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Your data matches 198 different statistics following compositions of up to 3 maps.
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Matching statistic: St000698
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Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,3),(2,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core.
For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$.
This statistic counts the $2$-rim hooks that are removed in this process to obtain a $2$-core.
Matching statistic: St000689
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000689: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 44%●distinct values known / distinct values provided: 17%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000689: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 44%●distinct values known / distinct values provided: 17%
Values
([],4)
=> [1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
([(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
([(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
([(3,4),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
([(2,3),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
([(1,5),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
([(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
([(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
([(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
([(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
([(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
([(1,5),(2,4),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
([(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
([(0,5),(1,5),(2,3),(2,4)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
([(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3)],6)
=> [2,2,2]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6)
=> [2,2,2]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1 - 1
([(1,5),(2,3),(2,4)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,1)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,1),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,6),(6,1)],7)
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,1)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,1),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(0,5),(4,6),(5,6),(6,1)],7)
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
([(0,3),(0,4),(0,5),(0,6),(5,2),(6,1)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(0,5),(4,6),(5,1),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,6),(3,6),(4,6),(6,5)],7)
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
([(1,3),(1,4),(1,5),(3,6),(4,6),(5,2)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,6),(3,5),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
([(1,3),(1,4),(1,5),(3,6),(4,6),(5,2),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
([(0,3),(0,4),(0,5),(3,6),(4,6),(5,1),(5,2)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
([(0,3),(0,4),(0,5),(3,6),(4,6),(5,1),(5,2),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,6),(3,5),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
([(1,2),(1,3),(1,4),(3,6),(4,6),(6,5)],7)
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
([(1,4),(1,5),(1,6),(5,3),(6,2)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
Description
The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid.
The correspondence between LNakayama algebras and Dyck paths is explained in [[St000684]]. A module $M$ is $n$-rigid, if $\operatorname{Ext}^i(M,M)=0$ for $1\leq i\leq n$.
This statistic gives the maximal $n$ such that the minimal generator-cogenerator module $A \oplus D(A)$ of the LNakayama algebra $A$ corresponding to a Dyck path is $n$-rigid.
An application is to check for maximal $n$-orthogonal objects in the module category in the sense of [2].
Matching statistic: St001195
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 34%●distinct values known / distinct values provided: 17%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 34%●distinct values known / distinct values provided: 17%
Values
([],4)
=> [1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
([(4,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(3,4),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(2,3),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(1,5),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
([(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
([(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
([(0,5),(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
([(1,5),(2,4),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
([(0,5),(1,5),(2,3),(2,4)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
([(0,5),(1,5),(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
([(0,5),(1,5),(2,4),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,1)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,1),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,1)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,1),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,3),(0,4),(0,5),(0,6),(5,2),(6,1)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,2),(0,3),(0,4),(0,5),(4,6),(5,1),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(1,3),(1,4),(1,5),(3,6),(4,6),(5,2)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(1,2),(1,3),(1,4),(2,6),(3,5),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(1,2),(1,3),(1,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(1,3),(1,4),(1,5),(3,6),(4,6),(5,2),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,3),(0,4),(0,5),(3,6),(4,6),(5,1),(5,2)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,3),(0,4),(0,5),(3,6),(4,6),(5,1),(5,2),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(1,4),(1,5),(1,6),(5,3),(6,2)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(1,3),(1,4),(1,5),(4,6),(5,2),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(1,2),(1,3),(1,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,4),(0,5),(0,6),(5,3),(6,1),(6,2)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,3),(0,4),(0,5),(4,6),(5,1),(5,2),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,3),(0,4),(0,5),(4,2),(4,6),(5,1),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,2),(0,3),(0,4),(3,5),(3,6),(4,1),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,1),(0,2),(0,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(2,5),(2,6),(5,4),(6,3)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(2,3),(2,4),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(1,5),(1,6),(5,4),(6,2),(6,3)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(1,4),(1,5),(4,6),(5,2),(5,3),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,5),(0,6),(5,4),(6,1),(6,2),(6,3)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(0,4),(0,5),(4,6),(5,1),(5,2),(5,3),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(1,4),(1,5),(4,3),(4,6),(5,2),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(1,3),(1,4),(3,5),(3,6),(4,2),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
([(1,2),(1,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Matching statistic: St001001
