
=> ? = 3 + 1
[(1,6),(2,7),(3,4),(5,8),(9,10)]
=> [4,6,7,3,8,1,2,5,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [4,5,7,3,1,8,2,6,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,4),(2,7),(3,5),(6,8),(9,10)]
=> [4,5,7,1,3,8,2,6,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,3),(2,7),(4,5),(6,8),(9,10)]
=> [3,5,1,7,4,8,2,6,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,2),(3,7),(4,5),(6,8),(9,10)]
=> [2,1,5,7,4,8,3,6,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,3),(2,8),(4,5),(6,7),(9,10)]
=> [3,5,1,7,4,8,6,2,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,4),(2,8),(3,5),(6,7),(9,10)]
=> [4,5,7,1,3,8,6,2,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,5),(2,8),(3,4),(6,7),(9,10)]
=> [4,5,7,3,1,8,6,2,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,6),(2,8),(3,4),(5,7),(9,10)]
=> [4,6,7,3,8,1,5,2,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,8),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,1,2,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,9),(2,7),(3,4),(5,6),(8,10)]
=> [4,6,7,3,9,5,2,10,1,8] => ?
=> ?
=> ? = 3 + 1
[(1,10),(2,8),(3,4),(5,6),(7,9)]
=> [4,6,8,3,9,5,10,2,7,1] => ?
=> ?
=> ? = 3 + 1
[(1,9),(2,8),(3,4),(5,6),(7,10)]
=> [4,6,8,3,9,5,10,2,1,7] => ?
=> ?
=> ? = 3 + 1
[(1,8),(2,9),(3,4),(5,6),(7,10)]
=> [4,6,8,3,9,5,10,1,2,7] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,9),(3,4),(5,6),(8,10)]
=> [4,6,7,3,9,5,1,10,2,8] => ?
=> ?
=> ? = 2 + 1
[(1,6),(2,9),(3,4),(5,7),(8,10)]
=> [4,6,7,3,9,1,5,10,2,8] => ?
=> ?
=> ? = 2 + 1
[(1,5),(2,9),(3,4),(6,7),(8,10)]
=> [4,5,7,3,1,9,6,10,2,8] => ?
=> ?
=> ? = 2 + 1
[(1,4),(2,9),(3,5),(6,7),(8,10)]
=> [4,5,7,1,3,9,6,10,2,8] => ?
=> ?
=> ? = 2 + 1
[(1,3),(2,9),(4,5),(6,7),(8,10)]
=> [3,5,1,7,4,9,6,10,2,8] => ?
=> ?
=> ? = 2 + 1
[(1,2),(3,9),(4,5),(6,7),(8,10)]
=> [2,1,5,7,4,9,6,10,3,8] => ?
=> ?
=> ? = 2 + 1
[(1,3),(2,10),(4,5),(6,7),(8,9)]
=> [3,5,1,7,4,9,6,10,8,2] => ?
=> ?
=> ? = 2 + 1
[(1,4),(2,10),(3,5),(6,7),(8,9)]
=> [4,5,7,1,3,9,6,10,8,2] => ?
=> ?
=> ? = 2 + 1
[(1,5),(2,10),(3,4),(6,7),(8,9)]
=> [4,5,7,3,1,9,6,10,8,2] => ?
=> ?
=> ? = 2 + 1
[(1,6),(2,10),(3,4),(5,7),(8,9)]
=> [4,6,7,3,9,1,5,10,8,2] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,10),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,1,10,8,2] => ?
=> ?
=> ? = 2 + 1
[(1,8),(2,10),(3,4),(5,6),(7,9)]
=> [4,6,8,3,9,5,10,1,7,2] => ?
=> ?
=> ? = 2 + 1
[(1,9),(2,10),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,1,2] => ?
=> ?
=> ? = 2 + 1
[(1,10),(2,9),(3,5),(4,6),(7,8)]
=> [5,6,8,9,3,4,10,7,2,1] => ?
=> ?
=> ? = 3 + 1
[(1,9),(2,10),(3,5),(4,6),(7,8)]
=> [5,6,8,9,3,4,10,7,1,2] => ?
=> ?
=> ? = 2 + 1
[(1,8),(2,10),(3,5),(4,6),(7,9)]
=> [5,6,8,9,3,4,10,1,7,2] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,10),(3,5),(4,6),(8,9)]
=> [5,6,7,9,3,4,1,10,8,2] => ?
=> ?
