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Your data matches 15 different statistics following compositions of up to 3 maps.
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Matching statistic: St001590
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
St001590: 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)]
=> 2
[(1,4),(2,3)]
=> 1
[(1,2),(3,4),(5,6)]
=> 1
[(1,3),(2,4),(5,6)]
=> 2
[(1,4),(2,3),(5,6)]
=> 1
[(1,5),(2,3),(4,6)]
=> 2
[(1,6),(2,3),(4,5)]
=> 1
[(1,6),(2,4),(3,5)]
=> 2
[(1,5),(2,4),(3,6)]
=> 2
[(1,4),(2,5),(3,6)]
=> 3
[(1,3),(2,5),(4,6)]
=> 2
[(1,2),(3,5),(4,6)]
=> 2
[(1,2),(3,6),(4,5)]
=> 1
[(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)]
=> 1
[(1,2),(3,4),(5,6),(7,8)]
=> 1
[(1,3),(2,4),(5,6),(7,8)]
=> 2
[(1,4),(2,3),(5,6),(7,8)]
=> 1
[(1,5),(2,3),(4,6),(7,8)]
=> 2
[(1,6),(2,3),(4,5),(7,8)]
=> 1
[(1,7),(2,3),(4,5),(6,8)]
=> 2
[(1,8),(2,3),(4,5),(6,7)]
=> 1
[(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)]
=> 3
[(1,3),(2,5),(4,6),(7,8)]
=> 2
[(1,2),(3,5),(4,6),(7,8)]
=> 2
[(1,2),(3,6),(4,5),(7,8)]
=> 1
[(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)]
=> 1
[(1,7),(2,5),(3,4),(6,8)]
=> 2
[(1,8),(2,5),(3,4),(6,7)]
=> 1
[(1,8),(2,6),(3,4),(5,7)]
=> 2
[(1,7),(2,6),(3,4),(5,8)]
=> 2
[(1,6),(2,7),(3,4),(5,8)]
=> 3
[(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)]
=> 1
[(1,3),(2,8),(4,5),(6,7)]
=> 2
[(1,4),(2,8),(3,5),(6,7)]
=> 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: St001589
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00165: Perfect matchings —Chen Deng Du Stanley Yan⟶ Perfect matchings
St001589: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001589: 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)]
=> 2
[(1,4),(2,3)]
=> [(1,3),(2,4)]
=> 1
[(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)]
=> 2
[(1,4),(2,3),(5,6)]
=> [(1,3),(2,4),(5,6)]
=> 1
[(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)]
=> 1
[(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)]
=> 3
[(1,3),(2,5),(4,6)]
=> [(1,6),(2,3),(4,5)]
=> 2
[(1,2),(3,5),(4,6)]
=> [(1,2),(3,6),(4,5)]
=> 2
[(1,2),(3,6),(4,5)]
=> [(1,2),(3,5),(4,6)]
=> 1
[(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)]
=> 1
[(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)]
=> 2
[(1,4),(2,3),(5,6),(7,8)]
=> [(1,3),(2,4),(5,6),(7,8)]
=> 1
[(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)]
=> 1
[(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)]
=> 1
[(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)]
=> 3
[(1,3),(2,5),(4,6),(7,8)]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 2
[(1,2),(3,5),(4,6),(7,8)]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 2
[(1,2),(3,6),(4,5),(7,8)]
=> [(1,2),(3,5),(4,6),(7,8)]
=> 1
[(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)]
=> 1
[(1,7),(2,5),(3,4),(6,8)]
=> [(1,4),(2,5),(3,8),(6,7)]
=> 2
[(1,8),(2,5),(3,4),(6,7)]
=> [(1,4),(2,5),(3,7),(6,8)]
=> 1
[(1,8),(2,6),(3,4),(5,7)]
=> [(1,4),(2,7),(3,8),(5,6)]
=> 2
[(1,7),(2,6),(3,4),(5,8)]
=> [(1,4),(2,6),(3,8),(5,7)]
=> 2
[(1,6),(2,7),(3,4),(5,8)]
=> [(1,8),(2,4),(3,7),(5,6)]
=> 3
[(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)]
=> 1
[(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 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: St000147
Mp00165: Perfect matchings —Chen Deng Du Stanley Yan⟶ Perfect matchings
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%
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%
Values
[(1,2)]
=> [(1,2)]
=> [2,1] => [2]
=> 2 = 1 + 1
[(1,2),(3,4)]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[(1,3),(2,4)]
=> [(1,4),(2,3)]
=> [3,4,2,1] => [3,1]
=> 3 = 2 + 1
[(1,4),(2,3)]
=> [(1,3),(2,4)]
=> [3,4,1,2] => [2,1,1]
=> 2 = 1 + 1
[(1,2),(3,4),(5,6)]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,2,2]
=> 2 = 1 + 1
[(1,3),(2,4),(5,6)]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [3,2,1]
=> 3 = 2 + 1
[(1,4),(2,3),(5,6)]
=> [(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [2,2,1,1]
=> 2 = 1 + 1
[(1,5),(2,3),(4,6)]
=> [(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [3,2,1]
=> 3 = 2 + 1
[(1,6),(2,3),(4,5)]
=> [(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [2,2,1,1]
=> 2 = 1 + 1
[(1,6),(2,4),(3,5)]
=> [(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [3,1,1,1]
=> 3 = 2 + 1
[(1,5),(2,4),(3,6)]
=> [(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [3,1,1,1]
=> 3 = 2 + 1
[(1,4),(2,5),(3,6)]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [4,1,1]
=> 4 = 3 + 1
[(1,3),(2,5),(4,6)]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [3,2,1]
=> 3 = 2 + 1
[(1,2),(3,5),(4,6)]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [3,2,1]
=> 3 = 2 + 1
[(1,2),(3,6),(4,5)]
=> [(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,2,1,1]
=> 2 = 1 + 1
[(1,3),(2,6),(4,5)]
=> [(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [3,1,1,1]
=> 3 = 2 + 1
[(1,4),(2,6),(3,5)]
=> [(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [3,1,1,1]
=> 3 = 2 + 1
[(1,5),(2,6),(3,4)]
=> [(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [3,1,1,1]
=> 3 = 2 + 1
[(1,6),(2,5),(3,4)]
=> [(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [2,1,1,1,1]
=> 2 = 1 + 1
[(1,2),(3,4),(5,6),(7,8)]
=> [(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)]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [3,2,2,1]
=> 3 = 2 + 1
[(1,4),(2,3),(5,6),(7,8)]
=> [(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [2,2,2,1,1]
=> 2 = 1 + 1
[(1,5),(2,3),(4,6),(7,8)]
=> [(1,3),(2,6),(4,5),(7,8)]
=> [3,5,1,6,4,2,8,7] => [3,2,2,1]
=> 3 = 2 + 1
[(1,6),(2,3),(4,5),(7,8)]
=> [(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [2,2,2,1,1]
=> 2 = 1 + 1
[(1,7),(2,3),(4,5),(6,8)]
=> [(1,3),(2,5),(4,8),(6,7)]
=> [3,5,1,7,2,8,6,4] => [3,2,2,1]
=> 3 = 2 + 1
[(1,8),(2,3),(4,5),(6,7)]
=> [(1,3),(2,5),(4,7),(6,8)]
=> [3,5,1,7,2,8,4,6] => [2,2,2,1,1]
=> 2 = 1 + 1
[(1,8),(2,4),(3,5),(6,7)]
=> [(1,5),(2,7),(3,4),(6,8)]
=> [4,5,7,3,1,8,2,6] => [3,2,1,1,1]
=> 3 = 2 + 1
[(1,7),(2,4),(3,5),(6,8)]
=> [(1,5),(2,8),(3,4),(6,7)]
=> [4,5,7,3,1,8,6,2] => [3,3,1,1]
=> 3 = 2 + 1
[(1,6),(2,4),(3,5),(7,8)]
=> [(1,5),(2,6),(3,4),(7,8)]
=> [4,5,6,3,1,2,8,7] => [3,2,1,1,1]
=> 3 = 2 + 1
[(1,5),(2,4),(3,6),(7,8)]
=> [(1,4),(2,6),(3,5),(7,8)]
=> [4,5,6,1,3,2,8,7] => [3,2,1,1,1]
=> 3 = 2 + 1
[(1,4),(2,5),(3,6),(7,8)]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [4,2,1,1]
=> 4 = 3 + 1
[(1,3),(2,5),(4,6),(7,8)]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [3,2,2,1]
=> 3 = 2 + 1
[(1,2),(3,5),(4,6),(7,8)]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [3,2,2,1]
=> 3 = 2 + 1
[(1,2),(3,6),(4,5),(7,8)]
=> [(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [2,2,2,1,1]
=> 2 = 1 + 1
[(1,3),(2,6),(4,5),(7,8)]
=> [(1,5),(2,3),(4,6),(7,8)]
=> [3,5,2,6,1,4,8,7] => [3,2,1,1,1]
=> 3 = 2 + 1
[(1,4),(2,6),(3,5),(7,8)]
=> [(1,5),(2,4),(3,6),(7,8)]
=> [4,5,6,2,1,3,8,7] => [3,2,1,1,1]
=> 3 = 2 + 1
[(1,5),(2,6),(3,4),(7,8)]
=> [(1,6),(2,4),(3,5),(7,8)]
=> [4,5,6,2,3,1,8,7] => [3,2,1,1,1]
=> 3 = 2 + 1
[(1,6),(2,5),(3,4),(7,8)]
=> [(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
[(1,7),(2,5),(3,4),(6,8)]
=> [(1,4),(2,5),(3,8),(6,7)]
=> [4,5,7,1,2,8,6,3] => [3,2,1,1,1]
=> 3 = 2 + 1
[(1,8),(2,5),(3,4),(6,7)]
=> [(1,4),(2,5),(3,7),(6,8)]
=> [4,5,7,1,2,8,3,6] => [2,2,1,1,1,1]
=> 2 = 1 + 1
[(1,8),(2,6),(3,4),(5,7)]
=> [(1,4),(2,7),(3,8),(5,6)]
=> [4,6,7,1,8,5,2,3] => [3,2,1,1,1]
=> 3 = 2 + 1
[(1,7),(2,6),(3,4),(5,8)]
=> [(1,4),(2,6),(3,8),(5,7)]
=> [4,6,7,1,8,2,5,3] => [3,2,1,1,1]
=> 3 = 2 + 1
[(1,6),(2,7),(3,4),(5,8)]
=> [(1,8),(2,4),(3,7),(5,6)]
=> [4,6,7,2,8,5,3,1] => [4,2,1,1]
=> 4 = 3 + 1
[(1,5),(2,7),(3,4),(6,8)]
=> [(1,8),(2,4),(3,5),(6,7)]
=> [4,5,7,2,3,8,6,1] => [3,2,1,1,1]
=> 3 = 2 + 1
[(1,4),(2,7),(3,5),(6,8)]
=> [(1,5),(2,4),(3,8),(6,7)]
=> [4,5,7,2,1,8,6,3] => [3,3,1,1]
=> 3 = 2 + 1
[(1,3),(2,7),(4,5),(6,8)]
=> [(1,5),(2,3),(4,8),(6,7)]
=> [3,5,2,7,1,8,6,4] => [3,3,1,1]
=> 3 = 2 + 1
[(1,2),(3,7),(4,5),(6,8)]
=> [(1,2),(3,5),(4,8),(6,7)]
=> [2,1,5,7,3,8,6,4] => [3,2,2,1]
=> 3 = 2 + 1
[(1,2),(3,8),(4,5),(6,7)]
=> [(1,2),(3,5),(4,7),(6,8)]
=> [2,1,5,7,3,8,4,6] => [2,2,2,1,1]
=> 2 = 1 + 1
[(1,3),(2,8),(4,5),(6,7)]
=> [(1,5),(2,3),(4,7),(6,8)]
=> [3,5,2,7,1,8,4,6] => [3,2,1,1,1]
=> 3 = 2 + 1
[(1,4),(2,8),(3,5),(6,7)]
=> [(1,5),(2,4),(3,7),(6,8)]
=> [4,5,7,2,1,8,3,6] => [3,2,1,1,1]
=> 3 = 2 + 1
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [3,5,1,6,4,2,8,7,10,9] => ?
=> ? = 2 + 1
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [(1,3),(2,5),(4,8),(6,7),(9,10)]
=> [3,5,1,7,2,8,6,4,10,9] => ?
=> ? = 2 + 1
[(1,9),(2,3),(4,5),(6,7),(8,10)]
=> [(1,3),(2,5),(4,7),(6,10),(8,9)]
=> [3,5,1,7,2,9,4,10,8,6] => ?
=> ? = 2 + 1
[(1,10),(2,4),(3,5),(6,7),(8,9)]
=> [(1,5),(2,7),(3,4),(6,9),(8,10)]
=> [4,5,7,3,1,9,2,10,6,8] => ?
=> ? = 2 + 1
[(1,9),(2,4),(3,5),(6,7),(8,10)]
=> [(1,5),(2,7),(3,4),(6,10),(8,9)]
=> [4,5,7,3,1,9,2,10,8,6] => ?
=> ? = 2 + 1
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [4,5,7,3,1,8,2,6,10,9] => ?
=> ? = 2 + 1
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [(1,5),(2,8),(3,4),(6,7),(9,10)]
=> [4,5,7,3,1,8,6,2,10,9] => ?
=> ? = 2 + 1
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [4,5,6,3,1,2,8,7,10,9] => ?
=> ? = 2 + 1
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [4,5,6,1,3,2,8,7,10,9] => ?
=> ? = 2 + 1
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [3,5,2,6,1,4,8,7,10,9] => ?
=> ? = 2 + 1
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [4,5,6,2,1,3,8,7,10,9] => ?
=> ? = 2 + 1
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [4,5,6,2,3,1,8,7,10,9] => ?
=> ? = 2 + 1
[(1,7),(2,5),(3,4),(6,8),(9,10)]
=> [(1,4),(2,5),(3,8),(6,7),(9,10)]
=> [4,5,7,1,2,8,6,3,10,9] => ?
=> ? = 2 + 1
[(1,9),(2,5),(3,4),(6,7),(8,10)]
=> [(1,4),(2,5),(3,7),(6,10),(8,9)]
=> [4,5,7,1,2,9,3,10,8,6] => ?
=> ? = 2 + 1
[(1,10),(2,6),(3,4),(5,7),(8,9)]
=> [(1,4),(2,7),(3,9),(5,6),(8,10)]
=> [4,6,7,1,9,5,2,10,3,8] => ?
=> ? = 2 + 1
[(1,9),(2,6),(3,4),(5,7),(8,10)]
=> [(1,4),(2,7),(3,10),(5,6),(8,9)]
=> [4,6,7,1,9,5,2,10,8,3] => ?
=> ? = 2 + 1
[(1,8),(2,6),(3,4),(5,7),(9,10)]
=> [(1,4),(2,7),(3,8),(5,6),(9,10)]
=> [4,6,7,1,8,5,2,3,10,9] => ?
=> ? = 2 + 1
[(1,7),(2,6),(3,4),(5,8),(9,10)]
=> [(1,4),(2,6),(3,8),(5,7),(9,10)]
=> [4,6,7,1,8,2,5,3,10,9] => ?
=> ? = 2 + 1
[(1,6),(2,7),(3,4),(5,8),(9,10)]
=> [(1,8),(2,4),(3,7),(5,6),(9,10)]
=> [4,6,7,2,8,5,3,1,10,9] => ?
=> ? = 3 + 1
[(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [4,5,7,2,3,8,6,1,10,9] => ?
=> ? = 2 + 1
[(1,4),(2,7),(3,5),(6,8),(9,10)]
=> [(1,5),(2,4),(3,8),(6,7),(9,10)]
=> [4,5,7,2,1,8,6,3,10,9] => ?
