- FindStat

Identifier

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000094: Ordered trees ⟶ ℤ (values match St000166The depth minus 1 of an ordered tree.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q^{2}$

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

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

$F_{4} = q^{2} + 7\ q^{3} + 10\ q^{4} + 6\ q^{5}$

$F_{5} = q^{2} + 15\ q^{3} + 38\ q^{4} + 42\ q^{5} + 24\ q^{6}$

$F_{6} = q^{2} + 31\ q^{3} + 130\ q^{4} + 222\ q^{5} + 216\ q^{6} + 120\ q^{7}$

Description

The depth of an ordered tree.

Map

left-to-right-maxima to Dyck path

Description

The left-to-right maxima of a permutation as a Dyck path.
Let

$(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are

$c_1, c_1+c_2, \dots, c_1+\dots+c_k$ .
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.

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.