=> ? = 2 + 1
[(1,6),(2,10),(3,5),(4,7),(8,9)]
=> [5,6,7,9,3,1,4,10,8,2] => ?
=> ?
=> ? = 2 + 1
Description
The diagonal inversion number of an integer partition.
The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$.
See also exercise 3.19 of [2].
This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000733
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[(1,2),(3,4)]
=> [2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> 2 = 1 + 1
[(1,3),(2,4)]
=> [3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> 2 = 1 + 1
[(1,4),(2,3)]
=> [3,4,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3 = 2 + 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [3,3]
=> [[1,2,3],[4,5,6]]
=> 2 = 1 + 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [3,3]
=> [[1,2,3],[4,5,6]]
=> 2 = 1 + 1
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3 = 2 + 1
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3 = 2 + 1
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3 = 2 + 1
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3 = 2 + 1
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3 = 2 + 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [3,3]
=> [[1,2,3],[4,5,6]]
=> 2 = 1 + 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [3,3]
=> [[1,2,3],[4,5,6]]
=> 2 = 1 + 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [3,3]
=> [[1,2,3],[4,5,6]]
=> 2 = 1 + 1
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3 = 2 + 1
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3 = 2 + 1
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3 = 2 + 1
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3 = 2 + 1
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 4 = 3 + 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 2 = 1 + 1
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 2 = 1 + 1
[(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 3 = 2 + 1
[(1,5),(2,3),(4,6),(7,8)]
=> [3,5,2,6,1,4,8,7] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 3 = 2 + 1
[(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 3 = 2 + 1
[(1,7),(2,3),(4,5),(6,8)]
=> [3,5,2,7,4,8,1,6] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 3 = 2 + 1
[(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 3 = 2 + 1
[(1,8),(2,4),(3,5),(6,7)]
=> [4,5,7,2,3,8,6,1] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 3 = 2 + 1
[(1,7),(2,4),(3,5),(6,8)]
=> [4,5,7,2,3,8,1,6] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 3 = 2 + 1
[(1,6),(2,4),(3,5),(7,8)]
=> [4,5,6,2,3,1,8,7] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 3 = 2 + 1
[(1,5),(2,4),(3,6),(7,8)]
=> [4,5,6,2,1,3,8,7] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 3 = 2 + 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 2 = 1 + 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 2 = 1 + 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 2 = 1 + 1
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 3 = 2 + 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,5,1,6,4,2,8,7] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 3 = 2 + 1
[(1,4),(2,6),(3,5),(7,8)]
=> [4,5,6,1,3,2,8,7] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 3 = 2 + 1
[(1,5),(2,6),(3,4),(7,8)]
=> [4,5,6,3,1,2,8,7] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 3 = 2 + 1
[(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 4 = 3 + 1
[(1,7),(2,5),(3,4),(6,8)]
=> [4,5,7,3,2,8,1,6] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 4 = 3 + 1
[(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 4 = 3 + 1
[(1,8),(2,6),(3,4),(5,7)]
=> [4,6,7,3,8,2,5,1] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 4 = 3 + 1
[(1,7),(2,6),(3,4),(5,8)]
=> [4,6,7,3,8,2,1,5] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 4 = 3 + 1
[(1,6),(2,7),(3,4),(5,8)]
=> [4,6,7,3,8,1,2,5] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 3 = 2 + 1
[(1,5),(2,7),(3,4),(6,8)]
=> [4,5,7,3,1,8,2,6] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 3 = 2 + 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,5,7,1,3,8,2,6] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 3 = 2 + 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,5,1,7,4,8,2,6] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 3 = 2 + 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,5,7,4,8,3,6] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 3 = 2 + 1
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 3 = 2 + 1
[(1,3),(2,8),(4,5),(6,7)]
=> [3,5,1,7,4,8,6,2] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 3 = 2 + 1
[(1,4),(2,8),(3,5),(6,7)]
=> [4,5,7,1,3,8,6,2] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 3 = 2 + 1
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [3,5,2,6,1,4,8,7,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [3,5,2,7,4,8,1,6,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,9),(2,3),(4,5),(6,7),(8,10)]
=> [3,5,2,7,4,9,6,10,1,8] => ?
=> ?
=> ? = 2 + 1
[(1,10),(2,4),(3,5),(6,7),(8,9)]
=> [4,5,7,2,3,9,6,10,8,1] => ?
=> ?
