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Your data matches 91 different statistics following compositions of up to 3 maps.
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Matching statistic: St000228
(load all 36 compositions to match this statistic)
(load all 36 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 1 = 2 - 1
([],2)
=> [1,1]
=> 2 = 3 - 1
([(0,1)],2)
=> [2]
=> 2 = 3 - 1
([(1,2)],3)
=> [2,1]
=> 3 = 4 - 1
([(0,1),(0,2)],3)
=> [2,1]
=> 3 = 4 - 1
([(0,2),(2,1)],3)
=> [3]
=> 3 = 4 - 1
([(0,2),(1,2)],3)
=> [2,1]
=> 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> 4 = 5 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 4 = 5 - 1
([(1,2),(2,3)],4)
=> [3,1]
=> 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> 4 = 5 - 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 4 = 5 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> 4 = 5 - 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 4 = 5 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 4 = 5 - 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 5 - 1
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 4 = 5 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 5 = 6 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> 5 = 6 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> 5 = 6 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> 5 = 6 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> 5 = 6 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 5 = 6 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> 5 = 6 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> 5 = 6 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> 5 = 6 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> 5 = 6 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> 5 = 6 - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> 5 = 6 - 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> 5 = 6 - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> 5 = 6 - 1
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> 5 = 6 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> 5 = 6 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 5 = 6 - 1
([(0,3),(1,4),(4,2)],5)
=> [3,2]
=> 5 = 6 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> 5 = 6 - 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> 5 = 6 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 6 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 5 = 6 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 5 = 6 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 5 = 6 - 1
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> [4,2]
=> 6 = 7 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 6 = 7 - 1
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5,1]
=> 6 = 7 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 7 - 1
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> [5,2]
=> 7 = 8 - 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 7 = 8 - 1
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000293
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 10 => 1 = 2 - 1
([],2)
=> [1,1]
=> 110 => 2 = 3 - 1
([(0,1)],2)
=> [2]
=> 100 => 2 = 3 - 1
([(1,2)],3)
=> [2,1]
=> 1010 => 3 = 4 - 1
([(0,1),(0,2)],3)
=> [2,1]
=> 1010 => 3 = 4 - 1
([(0,2),(2,1)],3)
=> [3]
=> 1000 => 3 = 4 - 1
([(0,2),(1,2)],3)
=> [2,1]
=> 1010 => 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> 10010 => 4 = 5 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 10010 => 4 = 5 - 1
([(1,2),(2,3)],4)
=> [3,1]
=> 10010 => 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> 10010 => 4 = 5 - 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 10010 => 4 = 5 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> 1100 => 4 = 5 - 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 1100 => 4 = 5 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 1100 => 4 = 5 - 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> 10000 => 4 = 5 - 1
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 10010 => 4 = 5 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 100010 => 5 = 6 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> 10100 => 5 = 6 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> 10100 => 5 = 6 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> 10100 => 5 = 6 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> 10100 => 5 = 6 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 100010 => 5 = 6 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> 10100 => 5 = 6 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> 10100 => 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> 10100 => 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> 10100 => 5 = 6 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> 10100 => 5 = 6 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> 100010 => 5 = 6 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> 100010 => 5 = 6 - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> 10100 => 5 = 6 - 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> 10100 => 5 = 6 - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> 10100 => 5 = 6 - 1
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> 100010 => 5 = 6 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> 100010 => 5 = 6 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 100010 => 5 = 6 - 1
([(0,3),(1,4),(4,2)],5)
=> [3,2]
=> 10100 => 5 = 6 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> 100010 => 5 = 6 - 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> 10100 => 5 = 6 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 100000 => 5 = 6 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 10100 => 5 = 6 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 100010 => 5 = 6 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 100010 => 5 = 6 - 1
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> [4,2]
=> 100100 => 6 = 7 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 100100 => 6 = 7 - 1
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5,1]
=> 1000010 => 6 = 7 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 1000000 => 6 = 7 - 1
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> [5,2]
=> 1000100 => 7 = 8 - 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 1000100 => 7 = 8 - 1
Description
The number of inversions of a binary word.