=> ? = 2 + 1
[(1,3),(2,7),(4,5),(6,8),(9,10)]
=> [(1,5),(2,3),(4,8),(6,7),(9,10)]
=> [3,5,2,7,1,8,6,4,10,9] => ?
=> ? = 2 + 1
[(1,2),(3,7),(4,5),(6,8),(9,10)]
=> [(1,2),(3,5),(4,8),(6,7),(9,10)]
=> [2,1,5,7,3,8,6,4,10,9] => ?
=> ? = 2 + 1
[(1,3),(2,8),(4,5),(6,7),(9,10)]
=> [(1,5),(2,3),(4,7),(6,8),(9,10)]
=> [3,5,2,7,1,8,4,6,10,9] => ?
=> ? = 2 + 1
[(1,4),(2,8),(3,5),(6,7),(9,10)]
=> [(1,5),(2,4),(3,7),(6,8),(9,10)]
=> [4,5,7,2,1,8,3,6,10,9] => ?
=> ? = 2 + 1
[(1,5),(2,8),(3,4),(6,7),(9,10)]
=> [(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [4,5,7,2,3,8,1,6,10,9] => ?
=> ? = 2 + 1
[(1,6),(2,8),(3,4),(5,7),(9,10)]
=> [(1,7),(2,4),(3,6),(5,8),(9,10)]
=> [4,6,7,2,8,3,1,5,10,9] => ?
=> ? = 2 + 1
[(1,7),(2,8),(3,4),(5,6),(9,10)]
=> [(1,8),(2,4),(3,6),(5,7),(9,10)]
=> [4,6,7,2,8,3,5,1,10,9] => ?
=> ? = 2 + 1
[(1,9),(2,7),(3,4),(5,6),(8,10)]
=> [(1,4),(2,6),(3,7),(5,10),(8,9)]
=> [4,6,7,1,9,2,3,10,8,5] => ?
=> ? = 2 + 1
[(1,10),(2,8),(3,4),(5,6),(7,9)]
=> [(1,4),(2,6),(3,9),(5,10),(7,8)]
=> [4,6,8,1,9,2,10,7,3,5] => ?
=> ? = 2 + 1
[(1,9),(2,8),(3,4),(5,6),(7,10)]
=> [(1,4),(2,6),(3,8),(5,10),(7,9)]
=> [4,6,8,1,9,2,10,3,7,5] => ?
=> ? = 2 + 1
[(1,8),(2,9),(3,4),(5,6),(7,10)]
=> [(1,10),(2,4),(3,6),(5,9),(7,8)]
=> [4,6,8,2,9,3,10,7,5,1] => ?
=> ? = 3 + 1
[(1,7),(2,9),(3,4),(5,6),(8,10)]
=> [(1,10),(2,4),(3,6),(5,7),(8,9)]
=> [4,6,7,2,9,3,5,10,8,1] => ?
=> ? = 2 + 1
[(1,6),(2,9),(3,4),(5,7),(8,10)]
=> [(1,7),(2,4),(3,6),(5,10),(8,9)]
=> [4,6,7,2,9,3,1,10,8,5] => ?
=> ? = 2 + 1
[(1,5),(2,9),(3,4),(6,7),(8,10)]
=> [(1,7),(2,4),(3,5),(6,10),(8,9)]
=> [4,5,7,2,3,9,1,10,8,6] => ?
=> ? = 2 + 1
[(1,4),(2,9),(3,5),(6,7),(8,10)]
=> [(1,5),(2,4),(3,7),(6,10),(8,9)]
=> [4,5,7,2,1,9,3,10,8,6] => ?
=> ? = 2 + 1
[(1,3),(2,9),(4,5),(6,7),(8,10)]
=> [(1,5),(2,3),(4,7),(6,10),(8,9)]
=> [3,5,2,7,1,9,4,10,8,6] => ?
=> ? = 2 + 1
[(1,2),(3,9),(4,5),(6,7),(8,10)]
=> [(1,2),(3,5),(4,7),(6,10),(8,9)]
=> [2,1,5,7,3,9,4,10,8,6] => ?
=> ? = 2 + 1
[(1,3),(2,10),(4,5),(6,7),(8,9)]
=> [(1,5),(2,3),(4,7),(6,9),(8,10)]
=> [3,5,2,7,1,9,4,10,6,8] => ?
=> ? = 2 + 1
[(1,4),(2,10),(3,5),(6,7),(8,9)]
=> [(1,5),(2,4),(3,7),(6,9),(8,10)]
=> [4,5,7,2,1,9,3,10,6,8] => ?
=> ? = 2 + 1
[(1,5),(2,10),(3,4),(6,7),(8,9)]
=> [(1,7),(2,4),(3,5),(6,9),(8,10)]
=> [4,5,7,2,3,9,1,10,6,8] => ?
=> ? = 2 + 1
[(1,6),(2,10),(3,4),(5,7),(8,9)]
=> [(1,7),(2,4),(3,6),(5,9),(8,10)]
=> [4,6,7,2,9,3,1,10,5,8] => ?
=> ? = 2 + 1
[(1,7),(2,10),(3,4),(5,6),(8,9)]
=> [(1,9),(2,4),(3,6),(5,7),(8,10)]
=> [4,6,7,2,9,3,5,10,1,8] => ?
=> ? = 2 + 1
[(1,8),(2,10),(3,4),(5,6),(7,9)]
=> [(1,9),(2,4),(3,6),(5,8),(7,10)]
=> [4,6,8,2,9,3,10,5,1,7] => ?
=> ? = 2 + 1
[(1,9),(2,10),(3,4),(5,6),(7,8)]
=> [(1,10),(2,4),(3,6),(5,8),(7,9)]
=> [4,6,8,2,9,3,10,5,7,1] => ?
=> ? = 2 + 1
[(1,10),(2,9),(3,5),(4,6),(7,8)]
=> [(1,6),(2,8),(3,9),(4,5),(7,10)]
=> [5,6,8,9,4,1,10,2,3,7] => ?
=> ? = 2 + 1
[(1,9),(2,10),(3,5),(4,6),(7,8)]
=> [(1,6),(2,5),(3,10),(4,8),(7,9)]
=> [5,6,8,9,2,1,10,4,7,3] => ?
=> ? = 2 + 1
[(1,8),(2,10),(3,5),(4,6),(7,9)]
=> [(1,6),(2,5),(3,9),(4,8),(7,10)]
=> [5,6,8,9,2,1,10,4,3,7] => ?
=> ? = 2 + 1
[(1,7),(2,10),(3,5),(4,6),(8,9)]
=> [(1,6),(2,5),(3,9),(4,7),(8,10)]
=> [5,6,7,9,2,1,4,10,3,8] => ?
=> ? = 2 + 1
[(1,6),(2,10),(3,5),(4,7),(8,9)]
=> [(1,7),(2,5),(3,9),(4,6),(8,10)]
=> [5,6,7,9,2,4,1,10,3,8] => ?
=> ? = 2 + 1
Description
The largest part of an integer partition.
Matching statistic: St000010
Mp00165: Perfect matchings —Chen Deng Du Stanley Yan⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [(1,2)]
=> [2,1] => [1,1]
=> 2 = 1 + 1
[(1,2),(3,4)]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[(1,3),(2,4)]
=> [(1,4),(2,3)]
=> [3,4,2,1] => [2,1,1]
=> 3 = 2 + 1
[(1,4),(2,3)]
=> [(1,3),(2,4)]
=> [3,4,1,2] => [2,2]
=> 2 = 1 + 1
[(1,2),(3,4),(5,6)]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [3,3]
=> 2 = 1 + 1
[(1,3),(2,4),(5,6)]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [3,2,1]
=> 3 = 2 + 1
[(1,4),(2,3),(5,6)]
=> [(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [3,3]
=> 2 = 1 + 1
[(1,5),(2,3),(4,6)]
=> [(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [3,2,1]
=> 3 = 2 + 1
[(1,6),(2,3),(4,5)]
=> [(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [3,3]
=> 2 = 1 + 1
[(1,6),(2,4),(3,5)]
=> [(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [3,2,1]
=> 3 = 2 + 1
[(1,5),(2,4),(3,6)]
=> [(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [3,2,1]
=> 3 = 2 + 1
[(1,4),(2,5),(3,6)]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [3,1,1,1]
=> 4 = 3 + 1
[(1,3),(2,5),(4,6)]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [3,2,1]
=> 3 = 2 + 1
[(1,2),(3,5),(4,6)]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [3,2,1]
=> 3 = 2 + 1
[(1,2),(3,6),(4,5)]
=> [(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [3,3]
=> 2 = 1 + 1
[(1,3),(2,6),(4,5)]
=> [(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [3,2,1]
=> 3 = 2 + 1
[(1,4),(2,6),(3,5)]
=> [(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [3,2,1]
=> 3 = 2 + 1
[(1,5),(2,6),(3,4)]
=> [(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [3,2,1]
=> 3 = 2 + 1
[(1,6),(2,5),(3,4)]
=> [(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [3,3]
=> 2 = 1 + 1
[(1,2),(3,4),(5,6),(7,8)]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [4,4]
=> 2 = 1 + 1
[(1,3),(2,4),(5,6),(7,8)]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [4,3,1]
=> 3 = 2 + 1
[(1,4),(2,3),(5,6),(7,8)]
=> [(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [4,4]
=> 2 = 1 + 1
[(1,5),(2,3),(4,6),(7,8)]
=> [(1,3),(2,6),(4,5),(7,8)]
=> [3,5,1,6,4,2,8,7] => [4,3,1]
=> 3 = 2 + 1
[(1,6),(2,3),(4,5),(7,8)]
=> [(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [4,4]
=> 2 = 1 + 1
[(1,7),(2,3),(4,5),(6,8)]
=> [(1,3),(2,5),(4,8),(6,7)]
=> [3,5,1,7,2,8,6,4] => [4,3,1]
=> 3 = 2 + 1
[(1,8),(2,3),(4,5),(6,7)]
=> [(1,3),(2,5),(4,7),(6,8)]
=> [3,5,1,7,2,8,4,6] => [4,4]
=> 2 = 1 + 1
[(1,8),(2,4),(3,5),(6,7)]
=> [(1,5),(2,7),(3,4),(6,8)]
=> [4,5,7,3,1,8,2,6] => [4,3,1]
=> 3 = 2 + 1
[(1,7),(2,4),(3,5),(6,8)]
=> [(1,5),(2,8),(3,4),(6,7)]
=> [4,5,7,3,1,8,6,2] => [4,2,2]
=> 3 = 2 + 1
[(1,6),(2,4),(3,5),(7,8)]
=> [(1,5),(2,6),(3,4),(7,8)]
=> [4,5,6,3,1,2,8,7] => [4,3,1]
=> 3 = 2 + 1
[(1,5),(2,4),(3,6),(7,8)]
=> [(1,4),(2,6),(3,5),(7,8)]
=> [4,5,6,1,3,2,8,7] => [4,3,1]
=> 3 = 2 + 1
[(1,4),(2,5),(3,6),(7,8)]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [4,2,1,1]
=> 4 = 3 + 1
[(1,3),(2,5),(4,6),(7,8)]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [4,3,1]
=> 3 = 2 + 1
[(1,2),(3,5),(4,6),(7,8)]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [4,3,1]
=> 3 = 2 + 1
[(1,2),(3,6),(4,5),(7,8)]
=> [(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [4,4]
=> 2 = 1 + 1
[(1,3),(2,6),(4,5),(7,8)]
=> [(1,5),(2,3),(4,6),(7,8)]
=> [3,5,2,6,1,4,8,7] => [4,3,1]
=> 3 = 2 + 1
[(1,4),(2,6),(3,5),(7,8)]
=> [(1,5),(2,4),(3,6),(7,8)]
=> [4,5,6,2,1,3,8,7] => [4,3,1]
=> 3 = 2 + 1
[(1,5),(2,6),(3,4),(7,8)]
=> [(1,6),(2,4),(3,5),(7,8)]
=> [4,5,6,2,3,1,8,7] => [4,3,1]
=> 3 = 2 + 1
[(1,6),(2,5),(3,4),(7,8)]
=> [(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [4,4]
=> 2 = 1 + 1
[(1,7),(2,5),(3,4),(6,8)]
=> [(1,4),(2,5),(3,8),(6,7)]
=> [4,5,7,1,2,8,6,3] => [4,3,1]
=> 3 = 2 + 1
[(1,8),(2,5),(3,4),(6,7)]
=> [(1,4),(2,5),(3,7),(6,8)]
=> [4,5,7,1,2,8,3,6] => [4,4]
=> 2 = 1 + 1
[(1,8),(2,6),(3,4),(5,7)]
=> [(1,4),(2,7),(3,8),(5,6)]
=> [4,6,7,1,8,5,2,3] => [4,3,1]
=> 3 = 2 + 1
[(1,7),(2,6),(3,4),(5,8)]
=> [(1,4),(2,6),(3,8),(5,7)]
=> [4,6,7,1,8,2,5,3] => [4,3,1]
=> 3 = 2 + 1
[(1,6),(2,7),(3,4),(5,8)]
=> [(1,8),(2,4),(3,7),(5,6)]
=> [4,6,7,2,8,5,3,1] => [4,2,1,1]
=> 4 = 3 + 1
[(1,5),(2,7),(3,4),(6,8)]
=> [(1,8),(2,4),(3,5),(6,7)]
=> [4,5,7,2,3,8,6,1] => [4,3,1]
=> 3 = 2 + 1
[(1,4),(2,7),(3,5),(6,8)]
=> [(1,5),(2,4),(3,8),(6,7)]
=> [4,5,7,2,1,8,6,3] => [4,2,2]
=> 3 = 2 + 1
[(1,3),(2,7),(4,5),(6,8)]
=> [(1,5),(2,3),(4,8),(6,7)]
=> [3,5,2,7,1,8,6,4] => [4,2,2]
=> 3 = 2 + 1
[(1,2),(3,7),(4,5),(6,8)]
=> [(1,2),(3,5),(4,8),(6,7)]
=> [2,1,5,7,3,8,6,4] => [4,3,1]
=> 3 = 2 + 1
[(1,2),(3,8),(4,5),(6,7)]
=> [(1,2),(3,5),(4,7),(6,8)]
=> [2,1,5,7,3,8,4,6] => [4,4]
=> 2 = 1 + 1
[(1,3),(2,8),(4,5),(6,7)]
=> [(1,5),(2,3),(4,7),(6,8)]
=> [3,5,2,7,1,8,4,6] => [4,3,1]
=> 3 = 2 + 1
[(1,4),(2,8),(3,5),(6,7)]
=> [(1,5),(2,4),(3,7),(6,8)]
=> [4,5,7,2,1,8,3,6] => [4,3,1]
=> 3 = 2 + 1
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [3,5,1,6,4,2,8,7,10,9] => ?
=> ? = 2 + 1
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [(1,3),(2,5),(4,8),(6,7),(9,10)]
=> [3,5,1,7,2,8,6,4,10,9] => ?
=> ? = 2 + 1
[(1,9),(2,3),(4,5),(6,7),(8,10)]
=> [(1,3),(2,5),(4,7),(6,10),(8,9)]
=> [3,5,1,7,2,9,4,10,8,6] => ?
=> ? = 2 + 1
[(1,10),(2,4),(3,5),(6,7),(8,9)]
=> [(1,5),(2,7),(3,4),(6,9),(8,10)]
=> [4,5,7,3,1,9,2,10,6,8] => ?
=> ? = 2 + 1
[(1,9),(2,4),(3,5),(6,7),(8,10)]
=> [(1,5),(2,7),(3,4),(6,10),(8,9)]
=> [4,5,7,3,1,9,2,10,8,6] => ?
=> ? = 2 + 1
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [4,5,7,3,1,8,2,6,10,9] => ?
=> ? = 2 + 1
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [(1,5),(2,8),(3,4),(6,7),(9,10)]
=> [4,5,7,3,1,8,6,2,10,9] => ?