=> ? = 2 + 1
[(1,9),(2,4),(3,5),(6,7),(8,10)]
=> [4,5,7,2,3,9,6,10,1,8] => ?
=> ?
=> ? = 2 + 1
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [4,5,7,2,3,8,6,1,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [4,5,7,2,3,8,1,6,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [4,5,6,2,3,1,8,7,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [4,5,6,2,1,3,8,7,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [3,5,1,6,4,2,8,7,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [4,5,6,1,3,2,8,7,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [4,5,6,3,1,2,8,7,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,5),(3,4),(6,8),(9,10)]
=> [4,5,7,3,2,8,1,6,10,9] => ?
=> ?
=> ? = 3 + 1
[(1,9),(2,5),(3,4),(6,7),(8,10)]
=> [4,5,7,3,2,9,6,10,1,8] => ?
=> ?
=> ? = 3 + 1
[(1,10),(2,6),(3,4),(5,7),(8,9)]
=> [4,6,7,3,9,2,5,10,8,1] => ?
=> ?
=> ? = 3 + 1
[(1,9),(2,6),(3,4),(5,7),(8,10)]
=> [4,6,7,3,9,2,5,10,1,8] => ?
=> ?
=> ? = 3 + 1
[(1,8),(2,6),(3,4),(5,7),(9,10)]
=> [4,6,7,3,8,2,5,1,10,9] => ?
=> ?
=> ? = 3 + 1
[(1,7),(2,6),(3,4),(5,8),(9,10)]
=> [4,6,7,3,8,2,1,5,10,9] => ?
=> ?
=> ? = 3 + 1
[(1,6),(2,7),(3,4),(5,8),(9,10)]
=> [4,6,7,3,8,1,2,5,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [4,5,7,3,1,8,2,6,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,4),(2,7),(3,5),(6,8),(9,10)]
=> [4,5,7,1,3,8,2,6,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,3),(2,7),(4,5),(6,8),(9,10)]
=> [3,5,1,7,4,8,2,6,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,2),(3,7),(4,5),(6,8),(9,10)]
=> [2,1,5,7,4,8,3,6,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,3),(2,8),(4,5),(6,7),(9,10)]
=> [3,5,1,7,4,8,6,2,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,4),(2,8),(3,5),(6,7),(9,10)]
=> [4,5,7,1,3,8,6,2,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,5),(2,8),(3,4),(6,7),(9,10)]
=> [4,5,7,3,1,8,6,2,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,6),(2,8),(3,4),(5,7),(9,10)]
=> [4,6,7,3,8,1,5,2,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,8),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,1,2,10,9] => ?
=> ?
=> ? = 2 + 1
[(1,9),(2,7),(3,4),(5,6),(8,10)]
=> [4,6,7,3,9,5,2,10,1,8] => ?
=> ?
=> ? = 3 + 1
[(1,10),(2,8),(3,4),(5,6),(7,9)]
=> [4,6,8,3,9,5,10,2,7,1] => ?
=> ?
=> ? = 3 + 1
[(1,9),(2,8),(3,4),(5,6),(7,10)]
=> [4,6,8,3,9,5,10,2,1,7] => ?
=> ?
=> ? = 3 + 1
[(1,8),(2,9),(3,4),(5,6),(7,10)]
=> [4,6,8,3,9,5,10,1,2,7] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,9),(3,4),(5,6),(8,10)]
=> [4,6,7,3,9,5,1,10,2,8] => ?
=> ?
=> ? = 2 + 1
[(1,6),(2,9),(3,4),(5,7),(8,10)]
=> [4,6,7,3,9,1,5,10,2,8] => ?
=> ?
=> ? = 2 + 1
[(1,5),(2,9),(3,4),(6,7),(8,10)]
=> [4,5,7,3,1,9,6,10,2,8] => ?
=> ?
=> ? = 2 + 1
[(1,4),(2,9),(3,5),(6,7),(8,10)]
=> [4,5,7,1,3,9,6,10,2,8] => ?
=> ?
=> ? = 2 + 1
[(1,3),(2,9),(4,5),(6,7),(8,10)]
=> [3,5,1,7,4,9,6,10,2,8] => ?
=> ?
=> ? = 2 + 1
[(1,2),(3,9),(4,5),(6,7),(8,10)]
=> [2,1,5,7,4,9,6,10,3,8] => ?
=> ?
=> ? = 2 + 1
[(1,3),(2,10),(4,5),(6,7),(8,9)]
=> [3,5,1,7,4,9,6,10,8,2] => ?
