Your data matches 27 different statistics following compositions of up to 3 maps.
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Mp00074: Posets to graphGraphs
Mp00275: Graphs to edge-partition of connected componentsInteger partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> []
=> 0
([],2)
=> ([],2)
=> []
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> [1]
=> 1
([],3)
=> ([],3)
=> []
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> [1]
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [2]
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [2]
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2]
=> 2
([],4)
=> ([],4)
=> []
=> 0
([(2,3)],4)
=> ([(2,3)],4)
=> [1]
=> 1
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [2]
=> 2
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [3]
=> 3
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 4
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2]
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [3]
=> 3
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2]
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [3]
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [3]
=> 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 2
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 4
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 3
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 3
([],5)
=> ([],5)
=> []
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> [1]
=> 1
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> 2
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [3]
=> 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 4
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> 6
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [3]
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> 4
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 5
([(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)
=> [6]
=> 6
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> 2
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [3]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 4
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> 2
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [3]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 4
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [3]
=> 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 4
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> 4
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St001251
Mp00074: Posets to graphGraphs
Mp00251: Graphs clique sizesInteger partitions
St001251: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> 0
([],2)
=> ([],2)
=> [1,1]
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 1
([],3)
=> ([],3)
=> [1,1,1]
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([],4)
=> ([],4)
=> [1,1,1,1]
=> 0
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 1
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 4
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 2
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 4
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 1
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> 6
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 4
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(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)
=> [2,2,2,2,2,2]
=> 6
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 4
Description
The number of parts of a partition that are not congruent 1 modulo 3.
Matching statistic: St001252
Mp00074: Posets to graphGraphs
Mp00251: Graphs clique sizesInteger partitions
St001252: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> 0
([],2)
=> ([],2)
=> [1,1]
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 1
([],3)
=> ([],3)
=> [1,1,1]
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([],4)
=> ([],4)
=> [1,1,1,1]
=> 0
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 1
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 4
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 2
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 4
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 1
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> 6
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 4
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(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)
=> [2,2,2,2,2,2]
=> 6
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 4
([(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,1,1]
=> ? = 6
([(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,1,1]
=> ? = 6
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,1,1]
=> ? = 6
([(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,1,1]
=> ? = 6
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,1,1]
=> ? = 6
Description
Half the sum of the even parts of a partition.
Matching statistic: St000935
Mp00074: Posets to graphGraphs
Mp00251: Graphs clique sizesInteger partitions
St000935: Integer partitions ⟶ ℤResult quality: 98% values known / values provided: 98%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> 1 = 0 + 1
([],2)
=> ([],2)
=> [1,1]
=> 1 = 0 + 1
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
([],3)
=> ([],3)
=> [1,1,1]
=> 1 = 0 + 1
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 2 = 1 + 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 3 = 2 + 1
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 3 = 2 + 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 3 = 2 + 1
([],4)
=> ([],4)
=> [1,1,1,1]
=> 1 = 0 + 1
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 2 = 1 + 1
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 3 = 2 + 1
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 5 = 4 + 1
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 3 = 2 + 1
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 3 = 2 + 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 3 = 2 + 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 5 = 4 + 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> 1 = 0 + 1
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 2 = 1 + 1
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 3 = 2 + 1
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 4 = 3 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 6 = 5 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> 7 = 6 + 1
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 4 = 3 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> 5 = 4 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 6 = 5 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 6 = 5 + 1
([(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)
=> [2,2,2,2,2,2]
=> 7 = 6 + 1
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 3 = 2 + 1
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 4 = 3 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 3 = 2 + 1
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 4 = 3 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 4 = 3 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(1,2),(1,3),(1,4),(1,5),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,1,1]
=> ? = 6 + 1
