- FindStat

Identifier

Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000400: Ordered trees ⟶ ℤ

Values

[1,0] => [[]] => 1

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

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

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

[1,0,1,1,0,0] => [[],[[]]] => 4

[1,1,0,0,1,0] => [[[]],[]] => 4

[1,1,0,1,0,0] => [[[],[]]] => 5

[1,1,1,0,0,0] => [[[[]]]] => 6

[1,0,1,0,1,0,1,0] => [[],[],[],[]] => 4

[1,0,1,0,1,1,0,0] => [[],[],[[]]] => 5

[1,0,1,1,0,0,1,0] => [[],[[]],[]] => 5

[1,0,1,1,0,1,0,0] => [[],[[],[]]] => 6

[1,0,1,1,1,0,0,0] => [[],[[[]]]] => 7

[1,1,0,0,1,0,1,0] => [[[]],[],[]] => 5

[1,1,0,0,1,1,0,0] => [[[]],[[]]] => 6

[1,1,0,1,0,0,1,0] => [[[],[]],[]] => 6

[1,1,0,1,0,1,0,0] => [[[],[],[]]] => 7

[1,1,0,1,1,0,0,0] => [[[],[[]]]] => 8

[1,1,1,0,0,0,1,0] => [[[[]]],[]] => 7

[1,1,1,0,0,1,0,0] => [[[[]],[]]] => 8

[1,1,1,0,1,0,0,0] => [[[[],[]]]] => 9

[1,1,1,1,0,0,0,0] => [[[[[]]]]] => 10

[1,0,1,0,1,0,1,0,1,0] => [[],[],[],[],[]] => 5

[1,0,1,0,1,0,1,1,0,0] => [[],[],[],[[]]] => 6

[1,0,1,0,1,1,0,0,1,0] => [[],[],[[]],[]] => 6

[1,0,1,0,1,1,0,1,0,0] => [[],[],[[],[]]] => 7

[1,0,1,0,1,1,1,0,0,0] => [[],[],[[[]]]] => 8

[1,0,1,1,0,0,1,0,1,0] => [[],[[]],[],[]] => 6

[1,0,1,1,0,0,1,1,0,0] => [[],[[]],[[]]] => 7

[1,0,1,1,0,1,0,0,1,0] => [[],[[],[]],[]] => 7

[1,0,1,1,0,1,0,1,0,0] => [[],[[],[],[]]] => 8

[1,0,1,1,0,1,1,0,0,0] => [[],[[],[[]]]] => 9

[1,0,1,1,1,0,0,0,1,0] => [[],[[[]]],[]] => 8

[1,0,1,1,1,0,0,1,0,0] => [[],[[[]],[]]] => 9

[1,0,1,1,1,0,1,0,0,0] => [[],[[[],[]]]] => 10

[1,0,1,1,1,1,0,0,0,0] => [[],[[[[]]]]] => 11

[1,1,0,0,1,0,1,0,1,0] => [[[]],[],[],[]] => 6

[1,1,0,0,1,0,1,1,0,0] => [[[]],[],[[]]] => 7

[1,1,0,0,1,1,0,0,1,0] => [[[]],[[]],[]] => 7

[1,1,0,0,1,1,0,1,0,0] => [[[]],[[],[]]] => 8

[1,1,0,0,1,1,1,0,0,0] => [[[]],[[[]]]] => 9

[1,1,0,1,0,0,1,0,1,0] => [[[],[]],[],[]] => 7

[1,1,0,1,0,0,1,1,0,0] => [[[],[]],[[]]] => 8

[1,1,0,1,0,1,0,0,1,0] => [[[],[],[]],[]] => 8

[1,1,0,1,0,1,0,1,0,0] => [[[],[],[],[]]] => 9

[1,1,0,1,0,1,1,0,0,0] => [[[],[],[[]]]] => 10

[1,1,0,1,1,0,0,0,1,0] => [[[],[[]]],[]] => 9

[1,1,0,1,1,0,0,1,0,0] => [[[],[[]],[]]] => 10

[1,1,0,1,1,0,1,0,0,0] => [[[],[[],[]]]] => 11

[1,1,0,1,1,1,0,0,0,0] => [[[],[[[]]]]] => 12

[1,1,1,0,0,0,1,0,1,0] => [[[[]]],[],[]] => 8

[1,1,1,0,0,0,1,1,0,0] => [[[[]]],[[]]] => 9

[1,1,1,0,0,1,0,0,1,0] => [[[[]],[]],[]] => 9

[1,1,1,0,0,1,0,1,0,0] => [[[[]],[],[]]] => 10

[1,1,1,0,0,1,1,0,0,0] => [[[[]],[[]]]] => 11

[1,1,1,0,1,0,0,0,1,0] => [[[[],[]]],[]] => 10

[1,1,1,0,1,0,0,1,0,0] => [[[[],[]],[]]] => 11

[1,1,1,0,1,0,1,0,0,0] => [[[[],[],[]]]] => 12

[1,1,1,0,1,1,0,0,0,0] => [[[[],[[]]]]] => 13

[1,1,1,1,0,0,0,0,1,0] => [[[[[]]]],[]] => 11

[1,1,1,1,0,0,0,1,0,0] => [[[[[]]],[]]] => 12

[1,1,1,1,0,0,1,0,0,0] => [[[[[]],[]]]] => 13

[1,1,1,1,0,1,0,0,0,0] => [[[[[],[]]]]] => 14

[1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => 15