=> ?
=> ? = 2 + 1
[(1,4),(2,10),(3,5),(6,7),(8,9)]
=> [4,5,7,1,3,9,6,10,8,2] => ?
=> ?
=> ? = 2 + 1
[(1,5),(2,10),(3,4),(6,7),(8,9)]
=> [4,5,7,3,1,9,6,10,8,2] => ?
=> ?
=> ? = 2 + 1
[(1,6),(2,10),(3,4),(5,7),(8,9)]
=> [4,6,7,3,9,1,5,10,8,2] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,10),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,1,10,8,2] => ?
=> ?
=> ? = 2 + 1
[(1,8),(2,10),(3,4),(5,6),(7,9)]
=> [4,6,8,3,9,5,10,1,7,2] => ?
=> ?
=> ? = 2 + 1
[(1,9),(2,10),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,1,2] => ?
=> ?
=> ? = 2 + 1
[(1,10),(2,9),(3,5),(4,6),(7,8)]
=> [5,6,8,9,3,4,10,7,2,1] => ?
=> ?
=> ? = 3 + 1
[(1,9),(2,10),(3,5),(4,6),(7,8)]
=> [5,6,8,9,3,4,10,7,1,2] => ?
=> ?
=> ? = 2 + 1
[(1,8),(2,10),(3,5),(4,6),(7,9)]
=> [5,6,8,9,3,4,10,1,7,2] => ?
=> ?
=> ? = 2 + 1
[(1,7),(2,10),(3,5),(4,6),(8,9)]
=> [5,6,7,9,3,4,1,10,8,2] => ?
=> ?
=> ? = 2 + 1
[(1,6),(2,10),(3,5),(4,7),(8,9)]
=> [5,6,7,9,3,1,4,10,8,2] => ?
=> ?
=> ? = 2 + 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St001587
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => [2]
=> 1
[(1,2),(3,4)]
=> [2,1,4,3] => [2,2]
=> 1
[(1,3),(2,4)]
=> [3,4,1,2] => [2,1,1]
=> 1
[(1,4),(2,3)]
=> [4,3,2,1] => [4]
=> 2
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,2,2]
=> 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [2,2,1,1]
=> 1
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [4,2]
=> 2
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [4,1,1]
=> 2
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [4,2]
=> 2
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [4,1,1]
=> 2
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [4,1,1]
=> 2
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [2,1,1,1,1]
=> 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [2,2,1,1]
=> 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,2,1,1]
=> 1
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [4,2]
=> 2
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [4,1,1]
=> 2
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [4,1,1]
=> 2
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [4,1,1]
=> 2
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [6]
=> 3
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,2,2,2]
=> 1
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [2,2,2,1,1]
=> 1
[(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [4,2,2]
=> 2
[(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => [4,2,1,1]
=> 2
[(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [4,2,2]
=> 2
[(1,7),(2,3),(4,5),(6,8)]
=> [7,3,2,5,4,8,1,6] => [4,2,1,1]
=> 2
[(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [4,2,2]
=> 2
[(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => [4,2,1,1]
=> 2
[(1,7),(2,4),(3,5),(6,8)]
=> [7,4,5,2,3,8,1,6] => [4,1,1,1,1]
=> 2
[(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => [4,2,1,1]
=> 2
[(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => [4,2,1,1]
=> 2
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [2,2,1,1,1,1]
=> 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [2,2,2,1,1]
=> 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [2,2,2,1,1]
=> 1
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [4,2,2]
=> 2
[(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => [4,2,1,1]
=> 2
[(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => [4,2,1,1]
=> 2
[(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => [4,2,1,1]
=> 2
[(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [6,2]
=> 3
[(1,7),(2,5),(3,4),(6,8)]
=> [7,5,4,3,2,8,1,6] => [6,1,1]
=> 3
[(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [6,2]
=> 3
[(1,8),(2,6),(3,4),(5,7)]
=> [8,6,4,3,7,2,5,1] => [6,1,1]
=> 3
[(1,7),(2,6),(3,4),(5,8)]
=> [7,6,4,3,8,2,1,5] => [6,1,1]
=> 3
[(1,6),(2,7),(3,4),(5,8)]
=> [6,7,4,3,8,1,2,5] => [4,1,1,1,1]
=> 2
[(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => [4,2,1,1]
=> 2
[(1,4),(2,7),(3,5),(6,8)]
=> [4,7,5,1,3,8,2,6] => [4,1,1,1,1]
=> 2
[(1,3),(2,7),(4,5),(6,8)]
=> [3,7,1,5,4,8,2,6] => [4,1,1,1,1]
=> 2
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,7,5,4,8,3,6] => [4,2,1,1]
=> 2
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [4,2,2]
=> 2
[(1,3),(2,8),(4,5),(6,7)]
=> [3,8,1,5,4,7,6,2] => [4,2,1,1]
=> 2
[(1,4),(2,8),(3,5),(6,7)]
=> [4,8,5,1,3,7,6,2] => [4,2,1,1]
=> 2
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [5,3,2,6,1,4,8,7,10,9] => ?
