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Your data matches 107 different statistics following compositions of up to 3 maps.
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Matching statistic: St000940
Mp00110: Posets āGreene-Kleitman invariantā¶ Integer partitions
Mp00313: Integer partitions āGlaisher-Franklin inverseā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000940: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00313: Integer partitions āGlaisher-Franklin inverseā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000940: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
([(1,2)],3)
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,1),(0,2)],3)
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,2),(1,2)],3)
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 0
([],4)
=> [1,1,1,1]
=> [2,2]
=> [2]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(1,2),(1,3)],4)
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(1,3),(2,3)],4)
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([],5)
=> [1,1,1,1,1]
=> [2,2,1]
=> [2,1]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
([(2,3),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
Description
The number of characters of the symmetric group whose value on the partition is zero.
The maximal value for any given size is recorded in [2].
Matching statistic: St001549
Mp00110: Posets āGreene-Kleitman invariantā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00201: Dyck paths āRingelā¶ Permutations
St001549: Permutations ā¶ ā¤Result quality: 15% āvalues known / values provided: 29%ādistinct values known / distinct values provided: 15%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00201: Dyck paths āRingelā¶ Permutations
St001549: Permutations ā¶ ā¤Result quality: 15% āvalues known / values provided: 29%ādistinct values known / distinct values provided: 15%
Values
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0
([(0,1),(0,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0
([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0
([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 0
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 0
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 0
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 0
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 0
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 0
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 0
([],6)
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ? = 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 1
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 1
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 1
([(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]
=> [6,1,7,2,3,4,5] => ? = 1
([(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]
=> [6,1,7,2,3,4,5] => ? = 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
([(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
([(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1
([(2,3),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1
([(2,3),(3,5),(5,4)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
([(1,4),(4,5),(5,2),(5,3)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
([(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 1
([(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1
([(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1
([(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]
=> [6,1,7,2,3,4,5] => ? = 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1
([(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]
=> [6,1,7,2,3,4,5] => ? = 1
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
([(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]
=> [7,1,2,6,3,4,5] => ? = 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1
([(1,5),(2,5),(3,4),(5,3)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
([(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]
=> [7,1,2,6,3,4,5] => ? = 1
([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
([(0,5),(1,5),(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1
([(0,5),(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
([(0,5),(1,5),(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]
=> [7,1,2,6,3,4,5] => ? = 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1
([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
Description
The number of restricted non-inversions between exceedances.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{nie}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(i) < \sigma(j) \}.$$
Matching statistic: St000486
Mp00110: Posets āGreene-Kleitman invariantā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00201: Dyck paths āRingelā¶ Permutations
St000486: Permutations ā¶ ā¤Result quality: 15% āvalues known / values provided: 29%ādistinct values known / distinct values provided: 15%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00201: Dyck paths āRingelā¶ Permutations
St000486: Permutations ā¶ ā¤Result quality: 15% āvalues known / values provided: 29%ādistinct values known / distinct values provided: 15%
Values
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1 = 0 + 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1 = 0 + 1
([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1 = 0 + 1
([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 1 = 0 + 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 2 = 1 + 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 1 = 0 + 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 1 = 0 + 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 1 = 0 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 1 = 0 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 1 = 0 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 1 = 0 + 1
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 1 = 0 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 1 = 0 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 1 = 0 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 1 = 0 + 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 1 = 0 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 1 = 0 + 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 1 = 0 + 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 1 = 0 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 1 = 0 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 1 = 0 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 1 = 0 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 1 = 0 + 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 1 = 0 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 1 = 0 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 1 = 0 + 1
([],6)
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ? = 0 + 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 1 + 1
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 1 + 1
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 1 + 1
([(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]
=> [6,1,7,2,3,4,5] => ? = 1 + 1
([(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]
=> [6,1,7,2,3,4,5] => ? = 1 + 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1 + 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1 + 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1 + 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1 + 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1 + 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1 + 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1 + 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 1
([(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1 + 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1 + 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1 + 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1 + 1
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 1
([(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1 + 1
([(2,3),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1 + 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1 + 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1 + 1
([(2,3),(3,5),(5,4)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 1
([(1,4),(4,5),(5,2),(5,3)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 1
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 1
([(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 1 + 1
([(2,5),(3,5),(5,4)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1 + 1
([(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 1 + 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1 + 1
([(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]
=> [6,1,7,2,3,4,5] => ? = 1 + 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1 + 1
([(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]
=> [6,1,7,2,3,4,5] => ? = 1 + 1
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1 + 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1 + 1
([(1,5),(2,5),(3,4),(5,3)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1 + 1
([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 1
([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 1
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 1
([(0,5),(1,5),(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1 + 1
([(0,5),(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 1
([(0,5),(1,5),(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]
=> [7,1,2,6,3,4,5] => ? = 1 + 1
([(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]
=> [7,1,2,6,3,4,5] => ? = 1 + 1
([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0 + 1
Description
The number of cycles of length at least 3 of a permutation.
