Identifier
-
Mp00251:
Graphs
—clique sizes⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001515: Dyck paths ⟶ ℤ
Values
([],1) => [1] => [1,0,1,0] => [1,0,1,0] => 1
([],2) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
([(0,1)],2) => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 2
([],3) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 3
([(1,2)],3) => [2,1] => [1,0,1,0,1,0] => [1,0,1,0,1,0] => 2
([(0,2),(1,2)],3) => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0] => 3
([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0] => 3
([],4) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 4
([(2,3)],4) => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,0] => 3
([(1,3),(2,3)],4) => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0] => 3
([(0,3),(1,3),(2,3)],4) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,0,0,0] => 4
([(0,3),(1,2)],4) => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0] => 3
([(0,3),(1,2),(2,3)],4) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,0,0,0] => 4
([(1,2),(1,3),(2,3)],4) => [3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 2
([(0,3),(1,2),(1,3),(2,3)],4) => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 4
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 4
([(3,4)],5) => [2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => 4
([(2,4),(3,4)],5) => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => 4
([(1,4),(2,4),(3,4)],5) => [2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => 4
([(1,4),(2,3)],5) => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0] => 3
([(1,4),(2,3),(3,4)],5) => [2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => 4
([(0,1),(2,4),(3,4)],5) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,0,0,0] => 4
([(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => 3
([(1,4),(2,3),(2,4),(3,4)],5) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 3
([(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] => [1,1,0,1,0,0,1,1,0,0] => 4
([(0,1),(2,3),(2,4),(3,4)],5) => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 3
([(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] => [1,1,1,0,0,0,1,1,0,0] => 4
([(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] => [1,1,0,1,0,0,1,1,0,0] => 4
([(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] => [1,0,1,1,1,0,0,1,0,0] => 3
([(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] => [1,1,1,0,0,1,0,0,1,0] => 4
([(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] => [1,1,1,0,0,1,0,1,0,0] => 4
([(2,5),(3,4)],6) => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => 4
([(1,2),(3,5),(4,5)],6) => [2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => 4
([(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => 4
([(2,5),(3,4),(3,5),(4,5)],6) => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,1,0,0] => 4
([(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] => [1,0,1,1,0,1,0,1,0,0] => 4
([(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] => [1,1,0,0,1,1,0,1,0,0] => 4
([(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] => [1,0,1,1,0,1,0,1,0,0] => 4
([(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] => [1,1,0,1,0,1,0,1,0,0] => 4
([(0,5),(1,4),(2,3)],6) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,0,0,0] => 4
([(1,2),(3,4),(3,5),(4,5)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => 3
([(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] => [1,0,1,1,0,1,0,1,0,0] => 4
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 3
([(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] => [1,1,0,0,1,0,1,1,0,0] => 4
([(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] => [1,1,0,1,0,0,1,1,0,0] => 4
([(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] => [1,1,0,1,0,1,0,1,0,0] => 4
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 3
([(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] => [1,1,0,1,0,0,1,1,0,0] => 4
([(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] => [1,1,0,1,0,0,1,1,0,0] => 4
([(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] => [1,1,0,1,1,0,0,1,0,0] => 4
([(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] => [1,0,1,0,1,1,0,0,1,0] => 3
([(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] => [1,1,0,0,1,1,0,0,1,0] => 3
([(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] => [1,0,1,1,0,0,1,0,1,0] => 3
([(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] => [1,1,0,0,1,1,0,0,1,0] => 3
([(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] => [1,1,0,0,1,0,1,0,1,0] => 4
([(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] => [1,1,0,0,1,0,1,0,1,0] => 4
([(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] => [1,1,1,0,0,0,1,1,0,0] => 4
([(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] => [1,1,0,1,0,0,1,1,0,0] => 4
([(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] => [1,1,1,0,0,1,0,0,1,0] => 4
([(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] => [1,1,0,0,1,1,0,0,1,0] => 3
([(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] => [1,1,1,0,0,1,0,1,0,0] => 4
([(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] => [1,1,0,0,1,0,1,0,1,0] => 4
([(1,6),(2,5),(3,4)],7) => [2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => 4
([(2,3),(4,5),(4,6),(5,6)],7) => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,1,0,0] => 4
([(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] => [1,0,1,1,0,1,0,1,0,0] => 4
([(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] => [1,1,0,0,1,1,0,1,0,0] => 4
([(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] => [1,1,0,1,0,1,0,1,0,0] => 4
([(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] => [1,0,1,1,0,1,0,1,0,0] => 4
([(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] => [1,1,0,1,0,1,0,1,0,0] => 4
([(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] => [1,1,0,1,0,1,0,1,0,0] => 4
([(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] => [1,1,1,0,1,0,0,1,0,0] => 4
([(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] => [1,0,1,1,0,1,0,0,1,0] => 4
([(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] => [1,1,0,1,0,1,0,0,1,0] => 4
([(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] => [1,1,0,1,0,0,1,0,1,0] => 4
([(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] => [1,1,0,1,0,1,0,0,1,0] => 4
([(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] => [1,0,1,0,1,0,1,0,1,0] => 4
([(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] => [1,0,1,0,1,0,1,0,1,0] => 4
([(0,3),(1,2),(4,5),(4,6),(5,6)],7) => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 3
([(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] => [1,1,0,1,0,0,1,1,0,0] => 4
([(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] => [1,1,0,0,1,0,1,1,0,0] => 4
([(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] => [1,1,0,1,0,1,0,1,0,0] => 4
([(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] => [1,1,0,1,0,0,1,1,0,0] => 4
([(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] => [1,0,1,0,1,1,0,0,1,0] => 3
([(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] => [1,1,0,1,0,1,0,0,1,0] => 4
([(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] => [1,1,0,0,1,1,0,0,1,0] => 3
([(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] => [1,0,1,1,0,0,1,0,1,0] => 3
([(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] => [1,1,0,0,1,0,1,0,1,0] => 4
([(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] => [1,0,1,0,1,0,1,0,1,0] => 4
([(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] => [1,1,0,0,1,0,1,0,1,0] => 4
([(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] => [1,1,0,0,1,0,1,0,1,0] => 4
([(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] => [1,1,0,0,1,1,0,0,1,0] => 3
([(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] => [1,1,0,0,1,0,1,0,1,0] => 4
([(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] => [1,1,1,0,0,1,0,1,0,0] => 4
([(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] => [1,1,0,0,1,0,1,0,1,0] => 4
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule).
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
Cori-Le Borgne involution
Description
The Cori-Le Borgne involution on Dyck paths.
Append an additional down step to the Dyck path and consider its (literal) reversal. The image of the involution is then the unique rotation of this word which is a Dyck word followed by an additional down step. Alternatively, it is the composite $\zeta\circ\mathrm{rev}\circ\zeta^{(-1)}$, where $\zeta$ is Mp00030zeta map.
Append an additional down step to the Dyck path and consider its (literal) reversal. The image of the involution is then the unique rotation of this word which is a Dyck word followed by an additional down step. Alternatively, it is the composite $\zeta\circ\mathrm{rev}\circ\zeta^{(-1)}$, where $\zeta$ is Mp00030zeta map.
Map
clique sizes
Description
The integer partition of the sizes of the maximal cliques of a graph.
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