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 34%●distinct values known / distinct values provided: 17%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 34%●distinct values known / distinct values provided: 17%
Values
([],4)
=> [1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
([(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(3,4),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(2,3),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(1,5),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(0,5),(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 - 1
([(1,5),(2,4),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 - 1
([(0,5),(1,5),(2,3),(2,4)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 - 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 - 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,1)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,1),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,1)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,1),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,3),(0,4),(0,5),(0,6),(5,2),(6,1)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(0,5),(4,6),(5,1),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(1,3),(1,4),(1,5),(3,6),(4,6),(5,2)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,6),(3,5),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(1,3),(1,4),(1,5),(3,6),(4,6),(5,2),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,3),(0,4),(0,5),(3,6),(4,6),(5,1),(5,2)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,3),(0,4),(0,5),(3,6),(4,6),(5,1),(5,2),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(1,4),(1,5),(1,6),(5,3),(6,2)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(1,3),(1,4),(1,5),(4,6),(5,2),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(1,2),(1,3),(1,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,4),(0,5),(0,6),(5,3),(6,1),(6,2)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,3),(0,4),(0,5),(4,6),(5,1),(5,2),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,3),(0,4),(0,5),(4,2),(4,6),(5,1),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(3,5),(3,6),(4,1),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(2,5),(2,6),(5,4),(6,3)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(2,3),(2,4),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(1,5),(1,6),(5,4),(6,2),(6,3)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(1,4),(1,5),(4,6),(5,2),(5,3),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,5),(0,6),(5,4),(6,1),(6,2),(6,3)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(0,4),(0,5),(4,6),(5,1),(5,2),(5,3),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(1,4),(1,5),(4,3),(4,6),(5,2),(5,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(1,3),(1,4),(3,5),(3,6),(4,2),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
([(1,2),(1,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
Description
The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001232
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 20%●distinct values known / distinct values provided: 17%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 20%●distinct values known / distinct values provided: 17%
Values
([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 2
([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([],6)
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 2
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(1,3),(1,4),(1,5),(5,2)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(1,4),(1,5),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(2,3),(2,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(2,3),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(1,5),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(0,5),(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
([(1,5),(2,4),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
([(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
([(0,5),(1,5),(2,3),(2,4)],6)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
([(0,4),(0,5),(1,4),(1,5),(2,3)],6)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
([(0,5),(1,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
([(0,5),(1,4),(2,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4)],6)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
([(0,5),(1,4),(1,5),(2,3),(2,5)],6)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
([(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,1)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,1),(4,6)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,1),(4,5),(4,6)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,4),(0,5),(0,6),(4,3),(5,2),(6,1)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,3),(0,4),(0,5),(3,6),(4,2),(5,1),(5,6)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,2),(0,3),(0,4),(2,6),(3,5),(4,1),(4,5),(4,6)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(3,6),(4,1),(4,6)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,1),(0,2),(0,3),(1,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,1),(0,2),(0,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,1),(0,2),(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,1),(4,5),(4,6)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,3),(0,4),(0,5),(3,6),(4,2),(4,6),(5,1),(5,6)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,6),(1,6),(2,6),(6,3),(6,4),(6,5)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,6),(1,5),(2,5),(2,6),(6,3),(6,4)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,6),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(6,3),(6,4)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,6),(1,5),(1,6),(2,5),(2,6),(6,3),(6,4)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,6),(1,6),(2,3),(6,4),(6,5)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,6),(1,6),(2,5),(6,3),(6,4),(6,5)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,6),(1,6),(2,3),(2,6),(6,4),(6,5)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,6),(1,6),(2,4),(2,5),(6,3),(6,5)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,6),(1,6),(2,4),(2,5),(6,3),(6,4),(6,5)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,4),(1,4),(2,3),(2,5),(2,6),(4,5),(4,6)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,6),(1,6),(2,3),(2,4),(2,5),(6,3),(6,4),(6,5)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,4),(1,5),(1,6),(2,4),(2,5),(2,6),(6,3)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,4),(1,5),(1,6),(2,4),(2,5),(2,6),(5,3),(6,3)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(5,3),(6,3)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(5,3)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(4,5)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
([(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(4,5),(6,5)],7)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000264
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Values
([],4)
=> ([],4)
=> [4] => ([],4)
=> ? = 1 + 2
([],5)
=> ([],5)
=> [5] => ([],5)
=> ? = 1 + 2
([(3,4)],5)
=> ([(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
([],6)
=> ([],6)
=> [6] => ([],6)
=> ? = 2 + 2
([(4,5)],6)
=> ([(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(3,4),(3,5)],6)
