Identifier
-
Mp00201:
Dyck paths
—Ringel⟶
Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000729: Set partitions ⟶ ℤ
Values
[1,0] => [2,1] => [2,1] => {{1,2}} => 1
[1,0,1,0] => [3,1,2] => [2,3,1] => {{1,2,3}} => 1
[1,1,0,0] => [2,3,1] => [3,1,2] => {{1,3},{2}} => 2
[1,0,1,0,1,0] => [4,1,2,3] => [2,3,4,1] => {{1,2,3,4}} => 1
[1,0,1,1,0,0] => [3,1,4,2] => [2,4,1,3] => {{1,2,4},{3}} => 1
[1,1,0,0,1,0] => [2,4,1,3] => [3,1,4,2] => {{1,3,4},{2}} => 1
[1,1,0,1,0,0] => [4,3,1,2] => [3,4,2,1] => {{1,3},{2,4}} => 2
[1,1,1,0,0,0] => [2,3,4,1] => [4,1,2,3] => {{1,4},{2},{3}} => 3
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [2,3,4,5,1] => {{1,2,3,4,5}} => 1
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [2,3,5,1,4] => {{1,2,3,5},{4}} => 1
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [2,4,1,5,3] => {{1,2,4,5},{3}} => 1
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [2,4,5,3,1] => {{1,2,4},{3,5}} => 1
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [2,5,1,3,4] => {{1,2,5},{3},{4}} => 1
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [3,1,4,5,2] => {{1,3,4,5},{2}} => 1
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [3,1,5,2,4] => {{1,3,5},{2},{4}} => 2
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [3,4,2,5,1] => {{1,3},{2,4,5}} => 1
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [3,4,5,2,1] => {{1,3,5},{2,4}} => 2
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [3,5,2,1,4] => {{1,3},{2,5},{4}} => 2
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [4,1,2,5,3] => {{1,4,5},{2},{3}} => 1
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [4,1,5,3,2] => {{1,4},{2},{3,5}} => 2
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [4,5,2,3,1] => {{1,4},{2,5},{3}} => 3
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => {{1,5},{2},{3},{4}} => 4
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [2,3,4,5,6,1] => {{1,2,3,4,5,6}} => 1
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [2,3,4,6,1,5] => {{1,2,3,4,6},{5}} => 1
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [2,3,5,1,6,4] => {{1,2,3,5,6},{4}} => 1
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [2,3,5,6,4,1] => {{1,2,3,5},{4,6}} => 1
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [2,3,6,1,4,5] => {{1,2,3,6},{4},{5}} => 1
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [2,4,1,5,6,3] => {{1,2,4,5,6},{3}} => 1
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [2,4,1,6,3,5] => {{1,2,4,6},{3},{5}} => 1
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [2,4,5,3,6,1] => {{1,2,4},{3,5,6}} => 1
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [2,4,5,6,3,1] => {{1,2,4,6},{3,5}} => 1
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [2,4,6,3,1,5] => {{1,2,4},{3,6},{5}} => 1
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [2,5,1,3,6,4] => {{1,2,5,6},{3},{4}} => 1
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [2,5,1,6,4,3] => {{1,2,5},{3},{4,6}} => 1
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [2,5,6,3,4,1] => {{1,2,5},{3,6},{4}} => 1
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [2,6,1,3,4,5] => {{1,2,6},{3},{4},{5}} => 1
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [3,1,4,5,6,2] => {{1,3,4,5,6},{2}} => 1
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [3,1,4,6,2,5] => {{1,3,4,6},{2},{5}} => 1
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [3,1,5,2,6,4] => {{1,3,5,6},{2},{4}} => 1
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [3,1,5,6,4,2] => {{1,3,5},{2},{4,6}} => 2
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [3,1,6,2,4,5] => {{1,3,6},{2},{4},{5}} => 2