=> ? = 2 + 1
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [4,5,6,3,1,2,8,7,10,9] => ?
=> ? = 2 + 1
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [4,5,6,1,3,2,8,7,10,9] => ?
=> ? = 2 + 1
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [3,5,2,6,1,4,8,7,10,9] => ?
=> ? = 2 + 1
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [4,5,6,2,1,3,8,7,10,9] => ?
=> ? = 2 + 1
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [4,5,6,2,3,1,8,7,10,9] => ?
=> ? = 2 + 1
[(1,7),(2,5),(3,4),(6,8),(9,10)]
=> [(1,4),(2,5),(3,8),(6,7),(9,10)]
=> [4,5,7,1,2,8,6,3,10,9] => ?
=> ? = 2 + 1
[(1,9),(2,5),(3,4),(6,7),(8,10)]
=> [(1,4),(2,5),(3,7),(6,10),(8,9)]
=> [4,5,7,1,2,9,3,10,8,6] => ?
=> ? = 2 + 1
[(1,10),(2,6),(3,4),(5,7),(8,9)]
=> [(1,4),(2,7),(3,9),(5,6),(8,10)]
=> [4,6,7,1,9,5,2,10,3,8] => ?
=> ? = 2 + 1
[(1,9),(2,6),(3,4),(5,7),(8,10)]
=> [(1,4),(2,7),(3,10),(5,6),(8,9)]
=> [4,6,7,1,9,5,2,10,8,3] => ?
=> ? = 2 + 1
[(1,8),(2,6),(3,4),(5,7),(9,10)]
=> [(1,4),(2,7),(3,8),(5,6),(9,10)]
=> [4,6,7,1,8,5,2,3,10,9] => ?
=> ? = 2 + 1
[(1,7),(2,6),(3,4),(5,8),(9,10)]
=> [(1,4),(2,6),(3,8),(5,7),(9,10)]
=> [4,6,7,1,8,2,5,3,10,9] => ?
=> ? = 2 + 1
[(1,6),(2,7),(3,4),(5,8),(9,10)]
=> [(1,8),(2,4),(3,7),(5,6),(9,10)]
=> [4,6,7,2,8,5,3,1,10,9] => ?
=> ? = 3 + 1
[(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [4,5,7,2,3,8,6,1,10,9] => ?
=> ? = 2 + 1
[(1,4),(2,7),(3,5),(6,8),(9,10)]
=> [(1,5),(2,4),(3,8),(6,7),(9,10)]
=> [4,5,7,2,1,8,6,3,10,9] => ?
=> ? = 2 + 1
[(1,3),(2,7),(4,5),(6,8),(9,10)]
=> [(1,5),(2,3),(4,8),(6,7),(9,10)]
=> [3,5,2,7,1,8,6,4,10,9] => ?
=> ? = 2 + 1
[(1,2),(3,7),(4,5),(6,8),(9,10)]
=> [(1,2),(3,5),(4,8),(6,7),(9,10)]
=> [2,1,5,7,3,8,6,4,10,9] => ?
=> ? = 2 + 1
[(1,3),(2,8),(4,5),(6,7),(9,10)]
=> [(1,5),(2,3),(4,7),(6,8),(9,10)]
=> [3,5,2,7,1,8,4,6,10,9] => ?
=> ? = 2 + 1
[(1,4),(2,8),(3,5),(6,7),(9,10)]
=> [(1,5),(2,4),(3,7),(6,8),(9,10)]
=> [4,5,7,2,1,8,3,6,10,9] => ?
=> ? = 2 + 1
[(1,5),(2,8),(3,4),(6,7),(9,10)]
=> [(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [4,5,7,2,3,8,1,6,10,9] => ?
=> ? = 2 + 1
[(1,6),(2,8),(3,4),(5,7),(9,10)]
=> [(1,7),(2,4),(3,6),(5,8),(9,10)]
=> [4,6,7,2,8,3,1,5,10,9] => ?
=> ? = 2 + 1
[(1,7),(2,8),(3,4),(5,6),(9,10)]
=> [(1,8),(2,4),(3,6),(5,7),(9,10)]
=> [4,6,7,2,8,3,5,1,10,9] => ?
=> ? = 2 + 1
[(1,9),(2,7),(3,4),(5,6),(8,10)]
=> [(1,4),(2,6),(3,7),(5,10),(8,9)]
=> [4,6,7,1,9,2,3,10,8,5] => ?
=> ? = 2 + 1
[(1,10),(2,8),(3,4),(5,6),(7,9)]
=> [(1,4),(2,6),(3,9),(5,10),(7,8)]
=> [4,6,8,1,9,2,10,7,3,5] => ?
=> ? = 2 + 1
[(1,9),(2,8),(3,4),(5,6),(7,10)]
=> [(1,4),(2,6),(3,8),(5,10),(7,9)]
=> [4,6,8,1,9,2,10,3,7,5] => ?
=> ? = 2 + 1
[(1,8),(2,9),(3,4),(5,6),(7,10)]
=> [(1,10),(2,4),(3,6),(5,9),(7,8)]
=> [4,6,8,2,9,3,10,7,5,1] => ?
=> ? = 3 + 1
[(1,7),(2,9),(3,4),(5,6),(8,10)]
=> [(1,10),(2,4),(3,6),(5,7),(8,9)]
=> [4,6,7,2,9,3,5,10,8,1] => ?
=> ? = 2 + 1
[(1,6),(2,9),(3,4),(5,7),(8,10)]
=> [(1,7),(2,4),(3,6),(5,10),(8,9)]
=> [4,6,7,2,9,3,1,10,8,5] => ?
=> ? = 2 + 1
[(1,5),(2,9),(3,4),(6,7),(8,10)]
=> [(1,7),(2,4),(3,5),(6,10),(8,9)]
=> [4,5,7,2,3,9,1,10,8,6] => ?
=> ? = 2 + 1
[(1,4),(2,9),(3,5),(6,7),(8,10)]
=> [(1,5),(2,4),(3,7),(6,10),(8,9)]
=> [4,5,7,2,1,9,3,10,8,6] => ?
=> ? = 2 + 1
[(1,3),(2,9),(4,5),(6,7),(8,10)]
=> [(1,5),(2,3),(4,7),(6,10),(8,9)]
=> [3,5,2,7,1,9,4,10,8,6] => ?
=> ? = 2 + 1
[(1,2),(3,9),(4,5),(6,7),(8,10)]
=> [(1,2),(3,5),(4,7),(6,10),(8,9)]
=> [2,1,5,7,3,9,4,10,8,6] => ?
=> ? = 2 + 1
[(1,3),(2,10),(4,5),(6,7),(8,9)]
=> [(1,5),(2,3),(4,7),(6,9),(8,10)]
=> [3,5,2,7,1,9,4,10,6,8] => ?
=> ? = 2 + 1
[(1,4),(2,10),(3,5),(6,7),(8,9)]
=> [(1,5),(2,4),(3,7),(6,9),(8,10)]
=> [4,5,7,2,1,9,3,10,6,8] => ?
=> ? = 2 + 1
[(1,5),(2,10),(3,4),(6,7),(8,9)]
=> [(1,7),(2,4),(3,5),(6,9),(8,10)]
=> [4,5,7,2,3,9,1,10,6,8] => ?
=> ? = 2 + 1
[(1,6),(2,10),(3,4),(5,7),(8,9)]
=> [(1,7),(2,4),(3,6),(5,9),(8,10)]
=> [4,6,7,2,9,3,1,10,5,8] => ?
=> ? = 2 + 1
[(1,7),(2,10),(3,4),(5,6),(8,9)]
=> [(1,9),(2,4),(3,6),(5,7),(8,10)]
=> [4,6,7,2,9,3,5,10,1,8] => ?
=> ? = 2 + 1
[(1,8),(2,10),(3,4),(5,6),(7,9)]
=> [(1,9),(2,4),(3,6),(5,8),(7,10)]
=> [4,6,8,2,9,3,10,5,1,7] => ?
=> ? = 2 + 1
[(1,9),(2,10),(3,4),(5,6),(7,8)]
=> [(1,10),(2,4),(3,6),(5,8),(7,9)]
=> [4,6,8,2,9,3,10,5,7,1] => ?
=> ? = 2 + 1
[(1,10),(2,9),(3,5),(4,6),(7,8)]
=> [(1,6),(2,8),(3,9),(4,5),(7,10)]
=> [5,6,8,9,4,1,10,2,3,7] => ?
=> ? = 2 + 1
[(1,9),(2,10),(3,5),(4,6),(7,8)]
=> [(1,6),(2,5),(3,10),(4,8),(7,9)]
=> [5,6,8,9,2,1,10,4,7,3] => ?
=> ? = 2 + 1
[(1,8),(2,10),(3,5),(4,6),(7,9)]
=> [(1,6),(2,5),(3,9),(4,8),(7,10)]
=> [5,6,8,9,2,1,10,4,3,7] => ?
=> ? = 2 + 1
[(1,7),(2,10),(3,5),(4,6),(8,9)]
=> [(1,6),(2,5),(3,9),(4,7),(8,10)]
=> [5,6,7,9,2,1,4,10,3,8] => ?
=> ? = 2 + 1
[(1,6),(2,10),(3,5),(4,7),(8,9)]
=> [(1,7),(2,5),(3,9),(4,6),(8,10)]
=> [5,6,7,9,2,4,1,10,3,8] => ?
=> ? = 2 + 1
Description
The length of the partition.
Matching statistic: St001587
Mp00165: Perfect matchings —Chen Deng Du Stanley Yan⟶ Perfect matchings
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%
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%
Values
[(1,2)]
=> [(1,2)]
=> [2,1] => [2]
=> 1
[(1,2),(3,4)]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,2]
=> 1
[(1,3),(2,4)]
=> [(1,4),(2,3)]
=> [4,3,2,1] => [4]
=> 2
[(1,4),(2,3)]
=> [(1,3),(2,4)]
=> [3,4,1,2] => [2,1,1]
=> 1
[(1,2),(3,4),(5,6)]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,2,2]
=> 1
[(1,3),(2,4),(5,6)]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [4,2]
=> 2
[(1,4),(2,3),(5,6)]
=> [(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [2,2,1,1]
=> 1
[(1,5),(2,3),(4,6)]
=> [(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [4,1,1]
=> 2
[(1,6),(2,3),(4,5)]
=> [(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [2,2,1,1]
=> 1
[(1,6),(2,4),(3,5)]
=> [(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [4,1,1]
=> 2
[(1,5),(2,4),(3,6)]
=> [(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [4,1,1]
=> 2
[(1,4),(2,5),(3,6)]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [6]
=> 3
[(1,3),(2,5),(4,6)]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [4,2]
=> 2
[(1,2),(3,5),(4,6)]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [4,2]
=> 2
[(1,2),(3,6),(4,5)]
=> [(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,2,1,1]
=> 1
[(1,3),(2,6),(4,5)]
=> [(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [4,1,1]
=> 2
[(1,4),(2,6),(3,5)]
=> [(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [4,1,1]
=> 2
[(1,5),(2,6),(3,4)]
=> [(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [4,1,1]
=> 2
[(1,6),(2,5),(3,4)]
=> [(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [2,1,1,1,1]
=> 1
[(1,2),(3,4),(5,6),(7,8)]
=> [(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)]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [4,2,2]
=> 2
[(1,4),(2,3),(5,6),(7,8)]
=> [(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [2,2,2,1,1]
=> 1
[(1,5),(2,3),(4,6),(7,8)]
=> [(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => [4,2,1,1]
=> 2
[(1,6),(2,3),(4,5),(7,8)]
=> [(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [2,2,2,1,1]
=> 1
[(1,7),(2,3),(4,5),(6,8)]
=> [(1,3),(2,5),(4,8),(6,7)]
=> [3,5,1,8,2,7,6,4] => [4,2,1,1]
=> 2
[(1,8),(2,3),(4,5),(6,7)]
=> [(1,3),(2,5),(4,7),(6,8)]
=> [3,5,1,7,2,8,4,6] => [2,2,2,1,1]
=> 1
[(1,8),(2,4),(3,5),(6,7)]
=> [(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => [4,2,1,1]
=> 2
[(1,7),(2,4),(3,5),(6,8)]
=> [(1,5),(2,8),(3,4),(6,7)]
=> [5,8,4,3,1,7,6,2] => [4,3,1]
=> 2
[(1,6),(2,4),(3,5),(7,8)]
=> [(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => [4,2,1,1]
=> 2
[(1,5),(2,4),(3,6),(7,8)]
=> [(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => [4,2,1,1]
=> 2
[(1,4),(2,5),(3,6),(7,8)]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [6,2]
=> 3
[(1,3),(2,5),(4,6),(7,8)]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [4,2,2]
=> 2
[(1,2),(3,5),(4,6),(7,8)]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [4,2,2]
=> 2
[(1,2),(3,6),(4,5),(7,8)]
=> [(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [2,2,2,1,1]
=> 1
[(1,3),(2,6),(4,5),(7,8)]
=> [(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => [4,2,1,1]
=> 2
[(1,4),(2,6),(3,5),(7,8)]
=> [(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => [4,2,1,1]
=> 2
[(1,5),(2,6),(3,4),(7,8)]
=> [(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => [4,2,1,1]
=> 2
[(1,6),(2,5),(3,4),(7,8)]
=> [(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [2,2,1,1,1,1]
=> 1
[(1,7),(2,5),(3,4),(6,8)]
=> [(1,4),(2,5),(3,8),(6,7)]
=> [4,5,8,1,2,7,6,3] => [4,1,1,1,1]
=> 2
[(1,8),(2,5),(3,4),(6,7)]
=> [(1,4),(2,5),(3,7),(6,8)]
=> [4,5,7,1,2,8,3,6] => [2,2,1,1,1,1]
=> 1
[(1,8),(2,6),(3,4),(5,7)]
=> [(1,4),(2,7),(3,8),(5,6)]
=> [4,7,8,1,6,5,2,3] => [4,1,1,1,1]
=> 2
[(1,7),(2,6),(3,4),(5,8)]
=> [(1,4),(2,6),(3,8),(5,7)]
=> [4,6,8,1,7,2,5,3] => [4,1,1,1,1]
=> 2
[(1,6),(2,7),(3,4),(5,8)]
=> [(1,8),(2,4),(3,7),(5,6)]
=> [8,4,7,2,6,5,3,1] => [6,1,1]
=> 3
[(1,5),(2,7),(3,4),(6,8)]
=> [(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => [4,2,1,1]
=> 2
[(1,4),(2,7),(3,5),(6,8)]
=> [(1,5),(2,4),(3,8),(6,7)]
=> [5,4,8,2,1,7,6,3] => [4,3,1]
=> 2
[(1,3),(2,7),(4,5),(6,8)]
=> [(1,5),(2,3),(4,8),(6,7)]
=> [5,3,2,8,1,7,6,4] => [4,3,1]
=> 2
[(1,2),(3,7),(4,5),(6,8)]
=> [(1,2),(3,5),(4,8),(6,7)]
=> [2,1,5,8,3,7,6,4] => [4,2,1,1]
=> 2
[(1,2),(3,8),(4,5),(6,7)]
=> [(1,2),(3,5),(4,7),(6,8)]
=> [2,1,5,7,3,8,4,6] => [2,2,2,1,1]
=> 1
[(1,3),(2,8),(4,5),(6,7)]
=> [(1,5),(2,3),(4,7),(6,8)]
=> [5,3,2,7,1,8,4,6] => [4,2,1,1]
=> 2
[(1,4),(2,8),(3,5),(6,7)]
=> [(1,5),(2,4),(3,7),(6,8)]
=> [5,4,7,2,1,8,3,6] => [4,2,1,1]
=> 2
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [3,6,1,5,4,2,8,7,10,9] => ?