Matching statistic: St001141
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00306: Posets ārowmotion cycle typeā¶ Integer partitions
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00032: Dyck paths āinverse zeta mapā¶ Dyck paths
St001141: Dyck paths ā¶ ā¤Result quality: 8% āvalues known / values provided: 9%ādistinct values known / distinct values provided: 8%
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00032: Dyck paths āinverse zeta mapā¶ Dyck paths
St001141: Dyck paths ā¶ ā¤Result quality: 8% āvalues known / values provided: 9%ādistinct values known / distinct values provided: 8%
Values
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 0
([(0,1),(0,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
([(0,2),(1,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
([],4)
=> [2,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
([(2,3)],4)
=> [6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 0
([(1,2),(1,3)],4)
=> [6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> 0
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 0
([(1,3),(2,3)],4)
=> [6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 0
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
([],5)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
([(3,4)],5)
=> [6,6,6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> 0
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 0
([(1,3),(1,4),(4,2)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 0
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 0
([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> 0
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 0
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 0
([(0,4),(1,2),(1,3),(3,4)],5)
=> [10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
([(0,3),(1,4),(4,2)],5)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 0
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 0
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 0
([],6)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 0
([(4,5)],6)
=> [6,6,6,6,6,6,6,6]
=> ?
=> ?
=> ? = 1
([(3,4),(3,5)],6)
=> [6,6,6,6,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1
([(2,3),(2,4),(2,5)],6)
=> [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [7,6,6,6]
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [7,6,2,2,2,2]
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [7,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [4,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [14,6,6]
=> ?
=> ?
=> ? = 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [7,6,2,2,2,2]
=> ?
=> ?
=> ? = 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [14,2,2,2,2]
=> ?
=> ?
=> ? = 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [4,4,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> [6,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> [8,4,2]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 0
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 0
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 0
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> [10,7]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0,1,0]
=> ?
=> ? = 0
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7,2,2,2,2]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 0
([(2,3),(2,4),(4,5)],6)
=> [14,14]
=> ?
=> ?
=> ? = 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> [14,2,2,2,2]
=> ?
=> ?
=> ? = 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [7,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 0
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> 0
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 0
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> [6,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> 0
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> [6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> 0
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 0
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> 0
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 0
([(1,5),(2,5),(5,3),(5,4)],6)
=> [4,4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> 0
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> 0
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> 0
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 0
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 0
Description
The number of occurrences of hills of size 3 in a Dyck path.
A hill of size three is a subpath beginning at height zero, consisting of three up steps followed by three down steps.
