Identifier
-
Mp00251:
Graphs
—clique sizes⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000689: Dyck paths ⟶ ℤ
Values
([],1) => [1] => [1,0,1,0] => [1,1,0,0] => 0
([],2) => [1,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 0
([(0,1)],2) => [2] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => 0
([],3) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => 0
([(1,2)],3) => [2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 0
([(0,2),(1,2)],3) => [2,2] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => 0
([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => 0
([],4) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => 0
([(2,3)],4) => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,0] => 0
([(1,3),(2,3)],4) => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 0
([(0,3),(1,3),(2,3)],4) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => 0
([(0,3),(1,2)],4) => [2,2] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => 0
([(0,3),(1,2),(2,3)],4) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => 0
([(1,2),(1,3),(2,3)],4) => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => 0
([(0,3),(1,2),(1,3),(2,3)],4) => [3,2] => [1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => 0
([(3,4)],5) => [2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => 0
([(2,4),(3,4)],5) => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,0,1,0] => 0
([(1,4),(2,4),(3,4)],5) => [2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => 0
([(1,4),(2,3)],5) => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 0
([(1,4),(2,3),(3,4)],5) => [2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => 0
([(0,1),(2,4),(3,4)],5) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => 0
([(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 0
([(1,4),(2,3),(2,4),(3,4)],5) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 0
([(0,1),(2,3),(2,4),(3,4)],5) => [3,2] => [1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => [3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,0,0] => 0
([(2,5),(3,4)],6) => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,0,1,0] => 0
([(1,2),(3,5),(4,5)],6) => [2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => 0
([(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => 0
([(2,5),(3,4),(3,5),(4,5)],6) => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 0
([(0,5),(1,4),(2,3)],6) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => 0
([(1,2),(3,4),(3,5),(4,5)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 0
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 0
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 0
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => 0
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 0
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => 0
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => 0
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => 0
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => 0
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => 0
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6) => [3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 0
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => 0
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => 0
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,0,0] => 0
([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => 0
([(1,6),(2,5),(3,4)],7) => [2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => 0
([(2,3),(4,5),(4,6),(5,6)],7) => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => 0
([(1,2),(3,6),(4,5),(4,6),(5,6)],7) => [3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 0
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => 0
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 0
([(1,6),(2,6),(3,4),(3,5),(4,5)],7) => [3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 0
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 0
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 0
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 0
([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => 0
([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 0
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 0
([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 0
([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 0
([(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 0
([(0,3),(1,2),(4,5),(4,6),(5,6)],7) => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 0
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 0
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7) => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 0
([(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7) => [3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 0
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7) => [3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 0
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 0
([(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 0
([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => 0
([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => 0
([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => 0
([(1,4),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 0
([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => 0
([(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => 0
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => 0
([(0,5),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => 0
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,0,0] => 0
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => 0
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Description
The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid.
The correspondence between LNakayama algebras and Dyck paths is explained in St000684The global dimension of the LNakayama algebra associated to a Dyck path.. A module $M$ is $n$-rigid, if $\operatorname{Ext}^i(M,M)=0$ for $1\leq i\leq n$.
This statistic gives the maximal $n$ such that the minimal generator-cogenerator module $A \oplus D(A)$ of the LNakayama algebra $A$ corresponding to a Dyck path is $n$-rigid.
An application is to check for maximal $n$-orthogonal objects in the module category in the sense of [2].
The correspondence between LNakayama algebras and Dyck paths is explained in St000684The global dimension of the LNakayama algebra associated to a Dyck path.. A module $M$ is $n$-rigid, if $\operatorname{Ext}^i(M,M)=0$ for $1\leq i\leq n$.
This statistic gives the maximal $n$ such that the minimal generator-cogenerator module $A \oplus D(A)$ of the LNakayama algebra $A$ corresponding to a Dyck path is $n$-rigid.
An application is to check for maximal $n$-orthogonal objects in the module category in the sense of [2].
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
Lalanne-Kreweras involution
Description
The Lalanne-Kreweras involution on Dyck paths.
Label the upsteps from left to right and record the labels on the first up step of each double rise. Do the same for the downsteps. Then form the Dyck path whose ascent lengths and descent lengths are the consecutives differences of the labels.
Label the upsteps from left to right and record the labels on the first up step of each double rise. Do the same for the downsteps. Then form the Dyck path whose ascent lengths and descent lengths are the consecutives differences of the labels.
Map
clique sizes
Description
The integer partition of the sizes of the maximal cliques of a graph.
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