=> ? = 2
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [(1,3),(2,5),(4,8),(6,7),(9,10)]
=> [3,5,1,8,2,7,6,4,10,9] => ?
=> ? = 2
[(1,9),(2,3),(4,5),(6,7),(8,10)]
=> [(1,3),(2,5),(4,7),(6,10),(8,9)]
=> [3,5,1,7,2,10,4,9,8,6] => ?
=> ? = 2
[(1,10),(2,4),(3,5),(6,7),(8,9)]
=> [(1,5),(2,7),(3,4),(6,9),(8,10)]
=> [5,7,4,3,1,9,2,10,6,8] => ?
=> ? = 2
[(1,9),(2,4),(3,5),(6,7),(8,10)]
=> [(1,5),(2,7),(3,4),(6,10),(8,9)]
=> [5,7,4,3,1,10,2,9,8,6] => ?
=> ? = 2
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [5,7,4,3,1,8,2,6,10,9] => ?
=> ? = 2
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [(1,5),(2,8),(3,4),(6,7),(9,10)]
=> [5,8,4,3,1,7,6,2,10,9] => ?
=> ? = 2
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [5,6,4,3,1,2,8,7,10,9] => ?
=> ? = 2
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [4,6,5,1,3,2,8,7,10,9] => ?
=> ? = 2
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [5,3,2,6,1,4,8,7,10,9] => ?
=> ? = 2
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [5,4,6,2,1,3,8,7,10,9] => ?
=> ? = 2
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [6,4,5,2,3,1,8,7,10,9] => ?
=> ? = 2
[(1,7),(2,5),(3,4),(6,8),(9,10)]
=> [(1,4),(2,5),(3,8),(6,7),(9,10)]
=> [4,5,8,1,2,7,6,3,10,9] => ?
=> ? = 2
[(1,9),(2,5),(3,4),(6,7),(8,10)]
=> [(1,4),(2,5),(3,7),(6,10),(8,9)]
=> [4,5,7,1,2,10,3,9,8,6] => ?
=> ? = 2
[(1,10),(2,6),(3,4),(5,7),(8,9)]
=> [(1,4),(2,7),(3,9),(5,6),(8,10)]
=> [4,7,9,1,6,5,2,10,3,8] => ?
=> ? = 2
[(1,9),(2,6),(3,4),(5,7),(8,10)]
=> [(1,4),(2,7),(3,10),(5,6),(8,9)]
=> [4,7,10,1,6,5,2,9,8,3] => ?
=> ? = 2
[(1,8),(2,6),(3,4),(5,7),(9,10)]
=> [(1,4),(2,7),(3,8),(5,6),(9,10)]
=> [4,7,8,1,6,5,2,3,10,9] => ?
=> ? = 2
[(1,7),(2,6),(3,4),(5,8),(9,10)]
=> [(1,4),(2,6),(3,8),(5,7),(9,10)]
=> [4,6,8,1,7,2,5,3,10,9] => ?
=> ? = 2
[(1,6),(2,7),(3,4),(5,8),(9,10)]
=> [(1,8),(2,4),(3,7),(5,6),(9,10)]
=> [8,4,7,2,6,5,3,1,10,9] => ?
=> ? = 3
[(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [8,4,5,2,3,7,6,1,10,9] => ?
=> ? = 2
[(1,4),(2,7),(3,5),(6,8),(9,10)]
=> [(1,5),(2,4),(3,8),(6,7),(9,10)]
=> [5,4,8,2,1,7,6,3,10,9] => ?
=> ? = 2
[(1,3),(2,7),(4,5),(6,8),(9,10)]
=> [(1,5),(2,3),(4,8),(6,7),(9,10)]
=> [5,3,2,8,1,7,6,4,10,9] => ?
=> ? = 2
[(1,2),(3,7),(4,5),(6,8),(9,10)]
=> [(1,2),(3,5),(4,8),(6,7),(9,10)]
=> [2,1,5,8,3,7,6,4,10,9] => ?
=> ? = 2
[(1,3),(2,8),(4,5),(6,7),(9,10)]
=> [(1,5),(2,3),(4,7),(6,8),(9,10)]
=> [5,3,2,7,1,8,4,6,10,9] => ?
=> ? = 2
[(1,4),(2,8),(3,5),(6,7),(9,10)]
=> [(1,5),(2,4),(3,7),(6,8),(9,10)]
=> [5,4,7,2,1,8,3,6,10,9] => ?
=> ? = 2
[(1,5),(2,8),(3,4),(6,7),(9,10)]
=> [(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [7,4,5,2,3,8,1,6,10,9] => ?
=> ? = 2
[(1,6),(2,8),(3,4),(5,7),(9,10)]
=> [(1,7),(2,4),(3,6),(5,8),(9,10)]
=> [7,4,6,2,8,3,1,5,10,9] => ?
=> ? = 2
[(1,7),(2,8),(3,4),(5,6),(9,10)]
=> [(1,8),(2,4),(3,6),(5,7),(9,10)]
=> [8,4,6,2,7,3,5,1,10,9] => ?
=> ? = 2
[(1,9),(2,7),(3,4),(5,6),(8,10)]
=> [(1,4),(2,6),(3,7),(5,10),(8,9)]
=> [4,6,7,1,10,2,3,9,8,5] => ?
=> ? = 2
[(1,10),(2,8),(3,4),(5,6),(7,9)]
=> [(1,4),(2,6),(3,9),(5,10),(7,8)]
=> [4,6,9,1,10,2,8,7,3,5] => ?
=> ? = 2
[(1,9),(2,8),(3,4),(5,6),(7,10)]
=> [(1,4),(2,6),(3,8),(5,10),(7,9)]
=> [4,6,8,1,10,2,9,3,7,5] => ?
=> ? = 2
[(1,8),(2,9),(3,4),(5,6),(7,10)]
=> [(1,10),(2,4),(3,6),(5,9),(7,8)]
=> [10,4,6,2,9,3,8,7,5,1] => ?
=> ? = 3
[(1,7),(2,9),(3,4),(5,6),(8,10)]
=> [(1,10),(2,4),(3,6),(5,7),(8,9)]
=> [10,4,6,2,7,3,5,9,8,1] => ?
=> ? = 2
[(1,6),(2,9),(3,4),(5,7),(8,10)]
=> [(1,7),(2,4),(3,6),(5,10),(8,9)]
=> [7,4,6,2,10,3,1,9,8,5] => ?
=> ? = 2
[(1,5),(2,9),(3,4),(6,7),(8,10)]
=> [(1,7),(2,4),(3,5),(6,10),(8,9)]
=> [7,4,5,2,3,10,1,9,8,6] => ?
=> ? = 2
[(1,4),(2,9),(3,5),(6,7),(8,10)]
=> [(1,5),(2,4),(3,7),(6,10),(8,9)]
=> [5,4,7,2,1,10,3,9,8,6] => ?
=> ? = 2
[(1,3),(2,9),(4,5),(6,7),(8,10)]
=> [(1,5),(2,3),(4,7),(6,10),(8,9)]
=> [5,3,2,7,1,10,4,9,8,6] => ?
=> ? = 2
[(1,2),(3,9),(4,5),(6,7),(8,10)]
=> [(1,2),(3,5),(4,7),(6,10),(8,9)]
=> [2,1,5,7,3,10,4,9,8,6] => ?
=> ? = 2
[(1,3),(2,10),(4,5),(6,7),(8,9)]
=> [(1,5),(2,3),(4,7),(6,9),(8,10)]
=> [5,3,2,7,1,9,4,10,6,8] => ?
=> ? = 2
[(1,4),(2,10),(3,5),(6,7),(8,9)]
=> [(1,5),(2,4),(3,7),(6,9),(8,10)]
=> [5,4,7,2,1,9,3,10,6,8] => ?
=> ? = 2
[(1,5),(2,10),(3,4),(6,7),(8,9)]
=> [(1,7),(2,4),(3,5),(6,9),(8,10)]
=> [7,4,5,2,3,9,1,10,6,8] => ?
=> ? = 2
[(1,6),(2,10),(3,4),(5,7),(8,9)]
=> [(1,7),(2,4),(3,6),(5,9),(8,10)]
=> [7,4,6,2,9,3,1,10,5,8] => ?
=> ? = 2
[(1,7),(2,10),(3,4),(5,6),(8,9)]
=> [(1,9),(2,4),(3,6),(5,7),(8,10)]
=> [9,4,6,2,7,3,5,10,1,8] => ?
=> ? = 2
[(1,8),(2,10),(3,4),(5,6),(7,9)]
=> [(1,9),(2,4),(3,6),(5,8),(7,10)]
=> [9,4,6,2,8,3,10,5,1,7] => ?
=> ? = 2
[(1,9),(2,10),(3,4),(5,6),(7,8)]
=> [(1,10),(2,4),(3,6),(5,8),(7,9)]
=> [10,4,6,2,8,3,9,5,7,1] => ?
=> ? = 2
[(1,10),(2,9),(3,5),(4,6),(7,8)]
=> [(1,6),(2,8),(3,9),(4,5),(7,10)]
=> [6,8,9,5,4,1,10,2,3,7] => ?
=> ? = 2
[(1,9),(2,10),(3,5),(4,6),(7,8)]
=> [(1,6),(2,5),(3,10),(4,8),(7,9)]
=> [6,5,10,8,2,1,9,4,7,3] => ?
=> ? = 2
[(1,8),(2,10),(3,5),(4,6),(7,9)]
=> [(1,6),(2,5),(3,9),(4,8),(7,10)]
=> [6,5,9,8,2,1,10,4,3,7] => ?
=> ? = 2
[(1,7),(2,10),(3,5),(4,6),(8,9)]
=> [(1,6),(2,5),(3,9),(4,7),(8,10)]
=> [6,5,9,7,2,1,4,10,3,8] => ?
=> ? = 2
[(1,6),(2,10),(3,5),(4,7),(8,9)]
=> [(1,7),(2,5),(3,9),(4,6),(8,10)]
=> [7,5,9,6,2,4,1,10,3,8] => ?
=> ? = 2
Description
Half of the largest even part of an integer partition.
The largest even part is recorded by [[St000995]].
Matching statistic: St000150
Mp00165: Perfect matchings —Chen Deng Du Stanley Yan⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 100%
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [(1,2)]
=> [2,1] => [1,1]
=> 1
[(1,2),(3,4)]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,2]
=> 1
[(1,3),(2,4)]
=> [(1,4),(2,3)]
=> [4,3,2,1] => [1,1,1,1]
=> 2
[(1,4),(2,3)]
=> [(1,3),(2,4)]
=> [3,4,1,2] => [2,2]
=> 1
[(1,2),(3,4),(5,6)]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [3,3]
=> 1
[(1,3),(2,4),(5,6)]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [2,2,1,1]
=> 2
[(1,4),(2,3),(5,6)]
=> [(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [3,3]
=> 1
[(1,5),(2,3),(4,6)]
=> [(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [2,2,1,1]
=> 2
[(1,6),(2,3),(4,5)]
=> [(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [3,3]
=> 1
[(1,6),(2,4),(3,5)]
=> [(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [2,2,1,1]
=> 2
[(1,5),(2,4),(3,6)]
=> [(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [2,2,1,1]
=> 2
[(1,4),(2,5),(3,6)]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [1,1,1,1,1,1]
=> 3
[(1,3),(2,5),(4,6)]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [2,2,1,1]
=> 2
[(1,2),(3,5),(4,6)]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [2,2,1,1]
=> 2
[(1,2),(3,6),(4,5)]
=> [(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [3,3]
=> 1
[(1,3),(2,6),(4,5)]
=> [(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [2,2,1,1]
=> 2
[(1,4),(2,6),(3,5)]
=> [(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [2,2,1,1]
=> 2
[(1,5),(2,6),(3,4)]
=> [(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [2,2,1,1]
=> 2
[(1,6),(2,5),(3,4)]
=> [(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [3,3]
=> 1
[(1,2),(3,4),(5,6),(7,8)]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [4,4]
=> 1
[(1,3),(2,4),(5,6),(7,8)]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [3,3,1,1]
=> 2
[(1,4),(2,3),(5,6),(7,8)]
=> [(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [4,4]
=> 1
[(1,5),(2,3),(4,6),(7,8)]
=> [(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => [3,3,1,1]
=> 2
[(1,6),(2,3),(4,5),(7,8)]
=> [(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [4,4]
=> 1
[(1,7),(2,3),(4,5),(6,8)]
=> [(1,3),(2,5),(4,8),(6,7)]
=> [3,5,1,8,2,7,6,4] => [3,3,1,1]
=> 2
[(1,8),(2,3),(4,5),(6,7)]
=> [(1,3),(2,5),(4,7),(6,8)]
=> [3,5,1,7,2,8,4,6] => [4,4]
=> 1
[(1,8),(2,4),(3,5),(6,7)]
=> [(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => [3,3,1,1]
=> 2
[(1,7),(2,4),(3,5),(6,8)]
=> [(1,5),(2,8),(3,4),(6,7)]
=> [5,8,4,3,1,7,6,2] => [2,2,2,2]
=> 2
[(1,6),(2,4),(3,5),(7,8)]
=> [(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => [3,3,1,1]
=> 2
[(1,5),(2,4),(3,6),(7,8)]
=> [(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => [3,3,1,1]
=> 2
[(1,4),(2,5),(3,6),(7,8)]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [2,2,1,1,1,1]
=> 3
[(1,3),(2,5),(4,6),(7,8)]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [3,3,1,1]
=> 2
[(1,2),(3,5),(4,6),(7,8)]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [3,3,1,1]
=> 2
[(1,2),(3,6),(4,5),(7,8)]
=> [(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [4,4]
=> 1
[(1,3),(2,6),(4,5),(7,8)]
=> [(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => [3,3,1,1]
=> 2
[(1,4),(2,6),(3,5),(7,8)]
=> [(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => [3,3,1,1]
=> 2
[(1,5),(2,6),(3,4),(7,8)]
=> [(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => [3,3,1,1]
=> 2
[(1,6),(2,5),(3,4),(7,8)]
=> [(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [4,4]
=> 1
[(1,7),(2,5),(3,4),(6,8)]
=> [(1,4),(2,5),(3,8),(6,7)]
=> [4,5,8,1,2,7,6,3] => [3,3,1,1]
=> 2
[(1,8),(2,5),(3,4),(6,7)]
=> [(1,4),(2,5),(3,7),(6,8)]
=> [4,5,7,1,2,8,3,6] => [4,4]
=> 1
[(1,8),(2,6),(3,4),(5,7)]
=> [(1,4),(2,7),(3,8),(5,6)]
=> [4,7,8,1,6,5,2,3] => [3,3,1,1]
=> 2
[(1,7),(2,6),(3,4),(5,8)]
=> [(1,4),(2,6),(3,8),(5,7)]
=> [4,6,8,1,7,2,5,3] => [3,3,1,1]
=> 2
[(1,6),(2,7),(3,4),(5,8)]
=> [(1,8),(2,4),(3,7),(5,6)]
=> [8,4,7,2,6,5,3,1] => [2,2,1,1,1,1]
=> 3
[(1,5),(2,7),(3,4),(6,8)]
=> [(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => [3,3,1,1]
=> 2
[(1,4),(2,7),(3,5),(6,8)]
=> [(1,5),(2,4),(3,8),(6,7)]
=> [5,4,8,2,1,7,6,3] => [2,2,2,2]
=> 2
[(1,3),(2,7),(4,5),(6,8)]
=> [(1,5),(2,3),(4,8),(6,7)]
=> [5,3,2,8,1,7,6,4] => [2,2,2,2]
=> 2
[(1,2),(3,7),(4,5),(6,8)]
=> [(1,2),(3,5),(4,8),(6,7)]
=> [2,1,5,8,3,7,6,4] => [3,3,1,1]
=> 2
[(1,2),(3,8),(4,5),(6,7)]
=> [(1,2),(3,5),(4,7),(6,8)]
=> [2,1,5,7,3,8,4,6] => [4,4]
=> 1
[(1,3),(2,8),(4,5),(6,7)]
=> [(1,5),(2,3),(4,7),(6,8)]
=> [5,3,2,7,1,8,4,6] => [3,3,1,1]
=> 2
[(1,4),(2,8),(3,5),(6,7)]
=> [(1,5),(2,4),(3,7),(6,8)]
=> [5,4,7,2,1,8,3,6] => [3,3,1,1]
=> 2
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [3,6,1,5,4,2,8,7,10,9] => ?
