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Your data matches 263 different statistics following compositions of up to 3 maps.
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Matching statistic: St000150
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000150: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> 0
([],2)
=> []
=> 0
([(0,1)],2)
=> [1]
=> 0
([],3)
=> []
=> 0
([(1,2)],3)
=> [1]
=> 0
([(0,2),(1,2)],3)
=> [2]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([],4)
=> []
=> 0
([(2,3)],4)
=> [1]
=> 0
([(1,3),(2,3)],4)
=> [2]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [3]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 0
([],5)
=> []
=> 0
([(3,4)],5)
=> [1]
=> 0
([(2,4),(3,4)],5)
=> [2]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [3]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> 0
Description
The floored half-sum of the multiplicities of a partition.
This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].
Matching statistic: St000257
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> 0
([],2)
=> []
=> 0
([(0,1)],2)
=> [1]
=> 0
([],3)
=> []
=> 0
([(1,2)],3)
=> [1]
=> 0
([(0,2),(1,2)],3)
=> [2]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
([],4)
=> []
=> 0
([(2,3)],4)
=> [1]
=> 0
([(1,3),(2,3)],4)
=> [2]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [3]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 0
([],5)
=> []
=> 0
([(3,4)],5)
=> [1]
=> 0
([(2,4),(3,4)],5)
=> [2]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [3]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> 0
Description
The number of distinct parts of a partition that occur at least twice.
See Section 3.3.1 of [2].
Matching statistic: St001085
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
([],2)
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
([(0,1)],2)
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0
([],3)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
([(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
([(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 0
([],4)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 0
([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 0
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 0
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 0
([],5)
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 0
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 0
Description
The number of occurrences of the vincular pattern |21-3 in a permutation.
This is the number of occurrences of the pattern $213$, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive.
In other words, this is the number of ascents whose bottom value is strictly smaller and the top value is strictly larger than the first entry of the permutation.
Matching statistic: St001092
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> ?
=> ? = 0
([],2)
=> []
=> ?
=> ? = 0
([(0,1)],2)
=> [1]
=> [1]
=> 0
([],3)
=> []
=> ?
=> ? = 0
([(1,2)],3)
=> [1]
=> [1]
=> 0
([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [3]
=> 0
([],4)
=> []
=> ?
=> ? = 0
([(2,3)],4)
=> [1]
=> [1]
=> 0
([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [3]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [3]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [3]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [5]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [3,3]
=> 0
([],5)
=> []
=> ?
=> ? = 0
([(3,4)],5)
=> [1]
=> [1]
=> 0
([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [3]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [3]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1,1,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [3]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [3,3]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [3,3]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [7]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [3,1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [5]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> [3,3]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [5]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [3,3]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [7]
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [3,3]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [3,3]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [7]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [7]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> [9]
=> 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> [5,5]
=> 0
([],6)
=> []
=> ?
=> ? = 0
([(4,5)],6)
=> [1]
=> [1]
=> 0
([(3,5),(4,5)],6)
=> [2]
=> [1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [3]
=> [3]
=> 0
([],7)
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0}
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0}
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0}
([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0}
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [19]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0}
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0}
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [18]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0}
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [19]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0}
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [20]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0}
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [21]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0}
Description
The number of distinct even parts of a partition.
See Section 3.3.1 of [1].
Matching statistic: St000149
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> []
=> 0
([],2)
=> []
=> []
=> 0
([(0,1)],2)
=> [1]
=> [1]
=> 0
([],3)
=> []
=> []
=> 0
([(1,2)],3)
=> [1]
=> [1]
=> 0
([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 0
([],4)
=> []
=> []
=> 0
([(2,3)],4)
=> [1]
=> [1]
=> 0
([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,1,1,1,1,1]
=> 0
([],5)
=> []
=> []
=> 0
([(3,4)],5)
=> [1]
=> [1]
=> 0
([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [2,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0
([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,1),(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [19]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [18]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [19]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [20]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [21]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
Description
The number of cells of the partition whose leg is zero and arm is odd.
This statistic is equidistributed with [[St000143]], see [1].
