- FindStat

Identifier

Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001480: Dyck paths ⟶ ℤ

Values

([],1) => [1] => [1,0,1,0] => 0

([],2) => [2] => [1,1,0,0,1,0] => 1

([(0,1)],2) => [1] => [1,0,1,0] => 0

([],3) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => 3

([(1,2)],3) => [3] => [1,1,1,0,0,0,1,0] => 2

([(0,1),(0,2)],3) => [2] => [1,1,0,0,1,0] => 1

([(0,2),(2,1)],3) => [1] => [1,0,1,0] => 0

([(0,2),(1,2)],3) => [2] => [1,1,0,0,1,0] => 1

([(0,1),(0,2),(0,3)],4) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => 3

([(0,2),(0,3),(3,1)],4) => [3] => [1,1,1,0,0,0,1,0] => 2

([(0,1),(0,2),(1,3),(2,3)],4) => [2] => [1,1,0,0,1,0] => 1

([(1,2),(2,3)],4) => [4] => [1,1,1,1,0,0,0,0,1,0] => 3

([(0,3),(3,1),(3,2)],4) => [2] => [1,1,0,0,1,0] => 1

([(0,3),(1,3),(3,2)],4) => [2] => [1,1,0,0,1,0] => 1

([(0,3),(1,3),(2,3)],4) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => 3

([(0,3),(1,2)],4) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => 3

([(0,3),(1,2),(1,3)],4) => [3,2] => [1,1,0,0,1,0,1,0] => 1

([(0,2),(0,3),(1,2),(1,3)],4) => [2,2] => [1,1,0,0,1,1,0,0] => 2

([(0,3),(2,1),(3,2)],4) => [1] => [1,0,1,0] => 0

([(0,3),(1,2),(2,3)],4) => [3] => [1,1,1,0,0,0,1,0] => 2

([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => 3

([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => [2] => [1,1,0,0,1,0] => 1

([(0,3),(0,4),(3,2),(4,1)],5) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => 3

([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => [3,2] => [1,1,0,0,1,0,1,0] => 1

([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => [2,2] => [1,1,0,0,1,1,0,0] => 2

([(0,4),(4,1),(4,2),(4,3)],5) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => 3

([(0,4),(1,4),(4,2),(4,3)],5) => [2,2] => [1,1,0,0,1,1,0,0] => 2

([(0,4),(1,4),(2,4),(4,3)],5) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => 3

([(0,4),(1,4),(2,3),(4,2)],5) => [2] => [1,1,0,0,1,0] => 1

([(0,4),(1,2),(1,4),(2,3)],5) => [5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => 3

([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => [3,2] => [1,1,0,0,1,0,1,0] => 1

([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => [2,2] => [1,1,0,0,1,1,0,0] => 2

([(0,2),(0,4),(3,1),(4,3)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => 3

([(0,4),(1,2),(1,3),(3,4)],5) => [4,4,3] => [1,1,1,0,0,0,1,0,1,1,0,0] => 3

([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => [3] => [1,1,1,0,0,0,1,0] => 2

([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => [5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => 4

([(0,3),(1,2),(1,4),(3,4)],5) => [5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => 3

([(1,4),(3,2),(4,3)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 4

([(0,3),(3,4),(4,1),(4,2)],5) => [2] => [1,1,0,0,1,0] => 1

([(0,4),(1,2),(2,4),(4,3)],5) => [3] => [1,1,1,0,0,0,1,0] => 2

([(0,4),(3,2),(4,1),(4,3)],5) => [3] => [1,1,1,0,0,0,1,0] => 2

([(0,4),(2,3),(3,1),(4,2)],5) => [1] => [1,0,1,0] => 0

([(0,3),(1,2),(2,4),(3,4)],5) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => 3

([(0,4),(1,2),(2,3),(3,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => 3

([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => [2] => [1,1,0,0,1,0] => 1

([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => 3

([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6) => [2,2] => [1,1,0,0,1,1,0,0] => 2

([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => [2,2] => [1,1,0,0,1,1,0,0] => 2

([(0,4),(4,5),(5,1),(5,2),(5,3)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => 3

([(0,5),(1,5),(2,5),(3,4),(5,3)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => 3

([(0,5),(1,5),(4,2),(4,3),(5,4)],6) => [2,2] => [1,1,0,0,1,1,0,0] => 2

([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => 3

([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => [2] => [1,1,0,0,1,0] => 1

([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => 3

([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6) => [2,2] => [1,1,0,0,1,1,0,0] => 2

([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6) => [3,2] => [1,1,0,0,1,0,1,0] => 1

([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6) => [4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 3

([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6) => [2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4

([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6) => [4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4

([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6) => [2,2] => [1,1,0,0,1,1,0,0] => 2

([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6) => [5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => 3

