Identifier
-
Mp00039:
Integer compositions
—complement⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000700: Ordered trees ⟶ ℤ
Values
[1] => [1] => [1,0] => [[]] => 1
[1,1] => [2] => [1,1,0,0] => [[[]]] => 2
[2] => [1,1] => [1,0,1,0] => [[],[]] => 1
[1,1,1] => [3] => [1,1,1,0,0,0] => [[[[]]]] => 3
[1,2] => [2,1] => [1,1,0,0,1,0] => [[[]],[]] => 1
[2,1] => [1,2] => [1,0,1,1,0,0] => [[],[[]]] => 1
[3] => [1,1,1] => [1,0,1,0,1,0] => [[],[],[]] => 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0] => [[[[[]]]]] => 4
[1,1,2] => [3,1] => [1,1,1,0,0,0,1,0] => [[[[]]],[]] => 1
[1,2,1] => [2,2] => [1,1,0,0,1,1,0,0] => [[[]],[[]]] => 2
[1,3] => [2,1,1] => [1,1,0,0,1,0,1,0] => [[[]],[],[]] => 1
[2,1,1] => [1,3] => [1,0,1,1,1,0,0,0] => [[],[[[]]]] => 1
[2,2] => [1,2,1] => [1,0,1,1,0,0,1,0] => [[],[[]],[]] => 1
[3,1] => [1,1,2] => [1,0,1,0,1,1,0,0] => [[],[],[[]]] => 1
[4] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [[],[],[],[]] => 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => 5
[1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [[[[[]]]],[]] => 1
[1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [[[[]]],[[]]] => 2
[1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [[[[]]],[],[]] => 1
[1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [[[]],[[[]]]] => 2
[1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [[[]],[[]],[]] => 1
[1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [[[]],[],[[]]] => 1
[1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [[[]],[],[],[]] => 1
[2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [[],[[[[]]]]] => 1
[2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [[],[[[]]],[]] => 1
[2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [[],[[]],[[]]] => 1
[2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [[],[[]],[],[]] => 1
[3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [[],[],[[[]]]] => 1
[3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [[],[],[[]],[]] => 1
[4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [[],[],[],[[]]] => 1
[5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [[],[],[],[],[]] => 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [[[[[[[]]]]]]] => 6
[1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [[[[[[]]]]],[]] => 1
[1,1,1,2,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [[[[[]]]],[[]]] => 2
[1,1,1,3] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [[[[[]]]],[],[]] => 1
[1,1,2,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [[[[]]],[[[]]]] => 3
[1,1,2,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [[[[]]],[[]],[]] => 1
[1,1,3,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [[[[]]],[],[[]]] => 1
[1,1,4] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [[[[]]],[],[],[]] => 1
[1,2,1,1,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [[[]],[[[[]]]]] => 2
[1,2,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [[[]],[[[]]],[]] => 1
[1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [[[]],[[]],[[]]] => 2
[1,2,3] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [[[]],[[]],[],[]] => 1
[1,3,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [[[]],[],[[[]]]] => 1
[1,3,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [[[]],[],[[]],[]] => 1
[1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [[[]],[],[],[[]]] => 1
[1,5] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [[[]],[],[],[],[]] => 1
[2,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [[],[[[[[]]]]]] => 1
[2,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [[],[[[[]]]],[]] => 1
[2,1,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [[],[[[]]],[[]]] => 1
[2,1,3] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [[],[[[]]],[],[]] => 1
[2,2,1,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [[],[[]],[[[]]]] => 1
[2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [[],[[]],[[]],[]] => 1
[2,3,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [[],[[]],[],[[]]] => 1
[2,4] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [[],[[]],[],[],[]] => 1
[3,1,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [[],[],[[[[]]]]] => 1
[3,1,2] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [[],[],[[[]]],[]] => 1
[3,2,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [[],[],[[]],[[]]] => 1
[3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [[],[],[[]],[],[]] => 1
[4,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [[],[],[],[[[]]]] => 1
[4,2] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [[],[],[],[[]],[]] => 1
[5,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [[],[],[],[],[[]]] => 1
[6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[],[],[],[],[],[]] => 1
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [[[[[[[[]]]]]]]] => 7