Matching statistic: St001175
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1]
=> []
=> ? = 0
([],2)
=> [1,1]
=> [2]
=> []
=> ? = 0
([(0,1)],2)
=> [2]
=> [1,1]
=> [1]
=> 0
([],3)
=> [1,1,1]
=> [2,1]
=> [1]
=> 0
([(1,2)],3)
=> [2,1]
=> [3]
=> []
=> ? = 0
([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> [1,1]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> [1,1]
=> 0
([],4)
=> [1,1,1,1]
=> [3,1]
=> [1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [2,2]
=> [2]
=> 0
([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [4]
=> []
=> ? = 1
([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([],5)
=> [1,1,1,1,1]
=> [3,2]
=> [2]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [4,1]
=> [1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [5]
=> []
=> ? ∊ {0,1}
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [4,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [5]
=> []
=> ? ∊ {0,1}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([],6)
=> [1,1,1,1,1,1]
=> [3,2,1]
=> [2,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [4,2]
=> [2]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [3,3]
=> [3]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1}
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1}
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1}
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> [4,3]
=> [7]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> [4,3]
=> [7]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [4,3]
=> [7]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [4,3]
=> [7]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> [4,3]
=> [7]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> [4,3]
=> [7]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> [4,3]
=> [7]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [4,3]
=> [7]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> [4,3]
=> [7]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3]
=> [7]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3]
=> [7]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,3]
=> [7]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1}
Description
The size of a partition minus the hook length of the base cell.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St001392
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> []
=> ?
=> ? = 0
([],2)
=> []
=> []
=> ?
=> ? ∊ {0,0}
([(0,1)],2)
=> [1]
=> [1]
=> []
=> ? ∊ {0,0}
([],3)
=> []
=> []
=> ?
=> ? ∊ {0,0}
([(1,2)],3)
=> [1]
=> [1]
=> []
=> ? ∊ {0,0}
([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> [1,1]
=> 0
([],4)
=> []
=> []
=> ?
=> ? ∊ {0,0,1}
([(2,3)],4)
=> [1]
=> [1]
=> []
=> ? ∊ {0,0,1}
([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> [1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> []
=> ? ∊ {0,0,1}
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([],5)
=> []
=> []
=> ?
=> ? ∊ {0,0,1}
([(3,4)],5)
=> [1]
=> [1]
=> []
=> ? ∊ {0,0,1}
([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> []
=> ? ∊ {0,0,1}
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1]
=> [1,1]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [2,1]
=> [1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> 0
([],6)
=> []
=> []
=> ?
=> ? ∊ {0,0,1,1}
([(4,5)],6)
=> [1]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
([(3,5),(4,5)],6)
=> [2]
=> [1,1]
=> [1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [3]
=> [1,1,1]
=> [1,1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(2,5),(3,4)],6)
=> [1,1]
=> [2]
=> []
=> ? ∊ {0,0,1,1}
([(2,5),(3,4),(4,5)],6)
=> [3]
=> [1,1,1]
=> [1,1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> [2,1]
=> [1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1]
=> [1,1]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [3]
=> []
=> ? ∊ {0,0,1,1}
([],7)
=> []
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
([(5,6)],7)
=> [1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
([(3,6),(4,5)],7)
=> [1,1]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
([(1,6),(2,5),(3,4)],7)
=> [1,1,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [19]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [18]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [19]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [20]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [21]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1}
Description
The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.
Matching statistic: St000142
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> ?
=> ? = 0
([],2)
=> []
=> ?
=> ? = 0
([(0,1)],2)
=> [1]
=> [1]
=> 0
([],3)
=> []
=> ?
=> ? = 0
([(1,2)],3)
=> [1]
=> [1]
=> 0
([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [3]
=> 0
([],4)
=> []
=> ?
=> ? = 0
([(2,3)],4)
=> [1]
=> [1]
=> 0
([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [3]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [3]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [3]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [5]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [3,3]
=> 0
([],5)
=> []
=> ?
=> ? = 0
([(3,4)],5)
=> [1]
=> [1]
=> 0
([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [3]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [3]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1,1,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [3]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [5]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [3,3]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [3,3]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [7]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [3,1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [5]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> [3,3]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [5]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [3,3]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [7]
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [3,3]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [3,3]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [7]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [7]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> [9]
=> 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> [5,5]
=> 0
([],6)
=> []
=> ?
=> ? = 0
([(4,5)],6)
=> [1]
=> [1]
=> 0
([(3,5),(4,5)],6)
=> [2]
=> [1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [3]
=> [3]
=> 0
([],7)
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,1),(0,2),(0,3),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,3),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,1),(0,3),(0,5),(0,6),(1,2),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,2),(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,2),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [19]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,1),(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [18]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [19]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [20]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [21]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}
Description
The number of even parts of a partition.