([(1,2),(1,3),(1,4),(2,6),(3,6),(4,6),(6,5)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,2),(1,3),(1,4),(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,3),(1,4),(1,5),(3,6),(4,6),(5,2),(5,6)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,2),(1,3),(1,4),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,2),(1,3),(1,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,1,1]
=> ? = 6 + 1
([(1,3),(1,4),(3,5),(3,6),(4,5),(4,6),(6,2)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,3),(1,4),(3,5),(3,6),(4,2),(4,5),(4,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,1,1]
=> ? = 6 + 1
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(6,4)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,4),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,1,1]
=> ? = 6 + 1
([(1,4),(1,5),(2,4),(2,5),(4,6),(5,6),(6,3)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,4),(1,6),(2,4),(2,6),(4,5),(6,3),(6,5)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,4),(1,5),(2,4),(2,5),(3,6),(4,6),(5,3)],7)
=> ([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(4,6)],7)
=> ([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,5),(1,6),(2,5),(2,6),(3,4),(4,5),(4,6)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,2)],7)
=> ([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,6),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,4),(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,3),(1,4),(2,6),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,3),(1,4),(2,5),(3,5),(3,6),(4,2),(4,6)],7)
=> ([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,4),(1,5),(2,6),(3,6),(4,6),(5,2),(5,3)],7)
=> ([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,4),(1,5),(2,3),(2,4),(2,5),(4,6),(5,6)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,1,1]
=> ? = 6 + 1
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(6,3)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(5,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,4),(1,6),(2,3),(2,4),(3,6),(4,5),(6,5)],7)
=> ([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,4),(1,5),(2,3),(2,4),(3,5),(3,6),(4,6)],7)
=> ([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,4),(1,5),(2,3),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,5),(1,6),(2,3),(2,4),(2,5),(3,6),(4,6)],7)
=> ([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,6)],7)
=> ([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,4),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6)],7)
=> ([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,2)],7)
=> ([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,5),(1,6),(2,3),(3,5),(3,6),(5,4),(6,4)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,5),(2,6),(3,6),(4,6),(5,2),(5,3),(5,4)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
Description
The number of ordered refinements of an integer partition. This is, for an integer partition $\mu = (\mu_1,\ldots,\mu_n)$ the number of integer partition $\lambda = (\lambda_1,\ldots,\lambda_m)$ such that there are indices $1 = a_0 < \ldots < a_n = m$ with $\mu_j = \lambda_{a_{j-1}} + \ldots + \lambda_{a_j-1}$.
Matching statistic: St000142
Mp00074: Posets to graphGraphs
Mp00251: Graphs clique sizesInteger partitions
St000142: Integer partitions ⟶ ℤResult quality: 97% values known / values provided: 97%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> 0
([],2)
=> ([],2)
=> [1,1]
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 1
([],3)
=> ([],3)
=> [1,1,1]
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([],4)
=> ([],4)
=> [1,1,1,1]
=> 0
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 1
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 4
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 2
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 4
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 1
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> 6
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 4
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(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)
=> [2,2,2,2,2,2]
=> 6
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 4
([(1,2),(1,3),(1,4),(1,5),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(2,3),(2,4),(2,5),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,1,1]
=> ? = 5
([(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,1,1]
=> ? = 6
([(1,2),(1,3),(1,4),(2,6),(3,6),(4,6),(6,5)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,2),(1,3),(1,4),(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,3),(1,4),(1,5),(3,6),(4,6),(5,2),(5,6)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,2),(1,3),(1,4),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,2),(1,3),(1,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(3,4),(3,5),(4,6),(5,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,1,1,1]
=> ? = 4
([(2,3),(2,4),(3,6),(4,6),(6,5)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,1,1]
=> ? = 5
([(2,3),(2,4),(3,6),(4,5),(4,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,1,1]
=> ? = 5
([(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,1,1]
=> ? = 6
([(1,3),(1,4),(3,5),(3,6),(4,5),(4,6),(6,2)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,3),(1,4),(3,5),(3,6),(4,2),(4,5),(4,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [2,2,2,2,2,1,1]
=> ? = 5
([(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,1,1]
=> ? = 5
([(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,1,1]
=> ? = 6
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(6,4)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,4),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,1,1]
=> ? = 5
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,1,1,1]
=> ? = 4
([(2,5),(2,6),(3,5),(3,6),(6,4)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,1,1]
=> ? = 5
([(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,1,1]
=> ? = 6
([(1,4),(1,5),(2,4),(2,5),(4,6),(5,6),(6,3)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,4),(1,6),(2,4),(2,6),(4,5),(6,3),(6,5)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,4),(1,5),(2,4),(2,5),(3,6),(4,6),(5,3)],7)
=> ([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(4,6)],7)
=> ([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,5),(1,6),(2,5),(2,6),(3,4),(4,5),(4,6)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,2)],7)
=> ([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,6),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,4),(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,3),(1,4),(2,6),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,3),(1,4),(2,5),(3,5),(3,6),(4,2),(4,6)],7)
=> ([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,1,1]
=> ? = 5
([(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,4),(1,5),(2,6),(3,6),(4,6),(5,2),(5,3)],7)
=> ([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,1,1]
=> ? = 5
([(1,4),(1,5),(2,3),(2,4),(2,5),(4,6),(5,6)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,1,1]
=> ? = 6
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(6,3)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(5,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(2,5),(2,6),(3,4),(3,5),(4,6)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [2,2,2,2,2,1,1]
=> ? = 5
([(1,4),(1,6),(2,3),(2,4),(3,6),(4,5),(6,5)],7)
=> ([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
Description
The number of even parts of a partition.