Matching statistic: St000685
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00306: Posets ārowmotion cycle typeā¶ Integer partitions
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
St000685: Dyck paths ā¶ ā¤Result quality: 8% āvalues known / values provided: 8%ādistinct values known / distinct values provided: 8%
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
St000685: Dyck paths ā¶ ā¤Result quality: 8% āvalues known / values provided: 8%ādistinct values known / distinct values provided: 8%
Values
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1 = 0 + 1
([(0,1),(0,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
([],4)
=> [2,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
([(2,3)],4)
=> [6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
([],5)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
([(3,4)],5)
=> [6,6,6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(1,3),(1,4),(4,2)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 1 = 0 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1 = 0 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 0 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0 + 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,3),(1,4),(4,2)],5)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1 = 0 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 1 = 0 + 1
([],6)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 0 + 1
([(4,5)],6)
=> [6,6,6,6,6,6,6,6]
=> ?
=> ? = 1 + 1
([(3,4),(3,5)],6)
=> [6,6,6,6,2,2,2,2,2,2,2,2]
=> ?
=> ? = 1 + 1
([(2,3),(2,4),(2,5)],6)
=> [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 1 + 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 1 + 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [7,6,6,6]
=> ?
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [7,6,2,2,2,2]
=> ?
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [7,2,2,2,2,2,2]
=> ?
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [4,2,2,2,2,2,2,2]
=> ?
=> ? = 1 + 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [14,6,6]
=> ?
=> ? = 1 + 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [7,6,2,2,2,2]
=> ?
=> ? = 1 + 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [14,2,2,2,2]
=> ?
=> ? = 1 + 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [4,4,2,2,2,2,2,2]
=> ?
=> ? = 1 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> [6,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> [8,4,2]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,1,0,0]
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> [10,7]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0,1,0]
=> ? = 0 + 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 1 = 0 + 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 1 = 0 + 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> [6,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> [6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
Description
The dominant dimension of the LNakayama algebra associated to a Dyck path.
To every Dyck path there is an LNakayama algebra associated as described in [[St000684]].
Matching statistic: St001501
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00306: Posets ārowmotion cycle typeā¶ Integer partitions
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
St001501: Dyck paths ā¶ ā¤Result quality: 8% āvalues known / values provided: 8%ādistinct values known / distinct values provided: 8%
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
St001501: Dyck paths ā¶ ā¤Result quality: 8% āvalues known / values provided: 8%ādistinct values known / distinct values provided: 8%
Values
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1 = 0 + 1
([(0,1),(0,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
([],4)
=> [2,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
([(2,3)],4)
=> [6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
([],5)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
([(3,4)],5)
=> [6,6,6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(1,3),(1,4),(4,2)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 1 = 0 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1 = 0 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 0 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0 + 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,3),(1,4),(4,2)],5)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1 = 0 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 1 = 0 + 1
([],6)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 0 + 1
([(4,5)],6)
=> [6,6,6,6,6,6,6,6]
=> ?
=> ? = 1 + 1
([(3,4),(3,5)],6)
=> [6,6,6,6,2,2,2,2,2,2,2,2]
=> ?
=> ? = 1 + 1
([(2,3),(2,4),(2,5)],6)
=> [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 1 + 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 1 + 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [7,6,6,6]
=> ?
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [7,6,2,2,2,2]
=> ?
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [7,2,2,2,2,2,2]
=> ?
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [4,2,2,2,2,2,2,2]
=> ?
=> ? = 1 + 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [14,6,6]
=> ?
=> ? = 1 + 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [7,6,2,2,2,2]
=> ?
=> ? = 1 + 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [14,2,2,2,2]
=> ?
=> ? = 1 + 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [4,4,2,2,2,2,2,2]
=> ?
=> ? = 1 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> [6,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> [8,4,2]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,1,0,0]
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> [10,7]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0,1,0]
=> ? = 0 + 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 1 = 0 + 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 1 = 0 + 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> [6,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> [6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
Description
The dominant dimension of magnitude 1 Nakayama algebras.
We use the code below to biject them to Dyck paths.