Matching statistic: St001593
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001593: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 93%●distinct values known / distinct values provided: 50%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001593: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 93%●distinct values known / distinct values provided: 50%
Values
([],1)
=> []
=> []
=> ?
=> ? = 0
([],2)
=> []
=> []
=> ?
=> ? ∊ {0,0}
([(0,1)],2)
=> [1]
=> [1]
=> []
=> ? ∊ {0,0}
([],3)
=> []
=> []
=> ?
=> ? ∊ {0,0,0}
([(1,2)],3)
=> [1]
=> [1]
=> []
=> ? ∊ {0,0,0}
([(0,2),(1,2)],3)
=> [1,1]
=> [2]
=> []
=> ? ∊ {0,0,0}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> [1,1]
=> 0
([],4)
=> []
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1}
([(2,3)],4)
=> [1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,1}
([(1,3),(2,3)],4)
=> [1,1]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,1}
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,1}
([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,1}
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,1}
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([],5)
=> []
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
([(3,4)],5)
=> [1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
([(2,4),(3,4)],5)
=> [1,1]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [2,2,2]
=> [2,2]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> 0
([],6)
=> []
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
([(4,5)],6)
=> [1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
([(3,5),(4,5)],6)
=> [1,1]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
([(2,5),(3,4)],6)
=> [1,1]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1]
=> [1,1]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1}
([],7)
=> []
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
([(5,6)],7)
=> [1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
([(4,6),(5,6)],7)
=> [1,1]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
([(3,6),(4,6),(5,6)],7)
=> [1,1,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
([(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1,1,1]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
([(3,6),(4,5)],7)
=> [1,1]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
Description
This is the number of standard Young tableaux of the given shifted shape.
For an integer partition $\lambda = (\lambda_1,\dots,\lambda_k)$, the shifted diagram is obtained by moving the $i$-th row in the diagram $i-1$ boxes to the right, i.e.,
$$\lambda^∗ = \{(i, j) | 1 \leq i \leq k, i \leq j \leq \lambda_i + i − 1 \}.$$
In particular, this statistic is zero if and only if $\lambda_{i+1} = \lambda_i$ for some $i$.
Matching statistic: St001518
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([],2)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,1)],2)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([],3)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
([(1,2)],3)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([],4)
=> ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(2,3)],4)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([],5)
=> ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(3,4)],5)
=> ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(1,4),(2,3)],5)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,0,0} + 1
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0} + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,0,0} + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([],6)
=> ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1} + 1
([(4,5)],6)
=> ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1} + 1
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1} + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1} + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1} + 1
([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1} + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1} + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1} + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1} + 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(0,6),(0,8),(1,7),(1,8),(2,7),(2,8),(3,4),(3,5),(3,8),(4,6),(4,8),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1} + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1} + 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1} + 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1} + 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(0,8),(1,7),(1,8),(2,6),(2,8),(3,4),(3,5),(3,8),(4,6),(4,8),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1} + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1} + 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(0,8),(1,5),(1,8),(2,7),(2,8),(3,5),(3,7),(3,8),(4,6),(4,7),(4,8),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1} + 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1} + 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1} + 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1} + 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(0,9),(1,7),(1,9),(2,3),(2,4),(2,9),(3,5),(3,9),(4,6),(4,9),(5,7),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1} + 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1} + 1
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1} + 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1} + 1
([],7)
=> ([],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1} + 1
([(5,6)],7)
=> ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1} + 1
([(4,6),(5,6)],7)
=> ([(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1} + 1
([(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1} + 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1} + 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1} + 1
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],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),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1} + 1
([(3,6),(4,5),(5,6)],7)
=> ([(3,7),(4,6),(5,6),(5,7)],8)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1} + 1
([(2,3),(4,6),(5,6)],7)
=> ([(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1} + 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1} + 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1} + 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1} + 1
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1} + 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1} + 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1} + 1
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1} + 1
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1} + 1
([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1} + 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1} + 1
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,7),(2,8),(3,8),(4,5),(4,6),(5,7),(6,8)],9)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1} + 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1} + 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1} + 1
([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1} + 1
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,8),(1,8),(2,8),(3,4),(4,6),(5,7),(5,8),(6,7)],9)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1} + 1
Description
The number of graphs with the same ordinary spectrum as the given graph.
