Identifier
-
Mp00284:
Standard tableaux
—rows⟶
Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000234: Permutations ⟶ ℤ (values match St000056The decomposition (or block) number of a permutation.)
Values
[[1]] => {{1}} => {{1}} => [1] => 0
[[1,2]] => {{1,2}} => {{1,2}} => [2,1] => 0
[[1],[2]] => {{1},{2}} => {{1},{2}} => [1,2] => 1
[[1,2,3]] => {{1,2,3}} => {{1,2,3}} => [2,3,1] => 0
[[1,3],[2]] => {{1,3},{2}} => {{1},{2,3}} => [1,3,2] => 1
[[1,2],[3]] => {{1,2},{3}} => {{1,2},{3}} => [2,1,3] => 1
[[1],[2],[3]] => {{1},{2},{3}} => {{1},{2},{3}} => [1,2,3] => 2
[[1,2,3,4]] => {{1,2,3,4}} => {{1,2,3,4}} => [2,3,4,1] => 0
[[1,3,4],[2]] => {{1,3,4},{2}} => {{1,4},{2,3}} => [4,3,2,1] => 0
[[1,2,4],[3]] => {{1,2,4},{3}} => {{1,2},{3,4}} => [2,1,4,3] => 1
[[1,2,3],[4]] => {{1,2,3},{4}} => {{1,2,3},{4}} => [2,3,1,4] => 1
[[1,3],[2,4]] => {{1,3},{2,4}} => {{1},{2,3,4}} => [1,3,4,2] => 1
[[1,2],[3,4]] => {{1,2},{3,4}} => {{1,2,4},{3}} => [2,4,3,1] => 0
[[1,4],[2],[3]] => {{1,4},{2},{3}} => {{1},{2},{3,4}} => [1,2,4,3] => 2
[[1,3],[2],[4]] => {{1,3},{2},{4}} => {{1},{2,3},{4}} => [1,3,2,4] => 2
[[1,2],[3],[4]] => {{1,2},{3},{4}} => {{1,2},{3},{4}} => [2,1,3,4] => 2
[[1],[2],[3],[4]] => {{1},{2},{3},{4}} => {{1},{2},{3},{4}} => [1,2,3,4] => 3
[[1,2,3,4,5]] => {{1,2,3,4,5}} => {{1,2,3,4,5}} => [2,3,4,5,1] => 0
[[1,3,4,5],[2]] => {{1,3,4,5},{2}} => {{1,4,5},{2,3}} => [4,3,2,5,1] => 0
[[1,2,4,5],[3]] => {{1,2,4,5},{3}} => {{1,2,5},{3,4}} => [2,5,4,3,1] => 0
[[1,2,3,5],[4]] => {{1,2,3,5},{4}} => {{1,2,3},{4,5}} => [2,3,1,5,4] => 1
[[1,2,3,4],[5]] => {{1,2,3,4},{5}} => {{1,2,3,4},{5}} => [2,3,4,1,5] => 1
[[1,3,5],[2,4]] => {{1,3,5},{2,4}} => {{1},{2,3,4,5}} => [1,3,4,5,2] => 1
[[1,2,5],[3,4]] => {{1,2,5},{3,4}} => {{1,2,4},{3,5}} => [2,4,5,1,3] => 0
[[1,3,4],[2,5]] => {{1,3,4},{2,5}} => {{1,4},{2,3,5}} => [4,3,5,1,2] => 0
[[1,2,4],[3,5]] => {{1,2,4},{3,5}} => {{1,2},{3,4,5}} => [2,1,4,5,3] => 1
[[1,2,3],[4,5]] => {{1,2,3},{4,5}} => {{1,2,3,5},{4}} => [2,3,5,4,1] => 0
[[1,4,5],[2],[3]] => {{1,4,5},{2},{3}} => {{1,5},{2},{3,4}} => [5,2,4,3,1] => 0
[[1,3,5],[2],[4]] => {{1,3,5},{2},{4}} => {{1},{2,3,5},{4}} => [1,3,5,4,2] => 1
[[1,2,5],[3],[4]] => {{1,2,5},{3},{4}} => {{1,2},{3},{4,5}} => [2,1,3,5,4] => 2
[[1,3,4],[2],[5]] => {{1,3,4},{2},{5}} => {{1,4},{2,3},{5}} => [4,3,2,1,5] => 1
[[1,2,4],[3],[5]] => {{1,2,4},{3},{5}} => {{1,2},{3,4},{5}} => [2,1,4,3,5] => 2