Matching statistic: St001125
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00306: Posets ārowmotion cycle typeā¶ Integer partitions
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00030: Dyck paths āzeta mapā¶ Dyck paths
St001125: Dyck paths ā¶ ā¤Result quality: 8% āvalues known / values provided: 8%ādistinct values known / distinct values provided: 8%
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00030: Dyck paths āzeta mapā¶ Dyck paths
St001125: Dyck paths ā¶ ā¤Result quality: 8% āvalues known / values provided: 8%ādistinct values known / distinct values provided: 8%
Values
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 0
([(0,1),(0,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
([(0,2),(1,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
([],4)
=> [2,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0
([(2,3)],4)
=> [6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0
([(1,2),(1,3)],4)
=> [6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> 0
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 0
([(1,3),(2,3)],4)
=> [6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 0
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 0
([],5)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
([(3,4)],5)
=> [6,6,6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 0
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 0
([(1,3),(1,4),(4,2)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 0
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 0
([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 0
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 0
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 0
([(0,4),(1,2),(1,3),(3,4)],5)
=> [10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
([(0,3),(1,4),(4,2)],5)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 0
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 0
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 0
([],6)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 0
([(4,5)],6)
=> [6,6,6,6,6,6,6,6]
=> ?
=> ?
=> ? = 1
([(3,4),(3,5)],6)
=> [6,6,6,6,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1
([(2,3),(2,4),(2,5)],6)
=> [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [7,6,6,6]
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [7,6,2,2,2,2]
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [7,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [4,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [14,6,6]
=> ?
=> ?
=> ? = 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [7,6,2,2,2,2]
=> ?
=> ?
=> ? = 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [14,2,2,2,2]
=> ?
=> ?
=> ? = 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [4,4,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> [6,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> [8,4,2]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 0
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 0
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 0
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> [10,7]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0,1,0]
=> ?
=> ? = 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 0
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 0
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> [6,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> 0
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> [6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 0
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 0
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> 0
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 0
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> 0
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> 0
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 0
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> 0
([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 0
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 0
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 0
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 0
([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> 0
Description
The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra.
Matching statistic: St000025
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00306: Posets ārowmotion cycle typeā¶ Integer partitions
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00120: Dyck paths āLalanne-Kreweras involutionā¶ Dyck paths
St000025: Dyck paths ā¶ ā¤Result quality: 8% āvalues known / values provided: 8%ādistinct values known / distinct values provided: 8%
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00120: Dyck paths āLalanne-Kreweras involutionā¶ Dyck paths
St000025: Dyck paths ā¶ ā¤Result quality: 8% āvalues known / values provided: 8%ādistinct values known / distinct values provided: 8%
Values
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
([(0,1),(0,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([],4)
=> [2,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 1
([(2,3)],4)
=> [6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 0 + 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
([],5)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
([(3,4)],5)
=> [6,6,6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 1
([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 0 + 1
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(1,3),(1,4),(4,2)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 0 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 1
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 0 + 1
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0 + 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
([(0,3),(1,4),(4,2)],5)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
([],6)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 0 + 1
([(4,5)],6)
=> [6,6,6,6,6,6,6,6]
=> ?
=> ?
=> ? = 1 + 1
([(3,4),(3,5)],6)
=> [6,6,6,6,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(2,3),(2,4),(2,5)],6)
=> [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [7,6,6,6]
=> ?
=> ?
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [7,6,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [7,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [4,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [14,6,6]
=> ?
=> ?
=> ? = 1 + 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [7,6,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [14,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [4,4,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> [6,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> [8,4,2]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> [10,7]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0,1,0]
=> ?
=> ? = 0 + 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> [6,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> [6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000026
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00306: Posets ārowmotion cycle typeā¶ Integer partitions
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00120: Dyck paths āLalanne-Kreweras involutionā¶ Dyck paths
St000026: Dyck paths ā¶ ā¤Result quality: 8% āvalues known / values provided: 8%ādistinct values known / distinct values provided: 8%
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00120: Dyck paths āLalanne-Kreweras involutionā¶ Dyck paths
St000026: Dyck paths ā¶ ā¤Result quality: 8% āvalues known / values provided: 8%ādistinct values known / distinct values provided: 8%
Values
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
([(0,1),(0,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([],4)
=> [2,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 1
([(2,3)],4)
=> [6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 0 + 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
([],5)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
([(3,4)],5)
=> [6,6,6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 1
([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 0 + 1
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(1,3),(1,4),(4,2)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 0 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 1
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 0 + 1
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0 + 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
([(0,3),(1,4),(4,2)],5)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
([],6)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 0 + 1
([(4,5)],6)
=> [6,6,6,6,6,6,6,6]
=> ?