[1,1,1,1,1,2] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [[[[[[[]]]]]],[]] => 1
[1,1,1,1,2,1] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [[[[[[]]]]],[[]]] => 2
[1,1,1,1,3] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [[[[[[]]]]],[],[]] => 1
[1,1,1,2,1,1] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [[[[[]]]],[[[]]]] => 3
[1,1,1,2,2] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [[[[[]]]],[[]],[]] => 1
[1,1,1,3,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [[[[[]]]],[],[[]]] => 1
[1,1,1,4] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [[[[[]]]],[],[],[]] => 1
[1,1,2,1,1,1] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [[[[]]],[[[[]]]]] => 3
[1,1,2,1,2] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [[[[]]],[[[]]],[]] => 1
[1,1,2,2,1] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [[[[]]],[[]],[[]]] => 2
[1,1,2,3] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [[[[]]],[[]],[],[]] => 1
[1,1,3,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [[[[]]],[],[[[]]]] => 1
[1,1,3,2] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [[[[]]],[],[[]],[]] => 1
[1,1,4,1] => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [[[[]]],[],[],[[]]] => 1
[1,1,5] => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [[[[]]],[],[],[],[]] => 1
[1,2,1,1,1,1] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [[[]],[[[[[]]]]]] => 2
[1,2,1,1,2] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [[[]],[[[[]]]],[]] => 1
[1,2,1,2,1] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [[[]],[[[]]],[[]]] => 2
[1,2,1,3] => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [[[]],[[[]]],[],[]] => 1
[1,2,2,1,1] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [[[]],[[]],[[[]]]] => 2
[1,2,2,2] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [[[]],[[]],[[]],[]] => 1
[1,2,3,1] => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [[[]],[[]],[],[[]]] => 1
[1,2,4] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [[[]],[[]],[],[],[]] => 1
[1,3,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [[[]],[],[[[[]]]]] => 1
[1,3,1,2] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => [[[]],[],[[[]]],[]] => 1
[1,3,2,1] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [[[]],[],[[]],[[]]] => 1
[1,3,3] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [[[]],[],[[]],[],[]] => 1
[1,4,1,1] => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [[[]],[],[],[[[]]]] => 1
[1,4,2] => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [[[]],[],[],[[]],[]] => 1
[1,5,1] => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [[[]],[],[],[],[[]]] => 1
[1,6] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [[[]],[],[],[],[],[]] => 1
[2,1,1,1,1,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [[],[[[[[[]]]]]]] => 1
[2,1,1,1,2] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [[],[[[[[]]]]],[]] => 1
[2,1,1,2,1] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [[],[[[[]]]],[[]]] => 1
[2,1,1,3] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [[],[[[[]]]],[],[]] => 1
[2,1,2,1,1] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [[],[[[]]],[[[]]]] => 1
[2,1,2,2] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [[],[[[]]],[[]],[]] => 1
>>> Load all 236 entries. <<<
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The protection number of an ordered tree.
This is the minimal distance from the root to a leaf.
This is the minimal distance from the root to a leaf.
Map
complement
Description
The complement of a composition.
The complement of a composition $I$ is defined as follows:
If $I$ is the empty composition, then the complement is also the empty composition. Otherwise, let $S$ be the descent set corresponding to $I=(i_1,\dots,i_k)$, that is, the subset
$$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$$
of $\{ 1, 2, \ldots, |I|-1 \}$. Then, the complement of $I$ is the composition of the same size as $I$, whose descent set is $\{ 1, 2, \ldots, |I|-1 \} \setminus S$.
The complement of a composition $I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to $I$.
The complement of a composition $I$ is defined as follows:
If $I$ is the empty composition, then the complement is also the empty composition. Otherwise, let $S$ be the descent set corresponding to $I=(i_1,\dots,i_k)$, that is, the subset
$$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$$
of $\{ 1, 2, \ldots, |I|-1 \}$. Then, the complement of $I$ is the composition of the same size as $I$, whose descent set is $\{ 1, 2, \ldots, |I|-1 \} \setminus S$.
The complement of a composition $I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to $I$.
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
bounce path
Description
The bounce path determined by an integer composition.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!