[[1,2,3],[4],[5]] => {{1,2,3},{4},{5}} => {{1,2,3},{4},{5}} => [2,3,1,4,5] => 2
[[1,4],[2,5],[3]] => {{1,4},{2,5},{3}} => {{1},{2},{3,4,5}} => [1,2,4,5,3] => 2
[[1,3],[2,5],[4]] => {{1,3},{2,5},{4}} => {{1},{2,3},{4,5}} => [1,3,2,5,4] => 2
[[1,2],[3,5],[4]] => {{1,2},{3,5},{4}} => {{1,2},{3,5},{4}} => [2,1,5,4,3] => 1
[[1,3],[2,4],[5]] => {{1,3},{2,4},{5}} => {{1},{2,3,4},{5}} => [1,3,4,2,5] => 2
[[1,2],[3,4],[5]] => {{1,2},{3,4},{5}} => {{1,2,4},{3},{5}} => [2,4,3,1,5] => 1
[[1,5],[2],[3],[4]] => {{1,5},{2},{3},{4}} => {{1},{2},{3},{4,5}} => [1,2,3,5,4] => 3
[[1,4],[2],[3],[5]] => {{1,4},{2},{3},{5}} => {{1},{2},{3,4},{5}} => [1,2,4,3,5] => 3
[[1,3],[2],[4],[5]] => {{1,3},{2},{4},{5}} => {{1},{2,3},{4},{5}} => [1,3,2,4,5] => 3
[[1,2],[3],[4],[5]] => {{1,2},{3},{4},{5}} => {{1,2},{3},{4},{5}} => [2,1,3,4,5] => 3
[[1],[2],[3],[4],[5]] => {{1},{2},{3},{4},{5}} => {{1},{2},{3},{4},{5}} => [1,2,3,4,5] => 4
[[1,2,3,4,5,6]] => {{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 0
[[1,3,4,5,6],[2]] => {{1,3,4,5,6},{2}} => {{1,4,5,6},{2,3}} => [4,3,2,5,6,1] => 0
[[1,2,4,5,6],[3]] => {{1,2,4,5,6},{3}} => {{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => 0
[[1,2,3,5,6],[4]] => {{1,2,3,5,6},{4}} => {{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => 0
[[1,2,3,4,6],[5]] => {{1,2,3,4,6},{5}} => {{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => 1
[[1,2,3,4,5],[6]] => {{1,2,3,4,5},{6}} => {{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => 1
[[1,3,5,6],[2,4]] => {{1,3,5,6},{2,4}} => {{1,6},{2,3,4,5}} => [6,3,4,5,2,1] => 0
[[1,2,5,6],[3,4]] => {{1,2,5,6},{3,4}} => {{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => 0
[[1,3,4,6],[2,5]] => {{1,3,4,6},{2,5}} => {{1,4},{2,3,5,6}} => [4,3,5,1,6,2] => 0
[[1,2,4,6],[3,5]] => {{1,2,4,6},{3,5}} => {{1,2},{3,4,5,6}} => [2,1,4,5,6,3] => 1
[[1,2,3,6],[4,5]] => {{1,2,3,6},{4,5}} => {{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => 0
[[1,3,4,5],[2,6]] => {{1,3,4,5},{2,6}} => {{1,4,5},{2,3,6}} => [4,3,6,5,1,2] => 0
[[1,2,4,5],[3,6]] => {{1,2,4,5},{3,6}} => {{1,2,5},{3,4,6}} => [2,5,4,6,1,3] => 0
[[1,2,3,5],[4,6]] => {{1,2,3,5},{4,6}} => {{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => 1
[[1,2,3,4],[5,6]] => {{1,2,3,4},{5,6}} => {{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => 0
[[1,4,5,6],[2],[3]] => {{1,4,5,6},{2},{3}} => {{1,5,6},{2},{3,4}} => [5,2,4,3,6,1] => 0
[[1,3,5,6],[2],[4]] => {{1,3,5,6},{2},{4}} => {{1,6},{2,3,5},{4}} => [6,3,5,4,2,1] => 0