=> ? = 2
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [(1,3),(2,5),(4,8),(6,7),(9,10)]
=> [3,5,1,8,2,7,6,4,10,9] => ?
=> ? = 2
[(1,9),(2,3),(4,5),(6,7),(8,10)]
=> [(1,3),(2,5),(4,7),(6,10),(8,9)]
=> [3,5,1,7,2,10,4,9,8,6] => ?
=> ? = 2
[(1,10),(2,4),(3,5),(6,7),(8,9)]
=> [(1,5),(2,7),(3,4),(6,9),(8,10)]
=> [5,7,4,3,1,9,2,10,6,8] => ?
=> ? = 2
[(1,9),(2,4),(3,5),(6,7),(8,10)]
=> [(1,5),(2,7),(3,4),(6,10),(8,9)]
=> [5,7,4,3,1,10,2,9,8,6] => ?
=> ? = 2
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [5,7,4,3,1,8,2,6,10,9] => ?
=> ? = 2
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [(1,5),(2,8),(3,4),(6,7),(9,10)]
=> [5,8,4,3,1,7,6,2,10,9] => ?
=> ? = 2
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [5,6,4,3,1,2,8,7,10,9] => ?
=> ? = 2
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [4,6,5,1,3,2,8,7,10,9] => ?
=> ? = 2
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [5,3,2,6,1,4,8,7,10,9] => ?
=> ? = 2
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [5,4,6,2,1,3,8,7,10,9] => ?
=> ? = 2
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [6,4,5,2,3,1,8,7,10,9] => ?
=> ? = 2
[(1,7),(2,5),(3,4),(6,8),(9,10)]
=> [(1,4),(2,5),(3,8),(6,7),(9,10)]
=> [4,5,8,1,2,7,6,3,10,9] => ?
=> ? = 2
[(1,9),(2,5),(3,4),(6,7),(8,10)]
=> [(1,4),(2,5),(3,7),(6,10),(8,9)]
=> [4,5,7,1,2,10,3,9,8,6] => ?
=> ? = 2
[(1,10),(2,6),(3,4),(5,7),(8,9)]
=> [(1,4),(2,7),(3,9),(5,6),(8,10)]
=> [4,7,9,1,6,5,2,10,3,8] => ?
=> ? = 2
[(1,9),(2,6),(3,4),(5,7),(8,10)]
=> [(1,4),(2,7),(3,10),(5,6),(8,9)]
=> [4,7,10,1,6,5,2,9,8,3] => ?
=> ? = 2
[(1,8),(2,6),(3,4),(5,7),(9,10)]
=> [(1,4),(2,7),(3,8),(5,6),(9,10)]
=> [4,7,8,1,6,5,2,3,10,9] => ?
=> ? = 2
[(1,7),(2,6),(3,4),(5,8),(9,10)]
=> [(1,4),(2,6),(3,8),(5,7),(9,10)]
=> [4,6,8,1,7,2,5,3,10,9] => ?
=> ? = 2
[(1,6),(2,7),(3,4),(5,8),(9,10)]
=> [(1,8),(2,4),(3,7),(5,6),(9,10)]
=> [8,4,7,2,6,5,3,1,10,9] => ?
=> ? = 3
[(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [8,4,5,2,3,7,6,1,10,9] => ?
=> ? = 2
[(1,4),(2,7),(3,5),(6,8),(9,10)]
=> [(1,5),(2,4),(3,8),(6,7),(9,10)]
=> [5,4,8,2,1,7,6,3,10,9] => ?
=> ? = 2
[(1,3),(2,7),(4,5),(6,8),(9,10)]
=> [(1,5),(2,3),(4,8),(6,7),(9,10)]
=> [5,3,2,8,1,7,6,4,10,9] => ?
=> ? = 2
[(1,2),(3,7),(4,5),(6,8),(9,10)]
=> [(1,2),(3,5),(4,8),(6,7),(9,10)]
=> [2,1,5,8,3,7,6,4,10,9] => ?
=> ? = 2
[(1,3),(2,8),(4,5),(6,7),(9,10)]
=> [(1,5),(2,3),(4,7),(6,8),(9,10)]
=> [5,3,2,7,1,8,4,6,10,9] => ?
=> ? = 2
[(1,4),(2,8),(3,5),(6,7),(9,10)]
=> [(1,5),(2,4),(3,7),(6,8),(9,10)]
=> [5,4,7,2,1,8,3,6,10,9] => ?
=> ? = 2
[(1,5),(2,8),(3,4),(6,7),(9,10)]
=> [(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [7,4,5,2,3,8,1,6,10,9] => ?
=> ? = 2
[(1,6),(2,8),(3,4),(5,7),(9,10)]
=> [(1,7),(2,4),(3,6),(5,8),(9,10)]
=> [7,4,6,2,8,3,1,5,10,9] => ?
=> ? = 2
[(1,7),(2,8),(3,4),(5,6),(9,10)]
=> [(1,8),(2,4),(3,6),(5,7),(9,10)]
=> [8,4,6,2,7,3,5,1,10,9] => ?
=> ? = 2
[(1,9),(2,7),(3,4),(5,6),(8,10)]
=> [(1,4),(2,6),(3,7),(5,10),(8,9)]
=> [4,6,7,1,10,2,3,9,8,5] => ?
=> ? = 2
[(1,10),(2,8),(3,4),(5,6),(7,9)]
=> [(1,4),(2,6),(3,9),(5,10),(7,8)]
=> [4,6,9,1,10,2,8,7,3,5] => ?
=> ? = 2
[(1,9),(2,8),(3,4),(5,6),(7,10)]
=> [(1,4),(2,6),(3,8),(5,10),(7,9)]
=> [4,6,8,1,10,2,9,3,7,5] => ?
=> ? = 2
[(1,8),(2,9),(3,4),(5,6),(7,10)]
=> [(1,10),(2,4),(3,6),(5,9),(7,8)]
=> [10,4,6,2,9,3,8,7,5,1] => ?
=> ? = 3
[(1,7),(2,9),(3,4),(5,6),(8,10)]
=> [(1,10),(2,4),(3,6),(5,7),(8,9)]
=> [10,4,6,2,7,3,5,9,8,1] => ?
=> ? = 2
[(1,6),(2,9),(3,4),(5,7),(8,10)]
=> [(1,7),(2,4),(3,6),(5,10),(8,9)]
=> [7,4,6,2,10,3,1,9,8,5] => ?
=> ? = 2
[(1,5),(2,9),(3,4),(6,7),(8,10)]
=> [(1,7),(2,4),(3,5),(6,10),(8,9)]
=> [7,4,5,2,3,10,1,9,8,6] => ?
=> ? = 2
[(1,4),(2,9),(3,5),(6,7),(8,10)]
=> [(1,5),(2,4),(3,7),(6,10),(8,9)]
=> [5,4,7,2,1,10,3,9,8,6] => ?
=> ? = 2
[(1,3),(2,9),(4,5),(6,7),(8,10)]
=> [(1,5),(2,3),(4,7),(6,10),(8,9)]
=> [5,3,2,7,1,10,4,9,8,6] => ?
=> ? = 2
[(1,2),(3,9),(4,5),(6,7),(8,10)]
=> [(1,2),(3,5),(4,7),(6,10),(8,9)]
=> [2,1,5,7,3,10,4,9,8,6] => ?
=> ? = 2
[(1,3),(2,10),(4,5),(6,7),(8,9)]
=> [(1,5),(2,3),(4,7),(6,9),(8,10)]
=> [5,3,2,7,1,9,4,10,6,8] => ?
=> ? = 2
[(1,4),(2,10),(3,5),(6,7),(8,9)]
=> [(1,5),(2,4),(3,7),(6,9),(8,10)]
=> [5,4,7,2,1,9,3,10,6,8] => ?
=> ? = 2
[(1,5),(2,10),(3,4),(6,7),(8,9)]
=> [(1,7),(2,4),(3,5),(6,9),(8,10)]
=> [7,4,5,2,3,9,1,10,6,8] => ?
=> ? = 2
[(1,6),(2,10),(3,4),(5,7),(8,9)]
=> [(1,7),(2,4),(3,6),(5,9),(8,10)]
=> [7,4,6,2,9,3,1,10,5,8] => ?
=> ? = 2
[(1,7),(2,10),(3,4),(5,6),(8,9)]
=> [(1,9),(2,4),(3,6),(5,7),(8,10)]
=> [9,4,6,2,7,3,5,10,1,8] => ?
=> ? = 2
[(1,8),(2,10),(3,4),(5,6),(7,9)]
=> [(1,9),(2,4),(3,6),(5,8),(7,10)]
=> [9,4,6,2,8,3,10,5,1,7] => ?
=> ? = 2
[(1,9),(2,10),(3,4),(5,6),(7,8)]
=> [(1,10),(2,4),(3,6),(5,8),(7,9)]
=> [10,4,6,2,8,3,9,5,7,1] => ?
=> ? = 2
[(1,10),(2,9),(3,5),(4,6),(7,8)]
=> [(1,6),(2,8),(3,9),(4,5),(7,10)]
=> [6,8,9,5,4,1,10,2,3,7] => ?
=> ? = 2
[(1,9),(2,10),(3,5),(4,6),(7,8)]
=> [(1,6),(2,5),(3,10),(4,8),(7,9)]
=> [6,5,10,8,2,1,9,4,7,3] => ?
=> ? = 2
[(1,8),(2,10),(3,5),(4,6),(7,9)]
=> [(1,6),(2,5),(3,9),(4,8),(7,10)]
=> [6,5,9,8,2,1,10,4,3,7] => ?
=> ? = 2
[(1,7),(2,10),(3,5),(4,6),(8,9)]
=> [(1,6),(2,5),(3,9),(4,7),(8,10)]
=> [6,5,9,7,2,1,4,10,3,8] => ?
=> ? = 2
[(1,6),(2,10),(3,5),(4,7),(8,9)]
=> [(1,7),(2,5),(3,9),(4,6),(8,10)]
=> [7,5,9,6,2,4,1,10,3,8] => ?
=> ? = 2
Description
The floored half-sum of the multiplicities of a partition.
This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].
Matching statistic: St000097
Mp00165: Perfect matchings —Chen Deng Du Stanley Yan⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 80%
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 80%
Values
[(1,2)]
=> [(1,2)]
=> [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[(1,2),(3,4)]
=> [(1,2),(3,4)]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[(1,3),(2,4)]
=> [(1,4),(2,3)]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[(1,4),(2,3)]
=> [(1,3),(2,4)]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[(1,2),(3,4),(5,6)]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
=> 2 = 1 + 1
[(1,3),(2,4),(5,6)]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[(1,4),(2,3),(5,6)]
=> [(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 1 + 1
[(1,5),(2,3),(4,6)]
=> [(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[(1,6),(2,3),(4,5)]
=> [(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
[(1,6),(2,4),(3,5)]
=> [(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
[(1,5),(2,4),(3,6)]
=> [(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[(1,4),(2,5),(3,6)]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[(1,3),(2,5),(4,6)]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
[(1,2),(3,5),(4,6)]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[(1,2),(3,6),(4,5)]
=> [(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 1 + 1
[(1,3),(2,6),(4,5)]
=> [(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[(1,4),(2,6),(3,5)]
=> [(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[(1,5),(2,6),(3,4)]
=> [(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
[(1,6),(2,5),(3,4)]
=> [(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 2 = 1 + 1
[(1,2),(3,4),(5,6),(7,8)]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => ([(0,7),(1,6),(2,5),(3,4)],8)
=> 2 = 1 + 1
[(1,3),(2,4),(5,6),(7,8)]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 3 = 2 + 1
[(1,4),(2,3),(5,6),(7,8)]
=> [(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => ([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
=> 2 = 1 + 1
[(1,5),(2,3),(4,6),(7,8)]
=> [(1,3),(2,6),(4,5),(7,8)]
=> [3,5,1,6,4,2,8,7] => ([(0,1),(2,3),(2,7),(3,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 3 = 2 + 1
[(1,6),(2,3),(4,5),(7,8)]
=> [(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => ([(0,1),(2,5),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 1 + 1
[(1,7),(2,3),(4,5),(6,8)]
=> [(1,3),(2,5),(4,8),(6,7)]
=> [3,5,1,7,2,8,6,4] => ([(0,4),(0,7),(1,2),(1,6),(2,5),(3,4),(3,5),(3,7),(4,7),(5,6),(6,7)],8)
=> 3 = 2 + 1
[(1,8),(2,3),(4,5),(6,7)]
=> [(1,3),(2,5),(4,7),(6,8)]
=> [3,5,1,7,2,8,4,6] => ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> 2 = 1 + 1
[(1,8),(2,4),(3,5),(6,7)]
=> [(1,5),(2,7),(3,4),(6,8)]
=> [4,5,7,3,1,8,2,6] => ([(0,1),(0,6),(1,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> 3 = 2 + 1
[(1,7),(2,4),(3,5),(6,8)]
=> [(1,5),(2,8),(3,4),(6,7)]
=> [4,5,7,3,1,8,6,2] => ([(0,3),(0,7),(1,4),(1,6),(1,7),(2,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,6),(5,7),(6,7)],8)
=> 3 = 2 + 1
[(1,6),(2,4),(3,5),(7,8)]
=> [(1,5),(2,6),(3,4),(7,8)]
=> [4,5,6,3,1,2,8,7] => ([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> 3 = 2 + 1
[(1,5),(2,4),(3,6),(7,8)]
=> [(1,4),(2,6),(3,5),(7,8)]
=> [4,5,6,1,3,2,8,7] => ([(0,1),(2,3),(2,4),(2,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 3 = 2 + 1
[(1,4),(2,5),(3,6),(7,8)]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 4 = 3 + 1
[(1,3),(2,5),(4,6),(7,8)]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ([(0,1),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> 3 = 2 + 1
[(1,2),(3,5),(4,6),(7,8)]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => ([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 3 = 2 + 1
[(1,2),(3,6),(4,5),(7,8)]
=> [(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => ([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
=> 2 = 1 + 1
[(1,3),(2,6),(4,5),(7,8)]
=> [(1,5),(2,3),(4,6),(7,8)]
=> [3,5,2,6,1,4,8,7] => ([(0,1),(2,3),(2,7),(3,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 3 = 2 + 1
[(1,4),(2,6),(3,5),(7,8)]
=> [(1,5),(2,4),(3,6),(7,8)]
=> [4,5,6,2,1,3,8,7] => ([(0,1),(2,3),(2,4),(2,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 3 = 2 + 1
[(1,5),(2,6),(3,4),(7,8)]
=> [(1,6),(2,4),(3,5),(7,8)]
=> [4,5,6,2,3,1,8,7] => ([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> 3 = 2 + 1
[(1,6),(2,5),(3,4),(7,8)]
=> [(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => ([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> 2 = 1 + 1
[(1,7),(2,5),(3,4),(6,8)]
=> [(1,4),(2,5),(3,8),(6,7)]
=> [4,5,7,1,2,8,6,3] => ([(0,5),(0,7),(1,3),(1,4),(1,7),(2,3),(2,4),(2,7),(3,6),(4,6),(5,6),(5,7),(6,7)],8)
=> 3 = 2 + 1
[(1,8),(2,5),(3,4),(6,7)]
=> [(1,4),(2,5),(3,7),(6,8)]
=> [4,5,7,1,2,8,3,6] => ([(0,1),(0,7),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,7),(5,7),(6,7)],8)
=> 2 = 1 + 1
[(1,8),(2,6),(3,4),(5,7)]
=> [(1,4),(2,7),(3,8),(5,6)]
=> [4,6,7,1,8,5,2,3] => ([(0,2),(0,5),(0,6),(1,5),(1,6),(1,7),(2,3),(2,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> 3 = 2 + 1
[(1,7),(2,6),(3,4),(5,8)]
=> [(1,4),(2,6),(3,8),(5,7)]
=> [4,6,7,1,8,2,5,3] => ([(0,3),(0,4),(0,7),(1,2),(1,5),(1,6),(2,4),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> 3 = 2 + 1
[(1,6),(2,7),(3,4),(5,8)]
=> [(1,8),(2,4),(3,7),(5,6)]
=> [4,6,7,2,8,5,3,1] => ([(0,4),(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> 4 = 3 + 1
[(1,5),(2,7),(3,4),(6,8)]
=> [(1,8),(2,4),(3,5),(6,7)]
=> [4,5,7,2,3,8,6,1] => ([(0,3),(0,7),(1,4),(1,5),(1,7),(2,4),(2,5),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 3 = 2 + 1
[(1,4),(2,7),(3,5),(6,8)]
=> [(1,5),(2,4),(3,8),(6,7)]
=> [4,5,7,2,1,8,6,3] => ([(0,3),(0,7),(1,4),(1,5),(1,7),(2,4),(2,5),(2,7),(3,6),(3,7),(4,5),(4,6),(5,6),(6,7)],8)
=> 3 = 2 + 1
[(1,3),(2,7),(4,5),(6,8)]
=> [(1,5),(2,3),(4,8),(6,7)]
=> [3,5,2,7,1,8,6,4] => ([(0,5),(0,7),(1,4),(1,6),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,6),(5,7)],8)
=> 3 = 2 + 1
[(1,2),(3,7),(4,5),(6,8)]
=> [(1,2),(3,5),(4,8),(6,7)]
=> [2,1,5,7,3,8,6,4] => ([(0,1),(2,3),(2,7),(3,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 3 = 2 + 1
[(1,2),(3,8),(4,5),(6,7)]
=> [(1,2),(3,5),(4,7),(6,8)]
=> [2,1,5,7,3,8,4,6] => ([(0,1),(2,5),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 1 + 1
[(1,3),(2,8),(4,5),(6,7)]
=> [(1,5),(2,3),(4,7),(6,8)]
=> [3,5,2,7,1,8,4,6] => ([(0,4),(0,7),(1,2),(1,6),(2,5),(3,4),(3,5),(3,7),(4,7),(5,6),(6,7)],8)
=> 3 = 2 + 1
[(1,4),(2,8),(3,5),(6,7)]
=> [(1,5),(2,4),(3,7),(6,8)]
=> [4,5,7,2,1,8,3,6] => ([(0,1),(0,7),(1,6),(2,4),(2,5),(2,7),(3,4),(3,5),(3,7),(4,5),(4,6),(5,6),(6,7)],8)
=> 3 = 2 + 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => ([(0,9),(1,8),(2,7),(3,6),(4,5)],10)
=> ? = 1 + 1
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [3,4,2,1,6,5,8,7,10,9] => ([(0,5),(1,4),(2,3),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 2 + 1
[(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [3,4,1,2,6,5,8,7,10,9] => ([(0,5),(1,4),(2,3),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 1 + 1
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [3,5,1,6,4,2,8,7,10,9] => ?