Matching statistic: St001034
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1,0]
=> 1 = 2 - 1
([],2)
=> [1,1]
=> [1,1,0,0]
=> 2 = 3 - 1
([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2 = 3 - 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 3 = 4 - 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 3 = 4 - 1
([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
([(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> 4 = 5 - 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> 4 = 5 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> 4 = 5 - 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5 = 6 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5 = 6 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5 = 6 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5 = 6 - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5 = 6 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5 = 6 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5 = 6 - 1
([(0,3),(1,4),(4,2)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5 = 6 - 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5 = 6 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5 = 6 - 1
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 6 = 7 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 6 = 7 - 1
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 6 = 7 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 7 - 1
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 7 = 8 - 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 7 = 8 - 1
Description
The area of the parallelogram polyomino associated with the Dyck path.
The (bivariate) generating function is given in [1].
Matching statistic: St001622
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 2 - 1
([],2)
=> ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 4 - 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 4 - 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 5 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 4 = 5 - 1
([(1,2),(2,3)],4)
=> ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 4 = 5 - 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 4 = 5 - 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 4 = 5 - 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> 4 = 5 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 4 = 5 - 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 4 = 5 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 5 = 6 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> 5 = 6 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> 5 = 6 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> 5 = 6 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> 5 = 6 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 5 = 6 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> 5 = 6 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> 5 = 6 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> 5 = 6 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 5 = 6 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> 5 = 6 - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> 5 = 6 - 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> 5 = 6 - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> 5 = 6 - 1
([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 5 = 6 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 5 = 6 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> 5 = 6 - 1
([(0,3),(1,4),(4,2)],5)
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> 5 = 6 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 5 = 6 - 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> 5 = 6 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> 5 = 6 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> 5 = 6 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 5 = 6 - 1
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> 6 = 7 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> 6 = 7 - 1
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> 6 = 7 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> 7 = 8 - 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> 7 = 8 - 1
Description
The number of join-irreducible elements of a lattice.
An element $j$ of a lattice $L$ is '''join irreducible''' if it is not the least element and if $j=x\vee y$, then $j\in\{x,y\}$ for all $x,y\in L$.
Matching statistic: St001746
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 1 = 2 - 1
([],2)
=> ([(0,1)],2)
=> ([],2)
=> 2 = 3 - 1
([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 2 = 3 - 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 3 = 4 - 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 4 = 5 - 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 4 = 5 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 5 - 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 5 = 6 - 1
([(1,4),(3,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(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)
=> 7 = 8 - 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(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)
=> 7 = 8 - 1
Description
The coalition number of a graph.
This is the maximal cardinality of a set partition such that each block is either a dominating set of cardinality one, or is not a dominating set but can be joined with a second block to form a dominating set.
Matching statistic: St000093
Values
([],1)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3
([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 4
([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 4
([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 4
([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 5
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 5
([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> 5
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 5
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 5
([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> 5
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(3,6),(4,5)],7)
=> 5
([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 5
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 6
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> 6
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> 6
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ([(4,7),(5,6)],8)
=> 6
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ([(4,7),(5,6)],8)
=> 6
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(5,6)],7)
=> 6
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ([(2,8),(3,4),(3,10),(4,9),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> 6
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> 6
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ([(3,4),(5,8),(6,7),(7,8)],9)
=> 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ([(4,7),(5,6)],8)
=> 6
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(2,9),(3,8),(4,5),(5,9),(6,7),(6,9),(7,8),(8,9)],10)
=> 6
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> 6
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> 6
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ([(2,9),(3,8),(4,6),(5,7),(6,8),(7,9),(8,9)],10)
=> 6
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ([(2,8),(3,4),(3,10),(4,9),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> 6
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ([(3,4),(5,8),(6,7),(7,8)],9)
=> 6
([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> 6
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 6
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> 6
([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> 6
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> 6
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ([(2,9),(3,8),(4,5),(5,9),(6,7),(6,9),(7,8),(8,9)],10)
=> 6
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> 6
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> 6
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 6
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> 7
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> 7
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> ([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> 7
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 7
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
=> 8
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
=> 8
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000147
Values
([],1)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 3
([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 4
([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 4
([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 4
([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 5
([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 5
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 5
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [5,1]
=> 5
([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> [5,3,1]
=> 5
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> [5,3]
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> [5,2]
=> 5
([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> [5,2]
=> 5
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [6,1]
=> 6
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [6,3,1]
=> 6
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> 6
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> [6,2]
=> 6
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> [6,2]
=> 6
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> [6,1]
=> 6
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> [6,4,1]
=> 6
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> [6,3]
=> 6
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> [6,3]
=> 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> [6,2]
=> 6
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> [6,4]
=> 6
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [6,3]
=> 6
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> [6,2]
=> 6
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> [6,4]
=> 6
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> [6,4,1]
=> 6
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> [6,3]
=> 6
([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> [6,4]
=> 6
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> [6,1]
=> 6
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> [6,2]
=> 6
([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> 6
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> [6,2]
=> 6
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> [6,4]
=> 6
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> [6,3,1]
=> 6
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> [6,3]
=> 6
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> [6,1]
=> 6
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> 7
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> 7
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [7,4]
=> 7
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [7]
=> 7
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> [8,3]
=> 8
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> [8,3]
=> 8
Description
The largest part of an integer partition.