[[1,2,5,6],[3],[4]] => {{1,2,5,6},{3},{4}} => {{1,2,6},{3},{4,5}} => [2,6,3,5,4,1] => 0
[[1,3,4,6],[2],[5]] => {{1,3,4,6},{2},{5}} => {{1,4},{2,3,6},{5}} => [4,3,6,1,5,2] => 0
[[1,2,4,6],[3],[5]] => {{1,2,4,6},{3},{5}} => {{1,2},{3,4,6},{5}} => [2,1,4,6,5,3] => 1
[[1,2,3,6],[4],[5]] => {{1,2,3,6},{4},{5}} => {{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => 2
[[1,3,4,5],[2],[6]] => {{1,3,4,5},{2},{6}} => {{1,4,5},{2,3},{6}} => [4,3,2,5,1,6] => 1
[[1,2,4,5],[3],[6]] => {{1,2,4,5},{3},{6}} => {{1,2,5},{3,4},{6}} => [2,5,4,3,1,6] => 1
[[1,2,3,5],[4],[6]] => {{1,2,3,5},{4},{6}} => {{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => 2
[[1,2,3,4],[5],[6]] => {{1,2,3,4},{5},{6}} => {{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => 2
[[1,3,5],[2,4,6]] => {{1,3,5},{2,4,6}} => {{1},{2,3,4,5,6}} => [1,3,4,5,6,2] => 1
[[1,2,5],[3,4,6]] => {{1,2,5},{3,4,6}} => {{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => 0
[[1,3,4],[2,5,6]] => {{1,3,4},{2,5,6}} => {{1,4,6},{2,3,5}} => [4,3,5,6,2,1] => 0
[[1,2,4],[3,5,6]] => {{1,2,4},{3,5,6}} => {{1,2,6},{3,4,5}} => [2,6,4,5,3,1] => 0
[[1,2,3],[4,5,6]] => {{1,2,3},{4,5,6}} => {{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => 0
[[1,4,6],[2,5],[3]] => {{1,4,6},{2,5},{3}} => {{1},{2,6},{3,4,5}} => [1,6,4,5,3,2] => 1
[[1,3,6],[2,5],[4]] => {{1,3,6},{2,5},{4}} => {{1},{2,3},{4,5,6}} => [1,3,2,5,6,4] => 2
[[1,2,6],[3,5],[4]] => {{1,2,6},{3,5},{4}} => {{1,2},{3,5},{4,6}} => [2,1,5,6,3,4] => 1
[[1,3,6],[2,4],[5]] => {{1,3,6},{2,4},{5}} => {{1},{2,3,4},{5,6}} => [1,3,4,2,6,5] => 2
[[1,2,6],[3,4],[5]] => {{1,2,6},{3,4},{5}} => {{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => 1
[[1,4,5],[2,6],[3]] => {{1,4,5},{2,6},{3}} => {{1,5},{2},{3,4,6}} => [5,2,4,6,1,3] => 0
[[1,3,5],[2,6],[4]] => {{1,3,5},{2,6},{4}} => {{1},{2,3,5},{4,6}} => [1,3,5,6,2,4] => 1
[[1,2,5],[3,6],[4]] => {{1,2,5},{3,6},{4}} => {{1,2},{3},{4,5,6}} => [2,1,3,5,6,4] => 2
[[1,3,4],[2,6],[5]] => {{1,3,4},{2,6},{5}} => {{1,4},{2,3},{5,6}} => [4,3,2,1,6,5] => 1
[[1,2,4],[3,6],[5]] => {{1,2,4},{3,6},{5}} => {{1,2},{3,4},{5,6}} => [2,1,4,3,6,5] => 2
[[1,2,3],[4,6],[5]] => {{1,2,3},{4,6},{5}} => {{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => 1
[[1,3,5],[2,4],[6]] => {{1,3,5},{2,4},{6}} => {{1},{2,3,4,5},{6}} => [1,3,4,5,2,6] => 2
[[1,2,5],[3,4],[6]] => {{1,2,5},{3,4},{6}} => {{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => 1