=> ? = 2
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [7,3,2,5,4,8,1,6,10,9] => ?
=> ? = 2
[(1,9),(2,3),(4,5),(6,7),(8,10)]
=> [9,3,2,5,4,7,6,10,1,8] => ?
=> ? = 2
[(1,10),(2,4),(3,5),(6,7),(8,9)]
=> [10,4,5,2,3,7,6,9,8,1] => ?
=> ? = 2
[(1,9),(2,4),(3,5),(6,7),(8,10)]
=> [9,4,5,2,3,7,6,10,1,8] => ?
=> ? = 2
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [8,4,5,2,3,7,6,1,10,9] => ?
=> ? = 2
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [7,4,5,2,3,8,1,6,10,9] => ?
=> ? = 2
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [6,4,5,2,3,1,8,7,10,9] => ?
=> ? = 2
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [5,4,6,2,1,3,8,7,10,9] => ?
=> ? = 2
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [3,6,1,5,4,2,8,7,10,9] => ?
=> ? = 2
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [4,6,5,1,3,2,8,7,10,9] => ?
=> ? = 2
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [5,6,4,3,1,2,8,7,10,9] => ?
=> ? = 2
[(1,7),(2,5),(3,4),(6,8),(9,10)]
=> [7,5,4,3,2,8,1,6,10,9] => ?
=> ? = 3
[(1,9),(2,5),(3,4),(6,7),(8,10)]
=> [9,5,4,3,2,7,6,10,1,8] => ?
=> ? = 3
[(1,10),(2,6),(3,4),(5,7),(8,9)]
=> [10,6,4,3,7,2,5,9,8,1] => ?
=> ? = 3
[(1,9),(2,6),(3,4),(5,7),(8,10)]
=> [9,6,4,3,7,2,5,10,1,8] => ?
=> ? = 3
[(1,8),(2,6),(3,4),(5,7),(9,10)]
=> [8,6,4,3,7,2,5,1,10,9] => ?
=> ? = 3
[(1,7),(2,6),(3,4),(5,8),(9,10)]
=> [7,6,4,3,8,2,1,5,10,9] => ?
=> ? = 3
[(1,6),(2,7),(3,4),(5,8),(9,10)]
=> [6,7,4,3,8,1,2,5,10,9] => ?
=> ? = 2
[(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [5,7,4,3,1,8,2,6,10,9] => ?
=> ? = 2
[(1,4),(2,7),(3,5),(6,8),(9,10)]
=> [4,7,5,1,3,8,2,6,10,9] => ?
=> ? = 2
[(1,3),(2,7),(4,5),(6,8),(9,10)]
=> [3,7,1,5,4,8,2,6,10,9] => ?
=> ? = 2
[(1,2),(3,7),(4,5),(6,8),(9,10)]
=> [2,1,7,5,4,8,3,6,10,9] => ?
=> ? = 2
[(1,3),(2,8),(4,5),(6,7),(9,10)]
=> [3,8,1,5,4,7,6,2,10,9] => ?
=> ? = 2
[(1,4),(2,8),(3,5),(6,7),(9,10)]
=> [4,8,5,1,3,7,6,2,10,9] => ?
=> ? = 2
[(1,5),(2,8),(3,4),(6,7),(9,10)]
=> [5,8,4,3,1,7,6,2,10,9] => ?
=> ? = 2
[(1,6),(2,8),(3,4),(5,7),(9,10)]
=> [6,8,4,3,7,1,5,2,10,9] => ?
=> ? = 2
[(1,7),(2,8),(3,4),(5,6),(9,10)]
=> [7,8,4,3,6,5,1,2,10,9] => ?
=> ? = 2
[(1,9),(2,7),(3,4),(5,6),(8,10)]
=> [9,7,4,3,6,5,2,10,1,8] => ?
