Identifier
  • Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
    St001224: Dyck paths ⟶ ℤ (values match St000024The number of double up and double down steps of a Dyck path., St000443The number of long tunnels of a Dyck path., St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path., St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra.)
Values
[[]] => [1,0] => 1
[[],[]] => [1,0,1,0] => 1
[[[]]] => [1,1,0,0] => 2
[[],[],[]] => [1,0,1,0,1,0] => 1
[[],[[]]] => [1,0,1,1,0,0] => 2
[[[]],[]] => [1,1,0,0,1,0] => 2
[[[],[]]] => [1,1,0,1,0,0] => 2
[[[[]]]] => [1,1,1,0,0,0] => 3
[[],[],[],[]] => [1,0,1,0,1,0,1,0] => 1
[[],[],[[]]] => [1,0,1,0,1,1,0,0] => 2
[[],[[]],[]] => [1,0,1,1,0,0,1,0] => 2
[[],[[],[]]] => [1,0,1,1,0,1,0,0] => 2
[[],[[[]]]] => [1,0,1,1,1,0,0,0] => 3
[[[]],[],[]] => [1,1,0,0,1,0,1,0] => 2
[[[]],[[]]] => [1,1,0,0,1,1,0,0] => 3
[[[],[]],[]] => [1,1,0,1,0,0,1,0] => 2
[[[[]]],[]] => [1,1,1,0,0,0,1,0] => 3
[[[],[],[]]] => [1,1,0,1,0,1,0,0] => 2
[[[],[[]]]] => [1,1,0,1,1,0,0,0] => 3
[[[[]],[]]] => [1,1,1,0,0,1,0,0] => 3
[[[[],[]]]] => [1,1,1,0,1,0,0,0] => 3
[[[[[]]]]] => [1,1,1,1,0,0,0,0] => 4
[[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0] => 1
[[],[],[],[[]]] => [1,0,1,0,1,0,1,1,0,0] => 2
[[],[],[[]],[]] => [1,0,1,0,1,1,0,0,1,0] => 2
[[],[],[[],[]]] => [1,0,1,0,1,1,0,1,0,0] => 2
[[],[],[[[]]]] => [1,0,1,0,1,1,1,0,0,0] => 3
[[],[[]],[],[]] => [1,0,1,1,0,0,1,0,1,0] => 2
[[],[[]],[[]]] => [1,0,1,1,0,0,1,1,0,0] => 3
[[],[[],[]],[]] => [1,0,1,1,0,1,0,0,1,0] => 2
[[],[[[]]],[]] => [1,0,1,1,1,0,0,0,1,0] => 3
[[],[[],[],[]]] => [1,0,1,1,0,1,0,1,0,0] => 2
[[],[[],[[]]]] => [1,0,1,1,0,1,1,0,0,0] => 3
[[],[[[]],[]]] => [1,0,1,1,1,0,0,1,0,0] => 3
[[],[[[],[]]]] => [1,0,1,1,1,0,1,0,0,0] => 3
[[],[[[[]]]]] => [1,0,1,1,1,1,0,0,0,0] => 4
[[[]],[],[],[]] => [1,1,0,0,1,0,1,0,1,0] => 2
[[[]],[],[[]]] => [1,1,0,0,1,0,1,1,0,0] => 3
[[[]],[[]],[]] => [1,1,0,0,1,1,0,0,1,0] => 3
[[[]],[[],[]]] => [1,1,0,0,1,1,0,1,0,0] => 3
[[[]],[[[]]]] => [1,1,0,0,1,1,1,0,0,0] => 4
[[[],[]],[],[]] => [1,1,0,1,0,0,1,0,1,0] => 2