[[1,3,4],[2,5],[6]] => {{1,3,4},{2,5},{6}} => {{1,4},{2,3,5},{6}} => [4,3,5,1,2,6] => 1
[[1,2,4],[3,5],[6]] => {{1,2,4},{3,5},{6}} => {{1,2},{3,4,5},{6}} => [2,1,4,5,3,6] => 2
[[1,2,3],[4,5],[6]] => {{1,2,3},{4,5},{6}} => {{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => 1
[[1,5,6],[2],[3],[4]] => {{1,5,6},{2},{3},{4}} => {{1,6},{2},{3},{4,5}} => [6,2,3,5,4,1] => 0
[[1,4,6],[2],[3],[5]] => {{1,4,6},{2},{3},{5}} => {{1},{2,6},{3,4},{5}} => [1,6,4,3,5,2] => 1
[[1,3,6],[2],[4],[5]] => {{1,3,6},{2},{4},{5}} => {{1},{2,3},{4,6},{5}} => [1,3,2,6,5,4] => 2
[[1,2,6],[3],[4],[5]] => {{1,2,6},{3},{4},{5}} => {{1,2},{3},{4},{5,6}} => [2,1,3,4,6,5] => 3
[[1,4,5],[2],[3],[6]] => {{1,4,5},{2},{3},{6}} => {{1,5},{2},{3,4},{6}} => [5,2,4,3,1,6] => 1
[[1,3,5],[2],[4],[6]] => {{1,3,5},{2},{4},{6}} => {{1},{2,3,5},{4},{6}} => [1,3,5,4,2,6] => 2
[[1,2,5],[3],[4],[6]] => {{1,2,5},{3},{4},{6}} => {{1,2},{3},{4,5},{6}} => [2,1,3,5,4,6] => 3
[[1,3,4],[2],[5],[6]] => {{1,3,4},{2},{5},{6}} => {{1,4},{2,3},{5},{6}} => [4,3,2,1,5,6] => 2
[[1,2,4],[3],[5],[6]] => {{1,2,4},{3},{5},{6}} => {{1,2},{3,4},{5},{6}} => [2,1,4,3,5,6] => 3
[[1,2,3],[4],[5],[6]] => {{1,2,3},{4},{5},{6}} => {{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => 3
[[1,4],[2,5],[3,6]] => {{1,4},{2,5},{3,6}} => {{1},{2},{3,4,5,6}} => [1,2,4,5,6,3] => 2
[[1,3],[2,5],[4,6]] => {{1,3},{2,5},{4,6}} => {{1},{2,3,6},{4,5}} => [1,3,6,5,4,2] => 1
>>> Load all 277 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 number of global ascents of a permutation.
The global ascents are the integers $i$ such that
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$$
Equivalently, by the pigeonhole principle,
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$$
For $n > 1$ it can also be described as an occurrence of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives oeis:A003319.
The global ascents are the integers $i$ such that
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$$
Equivalently, by the pigeonhole principle,
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$$
For $n > 1$ it can also be described as an occurrence of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives oeis:A003319.
Map
rows
Description
The set partition whose blocks are the rows of the tableau.
Map
intertwining number to dual major index
Description
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
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!