Identifier
-
Mp00150:
Perfect matchings
—to Dyck path⟶
Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001553: Dyck paths ⟶ ℤ
Values
[(1,2)] => [1,0] => [1,1,0,0] => [1,1,0,0] => 1
[(1,2),(3,4)] => [1,0,1,0] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => 1
[(1,3),(2,4)] => [1,1,0,0] => [1,1,1,0,0,0] => [1,1,1,0,0,0] => 1
[(1,4),(2,3)] => [1,1,0,0] => [1,1,1,0,0,0] => [1,1,1,0,0,0] => 1
[(1,2),(3,4),(5,6)] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => 2
[(1,3),(2,4),(5,6)] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0] => 2
[(1,4),(2,3),(5,6)] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0] => 2
[(1,5),(2,3),(4,6)] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0] => 1
[(1,6),(2,3),(4,5)] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0] => 1
[(1,6),(2,4),(3,5)] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => 1
[(1,5),(2,4),(3,6)] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => 1
[(1,4),(2,5),(3,6)] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => 1
[(1,3),(2,5),(4,6)] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0] => 1
[(1,2),(3,5),(4,6)] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0] => 1
[(1,2),(3,6),(4,5)] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0] => 1
[(1,3),(2,6),(4,5)] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0] => 1
[(1,4),(2,6),(3,5)] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => 1
[(1,5),(2,6),(3,4)] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => 1
[(1,6),(2,5),(3,4)] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => 1
[(1,2),(3,4),(5,6),(7,8)] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 2
[(1,3),(2,4),(5,6),(7,8)] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 2
[(1,4),(2,3),(5,6),(7,8)] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 2
[(1,5),(2,3),(4,6),(7,8)] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 2
[(1,6),(2,3),(4,5),(7,8)] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 2
[(1,7),(2,3),(4,5),(6,8)] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,8),(2,3),(4,5),(6,7)] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,8),(2,4),(3,5),(6,7)] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,7),(2,4),(3,5),(6,8)] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,6),(2,4),(3,5),(7,8)] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 2
[(1,5),(2,4),(3,6),(7,8)] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 2
[(1,4),(2,5),(3,6),(7,8)] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 2
[(1,3),(2,5),(4,6),(7,8)] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 2
[(1,2),(3,5),(4,6),(7,8)] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 2
[(1,2),(3,6),(4,5),(7,8)] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 2
[(1,3),(2,6),(4,5),(7,8)] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 2
[(1,4),(2,6),(3,5),(7,8)] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 2
[(1,5),(2,6),(3,4),(7,8)] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 2
[(1,6),(2,5),(3,4),(7,8)] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 2
[(1,7),(2,5),(3,4),(6,8)] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,8),(2,5),(3,4),(6,7)] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,8),(2,6),(3,4),(5,7)] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,7),(2,6),(3,4),(5,8)] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,6),(2,7),(3,4),(5,8)] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,5),(2,7),(3,4),(6,8)] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,4),(2,7),(3,5),(6,8)] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,3),(2,7),(4,5),(6,8)] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,2),(3,7),(4,5),(6,8)] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,2),(3,8),(4,5),(6,7)] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,3),(2,8),(4,5),(6,7)] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,4),(2,8),(3,5),(6,7)] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,5),(2,8),(3,4),(6,7)] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,6),(2,8),(3,4),(5,7)] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,7),(2,8),(3,4),(5,6)] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,8),(2,7),(3,4),(5,6)] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,8),(2,7),(3,5),(4,6)] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[(1,7),(2,8),(3,5),(4,6)] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[(1,6),(2,8),(3,5),(4,7)] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[(1,5),(2,8),(3,6),(4,7)] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[(1,4),(2,8),(3,6),(5,7)] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,3),(2,8),(4,6),(5,7)] => [1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,2),(3,8),(4,6),(5,7)] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,2),(3,7),(4,6),(5,8)] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,3),(2,7),(4,6),(5,8)] => [1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,4),(2,7),(3,6),(5,8)] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,5),(2,7),(3,6),(4,8)] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[(1,6),(2,7),(3,5),(4,8)] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[(1,7),(2,6),(3,5),(4,8)] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[(1,8),(2,6),(3,5),(4,7)] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[(1,8),(2,5),(3,6),(4,7)] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[(1,7),(2,5),(3,6),(4,8)] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[(1,6),(2,5),(3,7),(4,8)] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[(1,5),(2,6),(3,7),(4,8)] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[(1,4),(2,6),(3,7),(5,8)] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,3),(2,6),(4,7),(5,8)] => [1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,2),(3,6),(4,7),(5,8)] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,2),(3,5),(4,7),(6,8)] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,3),(2,5),(4,7),(6,8)] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,4),(2,5),(3,7),(6,8)] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,5),(2,4),(3,7),(6,8)] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,6),(2,4),(3,7),(5,8)] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,7),(2,4),(3,6),(5,8)] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,8),(2,4),(3,6),(5,7)] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,8),(2,3),(4,6),(5,7)] => [1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,7),(2,3),(4,6),(5,8)] => [1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,6),(2,3),(4,7),(5,8)] => [1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,5),(2,3),(4,7),(6,8)] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,4),(2,3),(5,7),(6,8)] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,3),(2,4),(5,7),(6,8)] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,2),(3,4),(5,7),(6,8)] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,2),(3,4),(5,8),(6,7)] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,3),(2,4),(5,8),(6,7)] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,4),(2,3),(5,8),(6,7)] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,5),(2,3),(4,8),(6,7)] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,6),(2,3),(4,8),(5,7)] => [1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,7),(2,3),(4,8),(5,6)] => [1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,8),(2,3),(4,7),(5,6)] => [1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,8),(2,4),(3,7),(5,6)] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,7),(2,4),(3,8),(5,6)] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,6),(2,4),(3,8),(5,7)] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[(1,5),(2,4),(3,8),(6,7)] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[(1,4),(2,5),(3,8),(6,7)] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
>>> Load all 124 entries. <<<
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 number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path.
The statistic returns zero in case that bimodule is the zero module.
The statistic returns zero in case that bimodule is the zero module.
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.
Map
bounce path
Description
Sends a Dyck path $D$ of length $2n$ to its bounce path.
This path is formed by starting at the endpoint $(n,n)$ of $D$ and travelling west until encountering the first vertical step of $D$, then south until hitting the diagonal, then west again to hit $D$, etc. until the point $(0,0)$ is reached.
This map is the first part of the zeta map Mp00030zeta map.
This path is formed by starting at the endpoint $(n,n)$ of $D$ and travelling west until encountering the first vertical step of $D$, then south until hitting the diagonal, then west again to hit $D$, etc. until the point $(0,0)$ is reached.
This map is the first part of the zeta map Mp00030zeta map.
Map
to Dyck path
Description
The Dyck path corresponding to the opener-closer sequence of the perfect matching.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!