[[[[]]],[],[]] => [1,1,1,0,0,0,1,0,1,0] => 3
[[[],[]],[[]]] => [1,1,0,1,0,0,1,1,0,0] => 3
[[[[]]],[[]]] => [1,1,1,0,0,0,1,1,0,0] => 4
[[[],[],[]],[]] => [1,1,0,1,0,1,0,0,1,0] => 2
[[[],[[]]],[]] => [1,1,0,1,1,0,0,0,1,0] => 3
[[[[]],[]],[]] => [1,1,1,0,0,1,0,0,1,0] => 3
[[[[],[]]],[]] => [1,1,1,0,1,0,0,0,1,0] => 3
[[[[[]]]],[]] => [1,1,1,1,0,0,0,0,1,0] => 4
[[[],[],[],[]]] => [1,1,0,1,0,1,0,1,0,0] => 2
[[[],[],[[]]]] => [1,1,0,1,0,1,1,0,0,0] => 3
[[[],[[]],[]]] => [1,1,0,1,1,0,0,1,0,0] => 3
[[[],[[],[]]]] => [1,1,0,1,1,0,1,0,0,0] => 3
[[[],[[[]]]]] => [1,1,0,1,1,1,0,0,0,0] => 4
[[[[]],[],[]]] => [1,1,1,0,0,1,0,1,0,0] => 3
[[[[]],[[]]]] => [1,1,1,0,0,1,1,0,0,0] => 4
[[[[],[]],[]]] => [1,1,1,0,1,0,0,1,0,0] => 3
[[[[[]]],[]]] => [1,1,1,1,0,0,0,1,0,0] => 4
[[[[],[],[]]]] => [1,1,1,0,1,0,1,0,0,0] => 3
[[[[],[[]]]]] => [1,1,1,0,1,1,0,0,0,0] => 4
[[[[[]],[]]]] => [1,1,1,1,0,0,1,0,0,0] => 4
[[[[[],[]]]]] => [1,1,1,1,0,1,0,0,0,0] => 4
[[[[[[]]]]]] => [1,1,1,1,1,0,0,0,0,0] => 5
[[],[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0,1,0] => 1
[[],[],[],[],[[]]] => [1,0,1,0,1,0,1,0,1,1,0,0] => 2
[[],[],[],[[]],[]] => [1,0,1,0,1,0,1,1,0,0,1,0] => 2
[[],[],[],[[],[]]] => [1,0,1,0,1,0,1,1,0,1,0,0] => 2
[[],[],[],[[[]]]] => [1,0,1,0,1,0,1,1,1,0,0,0] => 3
[[],[],[[]],[],[]] => [1,0,1,0,1,1,0,0,1,0,1,0] => 2
[[],[],[[]],[[]]] => [1,0,1,0,1,1,0,0,1,1,0,0] => 3
[[],[],[[],[]],[]] => [1,0,1,0,1,1,0,1,0,0,1,0] => 2
[[],[],[[[]]],[]] => [1,0,1,0,1,1,1,0,0,0,1,0] => 3
[[],[],[[],[],[]]] => [1,0,1,0,1,1,0,1,0,1,0,0] => 2
[[],[],[[],[[]]]] => [1,0,1,0,1,1,0,1,1,0,0,0] => 3
[[],[],[[[]],[]]] => [1,0,1,0,1,1,1,0,0,1,0,0] => 3
[[],[],[[[],[]]]] => [1,0,1,0,1,1,1,0,1,0,0,0] => 3
[[],[],[[[[]]]]] => [1,0,1,0,1,1,1,1,0,0,0,0] => 4
[[],[[]],[],[],[]] => [1,0,1,1,0,0,1,0,1,0,1,0] => 2
[[],[[]],[],[[]]] => [1,0,1,1,0,0,1,0,1,1,0,0] => 3
[[],[[]],[[]],[]] => [1,0,1,1,0,0,1,1,0,0,1,0] => 3
[[],[[]],[[],[]]] => [1,0,1,1,0,0,1,1,0,1,0,0] => 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] => 2
[[],[[[]]],[],[]] => [1,0,1,1,1,0,0,0,1,0,1,0] => 3