Matching statistic: St001280
Mp00074: Posets to graphGraphs
Mp00251: Graphs clique sizesInteger partitions
St001280: Integer partitions ⟶ ℤResult quality: 82% values known / values provided: 89%distinct values known / distinct values provided: 82%
Values
([],1)
=> ([],1)
=> [1]
=> 0
([],2)
=> ([],2)
=> [1,1]
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 1
([],3)
=> ([],3)
=> [1,1,1]
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([],4)
=> ([],4)
=> [1,1,1,1]
=> 0
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 1
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 4
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 2
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 4
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 1
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> 6
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 4
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(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)
=> [2,2,2,2,2,2]
=> 6
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 4
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,1),(5,6)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(0,6),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(6,1)],7)
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,1)],7)
=> ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,5),(4,6),(6,1)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,1),(4,6)],7)
=> ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,1),(4,5),(4,6)],7)
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(4,5)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,1),(0,2),(0,3),(1,6),(2,4),(2,5),(3,4),(3,5),(5,6)],7)
=> ([(0,3),(0,4),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(4,5)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(1,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1),(0,6),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,1),(4,5),(4,6)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,5),(5,6)],7)
=> ([(0,4),(0,6),(1,3),(1,5),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
=> ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,2),(0,3),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1),(6,5)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,2),(0,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(6,1)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(5,3),(6,3)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,2),(0,3),(2,4),(2,5),(2,6),(3,1),(3,4),(3,5),(3,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,6),(1,3),(1,6),(2,3),(2,6),(3,4),(3,5),(6,4),(6,5)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(5,4),(6,3),(6,4)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(4,3),(5,4),(6,4)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(5,4),(6,3)],7)
=> ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,6),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(5,4)],7)
=> ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001657
Mp00074: Posets to graphGraphs
Mp00251: Graphs clique sizesInteger partitions
St001657: Integer partitions ⟶ ℤResult quality: 82% values known / values provided: 89%distinct values known / distinct values provided: 82%
Values
([],1)
=> ([],1)
=> [1]
=> 0
([],2)
=> ([],2)
=> [1,1]
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 1
([],3)
=> ([],3)
=> [1,1,1]
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([],4)
=> ([],4)
=> [1,1,1,1]
=> 0
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 1
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 4
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 2
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 4
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 1
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> 6
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 4
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(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)
=> [2,2,2,2,2,2]
=> 6
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 4
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,1),(5,6)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(0,6),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(6,1)],7)
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,1)],7)
=> ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,5),(4,6),(6,1)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,1),(4,6)],7)
=> ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,1),(4,5),(4,6)],7)
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(4,5)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,1),(0,2),(0,3),(1,6),(2,4),(2,5),(3,4),(3,5),(5,6)],7)
=> ([(0,3),(0,4),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(4,5)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(1,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1),(0,6),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,1),(4,5),(4,6)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,5),(5,6)],7)
=> ([(0,4),(0,6),(1,3),(1,5),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
=> ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,2),(0,3),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1),(6,5)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,2),(0,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(6,1)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(5,3),(6,3)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,2),(0,3),(2,4),(2,5),(2,6),(3,1),(3,4),(3,5),(3,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,6),(1,3),(1,6),(2,3),(2,6),(3,4),(3,5),(6,4),(6,5)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(5,4),(6,3),(6,4)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(4,3),(5,4),(6,4)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(5,4),(6,3)],7)
=> ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,6),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(5,4)],7)
=> ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
Description
The number of twos in an integer partition. The total number of twos in all partitions of $n$ is equal to the total number of singletons [[St001484]] in all partitions of $n-1$, see [1].