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [3,4,2,5,6,1] => {{1,3},{2,4,5,6}} => 1
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [3,4,2,6,1,5] => {{1,3},{2,4,6},{5}} => 2
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [3,4,5,2,6,1] => {{1,3,5,6},{2,4}} => 1
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [3,4,5,6,1,2] => {{1,3,5},{2,4,6}} => 2
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [3,4,6,2,1,5] => {{1,3,6},{2,4},{5}} => 2
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [3,5,2,1,6,4] => {{1,3},{2,5,6},{4}} => 1
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [3,5,2,6,4,1] => {{1,3},{2,5},{4,6}} => 2
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [3,5,6,2,4,1] => {{1,3,6},{2,5},{4}} => 2
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [3,6,2,1,4,5] => {{1,3},{2,6},{4},{5}} => 2
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [4,1,2,5,6,3] => {{1,4,5,6},{2},{3}} => 1
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [4,1,2,6,3,5] => {{1,4,6},{2},{3},{5}} => 2
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [4,1,5,3,6,2] => {{1,4},{2},{3,5,6}} => 1
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [4,1,5,6,3,2] => {{1,4,6},{2},{3,5}} => 2
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [4,1,6,3,2,5] => {{1,4},{2},{3,6},{5}} => 3
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [4,5,2,3,6,1] => {{1,4},{2,5,6},{3}} => 1
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [4,5,2,6,3,1] => {{1,4,6},{2,5},{3}} => 2
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [4,5,6,3,2,1] => {{1,4},{2,5},{3,6}} => 3
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [4,6,2,3,1,5] => {{1,4},{2,6},{3},{5}} => 3
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [5,1,2,3,6,4] => {{1,5,6},{2},{3},{4}} => 1
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [5,1,2,6,4,3] => {{1,5},{2},{3},{4,6}} => 2
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [5,1,6,3,4,2] => {{1,5},{2},{3,6},{4}} => 3
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [5,6,2,3,4,1] => {{1,5},{2,6},{3},{4}} => 4
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => {{1,6},{2},{3},{4},{5}} => 5
[1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => {{1,2,3,4,5,6,7}} => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => [2,3,4,5,7,1,6] => {{1,2,3,4,5,7},{6}} => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [5,1,2,3,7,4,6] => [2,3,4,6,1,7,5] => {{1,2,3,4,6,7},{5}} => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [7,1,2,3,6,4,5] => [2,3,4,6,7,5,1] => {{1,2,3,4,6},{5,7}} => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => [2,3,4,7,1,5,6] => {{1,2,3,4,7},{5},{6}} => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [4,1,2,7,3,5,6] => [2,3,5,1,6,7,4] => {{1,2,3,5,6,7},{4}} => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [4,1,2,6,3,7,5] => [2,3,5,1,7,4,6] => {{1,2,3,5,7},{4},{6}} => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [7,1,2,5,3,4,6] => [2,3,5,6,4,7,1] => {{1,2,3,5},{4,6,7}} => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [7,1,2,6,3,4,5] => [2,3,5,6,7,4,1] => {{1,2,3,5,7},{4,6}} => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [6,1,2,5,3,7,4] => [2,3,5,7,4,1,6] => {{1,2,3,5},{4,7},{6}} => 1