The following 253 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000007The number of saliances of the permutation. St000929The constant term of the character polynomial of an integer partition. St000379The number of Hamiltonian cycles in a graph. St000256The number of parts from which one can substract 2 and still get an integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001961The sum of the greatest common divisors of all pairs of parts. St001570The minimal number of edges to add to make a graph Hamiltonian. St000296The length of the symmetric border of a binary word. St000297The number of leading ones in a binary word. St000629The defect of a binary word. St000052The number of valleys of a Dyck path not on the x-axis. St000386The number of factors DDU in a Dyck path. St001330The hat guessing number of a graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000567The sum of the products of all pairs of parts. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000699The toughness times the least common multiple of 1,. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001307The number of induced stars on four vertices in a graph. St000225Difference between largest and smallest parts in a partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001846The number of elements which do not have a complement in the lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001172The number of 1-rises at odd height of a Dyck path. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000661The number of rises of length 3 of a Dyck path. St001139The number of occurrences of hills of size 2 in a Dyck path. St000322The skewness of a graph. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000376The bounce deficit of a Dyck path. St000455The second largest eigenvalue of a graph if it is integral. St001578The minimal number of edges to add or remove to make a graph a line graph. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000658The number of rises of length 2 of a Dyck path. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St001584The area statistic between a Dyck path and its bounce path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St000091The descent variation of a composition. St000761The number of ascents in an integer composition. St000768The number of peaks in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St001309The number of four-cliques in a graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St000449The number of pairs of vertices of a graph with distance 4. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000017The number of inversions of a standard tableau. St000290The major index of a binary word. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000347The inversion sum of a binary word. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St001214The aft of an integer partition. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001596The number of two-by-two squares inside a skew partition. St001730The number of times the path corresponding to a binary word crosses the base line. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000447The number of pairs of vertices of a graph with distance 3. St000552The number of cut vertices of a graph. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St000478Another weight of a partition according to Alladi. St000311The number of vertices of odd degree in a graph. St000323The minimal crossing number of a graph. St000370The genus of a graph. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001323The independence gap of a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001395The number of strictly unfriendly partitions of a graph. St001479The number of bridges of a graph. St001521Half the total irregularity of a graph. St001522The total irregularity of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001647The number of edges that can be added without increasing the clique number. St001689The number of celebrities in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001708The number of pairs of vertices of different degree in a graph. St001742The difference of the maximal and the minimal degree in a graph. St001826The maximal number of leaves on a vertex of a graph. St000095The number of triangles of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000671The maximin edge-connectivity for choosing a subgraph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001071The beta invariant of the graph. St001305The number of induced cycles on four vertices in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001638The book thickness of a graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001736The total number of cycles in a graph. St001797The number of overfull subgraphs of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001871The number of triconnected components of a graph. St001970The signature of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St001829The common independence number of a graph. St000258The burning number of a graph. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001322The size of a minimal independent dominating set in a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001339The irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St000274The number of perfect matchings of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001969The difference in the number of possibilities of choosing a pair of negative eigenvalues and the signature of a graph. St001737The number of descents of type 2 in a permutation. St001481The minimal height of a peak of a Dyck path. St000422The energy of a graph, if it is integral. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001881The number of factors of a lattice as a Cartesian product of lattices. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001363The Euler characteristic of a graph according to Knill. St001496The number of graphs with the same Laplacian spectrum as the given graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000934The 2-degree of an integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St000287The number of connected components of a graph. St001964The interval resolution global dimension of a poset. St001845The number of join irreducibles minus the rank of a lattice. St000068The number of minimal elements in a poset. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001651The Frankl number of a lattice. St000069The number of maximal elements of a poset. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000944The 3-degree of an integer partition. St001280The number of parts of an integer partition that are at least two. St001587Half of the largest even part of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001541The Gini index of an integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001568The smallest positive integer that does not appear twice in the partition. St001271The competition number of a graph. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001490The number of connected components of a skew partition. St001301The first Betti number of the order complex associated with the poset. St000181The number of connected components of the Hasse diagram for the poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even. St001487The number of inner corners of a skew partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St000553The number of blocks of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000916The packing number of a graph. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph.
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