=> ? = 2 + 1
[(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [(1,3),(2,5),(4,6),(7,8),(9,10)]
=> [3,5,1,6,2,4,8,7,10,9] => ([(0,3),(1,2),(4,7),(4,9),(5,6),(5,9),(6,8),(7,8),(8,9)],10)
=> ? = 1 + 1
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [(1,3),(2,5),(4,8),(6,7),(9,10)]
=> [3,5,1,7,2,8,6,4,10,9] => ?
=> ? = 2 + 1
[(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [(1,3),(2,5),(4,7),(6,8),(9,10)]
=> [3,5,1,7,2,8,4,6,10,9] => ([(0,1),(2,5),(2,9),(3,4),(3,6),(4,7),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1 + 1
[(1,9),(2,3),(4,5),(6,7),(8,10)]
=> [(1,3),(2,5),(4,7),(6,10),(8,9)]
=> [3,5,1,7,2,9,4,10,8,6] => ?
=> ? = 2 + 1
[(1,10),(2,4),(3,5),(6,7),(8,9)]
=> [(1,5),(2,7),(3,4),(6,9),(8,10)]
=> [4,5,7,3,1,9,2,10,6,8] => ?
=> ? = 2 + 1
[(1,9),(2,4),(3,5),(6,7),(8,10)]
=> [(1,5),(2,7),(3,4),(6,10),(8,9)]
=> [4,5,7,3,1,9,2,10,8,6] => ?
=> ? = 2 + 1
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [4,5,7,3,1,8,2,6,10,9] => ?
=> ? = 2 + 1
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [(1,5),(2,8),(3,4),(6,7),(9,10)]
=> [4,5,7,3,1,8,6,2,10,9] => ?
=> ? = 2 + 1
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [4,5,6,3,1,2,8,7,10,9] => ?
=> ? = 2 + 1
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [4,5,6,1,3,2,8,7,10,9] => ?
=> ? = 2 + 1
[(1,4),(2,5),(3,6),(7,8),(9,10)]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => ([(0,3),(1,2),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 3 + 1
[(1,3),(2,5),(4,6),(7,8),(9,10)]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [3,5,2,6,4,1,8,7,10,9] => ([(0,3),(1,2),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 2 + 1
[(1,2),(3,5),(4,6),(7,8),(9,10)]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,5,6,4,3,8,7,10,9] => ([(0,5),(1,4),(2,3),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 2 + 1
[(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [(1,2),(3,5),(4,6),(7,8),(9,10)]
=> [2,1,5,6,3,4,8,7,10,9] => ([(0,5),(1,4),(2,3),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 1 + 1
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [3,5,2,6,1,4,8,7,10,9] => ?
=> ? = 2 + 1
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [4,5,6,2,1,3,8,7,10,9] => ?
=> ? = 2 + 1
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [4,5,6,2,3,1,8,7,10,9] => ?
=> ? = 2 + 1
[(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [(1,4),(2,5),(3,6),(7,8),(9,10)]
=> [4,5,6,1,2,3,8,7,10,9] => ([(0,3),(1,2),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9)],10)
=> ? = 1 + 1
[(1,7),(2,5),(3,4),(6,8),(9,10)]
=> [(1,4),(2,5),(3,8),(6,7),(9,10)]
=> [4,5,7,1,2,8,6,3,10,9] => ?
=> ? = 2 + 1
[(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [(1,4),(2,5),(3,7),(6,8),(9,10)]
=> [4,5,7,1,2,8,3,6,10,9] => ([(0,1),(2,3),(2,9),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 1 + 1
[(1,9),(2,5),(3,4),(6,7),(8,10)]
=> [(1,4),(2,5),(3,7),(6,10),(8,9)]
=> [4,5,7,1,2,9,3,10,8,6] => ?
=> ? = 2 + 1
[(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [(1,4),(2,5),(3,7),(6,9),(8,10)]
=> [4,5,7,1,2,9,3,10,6,8] => ([(0,1),(0,7),(1,6),(2,4),(2,5),(2,8),(3,4),(3,5),(3,8),(4,9),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1 + 1
[(1,10),(2,6),(3,4),(5,7),(8,9)]
=> [(1,4),(2,7),(3,9),(5,6),(8,10)]
=> [4,6,7,1,9,5,2,10,3,8] => ?
=> ? = 2 + 1
[(1,9),(2,6),(3,4),(5,7),(8,10)]
=> [(1,4),(2,7),(3,10),(5,6),(8,9)]
=> [4,6,7,1,9,5,2,10,8,3] => ?
=> ? = 2 + 1
[(1,8),(2,6),(3,4),(5,7),(9,10)]
=> [(1,4),(2,7),(3,8),(5,6),(9,10)]
=> [4,6,7,1,8,5,2,3,10,9] => ?
=> ? = 2 + 1
[(1,7),(2,6),(3,4),(5,8),(9,10)]
=> [(1,4),(2,6),(3,8),(5,7),(9,10)]
=> [4,6,7,1,8,2,5,3,10,9] => ?
=> ? = 2 + 1
[(1,6),(2,7),(3,4),(5,8),(9,10)]
=> [(1,8),(2,4),(3,7),(5,6),(9,10)]
=> [4,6,7,2,8,5,3,1,10,9] => ?
=> ? = 3 + 1
[(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [4,5,7,2,3,8,6,1,10,9] => ?
=> ? = 2 + 1
[(1,4),(2,7),(3,5),(6,8),(9,10)]
=> [(1,5),(2,4),(3,8),(6,7),(9,10)]
=> [4,5,7,2,1,8,6,3,10,9] => ?
=> ? = 2 + 1
[(1,3),(2,7),(4,5),(6,8),(9,10)]
=> [(1,5),(2,3),(4,8),(6,7),(9,10)]
=> [3,5,2,7,1,8,6,4,10,9] => ?
=> ? = 2 + 1
[(1,2),(3,7),(4,5),(6,8),(9,10)]
=> [(1,2),(3,5),(4,8),(6,7),(9,10)]
=> [2,1,5,7,3,8,6,4,10,9] => ?
=> ? = 2 + 1
[(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [(1,2),(3,5),(4,7),(6,8),(9,10)]
=> [2,1,5,7,3,8,4,6,10,9] => ([(0,3),(1,2),(4,7),(4,9),(5,6),(5,9),(6,8),(7,8),(8,9)],10)
=> ? = 1 + 1
[(1,3),(2,8),(4,5),(6,7),(9,10)]
=> [(1,5),(2,3),(4,7),(6,8),(9,10)]
=> [3,5,2,7,1,8,4,6,10,9] => ?
=> ? = 2 + 1
[(1,4),(2,8),(3,5),(6,7),(9,10)]
=> [(1,5),(2,4),(3,7),(6,8),(9,10)]
=> [4,5,7,2,1,8,3,6,10,9] => ?
=> ? = 2 + 1
[(1,5),(2,8),(3,4),(6,7),(9,10)]
=> [(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [4,5,7,2,3,8,1,6,10,9] => ?
=> ? = 2 + 1
[(1,6),(2,8),(3,4),(5,7),(9,10)]
=> [(1,7),(2,4),(3,6),(5,8),(9,10)]
=> [4,6,7,2,8,3,1,5,10,9] => ?
=> ? = 2 + 1
[(1,7),(2,8),(3,4),(5,6),(9,10)]
=> [(1,8),(2,4),(3,6),(5,7),(9,10)]
=> [4,6,7,2,8,3,5,1,10,9] => ?
=> ? = 2 + 1
[(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [(1,4),(2,6),(3,7),(5,8),(9,10)]
=> [4,6,7,1,8,2,3,5,10,9] => ([(0,1),(2,5),(2,8),(2,9),(3,4),(3,6),(3,7),(4,8),(4,9),(5,6),(5,7),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 1 + 1
[(1,9),(2,7),(3,4),(5,6),(8,10)]
=> [(1,4),(2,6),(3,7),(5,10),(8,9)]
=> [4,6,7,1,9,2,3,10,8,5] => ?
=> ? = 2 + 1
[(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [(1,4),(2,6),(3,7),(5,9),(8,10)]
=> [4,6,7,1,9,2,3,10,5,8] => ([(0,1),(0,9),(1,8),(2,3),(2,4),(2,5),(3,6),(3,7),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 1 + 1
[(1,10),(2,8),(3,4),(5,6),(7,9)]
=> [(1,4),(2,6),(3,9),(5,10),(7,8)]
=> [4,6,8,1,9,2,10,7,3,5] => ?
=> ? = 2 + 1
[(1,9),(2,8),(3,4),(5,6),(7,10)]
=> [(1,4),(2,6),(3,8),(5,10),(7,9)]
=> [4,6,8,1,9,2,10,3,7,5] => ?
=> ? = 2 + 1
[(1,8),(2,9),(3,4),(5,6),(7,10)]
=> [(1,10),(2,4),(3,6),(5,9),(7,8)]
=> [4,6,8,2,9,3,10,7,5,1] => ?
=> ? = 3 + 1
[(1,7),(2,9),(3,4),(5,6),(8,10)]
=> [(1,10),(2,4),(3,6),(5,7),(8,9)]
=> [4,6,7,2,9,3,5,10,8,1] => ?
=> ? = 2 + 1
[(1,6),(2,9),(3,4),(5,7),(8,10)]
=> [(1,7),(2,4),(3,6),(5,10),(8,9)]
=> [4,6,7,2,9,3,1,10,8,5] => ?
=> ? = 2 + 1
[(1,5),(2,9),(3,4),(6,7),(8,10)]
=> [(1,7),(2,4),(3,5),(6,10),(8,9)]
=> [4,5,7,2,3,9,1,10,8,6] => ?
