Identifier
-
Mp00080:
Set partitions
—to permutation⟶
Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000166: Ordered trees ⟶ ℤ (values match St000094The depth of an ordered tree.)
Values
{{1}} => [1] => [1,0] => [[]] => 1
{{1,2}} => [2,1] => [1,1,0,0] => [[[]]] => 2
{{1},{2}} => [1,2] => [1,0,1,0] => [[],[]] => 1
{{1,2,3}} => [2,3,1] => [1,1,0,1,0,0] => [[[],[]]] => 2
{{1,2},{3}} => [2,1,3] => [1,1,0,0,1,0] => [[[]],[]] => 2
{{1,3},{2}} => [3,2,1] => [1,1,1,0,0,0] => [[[[]]]] => 3
{{1},{2,3}} => [1,3,2] => [1,0,1,1,0,0] => [[],[[]]] => 2
{{1},{2},{3}} => [1,2,3] => [1,0,1,0,1,0] => [[],[],[]] => 1
{{1,2,3,4}} => [2,3,4,1] => [1,1,0,1,0,1,0,0] => [[[],[],[]]] => 2
{{1,2,3},{4}} => [2,3,1,4] => [1,1,0,1,0,0,1,0] => [[[],[]],[]] => 2
{{1,2,4},{3}} => [2,4,3,1] => [1,1,0,1,1,0,0,0] => [[[],[[]]]] => 3
{{1,2},{3,4}} => [2,1,4,3] => [1,1,0,0,1,1,0,0] => [[[]],[[]]] => 2
{{1,2},{3},{4}} => [2,1,3,4] => [1,1,0,0,1,0,1,0] => [[[]],[],[]] => 2
{{1,3,4},{2}} => [3,2,4,1] => [1,1,1,0,0,1,0,0] => [[[[]],[]]] => 3
{{1,3},{2,4}} => [3,4,1,2] => [1,1,1,0,1,0,0,0] => [[[[],[]]]] => 3
{{1,3},{2},{4}} => [3,2,1,4] => [1,1,1,0,0,0,1,0] => [[[[]]],[]] => 3
{{1,4},{2,3}} => [4,3,2,1] => [1,1,1,1,0,0,0,0] => [[[[[]]]]] => 4
{{1},{2,3,4}} => [1,3,4,2] => [1,0,1,1,0,1,0,0] => [[],[[],[]]] => 2
{{1},{2,3},{4}} => [1,3,2,4] => [1,0,1,1,0,0,1,0] => [[],[[]],[]] => 2
{{1,4},{2},{3}} => [4,2,3,1] => [1,1,1,1,0,0,0,0] => [[[[[]]]]] => 4
{{1},{2,4},{3}} => [1,4,3,2] => [1,0,1,1,1,0,0,0] => [[],[[[]]]] => 3
{{1},{2},{3,4}} => [1,2,4,3] => [1,0,1,0,1,1,0,0] => [[],[],[[]]] => 2
{{1},{2},{3},{4}} => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [[],[],[],[]] => 1
{{1,2,3,4,5}} => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => [[[],[],[],[]]] => 2
{{1,2,3,4},{5}} => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0] => [[[],[],[]],[]] => 2
{{1,2,3,5},{4}} => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0] => [[[],[],[[]]]] => 3
{{1,2,3},{4,5}} => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0] => [[[],[]],[[]]] => 2
{{1,2,3},{4},{5}} => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => [[[],[]],[],[]] => 2
{{1,2,4,5},{3}} => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0] => [[[],[[]],[]]] => 3
{{1,2,4},{3,5}} => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0] => [[[],[[],[]]]] => 3
{{1,2,4},{3},{5}} => [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0] => [[[],[[]]],[]] => 3
{{1,2,5},{3,4}} => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0] => [[[],[[[]]]]] => 4
{{1,2},{3,4,5}} => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0] => [[[]],[[],[]]] => 2
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0] => [[[]],[[]],[]] => 2
{{1,2,5},{3},{4}} => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0] => [[[],[[[]]]]] => 4
{{1,2},{3,5},{4}} => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0] => [[[]],[[[]]]] => 3
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0] => [[[]],[],[[]]] => 2
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [[[]],[],[],[]] => 2
{{1,3,4,5},{2}} => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0] => [[[[]],[],[]]] => 3
{{1,3,4},{2,5}} => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0] => [[[[],[[]]]]] => 4
{{1,3,4},{2},{5}} => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0] => [[[[]],[]],[]] => 3
{{1,3,5},{2,4}} => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0] => [[[[],[],[]]]] => 3
{{1,3},{2,4,5}} => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0] => [[[[],[]],[]]] => 3
{{1,3},{2,4},{5}} => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0] => [[[[],[]]],[]] => 3
{{1,3,5},{2},{4}} => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0] => [[[[]],[[]]]] => 3
{{1,3},{2,5},{4}} => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => [[[[],[[]]]]] => 4
{{1,3},{2},{4,5}} => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0] => [[[[]]],[[]]] => 3
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => [[[[]]],[],[]] => 3
{{1,4,5},{2,3}} => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0] => [[[[[]]],[]]] => 4
{{1,4},{2,3,5}} => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0] => [[[[[]],[]]]] => 4
{{1,4},{2,3},{5}} => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => [[[[[]]]],[]] => 4
{{1,5},{2,3,4}} => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => 5
{{1},{2,3,4,5}} => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => [[],[[],[],[]]] => 2
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => [[],[[],[]],[]] => 2
{{1,5},{2,3},{4}} => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => 5
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0] => [[],[[],[[]]]] => 3