[1,0,1,0,1,1,1,0,0,0,1,0] => [4,1,2,5,7,3,6] => [2,3,6,1,4,7,5] => {{1,2,3,6,7},{4},{5}} => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [4,1,2,7,6,3,5] => [2,3,6,1,7,5,4] => {{1,2,3,6},{4},{5,7}} => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [7,1,2,5,6,3,4] => [2,3,6,7,4,5,1] => {{1,2,3,6},{4,7},{5}} => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [4,1,2,5,6,7,3] => [2,3,7,1,4,5,6] => {{1,2,3,7},{4},{5},{6}} => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [3,1,7,2,4,5,6] => [2,4,1,5,6,7,3] => {{1,2,4,5,6,7},{3}} => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [3,1,6,2,4,7,5] => [2,4,1,5,7,3,6] => {{1,2,4,5,7},{3},{6}} => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [3,1,5,2,7,4,6] => [2,4,1,6,3,7,5] => {{1,2,4,6,7},{3},{5}} => 1
[1,0,1,1,0,0,1,1,0,1,0,0] => [3,1,7,2,6,4,5] => [2,4,1,6,7,5,3] => {{1,2,4,6},{3},{5,7}} => 1
[1,0,1,1,0,0,1,1,1,0,0,0] => [3,1,5,2,6,7,4] => [2,4,1,7,3,5,6] => {{1,2,4,7},{3},{5},{6}} => 1
[1,0,1,1,0,1,0,0,1,0,1,0] => [7,1,4,2,3,5,6] => [2,4,5,3,6,7,1] => {{1,2,4},{3,5,6,7}} => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [6,1,4,2,3,7,5] => [2,4,5,3,7,1,6] => {{1,2,4},{3,5,7},{6}} => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [7,1,5,2,3,4,6] => [2,4,5,6,3,7,1] => {{1,2,4,6,7},{3,5}} => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [6,1,7,2,3,4,5] => [2,4,5,6,7,1,3] => {{1,2,4,6},{3,5,7}} => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [6,1,5,2,3,7,4] => [2,4,5,7,3,1,6] => {{1,2,4,7},{3,5},{6}} => 1
[1,0,1,1,0,1,1,0,0,0,1,0] => [5,1,4,2,7,3,6] => [2,4,6,3,1,7,5] => {{1,2,4},{3,6,7},{5}} => 1
[1,0,1,1,0,1,1,0,0,1,0,0] => [7,1,4,2,6,3,5] => [2,4,6,3,7,5,1] => {{1,2,4},{3,6},{5,7}} => 1
[1,0,1,1,0,1,1,0,1,0,0,0] => [7,1,5,2,6,3,4] => [2,4,6,7,3,5,1] => {{1,2,4,7},{3,6},{5}} => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [5,1,4,2,6,7,3] => [2,4,7,3,1,5,6] => {{1,2,4},{3,7},{5},{6}} => 1
[1,0,1,1,1,0,0,0,1,0,1,0] => [3,1,4,7,2,5,6] => [2,5,1,3,6,7,4] => {{1,2,5,6,7},{3},{4}} => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [3,1,4,6,2,7,5] => [2,5,1,3,7,4,6] => {{1,2,5,7},{3},{4},{6}} => 1
[1,0,1,1,1,0,0,1,0,0,1,0] => [3,1,7,5,2,4,6] => [2,5,1,6,4,7,3] => {{1,2,5},{3},{4,6,7}} => 1
[1,0,1,1,1,0,0,1,0,1,0,0] => [3,1,7,6,2,4,5] => [2,5,1,6,7,4,3] => {{1,2,5,7},{3},{4,6}} => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => [3,1,6,5,2,7,4] => [2,5,1,7,4,3,6] => {{1,2,5},{3},{4,7},{6}} => 1
[1,0,1,1,1,0,1,0,0,0,1,0] => [7,1,4,5,2,3,6] => [2,5,6,3,4,7,1] => {{1,2,5},{3,6,7},{4}} => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [7,1,4,6,2,3,5] => [2,5,6,3,7,4,1] => {{1,2,5,7},{3,6},{4}} => 1
[1,0,1,1,1,0,1,0,1,0,0,0] => [7,1,6,5,2,3,4] => [2,5,6,7,4,3,1] => {{1,2,5},{3,6},{4,7}} => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [6,1,4,5,2,7,3] => [2,5,7,3,4,1,6] => {{1,2,5},{3,7},{4},{6}} => 1
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Description
The minimal arc length of a set partition.
The arcs of a set partition are those $i < j$ that are consecutive elements in the blocks. If there are no arcs, the minimal arc length is the size of the ground set (as the minimum of the empty set in the universe of arcs of length less than the size of the ground set).
The arcs of a set partition are those $i < j$ that are consecutive elements in the blocks. If there are no arcs, the minimal arc length is the size of the ground set (as the minimum of the empty set in the universe of arcs of length less than the size of the ground set).
Map
inverse
Description
Sends a permutation to its inverse.
Map
weak exceedance partition
Description
The set partition induced by the weak exceedances of a permutation.
This is the coarsest set partition that contains all arcs $(i, \pi(i))$ with $i\leq\pi(i)$.
This is the coarsest set partition that contains all arcs $(i, \pi(i))$ with $i\leq\pi(i)$.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
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