=> ([(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(2,3),(2,4),(2,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(1,2),(1,3),(1,4),(1,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(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)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(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)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(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)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(2,3),(2,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(1,4),(1,5),(5,2),(5,3)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(2,3),(2,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(3,4),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(2,3),(3,4),(3,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(1,5),(5,2),(5,3),(5,4)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(2,5),(3,5),(5,4)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(1,5),(2,5),(3,5),(5,4)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(0,5),(1,5),(2,5),(3,4)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(1,5),(2,5),(3,4)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(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)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(1,5),(2,5),(3,4),(3,5)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(0,5),(1,5),(2,3),(2,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(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)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(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)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,6),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(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)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(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)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(4,6)],7)
=> ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(6,4)],7)
=> ([(0,6),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(4,6)],7)
=> ([(0,6),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,6),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,6),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,5),(1,6),(2,4),(3,4),(3,6),(4,5)],7)
=> ([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(4,6)],7)
=> ([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,2),(1,4),(1,5),(3,6),(4,6),(5,3)],7)
=> ([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,3),(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2)],7)
=> ([(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(5,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,2)],7)
=> ([(0,6),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,6),(2,3),(2,4),(3,6),(4,5),(6,5)],7)
=> ([(0,6),(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,4),(1,5),(3,6),(4,6),(5,2),(5,3)],7)
=> ([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,4),(0,5),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,6),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,6),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,4),(1,5),(2,6),(3,6),(4,6),(5,2),(5,3)],7)
=> ([(0,6),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,6),(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,6),(2,4),(2,6),(3,4),(3,5),(6,5)],7)
=> ([(0,6),(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(4,6)],7)
=> ([(0,6),(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,5),(0,6),(1,5),(1,6),(2,4),(3,4),(3,5),(4,6)],7)
=> ([(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(5,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,5),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6)],7)
=> ([(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(5,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(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)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(5,3),(5,4)],7)
=> ([(0,6),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(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)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6)],7)
=> ([(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(5,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,4),(0,6),(1,4),(1,6),(2,3),(2,5),(2,6),(4,5)],7)
=> ([(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(5,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,3),(0,6),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(4,6)],7)
=> ([(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(5,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(3,6),(4,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(4,6),(5,1)],7)
=> ([(0,6),(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1)],7)
=> ([(0,6),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(1,2),(1,4),(1,5),(3,6),(4,6),(5,3)],7)
=> ([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(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)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,2),(1,3),(1,4),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,6),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(5,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,2),(1,3),(1,4),(3,6),(4,5),(6,5)],7)
=> ([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,3),(0,4),(0,5),(2,6),(4,6),(5,1),(5,2)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,5),(1,2),(1,3),(1,4),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(5,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,6),(1,2),(1,3),(1,4),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(5,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(1,4),(1,5),(3,6),(4,6),(5,2),(5,3)],7)
=> ([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St000697
Values
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> 2
([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(2,3),(2,4),(2,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(2,3),(2,4),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(2,3),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,5),(5,2),(5,3),(5,4)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(2,5),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(1,5),(2,5),(3,5),(5,4)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,5),(1,5),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,5),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,5),(1,5),(2,3),(2,4)],6)
=> ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 2
([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 2
([(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 2
([(3,4),(3,5),(3,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 2
([(2,3),(2,4),(2,5),(2,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 2
([(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6)],7)
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(6,1)],7)
=> ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(1,3),(1,4),(1,5),(1,6),(6,2)],7)
=> ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,3),(0,4),(0,5),(0,6),(6,1),(6,2)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(1,2),(1,3),(1,4),(1,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(1,2),(1,3),(1,4),(1,5),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(1,2),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ?