[1,0,1,0,1,0,1,0,1,0,1,0] => [[],[],[],[],[],[]] => 6

[1,0,1,0,1,0,1,0,1,1,0,0] => [[],[],[],[],[[]]] => 7

[1,0,1,0,1,0,1,1,0,0,1,0] => [[],[],[],[[]],[]] => 7

[1,0,1,0,1,0,1,1,0,1,0,0] => [[],[],[],[[],[]]] => 8

[1,0,1,0,1,0,1,1,1,0,0,0] => [[],[],[],[[[]]]] => 9

[1,0,1,0,1,1,0,0,1,0,1,0] => [[],[],[[]],[],[]] => 7

[1,0,1,0,1,1,0,0,1,1,0,0] => [[],[],[[]],[[]]] => 8

[1,0,1,0,1,1,0,1,0,0,1,0] => [[],[],[[],[]],[]] => 8

[1,0,1,0,1,1,0,1,0,1,0,0] => [[],[],[[],[],[]]] => 9

[1,0,1,0,1,1,0,1,1,0,0,0] => [[],[],[[],[[]]]] => 10

[1,0,1,0,1,1,1,0,0,0,1,0] => [[],[],[[[]]],[]] => 9

[1,0,1,0,1,1,1,0,0,1,0,0] => [[],[],[[[]],[]]] => 10

[1,0,1,0,1,1,1,0,1,0,0,0] => [[],[],[[[],[]]]] => 11

[1,0,1,0,1,1,1,1,0,0,0,0] => [[],[],[[[[]]]]] => 12

[1,0,1,1,0,0,1,0,1,0,1,0] => [[],[[]],[],[],[]] => 7

[1,0,1,1,0,0,1,0,1,1,0,0] => [[],[[]],[],[[]]] => 8

[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] => [[],[[]],[[],[]]] => 9

[1,0,1,1,0,0,1,1,1,0,0,0] => [[],[[]],[[[]]]] => 10

[1,0,1,1,0,1,0,0,1,0,1,0] => [[],[[],[]],[],[]] => 8

[1,0,1,1,0,1,0,0,1,1,0,0] => [[],[[],[]],[[]]] => 9

[1,0,1,1,0,1,0,1,0,0,1,0] => [[],[[],[],[]],[]] => 9

[1,0,1,1,0,1,0,1,0,1,0,0] => [[],[[],[],[],[]]] => 10

[1,0,1,1,0,1,0,1,1,0,0,0] => [[],[[],[],[[]]]] => 11

[1,0,1,1,0,1,1,0,0,0,1,0] => [[],[[],[[]]],[]] => 10

[1,0,1,1,0,1,1,0,0,1,0,0] => [[],[[],[[]],[]]] => 11

[1,0,1,1,0,1,1,0,1,0,0,0] => [[],[[],[[],[]]]] => 12

[1,0,1,1,0,1,1,1,0,0,0,0] => [[],[[],[[[]]]]] => 13

[1,0,1,1,1,0,0,0,1,0,1,0] => [[],[[[]]],[],[]] => 9

[1,0,1,1,1,0,0,0,1,1,0,0] => [[],[[[]]],[[]]] => 10

[1,0,1,1,1,0,0,1,0,0,1,0] => [[],[[[]],[]],[]] => 10

[1,0,1,1,1,0,0,1,0,1,0,0] => [[],[[[]],[],[]]] => 11

[1,0,1,1,1,0,0,1,1,0,0,0] => [[],[[[]],[[]]]] => 12

[1,0,1,1,1,0,1,0,0,0,1,0] => [[],[[[],[]]],[]] => 11

[1,0,1,1,1,0,1,0,0,1,0,0] => [[],[[[],[]],[]]] => 12

[1,0,1,1,1,0,1,0,1,0,0,0] => [[],[[[],[],[]]]] => 13

[1,0,1,1,1,0,1,1,0,0,0,0] => [[],[[[],[[]]]]] => 14

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Description

The path length of an ordered tree.
This is the sum of the lengths of all paths from the root to a node, see Section 2.3.4.5 of [1].

Map

to ordered tree

Description

Sends a Dyck path to the ordered tree encoding the heights of the path.
This map is recursively defined as follows: A Dyck path $D$ of semilength $n$ may be decomposed, according to its returns (St000011The number of touch points (or returns) of a Dyck path.), into smaller paths $D_1,\dots,D_k$ of respective semilengths $n_1,\dots,n_k$ (so one has $n = n_1 + \dots n_k$) each of which has no returns.
Denote by $\tilde D_i$ the path of semilength $n_i-1$ obtained from $D_i$ by removing the initial up- and the final down-step.
This map then sends $D$ to the tree $T$ having a root note with ordered children $T_1,\dots,T_k$ which are again ordered trees computed from $D_1,\dots,D_k$ respectively.
The unique path of semilength $1$ is sent to the tree consisting of a single node.