=> ? = 3
[(1,10),(2,8),(3,4),(5,6),(7,9)]
=> [10,8,4,3,6,5,9,2,7,1] => ?
=> ? = 3
[(1,9),(2,8),(3,4),(5,6),(7,10)]
=> [9,8,4,3,6,5,10,2,1,7] => ?
=> ? = 3
[(1,8),(2,9),(3,4),(5,6),(7,10)]
=> [8,9,4,3,6,5,10,1,2,7] => ?
=> ? = 2
[(1,7),(2,9),(3,4),(5,6),(8,10)]
=> [7,9,4,3,6,5,1,10,2,8] => ?
=> ? = 2
[(1,6),(2,9),(3,4),(5,7),(8,10)]
=> [6,9,4,3,7,1,5,10,2,8] => ?
=> ? = 2
[(1,5),(2,9),(3,4),(6,7),(8,10)]
=> [5,9,4,3,1,7,6,10,2,8] => ?
=> ? = 2
[(1,4),(2,9),(3,5),(6,7),(8,10)]
=> [4,9,5,1,3,7,6,10,2,8] => ?
=> ? = 2
[(1,3),(2,9),(4,5),(6,7),(8,10)]
=> [3,9,1,5,4,7,6,10,2,8] => ?
=> ? = 2
[(1,2),(3,9),(4,5),(6,7),(8,10)]
=> [2,1,9,5,4,7,6,10,3,8] => ?
=> ? = 2
[(1,3),(2,10),(4,5),(6,7),(8,9)]
=> [3,10,1,5,4,7,6,9,8,2] => ?
=> ? = 2
[(1,4),(2,10),(3,5),(6,7),(8,9)]
=> [4,10,5,1,3,7,6,9,8,2] => ?
=> ? = 2
[(1,5),(2,10),(3,4),(6,7),(8,9)]
=> [5,10,4,3,1,7,6,9,8,2] => ?
=> ? = 2
[(1,6),(2,10),(3,4),(5,7),(8,9)]
=> [6,10,4,3,7,1,5,9,8,2] => ?
=> ? = 2
[(1,7),(2,10),(3,4),(5,6),(8,9)]
=> [7,10,4,3,6,5,1,9,8,2] => ?
=> ? = 2
[(1,8),(2,10),(3,4),(5,6),(7,9)]
=> [8,10,4,3,6,5,9,1,7,2] => ?
=> ? = 2
[(1,9),(2,10),(3,4),(5,6),(7,8)]
=> [9,10,4,3,6,5,8,7,1,2] => ?
=> ? = 2
[(1,10),(2,9),(3,5),(4,6),(7,8)]
=> [10,9,5,6,3,4,8,7,2,1] => ?
=> ? = 3
[(1,9),(2,10),(3,5),(4,6),(7,8)]
=> [9,10,5,6,3,4,8,7,1,2] => ?
=> ? = 2
[(1,8),(2,10),(3,5),(4,6),(7,9)]
=> [8,10,5,6,3,4,9,1,7,2] => ?
=> ? = 2
[(1,7),(2,10),(3,5),(4,6),(8,9)]
=> [7,10,5,6,3,4,1,9,8,2] => ?
=> ? = 2
[(1,6),(2,10),(3,5),(4,7),(8,9)]
=> [6,10,5,7,3,1,4,9,8,2] => ?
=> ? = 2
Description
Half of the largest even part of an integer partition.
The largest even part is recorded by [[St000995]].
The following 34 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000143The largest repeated part of a partition. St000150The floored half-sum of the multiplicities of a partition. St000142The number of even parts of a partition. St000146The Andrews-Garvan crank of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000473The number of parts of a partition that are strictly bigger than the number of ones. St001280The number of parts of an integer partition that are at least two. St001424The number of distinct squares in a binary word. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000676The number of odd rises of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000097The order of the largest clique of the graph. St000451The length of the longest pattern of the form k 1 2. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001029The size of the core of a graph. St000098The chromatic number of a graph. St000741The Colin de Verdière graph invariant. St001644The dimension of a graph. St000454The largest eigenvalue of a graph if it is integral. St000093The cardinality of a maximal independent set of vertices of a graph. St001330The hat guessing number of a graph. St000527The width of the poset. St000223The number of nestings in the permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000528The height of a poset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000068The number of minimal elements in a poset. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000441The number of successions of a permutation. St001728The number of invisible descents of a permutation.
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