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,1),(5,6)],7)
=> ([(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,1)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,1),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(4,6),(5,6),(6,1)],7)
=> ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,3),(0,4),(0,5),(0,6),(5,2),(6,1)],7)
=> ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(4,6),(5,1),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(2,4),(2,5),(2,6),(6,3)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(1,4),(1,5),(1,6),(6,2),(6,3)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,4),(0,5),(0,6),(6,1),(6,2),(6,3)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(2,3),(2,4),(2,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(1,2),(1,3),(1,4),(2,6),(3,6),(4,6),(6,5)],7)
=> ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(1,3),(1,4),(1,5),(3,6),(4,6),(5,2)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(1,2),(1,3),(1,4),(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(1,2),(1,3),(1,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(1,3),(1,4),(1,5),(3,6),(4,6),(5,2),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
Description
The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core.
For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$.
This statistic counts the $3$-rim hooks that are removed in this process to obtain a $3$-core.
Matching statistic: St000944
Values
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> 2
([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(2,3),(2,4),(2,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(2,3),(2,4),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(2,3),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,5),(5,2),(5,3),(5,4)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(2,5),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(1,5),(2,5),(3,5),(5,4)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,5),(1,5),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,5),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,5),(1,5),(2,3),(2,4)],6)
=> ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 2
([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 2
([(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 2
([(3,4),(3,5),(3,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 2
([(2,3),(2,4),(2,5),(2,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 2
([(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6)],7)
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(6,1)],7)
=> ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(1,3),(1,4),(1,5),(1,6),(6,2)],7)
=> ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,3),(0,4),(0,5),(0,6),(6,1),(6,2)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(1,2),(1,3),(1,4),(1,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(1,2),(1,3),(1,4),(1,5),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(1,2),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ?
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,1),(5,6)],7)
=> ([(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,1)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,1),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(4,6),(5,6),(6,1)],7)
=> ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,3),(0,4),(0,5),(0,6),(5,2),(6,1)],7)
=> ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(4,6),(5,1),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(2,4),(2,5),(2,6),(6,3)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(1,4),(1,5),(1,6),(6,2),(6,3)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(0,4),(0,5),(0,6),(6,1),(6,2),(6,3)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(2,3),(2,4),(2,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(1,2),(1,3),(1,4),(2,6),(3,6),(4,6),(6,5)],7)
=> ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(1,3),(1,4),(1,5),(3,6),(4,6),(5,2)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(1,2),(1,3),(1,4),(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(1,2),(1,3),(1,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
([(1,3),(1,4),(1,5),(3,6),(4,6),(5,2),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1
Description
The 3-degree of an integer partition.
For an integer partition $\lambda$, this is given by the exponent of 3 in the Gram determinant of the integal Specht module of the symmetric group indexed by $\lambda$.
This stupid comment should not be accepted as an edit!