=> ?
=> ? = 1 + 1
([(3,4),(3,5)],6)
=> [6,6,6,6,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(2,3),(2,4),(2,5)],6)
=> [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [7,6,6,6]
=> ?
=> ?
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [7,6,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [7,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [4,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [14,6,6]
=> ?
=> ?
=> ? = 1 + 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [7,6,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [14,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [4,4,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> [6,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> [8,4,2]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> [10,7]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0,1,0]
=> ?
=> ? = 0 + 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> [6,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> [6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
Description
The position of the first return of a Dyck path.
Matching statistic: St001011
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00306: Posets ārowmotion cycle typeā¶ Integer partitions
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00030: Dyck paths āzeta mapā¶ Dyck paths
St001011: Dyck paths ā¶ ā¤Result quality: 8% āvalues known / values provided: 8%ādistinct values known / distinct values provided: 8%
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00030: Dyck paths āzeta mapā¶ Dyck paths
St001011: Dyck paths ā¶ ā¤Result quality: 8% āvalues known / values provided: 8%ādistinct values known / distinct values provided: 8%
Values
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(0,1),(0,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
([],4)
=> [2,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 1
([(2,3)],4)
=> [6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([],5)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 1
([(3,4)],5)
=> [6,6,6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
([(1,3),(1,4),(4,2)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 0 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 0 + 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0 + 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0 + 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0 + 1
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0 + 1
([(0,3),(1,4),(4,2)],5)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0 + 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0 + 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([],6)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 0 + 1
([(4,5)],6)
=> [6,6,6,6,6,6,6,6]
=> ?
=> ?
=> ? = 1 + 1
([(3,4),(3,5)],6)
=> [6,6,6,6,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(2,3),(2,4),(2,5)],6)
=> [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [7,6,6,6]
=> ?
=> ?
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [7,6,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [7,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [4,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [14,6,6]
=> ?
=> ?
=> ? = 1 + 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [7,6,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [14,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [4,4,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 1 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> [6,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> [8,4,2]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> [10,7]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0,1,0]
=> ?
=> ? = 0 + 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> [6,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> [6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1 = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> 1 = 0 + 1
Description
Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path.
The following 97 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001520The number of strict 3-descents. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St001139The number of occurrences of hills of size 2 in a Dyck path. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St000068The number of minimal elements in a poset. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001256Number of simple reflexive modules that are 2-stable reflexive. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. 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 $eA$. St000661The number of rises of length 3 of a Dyck path. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000335The difference of lower and upper interactions. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{nā1}]$ by adding $c_0$ to $c_{nā1}$. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001481The minimal height of a peak of a Dyck path. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001732The number of peaks visible from the left. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000658The number of rises of length 2 of a Dyck path. 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. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000674The number of hills of a Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St000322The skewness of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000929The constant term of the character polynomial of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St000993The multiplicity of the largest part of an integer partition. St001578The minimal number of edges to add or remove to make a graph a line graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001568The smallest positive integer that does not appear twice in the partition. St000097The order of the largest clique of the graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. St000264The girth of a graph, which is not a tree. St000699The toughness times the least common multiple of 1,. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000456The monochromatic index of a connected graph. St001118The acyclic chromatic index of a graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001545The second Elser number of a connected graph. St000464The Schultz index of a connected graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001613The binary logarithm of the size of the center of a lattice. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001274The number of indecomposable injective modules with projective dimension equal to two. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St000098The chromatic number of a graph.
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