- FindStat

Identifier

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000166: Ordered trees ⟶ ℤ (values match St000094The depth of an ordered tree.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

$F_{2} = q + q^{2}$

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

$F_{4} = q + 7\ q^{2} + 5\ q^{3} + q^{4}$

$F_{5} = q + 15\ q^{2} + 18\ q^{3} + 7\ q^{4} + q^{5}$

$F_{6} = q + 31\ q^{2} + 57\ q^{3} + 33\ q^{4} + 9\ q^{5} + q^{6}$

Description

The depth minus 1 of an ordered tree.
The ordered trees of size

$n$ are bijection with the Dyck paths of size

$n-1$ , and this statistic then corresponds to St000013The height of a Dyck path..

Map

Lalanne-Kreweras involution

Description

The Lalanne-Kreweras involution on Dyck paths.
Label the upsteps from left to right and record the labels on the first up step of each double rise. Do the same for the downsteps. Then form the Dyck path whose ascent lengths and descent lengths are the consecutives differences of the labels.

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.