Matching statistic: St001928
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 33%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 33%
Values
([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 2 = 1 + 1
([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 2 = 1 + 1
([(3,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 2 = 1 + 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 2 = 1 + 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 2 = 1 + 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 2 = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 2 = 1 + 1
([],6)
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 3 = 2 + 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 2 = 1 + 1
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 2 = 1 + 1
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 2 = 1 + 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 2 = 1 + 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 2 = 1 + 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 2 = 1 + 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 2 = 1 + 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 2 = 1 + 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 2 = 1 + 1
([(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 2 = 1 + 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 2 = 1 + 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 2 = 1 + 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 2 = 1 + 1
([(3,4),(4,5)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 2 = 1 + 1
([(2,3),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 2 = 1 + 1
([(1,5),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 2 = 1 + 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 2 = 1 + 1
([(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 2 = 1 + 1
([(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 2 = 1 + 1
([(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 2 = 1 + 1
([(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 2 = 1 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 2 = 1 + 1
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 2 = 1 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 2 = 1 + 1
([(0,5),(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 2 = 1 + 1
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 2 = 1 + 1
([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 2 = 1 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 2 = 1 + 1
([(1,5),(2,5),(3,4)],6)
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 2 = 1 + 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 2 = 1 + 1
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 2 = 1 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 2 = 1 + 1
([(1,5),(2,4),(3,4),(4,5)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 2 = 1 + 1
([(1,5),(2,5),(3,4),(3,5)],6)
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 2 = 1 + 1
([(0,5),(1,5),(2,3),(2,4)],6)
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 2 = 1 + 1
([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 2 = 1 + 1
([],7)
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 2 + 1
([(5,6)],7)
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 2 + 1
([(4,5),(4,6)],7)
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 2 + 1
([(3,4),(3,5),(3,6)],7)
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 2 + 1
([(2,3),(2,4),(2,5),(2,6)],7)
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 2 + 1
([(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 2 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6)],7)
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 2 + 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(6,1)],7)
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 1 + 1
([(1,3),(1,4),(1,5),(1,6),(6,2)],7)
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 1 + 1
([(0,3),(0,4),(0,5),(0,6),(6,1),(6,2)],7)
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 1 + 1
([(1,2),(1,3),(1,4),(1,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 1 + 1
([(1,2),(1,3),(1,4),(1,5),(3,6),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 1 + 1
([(1,2),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 1 + 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 1 + 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,1)],7)
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1 + 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,1),(5,6)],7)
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 1 + 1
([(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,6),(6,1)],7)
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 1 + 1
([(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,1)],7)
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1 + 1
([(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,1),(5,6)],7)
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5)],7)
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 1 + 1
([(0,2),(0,3),(0,4),(0,5),(4,6),(5,6),(6,1)],7)
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 1 + 1
([(0,3),(0,4),(0,5),(0,6),(5,2),(6,1)],7)
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1 + 1
([(0,2),(0,3),(0,4),(0,5),(4,6),(5,1),(5,6)],7)
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1 + 1
([(2,4),(2,5),(2,6),(6,3)],7)
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 1 + 1
([(1,4),(1,5),(1,6),(6,2),(6,3)],7)
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 1 + 1
([(0,4),(0,5),(0,6),(6,1),(6,2),(6,3)],7)
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 1 + 1
([(2,3),(2,4),(2,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 1 + 1
([(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 1 + 1
([(1,2),(1,3),(1,4),(2,6),(3,6),(4,6),(6,5)],7)
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 1 + 1
([(1,3),(1,4),(1,5),(3,6),(4,6),(5,2)],7)
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1 + 1
([(1,2),(1,3),(1,4),(2,6),(3,5),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1 + 1
([(1,2),(1,3),(1,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1 + 1
([(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1 + 1
([(1,3),(1,4),(1,5),(3,6),(4,6),(5,2),(5,6)],7)
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1 + 1
Description
The number of non-overlapping descents in a permutation.
In other words, any maximal descending subsequence $\pi_i,\pi_{i+1},\dots,\pi_k$ contributes $\lfloor\frac{k-i+1}{2}\rfloor$ to the total count.
Matching statistic: St001316
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
([(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(2,3),(2,4),(2,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(2,3),(2,4),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(2,3),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,5),(5,2),(5,3),(5,4)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(2,5),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,5),(2,5),(3,5),(5,4)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,5),(1,5),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 1
([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,5),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,5),(1,5),(2,3),(2,4)],6)
=> ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([(0,5),(1,5),(2,3),(2,4),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
([(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
([(3,4),(3,5),(3,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
([(2,3),(2,4),(2,5),(2,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
([(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6)],7)
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(6,1)],7)
=> ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(1,3),(1,4),(1,5),(1,6),(6,2)],7)
=> ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,3),(0,4),(0,5),(0,6),(6,1),(6,2)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(1,2),(1,3),(1,4),(1,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(1,2),(1,3),(1,4),(1,5),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(1,2),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,1)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,1),(5,6)],7)
=> ([(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,7),(3,4),(3,5),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,6),(6,1)],7)
=> ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,1)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,7),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(3,6),(4,6),(5,1),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,6),(2,7),(3,4),(3,5),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,7),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(4,6),(5,6),(6,1)],7)
=> ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,3),(0,4),(0,5),(0,6),(5,2),(6,1)],7)
=> ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(4,6),(5,1),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(2,4),(2,5),(2,6),(6,3)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(1,4),(1,5),(1,6),(6,2),(6,3)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(0,4),(0,5),(0,6),(6,1),(6,2),(6,3)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(2,3),(2,4),(2,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(1,2),(1,3),(1,4),(2,6),(3,6),(4,6),(6,5)],7)
=> ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(1,3),(1,4),(1,5),(3,6),(4,6),(5,2)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1
([(1,2),(1,3),(1,4),(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(1,2),(1,3),(1,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,5),(2,7),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
([(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,3),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
Description
The domatic number of a graph.