([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [3,2] => [1,1,0,0,1,0,1,0] => 1

([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6) => [5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => 3

([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6) => [5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => 4

([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => [3] => [1,1,1,0,0,0,1,0] => 2

([(0,4),(1,2),(1,3),(2,5),(3,4),(4,5)],6) => [4,4,3] => [1,1,1,0,0,0,1,0,1,1,0,0] => 3

([(0,3),(0,4),(2,5),(3,5),(4,1),(4,2)],6) => [4,4,3] => [1,1,1,0,0,0,1,0,1,1,0,0] => 3

([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => 3

([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => [2] => [1,1,0,0,1,0] => 1

([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6) => [5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => 4

([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6) => [3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 4

([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6) => [5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => 3

([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 4

([(0,5),(1,4),(4,2),(4,5),(5,3)],6) => [4,3,3] => [1,1,1,0,0,0,1,1,0,1,0,0] => 4

([(0,4),(3,5),(4,3),(5,1),(5,2)],6) => [2] => [1,1,0,0,1,0] => 1

([(0,4),(1,3),(3,5),(4,5),(5,2)],6) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => 3

([(0,5),(3,4),(4,2),(5,1),(5,3)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => 3

([(0,3),(1,2),(2,4),(2,5),(3,4),(3,5)],6) => [4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 3

([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => [2] => [1,1,0,0,1,0] => 1

([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => 3

([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => [5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => 3

([(0,5),(1,3),(1,5),(4,2),(5,4)],6) => [5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => 3

([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6) => [4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4

([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6) => [2,2] => [1,1,0,0,1,1,0,0] => 2

([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6) => [3,2] => [1,1,0,0,1,0,1,0] => 1

([(0,5),(3,2),(4,1),(5,3),(5,4)],6) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => 3

([(0,4),(3,2),(4,5),(5,1),(5,3)],6) => [3] => [1,1,1,0,0,0,1,0] => 2

([(0,5),(1,3),(3,4),(4,2),(4,5)],6) => [5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => 3

([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [1] => [1,0,1,0] => 0

([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => [3] => [1,1,1,0,0,0,1,0] => 2

([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 4

([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => [2] => [1,1,0,0,1,0] => 1

([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => [3] => [1,1,1,0,0,0,1,0] => 2

([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7) => [2,2] => [1,1,0,0,1,1,0,0] => 2

([(0,3),(0,4),(3,5),(3,6),(4,5),(4,6),(5,2),(6,1)],7) => [4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 3

([(0,1),(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7) => [2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4

([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => 3

([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => 3

([(0,6),(1,6),(2,5),(3,5),(4,2),(4,3),(6,4)],7) => [2,2] => [1,1,0,0,1,1,0,0] => 2

([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => [2,2] => [1,1,0,0,1,1,0,0] => 2

([(0,6),(1,6),(4,2),(5,4),(6,3),(6,5)],7) => [4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4

>>> Load all 173 entries. <<<

([(0,6),(1,6),(4,5),(5,2),(5,3),(6,4)],7) => [2,2] => [1,1,0,0,1,1,0,0] => 2

([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 4

([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => [2] => [1,1,0,0,1,0] => 1

([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7) => [2] => [1,1,0,0,1,0] => 1

([(0,6),(1,6),(2,5),(3,5),(5,4),(6,2),(6,3)],7) => [2,2] => [1,1,0,0,1,1,0,0] => 2

([(0,6),(1,6),(4,3),(5,2),(6,4),(6,5)],7) => [4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 3

([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(6,2),(6,3)],7) => [2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4

([(0,5),(0,6),(1,5),(1,6),(2,4),(3,4),(5,3),(6,2)],7) => [4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 3

([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7) => [2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4

([(0,5),(1,4),(1,5),(3,6),(4,3),(5,6),(6,2)],7) => [5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => 3

([(0,5),(0,6),(1,5),(1,6),(4,2),(4,3),(5,4),(6,4)],7) => [2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4

([(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1)],7) => [4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 3

([(0,5),(0,6),(1,5),(1,6),(2,3),(4,2),(5,4),(6,4)],7) => [2,2] => [1,1,0,0,1,1,0,0] => 2

([(0,5),(0,6),(1,5),(1,6),(2,3),(3,4),(5,2),(6,4)],7) => [4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4

([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => 3

([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => [3,2] => [1,1,0,0,1,0,1,0] => 1

([(0,3),(0,5),(3,6),(4,1),(4,6),(5,4),(6,2)],7) => [4,3,3] => [1,1,1,0,0,0,1,1,0,1,0,0] => 4

([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5),(4,6),(5,6)],7) => [5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => 4

([(0,4),(1,3),(1,5),(3,6),(4,5),(5,6),(6,2)],7) => [5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => 3