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0] => [[],[[]],[[]]] => 2
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [[],[[]],[],[]] => 2
{{1,4,5},{2},{3}} => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0] => [[[[[]]],[]]] => 4
{{1,4},{2,5},{3}} => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => [[[[[],[]]]]] => 4
{{1,4},{2},{3,5}} => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0] => [[[[[]],[]]]] => 4
{{1,4},{2},{3},{5}} => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0] => [[[[[]]]],[]] => 4
{{1,5},{2,4},{3}} => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => 5
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0] => [[],[[[]],[]]] => 3
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0] => [[],[[[],[]]]] => 3
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0] => [[],[[[]]],[]] => 3
{{1,5},{2},{3,4}} => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => 5
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => [[],[[[[]]]]] => 4
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => [[],[],[[],[]]] => 2
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [[],[],[[]],[]] => 2
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => 5
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0] => [[],[[[[]]]]] => 4
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => [[],[],[[[]]]] => 3
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [[],[],[],[[]]] => 2
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[],[],[],[],[]] => 1
{{1,2,3,4,5,6}} => [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [[[],[],[],[],[]]] => 2
{{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => [1,1,0,1,0,1,0,1,0,0,1,0] => [[[],[],[],[]],[]] => 2
{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [1,1,0,1,0,1,0,1,1,0,0,0] => [[[],[],[],[[]]]] => 3
{{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => [1,1,0,1,0,1,0,0,1,1,0,0] => [[[],[],[]],[[]]] => 2
{{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => [1,1,0,1,0,1,0,0,1,0,1,0] => [[[],[],[]],[],[]] => 2
{{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => [1,1,0,1,0,1,1,0,0,1,0,0] => [[[],[],[[]],[]]] => 3
{{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => [1,1,0,1,0,1,1,0,1,0,0,0] => [[[],[],[[],[]]]] => 3
{{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => [1,1,0,1,0,1,1,0,0,0,1,0] => [[[],[],[[]]],[]] => 3
{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [1,1,0,1,0,1,1,1,0,0,0,0] => [[[],[],[[[]]]]] => 4
{{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => [1,1,0,1,0,0,1,1,0,1,0,0] => [[[],[]],[[],[]]] => 2
{{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => [1,1,0,1,0,0,1,1,0,0,1,0] => [[[],[]],[[]],[]] => 2
{{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => [1,1,0,1,0,1,1,1,0,0,0,0] => [[[],[],[[[]]]]] => 4
{{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => [1,1,0,1,0,0,1,1,1,0,0,0] => [[[],[]],[[[]]]] => 3
{{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => [1,1,0,1,0,0,1,0,1,1,0,0] => [[[],[]],[],[[]]] => 2
{{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => [1,1,0,1,0,0,1,0,1,0,1,0] => [[[],[]],[],[],[]] => 2
{{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => [1,1,0,1,1,0,0,1,0,1,0,0] => [[[],[[]],[],[]]] => 3
{{1,2,4,5},{3,6}} => [2,4,6,5,1,3] => [1,1,0,1,1,0,1,1,0,0,0,0] => [[[],[[],[[]]]]] => 4
{{1,2,4,5},{3},{6}} => [2,4,3,5,1,6] => [1,1,0,1,1,0,0,1,0,0,1,0] => [[[],[[]],[]],[]] => 3
{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => [1,1,0,1,1,0,1,0,1,0,0,0] => [[[],[[],[],[]]]] => 3
{{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => [1,1,0,1,1,0,1,0,0,1,0,0] => [[[],[[],[]],[]]] => 3
{{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => [1,1,0,1,1,0,1,0,0,0,1,0] => [[[],[[],[]]],[]] => 3
{{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => [1,1,0,1,1,0,0,1,1,0,0,0] => [[[],[[]],[[]]]] => 3
{{1,2,4},{3,6},{5}} => [2,4,6,1,5,3] => [1,1,0,1,1,0,1,1,0,0,0,0] => [[[],[[],[[]]]]] => 4
{{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0] => [[[],[[]]],[[]]] => 3
{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0] => [[[],[[]]],[],[]] => 3
{{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => [1,1,0,1,1,1,0,0,0,1,0,0] => [[[],[[[]]],[]]] => 4
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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..
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
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.
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.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
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.
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.
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