Matching statistic: St001389
Mp00074: Posets to graphGraphs
Mp00251: Graphs clique sizesInteger partitions
St001389: Integer partitions ⟶ ℤResult quality: 82% values known / values provided: 89%distinct values known / distinct values provided: 82%
Values
([],1)
=> ([],1)
=> [1]
=> 1 = 0 + 1
([],2)
=> ([],2)
=> [1,1]
=> 1 = 0 + 1
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
([],3)
=> ([],3)
=> [1,1,1]
=> 1 = 0 + 1
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 2 = 1 + 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 3 = 2 + 1
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 3 = 2 + 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 3 = 2 + 1
([],4)
=> ([],4)
=> [1,1,1,1]
=> 1 = 0 + 1
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 2 = 1 + 1
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 3 = 2 + 1
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 5 = 4 + 1
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 3 = 2 + 1
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 3 = 2 + 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 3 = 2 + 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 5 = 4 + 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> 1 = 0 + 1
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 2 = 1 + 1
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 3 = 2 + 1
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 4 = 3 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 6 = 5 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> 7 = 6 + 1
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 4 = 3 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> 5 = 4 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 6 = 5 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 6 = 5 + 1
([(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)
=> [2,2,2,2,2,2]
=> 7 = 6 + 1
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 3 = 2 + 1
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 4 = 3 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 3 = 2 + 1
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 4 = 3 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 4 = 3 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10 + 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,1),(5,6)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10 + 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(0,6),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10 + 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10 + 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(6,1)],7)
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10 + 1
([(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,1)],7)
=> ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10 + 1
([(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,5),(4,6),(6,1)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,1),(4,6)],7)
=> ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,1),(4,5),(4,6)],7)
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10 + 1
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(4,5)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10 + 1
([(0,1),(0,2),(0,3),(1,6),(2,4),(2,5),(3,4),(3,5),(5,6)],7)
=> ([(0,3),(0,4),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,1),(0,2),(0,3),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(4,5)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10 + 1
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,1),(0,2),(0,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,1),(0,2),(0,3),(1,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1),(0,6),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,1),(0,2),(0,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10 + 1
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10 + 1
([(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,1),(4,5),(4,6)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,5),(5,6)],7)
=> ([(0,4),(0,6),(1,3),(1,5),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
=> ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,2),(0,3),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1),(6,5)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,1),(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10 + 1
([(0,2),(0,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(6,1)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,1),(0,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(5,3),(6,3)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10 + 1
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,2),(0,3),(2,4),(2,5),(2,6),(3,1),(3,4),(3,5),(3,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,1),(0,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10 + 1
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10 + 1
([(0,6),(1,3),(1,6),(2,3),(2,6),(3,4),(3,5),(6,4),(6,5)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(5,4),(6,3),(6,4)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 10 + 1
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(4,3),(5,4),(6,4)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(5,4),(6,3)],7)
=> ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,6),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(5,4)],7)
=> ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
Description
The number of partitions of the same length below the given integer partition. For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is $$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St001176
Mp00074: Posets to graphGraphs
Mp00251: Graphs clique sizesInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St001176: Integer partitions ⟶ ℤResult quality: 82% values known / values provided: 89%distinct values known / distinct values provided: 82%
Values
([],1)
=> ([],1)
=> [1]
=> [1]
=> 0
([],2)
=> ([],2)
=> [1,1]
=> [2]
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 1
([],3)
=> ([],3)
=> [1,1,1]
=> [3]
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> [2,1]
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> [2,2]
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> [2,2]
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> [2,2]
=> 2
([],4)
=> ([],4)
=> [1,1,1,1]
=> [4]
=> 0
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 1
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> [3,2]
=> 2
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [3,3]
=> 3
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [3,3]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [4,4]
=> 4
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> [3,2]
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [3,3]
=> 3
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> [3,2]
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [3,3]
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [3,3]
=> 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> 2
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [3,3]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [4,4]
=> 4
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [3,3]
=> 3
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [3,3]
=> 3
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> [5]
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 1
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [4,2]
=> 2
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [4,3]
=> 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [4,4]
=> 4
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [4,4]
=> 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> [5,5]
=> 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> [6,6]
=> 6
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [4,3]
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [4,4]
=> 4
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [5,4]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> [5,5]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [4,4]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> [5,5]
=> 5
([(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)
=> [2,2,2,2,2,2]
=> [6,6]
=> 6
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [4,2]
=> 2
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [4,3]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [4,4]
=> 4
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [4,2]
=> 2