[[],[[],[]],[[]]] => [1,0,1,1,0,1,0,0,1,1,0,0] => 3
[[],[[[]]],[[]]] => [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] => 2
[[],[[],[[]]],[]] => [1,0,1,1,0,1,1,0,0,0,1,0] => 3
[[],[[[]],[]],[]] => [1,0,1,1,1,0,0,1,0,0,1,0] => 3
[[],[[[],[]]],[]] => [1,0,1,1,1,0,1,0,0,0,1,0] => 3
[[],[[[[]]]],[]] => [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] => 2
[[],[[],[],[[]]]] => [1,0,1,1,0,1,0,1,1,0,0,0] => 3
[[],[[],[[]],[]]] => [1,0,1,1,0,1,1,0,0,1,0,0] => 3
[[],[[],[[],[]]]] => [1,0,1,1,0,1,1,0,1,0,0,0] => 3
[[],[[],[[[]]]]] => [1,0,1,1,0,1,1,1,0,0,0,0] => 4
[[],[[[]],[],[]]] => [1,0,1,1,1,0,0,1,0,1,0,0] => 3
[[],[[[]],[[]]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => 4
[[],[[[],[]],[]]] => [1,0,1,1,1,0,1,0,0,1,0,0] => 3
[[],[[[[]]],[]]] => [1,0,1,1,1,1,0,0,0,1,0,0] => 4
>>> Load all 196 entries. <<<
[[],[[[],[],[]]]] => [1,0,1,1,1,0,1,0,1,0,0,0] => 3
[[],[[[],[[]]]]] => [1,0,1,1,1,0,1,1,0,0,0,0] => 4
[[],[[[[]],[]]]] => [1,0,1,1,1,1,0,0,1,0,0,0] => 4
[[],[[[[],[]]]]] => [1,0,1,1,1,1,0,1,0,0,0,0] => 4
[[],[[[[[]]]]]] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[[[]],[],[],[],[]] => [1,1,0,0,1,0,1,0,1,0,1,0] => 2
[[[]],[],[],[[]]] => [1,1,0,0,1,0,1,0,1,1,0,0] => 3
[[[]],[],[[]],[]] => [1,1,0,0,1,0,1,1,0,0,1,0] => 3
[[[]],[],[[],[]]] => [1,1,0,0,1,0,1,1,0,1,0,0] => 3
[[[]],[],[[[]]]] => [1,1,0,0,1,0,1,1,1,0,0,0] => 4
[[[]],[[]],[],[]] => [1,1,0,0,1,1,0,0,1,0,1,0] => 3
[[[]],[[]],[[]]] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4
[[[]],[[],[]],[]] => [1,1,0,0,1,1,0,1,0,0,1,0] => 3
[[[]],[[[]]],[]] => [1,1,0,0,1,1,1,0,0,0,1,0] => 4
[[[]],[[],[],[]]] => [1,1,0,0,1,1,0,1,0,1,0,0] => 3
[[[]],[[],[[]]]] => [1,1,0,0,1,1,0,1,1,0,0,0] => 4
[[[]],[[[]],[]]] => [1,1,0,0,1,1,1,0,0,1,0,0] => 4
[[[]],[[[],[]]]] => [1,1,0,0,1,1,1,0,1,0,0,0] => 4
[[[]],[[[[]]]]] => [1,1,0,0,1,1,1,1,0,0,0,0] => 5
[[[],[]],[],[],[]] => [1,1,0,1,0,0,1,0,1,0,1,0] => 2
[[[[]]],[],[],[]] => [1,1,1,0,0,0,1,0,1,0,1,0] => 3
[[[],[]],[],[[]]] => [1,1,0,1,0,0,1,0,1,1,0,0] => 3
