Identifier
-
Mp00051:
Ordered trees
—to Dyck path⟶
Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001299: Dyck paths ⟶ ℤ
Values
[[]] => [1,0] => [1] => [1,0] => 1
[[],[]] => [1,0,1,0] => [1,1] => [1,0,1,0] => 2
[[[]]] => [1,1,0,0] => [2] => [1,1,0,0] => 1
[[],[],[]] => [1,0,1,0,1,0] => [1,1,1] => [1,0,1,0,1,0] => 6
[[],[[]]] => [1,0,1,1,0,0] => [1,2] => [1,0,1,1,0,0] => 2
[[[]],[]] => [1,1,0,0,1,0] => [2,1] => [1,1,0,0,1,0] => 2
[[[],[]]] => [1,1,0,1,0,0] => [2,1] => [1,1,0,0,1,0] => 2
[[[[]]]] => [1,1,1,0,0,0] => [3] => [1,1,1,0,0,0] => 1
[[],[],[],[]] => [1,0,1,0,1,0,1,0] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 24
[[],[],[[]]] => [1,0,1,0,1,1,0,0] => [1,1,2] => [1,0,1,0,1,1,0,0] => 6
[[],[[]],[]] => [1,0,1,1,0,0,1,0] => [1,2,1] => [1,0,1,1,0,0,1,0] => 4
[[],[[],[]]] => [1,0,1,1,0,1,0,0] => [1,2,1] => [1,0,1,1,0,0,1,0] => 4
[[],[[[]]]] => [1,0,1,1,1,0,0,0] => [1,3] => [1,0,1,1,1,0,0,0] => 2
[[[]],[],[]] => [1,1,0,0,1,0,1,0] => [2,1,1] => [1,1,0,0,1,0,1,0] => 6
[[[]],[[]]] => [1,1,0,0,1,1,0,0] => [2,2] => [1,1,0,0,1,1,0,0] => 2
[[[],[]],[]] => [1,1,0,1,0,0,1,0] => [2,1,1] => [1,1,0,0,1,0,1,0] => 6
[[[[]]],[]] => [1,1,1,0,0,0,1,0] => [3,1] => [1,1,1,0,0,0,1,0] => 2
[[[],[],[]]] => [1,1,0,1,0,1,0,0] => [2,1,1] => [1,1,0,0,1,0,1,0] => 6
[[[],[[]]]] => [1,1,0,1,1,0,0,0] => [2,2] => [1,1,0,0,1,1,0,0] => 2
[[[[]],[]]] => [1,1,1,0,0,1,0,0] => [3,1] => [1,1,1,0,0,0,1,0] => 2
[[[[],[]]]] => [1,1,1,0,1,0,0,0] => [3,1] => [1,1,1,0,0,0,1,0] => 2
[[[[[]]]]] => [1,1,1,1,0,0,0,0] => [4] => [1,1,1,1,0,0,0,0] => 1
[[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 120
[[],[],[],[[]]] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 24
[[],[],[[]],[]] => [1,0,1,0,1,1,0,0,1,0] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 12
[[],[],[[],[]]] => [1,0,1,0,1,1,0,1,0,0] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 12
[[],[],[[[]]]] => [1,0,1,0,1,1,1,0,0,0] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 6
[[],[[]],[],[]] => [1,0,1,1,0,0,1,0,1,0] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 12
[[],[[]],[[]]] => [1,0,1,1,0,0,1,1,0,0] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4
[[],[[],[]],[]] => [1,0,1,1,0,1,0,0,1,0] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 12
[[],[[[]]],[]] => [1,0,1,1,1,0,0,0,1,0] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4
[[],[[],[],[]]] => [1,0,1,1,0,1,0,1,0,0] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 12
[[],[[],[[]]]] => [1,0,1,1,0,1,1,0,0,0] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4
[[],[[[]],[]]] => [1,0,1,1,1,0,0,1,0,0] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4
[[],[[[],[]]]] => [1,0,1,1,1,0,1,0,0,0] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4
[[],[[[[]]]]] => [1,0,1,1,1,1,0,0,0,0] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 2
[[[]],[],[],[]] => [1,1,0,0,1,0,1,0,1,0] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 24
[[[]],[],[[]]] => [1,1,0,0,1,0,1,1,0,0] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 6
[[[]],[[]],[]] => [1,1,0,0,1,1,0,0,1,0] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 4
[[[]],[[],[]]] => [1,1,0,0,1,1,0,1,0,0] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 4
[[[]],[[[]]]] => [1,1,0,0,1,1,1,0,0,0] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2
[[[],[]],[],[]] => [1,1,0,1,0,0,1,0,1,0] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 24
[[[[]]],[],[]] => [1,1,1,0,0,0,1,0,1,0] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 6
[[[],[]],[[]]] => [1,1,0,1,0,0,1,1,0,0] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 6
[[[[]]],[[]]] => [1,1,1,0,0,0,1,1,0,0] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2
[[[],[],[]],[]] => [1,1,0,1,0,1,0,0,1,0] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 24
[[[],[[]]],[]] => [1,1,0,1,1,0,0,0,1,0] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 4
[[[[]],[]],[]] => [1,1,1,0,0,1,0,0,1,0] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 6
[[[[],[]]],[]] => [1,1,1,0,1,0,0,0,1,0] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 6
[[[[[]]]],[]] => [1,1,1,1,0,0,0,0,1,0] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 2
[[[],[],[],[]]] => [1,1,0,1,0,1,0,1,0,0] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 24
[[[],[],[[]]]] => [1,1,0,1,0,1,1,0,0,0] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 6
[[[],[[]],[]]] => [1,1,0,1,1,0,0,1,0,0] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 4
[[[],[[],[]]]] => [1,1,0,1,1,0,1,0,0,0] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 4