=> ? = 2 + 1
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000451
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 80%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 80%
Values
[(1,2)]
=> [2,1] => [2,1] => 2 = 1 + 1
[(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[(1,3),(2,4)]
=> [3,4,1,2] => [3,1,4,2] => 3 = 2 + 1
[(1,4),(2,3)]
=> [4,3,2,1] => [3,2,4,1] => 2 = 1 + 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => 2 = 1 + 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [3,1,4,2,6,5] => 3 = 2 + 1
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [3,2,4,1,6,5] => 2 = 1 + 1
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [3,2,5,1,6,4] => 3 = 2 + 1
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [3,2,5,4,6,1] => 2 = 1 + 1
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [4,2,5,3,6,1] => 3 = 2 + 1
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [4,2,5,1,6,3] => 3 = 2 + 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => 4 = 3 + 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [3,1,5,2,6,4] => 3 = 2 + 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,1,5,3,6,4] => 3 = 2 + 1
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [2,1,5,4,6,3] => 2 = 1 + 1
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [3,1,5,4,6,2] => 3 = 2 + 1
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [4,1,5,3,6,2] => 3 = 2 + 1
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [4,3,5,1,6,2] => 3 = 2 + 1
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [4,3,5,2,6,1] => 2 = 1 + 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 2 = 1 + 1
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [3,1,4,2,6,5,8,7] => 3 = 2 + 1
[(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [3,2,4,1,6,5,8,7] => 2 = 1 + 1
[(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => [3,2,5,1,6,4,8,7] => 3 = 2 + 1
[(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [3,2,5,4,6,1,8,7] => 2 = 1 + 1
[(1,7),(2,3),(4,5),(6,8)]
=> [7,3,2,5,4,8,1,6] => [3,2,5,4,7,1,8,6] => 3 = 2 + 1
[(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [3,2,5,4,7,6,8,1] => 2 = 1 + 1
[(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => [4,2,5,3,7,6,8,1] => 3 = 2 + 1
[(1,7),(2,4),(3,5),(6,8)]
=> [7,4,5,2,3,8,1,6] => [4,2,5,3,7,1,8,6] => 3 = 2 + 1
[(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => [4,2,5,3,6,1,8,7] => 3 = 2 + 1
[(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => [4,2,5,1,6,3,8,7] => 3 = 2 + 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [4,1,5,2,6,3,8,7] => 4 = 3 + 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [3,1,5,2,6,4,8,7] => 3 = 2 + 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [2,1,5,3,6,4,8,7] => 3 = 2 + 1
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [2,1,5,4,6,3,8,7] => 2 = 1 + 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => [3,1,5,4,6,2,8,7] => 3 = 2 + 1
[(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => [4,1,5,3,6,2,8,7] => 3 = 2 + 1
[(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => [4,3,5,1,6,2,8,7] => 3 = 2 + 1
[(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [4,3,5,2,6,1,8,7] => 2 = 1 + 1
[(1,7),(2,5),(3,4),(6,8)]
=> [7,5,4,3,2,8,1,6] => [4,3,5,2,7,1,8,6] => 3 = 2 + 1
[(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [4,3,5,2,7,6,8,1] => 2 = 1 + 1
[(1,8),(2,6),(3,4),(5,7)]
=> [8,6,4,3,7,2,5,1] => [4,3,6,2,7,5,8,1] => 3 = 2 + 1
[(1,7),(2,6),(3,4),(5,8)]
=> [7,6,4,3,8,2,1,5] => [4,3,6,2,7,1,8,5] => 3 = 2 + 1
[(1,6),(2,7),(3,4),(5,8)]
=> [6,7,4,3,8,1,2,5] => [4,3,6,1,7,2,8,5] => 4 = 3 + 1
[(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => [4,3,5,1,7,2,8,6] => 3 = 2 + 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,7,5,1,3,8,2,6] => [4,1,5,3,7,2,8,6] => 3 = 2 + 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,7,1,5,4,8,2,6] => [3,1,5,4,7,2,8,6] => 3 = 2 + 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,7,5,4,8,3,6] => [2,1,5,4,7,3,8,6] => 3 = 2 + 1
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [2,1,5,4,7,6,8,3] => 2 = 1 + 1
[(1,3),(2,8),(4,5),(6,7)]
=> [3,8,1,5,4,7,6,2] => [3,1,5,4,7,6,8,2] => 3 = 2 + 1
[(1,4),(2,8),(3,5),(6,7)]
=> [4,8,5,1,3,7,6,2] => [4,1,5,3,7,6,8,2] => 3 = 2 + 1
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [3,4,1,2,6,5,8,7,10,9] => [3,1,4,2,6,5,8,7,10,9] => ? = 2 + 1
[(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [4,3,2,1,6,5,8,7,10,9] => [3,2,4,1,6,5,8,7,10,9] => ? = 1 + 1
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [5,3,2,6,1,4,8,7,10,9] => ? => ? = 2 + 1
[(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [6,3,2,5,4,1,8,7,10,9] => [3,2,5,4,6,1,8,7,10,9] => ? = 1 + 1
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [7,3,2,5,4,8,1,6,10,9] => ? => ? = 2 + 1
[(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [8,3,2,5,4,7,6,1,10,9] => [3,2,5,4,7,6,8,1,10,9] => ? = 1 + 1
[(1,9),(2,3),(4,5),(6,7),(8,10)]
=> [9,3,2,5,4,7,6,10,1,8] => ? => ? = 2 + 1
[(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [10,3,2,5,4,7,6,9,8,1] => [3,2,5,4,7,6,9,8,10,1] => ? = 1 + 1
[(1,10),(2,4),(3,5),(6,7),(8,9)]
=> [10,4,5,2,3,7,6,9,8,1] => ? => ? = 2 + 1
[(1,9),(2,4),(3,5),(6,7),(8,10)]
=> [9,4,5,2,3,7,6,10,1,8] => ? => ? = 2 + 1
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [8,4,5,2,3,7,6,1,10,9] => ? => ? = 2 + 1
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [7,4,5,2,3,8,1,6,10,9] => ? => ? = 2 + 1
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [6,4,5,2,3,1,8,7,10,9] => [4,2,5,3,6,1,8,7,10,9] => ? = 2 + 1
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [5,4,6,2,1,3,8,7,10,9] => ? => ? = 2 + 1
[(1,4),(2,5),(3,6),(7,8),(9,10)]
=> [4,5,6,1,2,3,8,7,10,9] => [4,1,5,2,6,3,8,7,10,9] => ? = 3 + 1
[(1,3),(2,5),(4,6),(7,8),(9,10)]
=> [3,5,1,6,2,4,8,7,10,9] => [3,1,5,2,6,4,8,7,10,9] => ? = 2 + 1
[(1,2),(3,5),(4,6),(7,8),(9,10)]
=> [2,1,5,6,3,4,8,7,10,9] => [2,1,5,3,6,4,8,7,10,9] => ? = 2 + 1
[(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,6,5,4,3,8,7,10,9] => [2,1,5,4,6,3,8,7,10,9] => ? = 1 + 1
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [3,6,1,5,4,2,8,7,10,9] => ? => ? = 2 + 1
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [4,6,5,1,3,2,8,7,10,9] => ? => ? = 2 + 1
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [5,6,4,3,1,2,8,7,10,9] => ? => ? = 2 + 1
[(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [6,5,4,3,2,1,8,7,10,9] => [4,3,5,2,6,1,8,7,10,9] => ? = 1 + 1
[(1,7),(2,5),(3,4),(6,8),(9,10)]
=> [7,5,4,3,2,8,1,6,10,9] => ? => ? = 2 + 1
[(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [8,5,4,3,2,7,6,1,10,9] => [4,3,5,2,7,6,8,1,10,9] => ? = 1 + 1
[(1,9),(2,5),(3,4),(6,7),(8,10)]
=> [9,5,4,3,2,7,6,10,1,8] => ? => ? = 2 + 1
[(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [10,5,4,3,2,7,6,9,8,1] => [4,3,5,2,7,6,9,8,10,1] => ? = 1 + 1
[(1,10),(2,6),(3,4),(5,7),(8,9)]
=> [10,6,4,3,7,2,5,9,8,1] => ? => ? = 2 + 1
[(1,9),(2,6),(3,4),(5,7),(8,10)]
=> [9,6,4,3,7,2,5,10,1,8] => ? => ? = 2 + 1
[(1,8),(2,6),(3,4),(5,7),(9,10)]
=> [8,6,4,3,7,2,5,1,10,9] => ? => ? = 2 + 1
[(1,7),(2,6),(3,4),(5,8),(9,10)]
=> [7,6,4,3,8,2,1,5,10,9] => ? => ? = 2 + 1
[(1,6),(2,7),(3,4),(5,8),(9,10)]
=> [6,7,4,3,8,1,2,5,10,9] => ? => ? = 3 + 1
[(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [5,7,4,3,1,8,2,6,10,9] => ? => ? = 2 + 1
[(1,4),(2,7),(3,5),(6,8),(9,10)]
=> [4,7,5,1,3,8,2,6,10,9] => ? => ? = 2 + 1
[(1,3),(2,7),(4,5),(6,8),(9,10)]
=> [3,7,1,5,4,8,2,6,10,9] => ? => ? = 2 + 1
[(1,2),(3,7),(4,5),(6,8),(9,10)]
=> [2,1,7,5,4,8,3,6,10,9] => ? => ? = 2 + 1
[(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,8,5,4,7,6,3,10,9] => [2,1,5,4,7,6,8,3,10,9] => ? = 1 + 1
[(1,3),(2,8),(4,5),(6,7),(9,10)]
=> [3,8,1,5,4,7,6,2,10,9] => ? => ? = 2 + 1
[(1,4),(2,8),(3,5),(6,7),(9,10)]
=> [4,8,5,1,3,7,6,2,10,9] => ? => ? = 2 + 1
[(1,5),(2,8),(3,4),(6,7),(9,10)]
=> [5,8,4,3,1,7,6,2,10,9] => ? => ? = 2 + 1
[(1,6),(2,8),(3,4),(5,7),(9,10)]
=> [6,8,4,3,7,1,5,2,10,9] => ? => ? = 2 + 1
[(1,7),(2,8),(3,4),(5,6),(9,10)]
=> [7,8,4,3,6,5,1,2,10,9] => ? => ? = 2 + 1
[(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [8,7,4,3,6,5,2,1,10,9] => [4,3,6,5,7,2,8,1,10,9] => ? = 1 + 1
[(1,9),(2,7),(3,4),(5,6),(8,10)]
=> [9,7,4,3,6,5,2,10,1,8] => ? => ? = 2 + 1
[(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [10,7,4,3,6,5,2,9,8,1] => [4,3,6,5,7,2,9,8,10,1] => ? = 1 + 1
[(1,10),(2,8),(3,4),(5,6),(7,9)]
=> [10,8,4,3,6,5,9,2,7,1] => ? => ? = 2 + 1
[(1,9),(2,8),(3,4),(5,6),(7,10)]
=> [9,8,4,3,6,5,10,2,1,7] => ? => ? = 2 + 1
[(1,8),(2,9),(3,4),(5,6),(7,10)]
=> [8,9,4,3,6,5,10,1,2,7] => ? => ? = 3 + 1
[(1,7),(2,9),(3,4),(5,6),(8,10)]
=> [7,9,4,3,6,5,1,10,2,8] => ? => ? = 2 + 1
[(1,6),(2,9),(3,4),(5,7),(8,10)]
=> [6,9,4,3,7,1,5,10,2,8] => ? => ? = 2 + 1
[(1,5),(2,9),(3,4),(6,7),(8,10)]
=> [5,9,4,3,1,7,6,10,2,8] => ? => ? = 2 + 1
Description
The length of the longest pattern of the form k 1 2...(k-1).
Matching statistic: St001624
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 40%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 40%
Values
[(1,2)]
=> [2,1] => ([],2)
=> ([],1)
=> 1
[(1,2),(3,4)]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1
[(1,3),(2,4)]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[(1,4),(2,3)]
=> [4,3,2,1] => ([],4)
=> ([],1)
=> 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => ([(0,3),(1,2),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 1
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,4),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> ([(0,1)],2)
=> 1
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => ([(0,4),(0,5),(1,2),(1,3)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => ([],6)
=> ([],1)
=> 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => ([(0,6),(0,7),(1,6),(1,7),(4,2),(4,3),(5,2),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => ([(0,3),(1,2),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,2),(2,1)],3)
=> 1
[(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => ([(0,7),(1,6),(2,6),(2,7),(3,6),(3,7),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => ([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,2),(2,1)],3)
=> 1
[(1,7),(2,3),(4,5),(6,8)]
=> [7,3,2,5,4,8,1,6] => ([(0,7),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => ([(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,2),(2,1)],3)
=> 1
[(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => ([(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[(1,7),(2,4),(3,5),(6,8)]
=> [7,4,5,2,3,8,1,6] => ([(0,7),(1,6),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => ([(0,5),(1,5),(2,4),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,2),(5,3)],8)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => ([(0,3),(0,7),(1,2),(1,6),(2,7),(3,6),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ? = 2
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => ([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(6,3),(7,2)],8)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,2),(2,1)],3)
=> 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1)],2)
=> 1
[(1,7),(2,5),(3,4),(6,8)]
=> [7,5,4,3,2,8,1,6] => ([(0,7),(1,6),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1)],2)
=> 1
[(1,8),(2,6),(3,4),(5,7)]
=> [8,6,4,3,7,2,5,1] => ([(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[(1,7),(2,6),(3,4),(5,8)]
=> [7,6,4,3,8,2,1,5] => ([(0,7),(1,7),(2,6),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[(1,6),(2,7),(3,4),(5,8)]
=> [6,7,4,3,8,1,2,5] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(4,6),(5,7)],8)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,6),(4,7),(5,6)],8)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[(1,4),(2,7),(3,5),(6,8)]
=> [4,7,5,1,3,8,2,6] => ([(0,3),(0,5),(1,2),(1,4),(2,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[(1,3),(2,7),(4,5),(6,8)]
=> [3,7,1,5,4,8,2,6] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,7,5,4,8,3,6] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(6,2),(6,3),(7,2),(7,3)],8)
=> ([(0,2),(2,1)],3)
=> 1
[(1,3),(2,8),(4,5),(6,7)]
=> [3,8,1,5,4,7,6,2] => ([(0,3),(0,6),(0,7),(1,2),(1,6),(1,7),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[(1,4),(2,8),(3,5),(6,7)]
=> [4,8,5,1,3,7,6,2] => ([(0,3),(0,5),(1,2),(1,4),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[(1,5),(2,8),(3,4),(6,7)]
=> [5,8,4,3,1,7,6,2] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7)],8)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[(1,6),(2,8),(3,4),(5,7)]
=> [6,8,4,3,7,1,5,2] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,6)],8)
=> ([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7)
=> 2
[(1,7),(2,8),(3,4),(5,6)]
=> [7,8,4,3,6,5,1,2] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
[(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => ([(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1)],2)
=> 1
[(1,8),(2,7),(3,5),(4,6)]
=> [8,7,5,6,3,4,2,1] => ([(4,7),(5,6)],8)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[(1,7),(2,8),(3,5),(4,6)]
=> [7,8,5,6,3,4,1,2] => ([(0,7),(1,6),(2,5),(3,4)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 2
[(1,6),(2,8),(3,5),(4,7)]
=> [6,8,5,7,3,1,4,2] => ([(0,7),(1,6),(2,4),(2,6),(3,5),(3,7)],8)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[(1,5),(2,8),(3,6),(4,7)]
=> [5,8,6,7,1,3,4,2] => ([(0,5),(0,7),(1,4),(1,6),(6,2),(7,3)],8)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[(1,5),(2,7),(3,6),(4,8)]
=> [5,7,6,8,1,3,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,7),(5,7)],8)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[(1,6),(2,7),(3,5),(4,8)]
=> [6,7,5,8,3,1,2,4] => ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7)],8)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[(1,8),(2,5),(3,6),(4,7)]
=> [8,5,6,7,2,3,4,1] => ([(2,5),(3,4),(4,6),(5,7)],8)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[(1,7),(2,5),(3,6),(4,8)]
=> [7,5,6,8,2,3,1,4] => ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7)],8)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[(1,6),(2,5),(3,7),(4,8)]
=> [6,5,7,8,2,1,3,4] => ([(0,6),(1,6),(2,7),(3,7),(6,5),(7,4)],8)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[(1,5),(2,6),(3,7),(4,8)]
=> [5,6,7,8,1,2,3,4] => ([(0,7),(1,6),(4,2),(5,3),(6,4),(7,5)],8)
=> ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ? = 4
[(1,4),(2,6),(3,7),(5,8)]
=> [4,6,7,1,8,2,3,5] => ([(0,5),(0,7),(1,4),(1,6),(2,7),(3,6),(4,2),(5,3)],8)
=> ([(0,5),(0,6),(1,9),(2,8),(3,10),(4,11),(5,3),(5,13),(6,4),(6,13),(8,7),(9,7),(10,2),(10,12),(11,1),(11,12),(12,8),(12,9),(13,10),(13,11)],14)
=> ? = 3
[(1,3),(2,6),(4,7),(5,8)]
=> [3,6,1,7,8,2,4,5] => ([(0,5),(0,7),(1,4),(1,6),(4,7),(5,6),(6,2),(7,3)],8)
=> ([(0,3),(0,4),(1,10),(2,9),(3,7),(4,7),(5,2),(5,8),(6,1),(6,8),(7,5),(7,6),(8,9),(8,10),(9,11),(10,11)],12)
=> ? = 3
[(1,2),(3,6),(4,7),(5,8)]
=> [2,1,6,7,8,3,4,5] => ([(0,6),(0,7),(1,6),(1,7),(4,3),(5,2),(6,4),(7,5)],8)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 3
[(1,2),(3,5),(4,7),(6,8)]
=> [2,1,5,7,3,8,4,6] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4),(6,3),(6,5),(7,2),(7,4)],8)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 2
[(1,3),(2,5),(4,7),(6,8)]
=> [3,5,1,7,2,8,4,6] => ([(0,3),(0,7),(1,2),(1,6),(2,4),(2,7),(3,5),(3,6),(6,4),(7,5)],8)
=> ([(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(8,1),(8,2),(9,3),(9,4)],10)
=> ? = 2
[(1,4),(2,5),(3,7),(6,8)]
=> [4,5,7,1,2,8,3,6] => ([(0,5),(1,4),(2,7),(3,6),(4,2),(4,6),(5,3),(5,7)],8)
=> ([(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,2),(5,8),(6,1),(6,8),(8,9),(8,10),(9,11),(10,11),(11,3),(11,4)],12)
=> ? = 3
[(1,6),(2,4),(3,7),(5,8)]
=> [6,4,7,2,8,1,3,5] => ([(0,5),(1,4),(2,4),(2,7),(3,5),(3,6),(4,6),(5,7)],8)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[(1,6),(2,3),(4,7),(5,8)]
=> [6,3,2,7,8,1,4,5] => ([(0,7),(1,6),(2,6),(2,7),(3,6),(3,7),(6,4),(7,5)],8)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[(1,3),(2,4),(5,7),(6,8)]
=> [3,4,1,2,7,8,5,6] => ([(0,5),(1,4),(4,6),(4,7),(5,6),(5,7),(6,3),(7,2)],8)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ? = 2
[(1,4),(2,5),(3,8),(6,7)]
=> [4,5,8,1,2,7,6,3] => ([(0,5),(1,4),(4,2),(4,6),(4,7),(5,3),(5,6),(5,7)],8)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[(1,4),(2,6),(3,8),(5,7)]
=> [4,6,8,1,7,2,5,3] => ([(0,5),(0,7),(1,4),(1,6),(4,2),(4,7),(5,3),(5,6)],8)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[(1,5),(2,6),(3,8),(4,7)]
=> [5,6,8,7,1,2,4,3] => ([(0,7),(1,6),(6,2),(6,3),(7,4),(7,5)],8)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[(1,7),(2,5),(3,8),(4,6)]
=> [7,5,8,6,2,4,1,3] => ([(0,7),(1,6),(2,4),(2,6),(3,5),(3,7)],8)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[(1,6),(2,7),(3,8),(4,5)]
=> [6,7,8,5,4,1,2,3] => ([(2,5),(3,4),(4,6),(5,7)],8)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[(1,5),(2,7),(3,8),(4,6)]
=> [5,7,8,6,1,4,2,3] => ([(0,5),(0,7),(1,4),(1,6),(6,2),(7,3)],8)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[(1,4),(2,7),(3,8),(5,6)]
=> [4,7,8,1,6,5,2,3] => ([(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(4,2),(5,3)],8)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => ([(0,8),(0,9),(1,8),(1,9),(4,6),(4,7),(5,6),(5,7),(6,2),(6,3),(7,2),(7,3),(8,4),(8,5),(9,4),(9,5)],10)
=> ?