[[[[]]],[],[[]]] => [1,1,1,0,0,0,1,0,1,1,0,0] => 4
[[[],[]],[[]],[]] => [1,1,0,1,0,0,1,1,0,0,1,0] => 3
[[[[]]],[[]],[]] => [1,1,1,0,0,0,1,1,0,0,1,0] => 4
[[[],[]],[[],[]]] => [1,1,0,1,0,0,1,1,0,1,0,0] => 3
[[[],[]],[[[]]]] => [1,1,0,1,0,0,1,1,1,0,0,0] => 4
[[[[]]],[[],[]]] => [1,1,1,0,0,0,1,1,0,1,0,0] => 4
[[[[]]],[[[]]]] => [1,1,1,0,0,0,1,1,1,0,0,0] => 5
[[[],[],[]],[],[]] => [1,1,0,1,0,1,0,0,1,0,1,0] => 2
[[[],[[]]],[],[]] => [1,1,0,1,1,0,0,0,1,0,1,0] => 3
[[[[]],[]],[],[]] => [1,1,1,0,0,1,0,0,1,0,1,0] => 3
[[[[],[]]],[],[]] => [1,1,1,0,1,0,0,0,1,0,1,0] => 3
[[[[[]]]],[],[]] => [1,1,1,1,0,0,0,0,1,0,1,0] => 4
[[[],[],[]],[[]]] => [1,1,0,1,0,1,0,0,1,1,0,0] => 3
[[[],[[]]],[[]]] => [1,1,0,1,1,0,0,0,1,1,0,0] => 4
[[[[]],[]],[[]]] => [1,1,1,0,0,1,0,0,1,1,0,0] => 4
[[[[],[]]],[[]]] => [1,1,1,0,1,0,0,0,1,1,0,0] => 4
[[[[[]]]],[[]]] => [1,1,1,1,0,0,0,0,1,1,0,0] => 5
[[[],[],[],[]],[]] => [1,1,0,1,0,1,0,1,0,0,1,0] => 2
[[[],[],[[]]],[]] => [1,1,0,1,0,1,1,0,0,0,1,0] => 3
[[[],[[]],[]],[]] => [1,1,0,1,1,0,0,1,0,0,1,0] => 3
[[[],[[],[]]],[]] => [1,1,0,1,1,0,1,0,0,0,1,0] => 3
[[[],[[[]]]],[]] => [1,1,0,1,1,1,0,0,0,0,1,0] => 4
[[[[]],[],[]],[]] => [1,1,1,0,0,1,0,1,0,0,1,0] => 3
[[[[]],[[]]],[]] => [1,1,1,0,0,1,1,0,0,0,1,0] => 4
[[[[],[]],[]],[]] => [1,1,1,0,1,0,0,1,0,0,1,0] => 3
[[[[[]]],[]],[]] => [1,1,1,1,0,0,0,1,0,0,1,0] => 4
[[[[],[],[]]],[]] => [1,1,1,0,1,0,1,0,0,0,1,0] => 3
[[[[],[[]]]],[]] => [1,1,1,0,1,1,0,0,0,0,1,0] => 4
[[[[[]],[]]],[]] => [1,1,1,1,0,0,1,0,0,0,1,0] => 4
[[[[[],[]]]],[]] => [1,1,1,1,0,1,0,0,0,0,1,0] => 4
[[[[[[]]]]],[]] => [1,1,1,1,1,0,0,0,0,0,1,0] => 5
[[[],[],[],[],[]]] => [1,1,0,1,0,1,0,1,0,1,0,0] => 2
[[[],[],[],[[]]]] => [1,1,0,1,0,1,0,1,1,0,0,0] => 3
[[[],[],[[]],[]]] => [1,1,0,1,0,1,1,0,0,1,0,0] => 3
[[[],[],[[],[]]]] => [1,1,0,1,0,1,1,0,1,0,0,0] => 3
[[[],[],[[[]]]]] => [1,1,0,1,0,1,1,1,0,0,0,0] => 4
[[[],[[]],[],[]]] => [1,1,0,1,1,0,0,1,0,1,0,0] => 3
[[[],[[]],[[]]]] => [1,1,0,1,1,0,0,1,1,0,0,0] => 4
[[[],[[],[]],[]]] => [1,1,0,1,1,0,1,0,0,1,0,0] => 3
[[[],[[[]]],[]]] => [1,1,0,1,1,1,0,0,0,1,0,0] => 4