[[[],[[[]]]]] => [1,1,0,1,1,1,0,0,0,0] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2
[[[[]],[],[]]] => [1,1,1,0,0,1,0,1,0,0] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 6
[[[[]],[[]]]] => [1,1,1,0,0,1,1,0,0,0] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2
[[[[],[]],[]]] => [1,1,1,0,1,0,0,1,0,0] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 6
[[[[[]]],[]]] => [1,1,1,1,0,0,0,1,0,0] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 2
[[[[],[],[]]]] => [1,1,1,0,1,0,1,0,0,0] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 6
[[[[],[[]]]]] => [1,1,1,0,1,1,0,0,0,0] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2
[[[[[]],[]]]] => [1,1,1,1,0,0,1,0,0,0] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 2
[[[[[],[]]]]] => [1,1,1,1,0,1,0,0,0,0] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 2
[[[[[[]]]]]] => [1,1,1,1,1,0,0,0,0,0] => [5] => [1,1,1,1,1,0,0,0,0,0] => 1
[[],[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 720
[[],[],[],[],[[]]] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 120
[[],[],[],[[]],[]] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 48
[[],[],[],[[],[]]] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 48
[[],[],[],[[[]]]] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 24
[[],[],[[]],[],[]] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 36
[[],[],[[]],[[]]] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 12
[[],[],[[],[]],[]] => [1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 36
[[],[],[[[]]],[]] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 12
[[],[],[[],[],[]]] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 36
[[],[],[[],[[]]]] => [1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 12
[[],[],[[[]],[]]] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 12
[[],[],[[[],[]]]] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 12
[[],[],[[[[]]]]] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 6
[[],[[]],[],[],[]] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 48
[[],[[]],[],[[]]] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 12
[[],[[]],[[]],[]] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 8
[[],[[]],[[],[]]] => [1,0,1,1,0,0,1,1,0,1,0,0] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 8
[[],[[]],[[[]]]] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 4
[[],[[],[]],[],[]] => [1,0,1,1,0,1,0,0,1,0,1,0] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 48
[[],[[[]]],[],[]] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 12
[[],[[],[]],[[]]] => [1,0,1,1,0,1,0,0,1,1,0,0] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 12
[[],[[[]]],[[]]] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 4
[[],[[],[],[]],[]] => [1,0,1,1,0,1,0,1,0,0,1,0] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 48
[[],[[],[[]]],[]] => [1,0,1,1,0,1,1,0,0,0,1,0] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 8
[[],[[[]],[]],[]] => [1,0,1,1,1,0,0,1,0,0,1,0] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 12
[[],[[[],[]]],[]] => [1,0,1,1,1,0,1,0,0,0,1,0] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 12
[[],[[[[]]]],[]] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 4
[[],[[],[],[],[]]] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 48
[[],[[],[],[[]]]] => [1,0,1,1,0,1,0,1,1,0,0,0] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 12
[[],[[],[[]],[]]] => [1,0,1,1,0,1,1,0,0,1,0,0] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 8
[[],[[],[[],[]]]] => [1,0,1,1,0,1,1,0,1,0,0,0] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 8
[[],[[],[[[]]]]] => [1,0,1,1,0,1,1,1,0,0,0,0] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 4
[[],[[[]],[],[]]] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 12
[[],[[[]],[[]]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 4
[[],[[[],[]],[]]] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 12
[[],[[[[]]],[]]] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 4
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searching the database for statistics with the same generating function
Description
The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to Dyck path
Description
Return the Dyck path of the corresponding ordered tree induced by the recurrence of the Catalan numbers, see wikipedia:Catalan_number.
This sends the maximal height of the Dyck path to the depth of the tree.
This sends the maximal height of the Dyck path to the depth of the tree.
Map
rise composition
Description
Send a Dyck path to the composition of sizes of its rises.
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