([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7) => [4] => [1,1,1,1,0,0,0,0,1,0] => 3

([(0,2),(0,4),(1,5),(1,6),(2,5),(2,6),(3,1),(4,3)],7) => [4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4

([(0,2),(0,5),(2,6),(3,4),(4,1),(4,6),(5,3)],7) => [5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => 3

([(0,5),(1,3),(1,4),(3,6),(4,5),(5,6),(6,2)],7) => [4,4,3] => [1,1,1,0,0,0,1,0,1,1,0,0] => 3

([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7) => [2] => [1,1,0,0,1,0] => 1

([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(6,4)],7) => [4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4

([(0,3),(0,5),(3,6),(4,2),(5,1),(5,6),(6,4)],7) => [5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => 3

([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => 3

([(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(6,1)],7) => [3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 4

([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7) => [2,2] => [1,1,0,0,1,1,0,0] => 2

([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7) => [5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => 3

([(0,5),(2,6),(3,1),(4,3),(4,6),(5,2),(5,4)],7) => [5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => 3

([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => [3,2] => [1,1,0,0,1,0,1,0] => 1

([(0,2),(1,5),(1,6),(2,3),(3,5),(3,6),(5,4),(6,4)],7) => [4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4

([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7) => [2,2] => [1,1,0,0,1,1,0,0] => 2

([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7) => [2,2] => [1,1,0,0,1,1,0,0] => 2

([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => 3

([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => 3

([(0,5),(1,4),(4,6),(5,6),(6,2),(6,3)],7) => [4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 3

([(0,3),(1,2),(2,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7) => [4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 3

([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => [3] => [1,1,1,0,0,0,1,0] => 2

([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7) => [4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4

([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7) => [3,2] => [1,1,0,0,1,0,1,0] => 1

([(0,6),(1,3),(1,6),(2,5),(3,5),(4,2),(6,4)],7) => [5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => 3

([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7) => [5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => 3

([(0,5),(2,6),(3,6),(4,1),(4,3),(5,2),(5,4)],7) => [4,4,3] => [1,1,1,0,0,0,1,0,1,1,0,0] => 3

([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 4

([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => [3] => [1,1,1,0,0,0,1,0] => 2

([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7) => [4,4,3] => [1,1,1,0,0,0,1,0,1,1,0,0] => 3

([(0,4),(1,5),(2,5),(2,6),(3,1),(3,6),(4,2),(4,3)],7) => [5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => 4

([(0,3),(0,6),(1,5),(1,6),(3,5),(4,2),(5,4),(6,4)],7) => [5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => 4

([(0,3),(1,4),(1,6),(2,5),(3,4),(3,6),(4,2),(6,5)],7) => [3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 4

([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7) => [2,2] => [1,1,0,0,1,1,0,0] => 2

([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7) => [3,2] => [1,1,0,0,1,0,1,0] => 1

([(0,5),(2,6),(3,2),(4,1),(4,6),(5,3),(5,4)],7) => [5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => 3

([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7) => [2] => [1,1,0,0,1,0] => 1

([(0,5),(3,4),(4,1),(5,6),(6,2),(6,3)],7) => [4] => [1,1,1,1,0,0,0,0,1,0] => 3

([(0,6),(3,5),(4,3),(5,1),(6,2),(6,4)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 4

([(0,6),(1,3),(3,6),(5,2),(6,4),(6,5)],7) => [3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 4

([(0,6),(1,4),(2,5),(3,5),(4,3),(4,6),(6,2)],7) => [4,3,3] => [1,1,1,0,0,0,1,1,0,1,0,0] => 4

([(0,5),(3,2),(4,1),(5,6),(6,3),(6,4)],7) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => 3

([(0,6),(1,4),(2,5),(3,2),(3,6),(4,3),(6,5)],7) => [5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => 3

([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7) => [3] => [1,1,1,0,0,0,1,0] => 2

([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => [1] => [1,0,1,0] => 0

([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7) => [3] => [1,1,1,0,0,0,1,0] => 2

([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => [3] => [1,1,1,0,0,0,1,0] => 2

([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => 3

([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => 3

([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => [2] => [1,1,0,0,1,0] => 1

([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7) => [4] => [1,1,1,1,0,0,0,0,1,0] => 3

([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => [2] => [1,1,0,0,1,0] => 1

([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7) => [4] => [1,1,1,1,0,0,0,0,1,0] => 3

([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8) => [3,2] => [1,1,0,0,1,0,1,0] => 1

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 2\ q + q^{2} + q^{3}$

Description

The number of simple summands of the module J^2/J^3. Here J is the Jacobson radical of the Nakayama algebra algebra corresponding to the Dyck path.

Map

promotion cycle type

Description

The cycle type of promotion on the linear extensions of a poset.

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.