This is the maximal size of a partition of the vertices into dominating sets.
The following 188 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001281The normalized isoperimetric number of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001395The number of strictly unfriendly partitions of a graph. St001703The villainy of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000287The number of connected components of a graph. St001272The number of graphs with the same degree sequence. St001282The number of graphs with the same chromatic polynomial. St001322The size of a minimal independent dominating set in a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001352The number of internal nodes in the modular decomposition of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001393The induced matching number of a graph. St001463The number of distinct columns in the nullspace of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001642The Prague dimension of a graph. St001734The lettericity of a graph. St001765The number of connected components of the friends and strangers graph. St001871The number of triconnected components of a graph. St001917The order of toric promotion on the set of labellings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000299The number of nonisomorphic vertex-induced subtrees. St000315The number of isolated vertices of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001308The number of induced paths on three vertices in a graph. St001310The number of induced diamond graphs in a graph. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001374The Padmakar-Ivan index of a graph. St001386The number of prime labellings of a graph. St001479The number of bridges of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001574The minimal number of edges to add or remove to make a graph regular. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001742The difference of the maximal and the minimal degree in a graph. St000456The monochromatic index of a connected graph. St001118The acyclic chromatic index of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000379The number of Hamiltonian cycles in a graph. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000681The Grundy value of Chomp on Ferrers diagrams. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000477The weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000699The toughness times the least common multiple of 1,. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000997The even-odd crank of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000762The sum of the positions of the weak records of an integer composition. St000806The semiperimeter of the associated bargraph. St001060The distinguishing index of a graph. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000069The number of maximal elements of a poset. St000068The number of minimal elements in a poset. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001593This is the number of standard Young tableaux of the given shifted shape. St000257The number of distinct parts of a partition that occur at least twice. St001092The number of distinct even parts of a partition. St000445The number of rises of length 1 of a Dyck path. St001484The number of singletons of an integer partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001309The number of four-cliques in a graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001793The difference between the clique number and the chromatic number of a graph. St000387The matching number of a graph. St000759The smallest missing part in an integer partition. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St000172The Grundy number of a graph. St001112The 3-weak dynamic number of a graph. St000439The position of the first down step of a Dyck path. St000785The number of distinct colouring schemes of a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000323The minimal crossing number of a graph. St000370The genus of a graph. St000552The number of cut vertices of a graph. St001826The maximal number of leaves on a vertex of a graph. St001957The number of Hasse diagrams with a given underlying undirected graph. St000422The energy of a graph, if it is integral. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St000655The length of the minimal rise of a Dyck path. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000298The order dimension or Dushnik-Miller dimension of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001510The number of self-evacuating linear extensions of a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001336The minimal number of vertices in a graph whose complement is triangle-free. St000660The number of rises of length at least 3 of a Dyck path. St000475The number of parts equal to 1 in a partition. St000322The skewness of a graph. St000260The radius of a connected graph. St000659The number of rises of length at least 2 of a Dyck path. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St000617The number of global maxima of a Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000142The number of even parts of a partition. St001252Half the sum of the even parts of a partition. St001330The hat guessing number of a graph. St001645The pebbling number of a connected graph. St000678The number of up steps after the last double rise of a Dyck path.
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