Matching statistic: St000998
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00074: Posets —to graph⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1] => [1,0]
=> 2
([],2)
=> ([],2)
=> [2] => [1,1,0,0]
=> 3
([(0,1)],2)
=> ([(0,1)],2)
=> [1,1] => [1,0,1,0]
=> 3
([(1,2)],3)
=> ([(1,2)],3)
=> [1,2] => [1,0,1,1,0,0]
=> 4
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [1,1,1] => [1,0,1,0,1,0]
=> 4
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [1,1,1] => [1,0,1,0,1,0]
=> 4
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [1,1,1] => [1,0,1,0,1,0]
=> 4
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 5
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 5
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 5
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 5
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 5
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 5
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 6
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 6
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 6
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6
([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 6
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 6
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 7
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 8
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 8
Description
Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001065
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001065: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001065: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1,0]
=> [1,0]
=> 2
([],2)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 3
([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 3
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4
([(0,1),(0,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4
([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4
([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5
([(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 5
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 5
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 6
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 6
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 6
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 6
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 6
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 6
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 6
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 6
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 6
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 6
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 6
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 6
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6
([(0,3),(1,4),(4,2)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 6
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6
([(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 6
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6
([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 6
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 7
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 7
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 7
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 8
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 8
Description
Number of indecomposable reflexive modules in the corresponding Nakayama algebra.
Matching statistic: St000018
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000018: 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
St000018: 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] => 1 = 2 - 1
([],2)
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2 = 3 - 1
([(0,1)],2)
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2 = 3 - 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3 = 4 - 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3 = 4 - 1
([(0,2),(2,1)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3 = 4 - 1
([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4 = 5 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4 = 5 - 1
([(1,2),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4 = 5 - 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4 = 5 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4 = 5 - 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4 = 5 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4 = 5 - 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4 = 5 - 1
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4 = 5 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 5 = 6 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5 = 6 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5 = 6 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5 = 6 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5 = 6 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 5 = 6 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5 = 6 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5 = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5 = 6 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5 = 6 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 5 = 6 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 5 = 6 - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5 = 6 - 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5 = 6 - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5 = 6 - 1
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 5 = 6 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 5 = 6 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 5 = 6 - 1
([(0,3),(1,4),(4,2)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5 = 6 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 5 = 6 - 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5 = 6 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5 = 6 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5 = 6 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 5 = 6 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 5 = 6 - 1
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 6 = 7 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 6 = 7 - 1
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => 6 = 7 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 6 = 7 - 1
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => 7 = 8 - 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => 7 = 8 - 1
Description
The number of inversions of a permutation.
This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.
The following 81 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000246The number of non-inversions of a permutation. St000290The major index of a binary word. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000395The sum of the heights of the peaks of a Dyck path. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000507The number of ascents of a standard tableau. St000734The last entry in the first row of a standard tableau. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001462The number of factors of a standard tableaux under concatenation. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001672The restrained domination number of a graph. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St000384The maximal part of the shifted composition of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000806The semiperimeter of the associated bargraph. St001759The Rajchgot index of a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000189The number of elements in the poset. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001342The number of vertices in the center of a graph. St000171The degree of the graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St000144The pyramid weight of the Dyck path. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001120The length of a longest path in a graph. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001725The harmonious chromatic number of a graph. St000719The number of alignments in a perfect matching. St001645The pebbling number of a connected graph. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000273The domination number of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000916The packing number of a graph. St001286The annihilation number of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001829The common independence number of a graph. St000259The diameter of a connected graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001875The number of simple modules with projective dimension at most 1. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000528The height of a poset. St000906The length of the shortest maximal chain in a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001820The size of the image of the pop stack sorting operator. St001720The minimal length of a chain of small intervals in a lattice. St000080The rank of the poset. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000454The largest eigenvalue of a graph if it is integral. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian.
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