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [4,3]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [4,4]
=> 4
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [4,3]
=> 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [4,4]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [4,4]
=> 4
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [3,3]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [4,4]
=> 4
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,1),(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 10
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,1),(5,6)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,1),(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 10
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(0,6),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 10
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 10
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(6,1)],7)
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 10
([(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,1)],7)
=> ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 10
([(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,5),(4,6),(6,1)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,1),(4,6)],7)
=> ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,1),(4,5),(4,6)],7)
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 10
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(4,5)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 10
([(0,1),(0,2),(0,3),(1,6),(2,4),(2,5),(3,4),(3,5),(5,6)],7)
=> ([(0,3),(0,4),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,1),(0,2),(0,3),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(4,5)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 10
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,1),(0,2),(0,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,1),(0,2),(0,3),(1,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1),(0,6),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,1),(0,2),(0,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 10
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 10
([(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,1),(4,5),(4,6)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,5),(5,6)],7)
=> ([(0,4),(0,6),(1,3),(1,5),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
=> ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,2),(0,3),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1),(6,5)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,1),(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 10
([(0,2),(0,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(6,1)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,1),(0,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(5,3),(6,3)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 10
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,2),(0,3),(2,4),(2,5),(2,6),(3,1),(3,4),(3,5),(3,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,1),(0,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 10
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 10
([(0,6),(1,3),(1,6),(2,3),(2,6),(3,4),(3,5),(6,4),(6,5)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(5,4),(6,3),(6,4)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 10
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(4,3),(5,4),(6,4)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(5,4),(6,3)],7)
=> ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,6),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(5,4)],7)
=> ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 9
Description
The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000345
Mp00074: Posets to graphGraphs
Mp00251: Graphs clique sizesInteger partitions
St000345: Integer partitions ⟶ ℤResult quality: 38% values known / values provided: 38%distinct values known / distinct values provided: 64%
Values
([],1)
=> ([],1)
=> [1]
=> 1 = 0 + 1
([],2)
=> ([],2)
=> [1,1]
=> 1 = 0 + 1
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
([],3)
=> ([],3)
=> [1,1,1]
=> 1 = 0 + 1
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 2 = 1 + 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 3 = 2 + 1
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 3 = 2 + 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 3 = 2 + 1
([],4)
=> ([],4)
=> [1,1,1,1]
=> 1 = 0 + 1
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 2 = 1 + 1
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 3 = 2 + 1
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 5 = 4 + 1
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 3 = 2 + 1
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 3 = 2 + 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 3 = 2 + 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 5 = 4 + 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> 1 = 0 + 1
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 2 = 1 + 1
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 3 = 2 + 1
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 4 = 3 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 6 = 5 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> 7 = 6 + 1
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 4 = 3 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> 5 = 4 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 6 = 5 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 6 = 5 + 1
([(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)
=> [2,2,2,2,2,2]
=> 7 = 6 + 1
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 3 = 2 + 1
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 4 = 3 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 3 = 2 + 1
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 4 = 3 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 4 = 3 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2,2]
=> ? = 8 + 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,1]
=> ? = 6 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2,2]
=> ? = 8 + 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,1]
=> ? = 6 + 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2,2]
=> ? = 8 + 1
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2,2]
=> ? = 8 + 1
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,1]
=> ? = 6 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2,2]
=> ? = 8 + 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2,2]
=> ? = 8 + 1
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,1]
=> ? = 6 + 1
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2,2]
=> ? = 8 + 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(3,4),(3,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,5),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(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)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,4),(0,5),(1,2),(1,4),(1,5),(4,3),(5,3)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,1]
=> ? = 6 + 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(4,2),(5,2)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2,2]
=> ? = 8 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9 + 1
([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(5,3)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2,2]
=> ? = 8 + 1
([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2,2,2,2,2]
=> ? = 7 + 1
Description
The number of refinements of a partition. A partition $\lambda$ refines a partition $\mu$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$.
The following 17 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000185The weighted size of a partition. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000081The number of edges of a graph. St001622The number of join-irreducible elements of a lattice. St001311The cyclomatic number of a graph. St001341The number of edges in the center of a graph. St000448The number of pairs of vertices of a graph with distance 2. St001646The number of edges that can be added without increasing the maximal degree of a graph. St000321The number of integer partitions of n that are dominated by an integer partition. St000095The number of triangles of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000327The number of cover relations in a poset. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001621The number of atoms of a lattice.