=> ? = 1
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [3,4,1,2,6,5,8,7,10,9] => ([(0,3),(1,2),(2,8),(2,9),(3,8),(3,9),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ?
=> ? = 2
[(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [4,3,2,1,6,5,8,7,10,9] => ([(0,8),(0,9),(1,8),(1,9),(2,8),(2,9),(3,8),(3,9),(4,6),(4,7),(5,6),(5,7),(8,4),(8,5),(9,4),(9,5)],10)
=> ?
=> ? = 1
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [5,3,2,6,1,4,8,7,10,9] => ?
=> ?
=> ? = 2
[(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [6,3,2,5,4,1,8,7,10,9] => ([(0,8),(0,9),(1,8),(1,9),(2,4),(2,5),(3,4),(3,5),(4,8),(4,9),(5,8),(5,9),(8,6),(8,7),(9,6),(9,7)],10)
=> ?
=> ? = 1
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [7,3,2,5,4,8,1,6,10,9] => ?
=> ?
=> ? = 2
[(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [8,3,2,5,4,7,6,1,10,9] => ([(0,8),(0,9),(1,8),(1,9),(2,4),(2,5),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7),(6,8),(6,9),(7,8),(7,9)],10)
=> ?
=> ? = 1
[(1,9),(2,3),(4,5),(6,7),(8,10)]
=> [9,3,2,5,4,7,6,10,1,8] => ?
=> ?
=> ? = 2
[(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [10,3,2,5,4,7,6,9,8,1] => ([(2,8),(2,9),(3,8),(3,9),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ?
=> ? = 1
[(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
[(1,4),(2,5),(3,6),(7,8),(9,10)]
=> [4,5,6,1,2,3,8,7,10,9] => ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,2),(5,3),(6,8),(6,9),(7,8),(7,9)],10)
=> ?
=> ? = 3
[(1,3),(2,5),(4,6),(7,8),(9,10)]
=> [3,5,1,6,2,4,8,7,10,9] => ([(0,3),(0,9),(1,2),(1,8),(2,9),(3,8),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ?
=> ? = 2
[(1,2),(3,5),(4,6),(7,8),(9,10)]
=> [2,1,5,6,3,4,8,7,10,9] => ([(0,6),(0,7),(1,6),(1,7),(2,8),(2,9),(3,8),(3,9),(6,3),(7,2),(8,4),(8,5),(9,4),(9,5)],10)
=> ?
=> ? = 2
[(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,6,5,4,3,8,7,10,9] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,2),(8,3),(9,2),(9,3)],10)
=> ?
=> ? = 1
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [3,6,1,5,4,2,8,7,10,9] => ?
=> ?
=> ? = 2
Description
The breadth of a lattice.
The '''breadth''' of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.
Matching statistic: St001491
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 20%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 20%
Values
[(1,2)]
=> [2,1] => [2]
=> 0 => ? = 1 - 1
[(1,2),(3,4)]
=> [2,1,4,3] => [2,2]
=> 00 => ? = 1 - 1
[(1,3),(2,4)]
=> [3,4,1,2] => [2,1,1]
=> 011 => 1 = 2 - 1
[(1,4),(2,3)]
=> [4,3,2,1] => [4]
=> 0 => ? = 1 - 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,2,2]
=> 000 => ? = 1 - 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [2,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [4,2]
=> 00 => ? = 1 - 1
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [4,1,1]
=> 011 => 1 = 2 - 1
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [4,2]
=> 00 => ? = 1 - 1
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [4,1,1]
=> 011 => 1 = 2 - 1
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [4,1,1]
=> 011 => 1 = 2 - 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [2,1,1,1,1]
=> 01111 => ? = 3 - 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [2,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [4,2]
=> 00 => ? = 1 - 1
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [4,1,1]
=> 011 => 1 = 2 - 1
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [4,1,1]
=> 011 => 1 = 2 - 1
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [4,1,1]
=> 011 => 1 = 2 - 1
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [6]
=> 0 => ? = 1 - 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,2,2,2]
=> 0000 => ? = 1 - 1
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [2,2,2,1,1]
=> 00011 => ? = 2 - 1
[(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [4,2,2]
=> 000 => ? = 1 - 1
[(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [4,2,2]
=> 000 => ? = 1 - 1
[(1,7),(2,3),(4,5),(6,8)]
=> [7,3,2,5,4,8,1,6] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [4,2,2]
=> 000 => ? = 1 - 1
[(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,7),(2,4),(3,5),(6,8)]
=> [7,4,5,2,3,8,1,6] => [4,1,1,1,1]
=> 01111 => ? = 2 - 1
[(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [2,2,1,1,1,1]
=> 001111 => ? = 3 - 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [2,2,2,1,1]
=> 00011 => ? = 2 - 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [2,2,2,1,1]
=> 00011 => ? = 2 - 1
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [4,2,2]
=> 000 => ? = 1 - 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [6,2]
=> 00 => ? = 1 - 1
[(1,7),(2,5),(3,4),(6,8)]
=> [7,5,4,3,2,8,1,6] => [6,1,1]
=> 011 => 1 = 2 - 1
[(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [6,2]
=> 00 => ? = 1 - 1
[(1,8),(2,6),(3,4),(5,7)]
=> [8,6,4,3,7,2,5,1] => [6,1,1]
=> 011 => 1 = 2 - 1
[(1,7),(2,6),(3,4),(5,8)]
=> [7,6,4,3,8,2,1,5] => [6,1,1]
=> 011 => 1 = 2 - 1
[(1,6),(2,7),(3,4),(5,8)]
=> [6,7,4,3,8,1,2,5] => [4,1,1,1,1]
=> 01111 => ? = 3 - 1
[(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,7,5,1,3,8,2,6] => [4,1,1,1,1]
=> 01111 => ? = 2 - 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,7,1,5,4,8,2,6] => [4,1,1,1,1]
=> 01111 => ? = 2 - 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,7,5,4,8,3,6] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [4,2,2]
=> 000 => ? = 1 - 1
[(1,3),(2,8),(4,5),(6,7)]
=> [3,8,1,5,4,7,6,2] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,4),(2,8),(3,5),(6,7)]
=> [4,8,5,1,3,7,6,2] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,5),(2,8),(3,4),(6,7)]
=> [5,8,4,3,1,7,6,2] => [4,3,1]
=> 011 => 1 = 2 - 1
[(1,6),(2,8),(3,4),(5,7)]
=> [6,8,4,3,7,1,5,2] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,7),(2,8),(3,4),(5,6)]
=> [7,8,4,3,6,5,1,2] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [6,2]
=> 00 => ? = 1 - 1
[(1,8),(2,7),(3,5),(4,6)]
=> [8,7,5,6,3,4,2,1] => [6,1,1]
=> 011 => 1 = 2 - 1
[(1,7),(2,8),(3,5),(4,6)]
=> [7,8,5,6,3,4,1,2] => [4,1,1,1,1]
=> 01111 => ? = 2 - 1
[(1,6),(2,8),(3,5),(4,7)]
=> [6,8,5,7,3,1,4,2] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,5),(2,8),(3,6),(4,7)]
=> [5,8,6,7,1,3,4,2] => [4,1,1,1,1]
=> 01111 => ? = 3 - 1
[(1,4),(2,8),(3,6),(5,7)]
=> [4,8,6,1,7,3,5,2] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,3),(2,8),(4,6),(5,7)]
=> [3,8,1,6,7,4,5,2] => [4,1,1,1,1]
=> 01111 => ? = 2 - 1
[(1,2),(3,8),(4,6),(5,7)]
=> [2,1,8,6,7,4,5,3] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,2),(3,7),(4,6),(5,8)]
=> [2,1,7,6,8,4,3,5] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,3),(2,7),(4,6),(5,8)]
=> [3,7,1,6,8,4,2,5] => [4,1,1,1,1]
=> 01111 => ? = 2 - 1
[(1,4),(2,7),(3,6),(5,8)]
=> [4,7,6,1,8,3,2,5] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,5),(2,7),(3,6),(4,8)]
=> [5,7,6,8,1,3,2,4] => [4,1,1,1,1]
=> 01111 => ? = 3 - 1
[(1,6),(2,7),(3,5),(4,8)]
=> [6,7,5,8,3,1,2,4] => [4,1,1,1,1]
=> 01111 => ? = 3 - 1
[(1,7),(2,6),(3,5),(4,8)]
=> [7,6,5,8,3,2,1,4] => [6,1,1]
=> 011 => 1 = 2 - 1
[(1,8),(2,6),(3,5),(4,7)]
=> [8,6,5,7,3,2,4,1] => [6,1,1]
=> 011 => 1 = 2 - 1
[(1,8),(2,5),(3,6),(4,7)]
=> [8,5,6,7,2,3,4,1] => [4,1,1,1,1]
=> 01111 => ? = 3 - 1
[(1,7),(2,5),(3,6),(4,8)]
=> [7,5,6,8,2,3,1,4] => [4,1,1,1,1]
=> 01111 => ? = 3 - 1
[(1,6),(2,5),(3,7),(4,8)]
=> [6,5,7,8,2,1,3,4] => [4,1,1,1,1]
=> 01111 => ? = 3 - 1
[(1,5),(2,6),(3,7),(4,8)]
=> [5,6,7,8,1,2,3,4] => [2,1,1,1,1,1,1]
=> 0111111 => ? = 4 - 1
[(1,4),(2,6),(3,7),(5,8)]
=> [4,6,7,1,8,2,3,5] => [2,2,1,1,1,1]
=> 001111 => ? = 3 - 1
[(1,3),(2,6),(4,7),(5,8)]
=> [3,6,1,7,8,2,4,5] => [2,2,1,1,1,1]
=> 001111 => ? = 3 - 1
[(1,2),(3,6),(4,7),(5,8)]
=> [2,1,6,7,8,3,4,5] => [2,2,1,1,1,1]
=> 001111 => ? = 3 - 1
[(1,2),(3,5),(4,7),(6,8)]
=> [2,1,5,7,3,8,4,6] => [2,2,2,1,1]
=> 00011 => ? = 2 - 1
[(1,3),(2,5),(4,7),(6,8)]
=> [3,5,1,7,2,8,4,6] => [2,2,2,1,1]
=> 00011 => ? = 2 - 1
[(1,4),(2,5),(3,7),(6,8)]
=> [4,5,7,1,2,8,3,6] => [2,2,1,1,1,1]
=> 001111 => ? = 3 - 1
[(1,5),(2,4),(3,7),(6,8)]
=> [5,4,7,2,1,8,3,6] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,6),(2,4),(3,7),(5,8)]
=> [6,4,7,2,8,1,3,5] => [4,1,1,1,1]
=> 01111 => ? = 3 - 1
[(1,7),(2,4),(3,6),(5,8)]
=> [7,4,6,2,8,3,1,5] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,8),(2,4),(3,6),(5,7)]
=> [8,4,6,2,7,3,5,1] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,8),(2,3),(4,6),(5,7)]
=> [8,3,2,6,7,4,5,1] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,7),(2,3),(4,6),(5,8)]
=> [7,3,2,6,8,4,1,5] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,6),(2,3),(4,7),(5,8)]
=> [6,3,2,7,8,1,4,5] => [4,1,1,1,1]
=> 01111 => ? = 3 - 1
[(1,5),(2,3),(4,7),(6,8)]
=> [5,3,2,7,1,8,4,6] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,4),(2,3),(5,7),(6,8)]
=> [4,3,2,1,7,8,5,6] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,3),(2,4),(5,7),(6,8)]
=> [3,4,1,2,7,8,5,6] => [2,2,1,1,1,1]
=> 001111 => ? = 2 - 1
[(1,2),(3,4),(5,7),(6,8)]
=> [2,1,4,3,7,8,5,6] => [2,2,2,1,1]
=> 00011 => ? = 2 - 1
[(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,8,7,6,5] => [4,2,2]
=> 000 => ? = 1 - 1
[(1,3),(2,4),(5,8),(6,7)]
=> [3,4,1,2,8,7,6,5] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => [4,4]
=> 00 => ? = 1 - 1
[(1,5),(2,3),(4,8),(6,7)]
=> [5,3,2,8,1,7,6,4] => [4,3,1]
=> 011 => 1 = 2 - 1
[(1,6),(2,3),(4,8),(5,7)]
=> [6,3,2,8,7,1,5,4] => [4,3,1]
=> 011 => 1 = 2 - 1
[(1,7),(2,3),(4,8),(5,6)]
=> [7,3,2,8,6,5,1,4] => [4,3,1]
=> 011 => 1 = 2 - 1
[(1,8),(2,3),(4,7),(5,6)]
=> [8,3,2,7,6,5,4,1] => [6,2]
=> 00 => ? = 1 - 1
[(1,8),(2,4),(3,7),(5,6)]
=> [8,4,7,2,6,5,3,1] => [6,1,1]
=> 011 => 1 = 2 - 1
[(1,7),(2,4),(3,8),(5,6)]
=> [7,4,8,2,6,5,1,3] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,6),(2,4),(3,8),(5,7)]
=> [6,4,8,2,7,1,5,3] => [4,2,1,1]
=> 0011 => 1 = 2 - 1
[(1,4),(2,5),(3,8),(6,7)]
=> [4,5,8,1,2,7,6,3] => [4,1,1,1,1]
=> 01111 => ? = 3 - 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000098The chromatic number of a graph. St000527The width of the poset. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000223The number of nestings in the permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation.
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