[[[],[[],[],[]]]] => [1,1,0,1,1,0,1,0,1,0,0,0] => 3
[[[],[[],[[]]]]] => [1,1,0,1,1,0,1,1,0,0,0,0] => 4
[[[],[[[]],[]]]] => [1,1,0,1,1,1,0,0,1,0,0,0] => 4
[[[],[[[],[]]]]] => [1,1,0,1,1,1,0,1,0,0,0,0] => 4
[[[],[[[[]]]]]] => [1,1,0,1,1,1,1,0,0,0,0,0] => 5
[[[[]],[],[],[]]] => [1,1,1,0,0,1,0,1,0,1,0,0] => 3
[[[[]],[],[[]]]] => [1,1,1,0,0,1,0,1,1,0,0,0] => 4
[[[[]],[[]],[]]] => [1,1,1,0,0,1,1,0,0,1,0,0] => 4
[[[[]],[[],[]]]] => [1,1,1,0,0,1,1,0,1,0,0,0] => 4
[[[[]],[[[]]]]] => [1,1,1,0,0,1,1,1,0,0,0,0] => 5
[[[[],[]],[],[]]] => [1,1,1,0,1,0,0,1,0,1,0,0] => 3
[[[[[]]],[],[]]] => [1,1,1,1,0,0,0,1,0,1,0,0] => 4
[[[[],[]],[[]]]] => [1,1,1,0,1,0,0,1,1,0,0,0] => 4
[[[[[]]],[[]]]] => [1,1,1,1,0,0,0,1,1,0,0,0] => 5
[[[[],[],[]],[]]] => [1,1,1,0,1,0,1,0,0,1,0,0] => 3
[[[[],[[]]],[]]] => [1,1,1,0,1,1,0,0,0,1,0,0] => 4
[[[[[]],[]],[]]] => [1,1,1,1,0,0,1,0,0,1,0,0] => 4
[[[[[],[]]],[]]] => [1,1,1,1,0,1,0,0,0,1,0,0] => 4
[[[[[[]]]],[]]] => [1,1,1,1,1,0,0,0,0,1,0,0] => 5
[[[[],[],[],[]]]] => [1,1,1,0,1,0,1,0,1,0,0,0] => 3
[[[[],[],[[]]]]] => [1,1,1,0,1,0,1,1,0,0,0,0] => 4
[[[[],[[]],[]]]] => [1,1,1,0,1,1,0,0,1,0,0,0] => 4
[[[[],[[],[]]]]] => [1,1,1,0,1,1,0,1,0,0,0,0] => 4
[[[[],[[[]]]]]] => [1,1,1,0,1,1,1,0,0,0,0,0] => 5
[[[[[]],[],[]]]] => [1,1,1,1,0,0,1,0,1,0,0,0] => 4
[[[[[]],[[]]]]] => [1,1,1,1,0,0,1,1,0,0,0,0] => 5
[[[[[],[]],[]]]] => [1,1,1,1,0,1,0,0,1,0,0,0] => 4
[[[[[[]]],[]]]] => [1,1,1,1,1,0,0,0,1,0,0,0] => 5
[[[[[],[],[]]]]] => [1,1,1,1,0,1,0,1,0,0,0,0] => 4
[[[[[],[[]]]]]] => [1,1,1,1,0,1,1,0,0,0,0,0] => 5
[[[[[[]],[]]]]] => [1,1,1,1,1,0,0,1,0,0,0,0] => 5
[[[[[[],[]]]]]] => [1,1,1,1,1,0,1,0,0,0,0,0] => 5
[[[[[[[]]]]]]] => [1,1,1,1,1,1,0,0,0,0,0,0] => 6
search for individual values
searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
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Description
Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. Then the statistic gives the vector space dimension of the first Ext-group between X and the regular module.
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.