Identifier
-
Mp00311:
Plane partitions
—to partition⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001640: Permutations ⟶ ℤ
Values
[[1]] => [1] => [1,0,1,0] => [3,1,2] => 1
[[1],[1]] => [1,1] => [1,0,1,1,0,0] => [3,1,4,2] => 0
[[2]] => [2] => [1,1,0,0,1,0] => [2,4,1,3] => 1
[[1,1]] => [2] => [1,1,0,0,1,0] => [2,4,1,3] => 1
[[1],[1],[1]] => [1,1,1] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 0
[[2],[1]] => [2,1] => [1,0,1,0,1,0] => [4,1,2,3] => 2
[[1,1],[1]] => [2,1] => [1,0,1,0,1,0] => [4,1,2,3] => 2
[[3]] => [3] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 1
[[2,1]] => [3] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 1
[[1,1,1]] => [3] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 1
[[1],[1],[1],[1]] => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 0
[[2],[1],[1]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 1
[[2],[2]] => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 0
[[1,1],[1],[1]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 1
[[1,1],[1,1]] => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 0
[[3],[1]] => [3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 2
[[2,1],[1]] => [3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 2
[[1,1,1],[1]] => [3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 2
[[4]] => [4] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 1
[[3,1]] => [4] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 1
[[2,2]] => [4] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 1
[[2,1,1]] => [4] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 1
[[1,1,1,1]] => [4] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 1
[[2],[1],[1],[1]] => [2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => 1
[[2],[2],[1]] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 1
[[1,1],[1],[1],[1]] => [2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => 1
[[1,1],[1,1],[1]] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 1
[[3],[1],[1]] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
[[3],[2]] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 2
[[2,1],[1],[1]] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
[[2,1],[2]] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 2
[[2,1],[1,1]] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 2
[[1,1,1],[1],[1]] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
[[1,1,1],[1,1]] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 2
[[4],[1]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => 2
[[3,1],[1]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => 2
[[2,2],[1]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => 2
[[2,1,1],[1]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => 2
[[1,1,1,1],[1]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => 2
[[2],[2],[1],[1]] => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => 0
[[2],[2],[2]] => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 0
[[1,1],[1,1],[1],[1]] => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => 0
[[1,1],[1,1],[1,1]] => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 0
[[3],[1],[1],[1]] => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => 1
[[3],[2],[1]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 3
[[3],[3]] => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 0
[[2,1],[1],[1],[1]] => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => 1
[[2,1],[2],[1]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 3
[[2,1],[1,1],[1]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 3
[[2,1],[2,1]] => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 0
[[1,1,1],[1],[1],[1]] => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => 1
[[1,1,1],[1,1],[1]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 3
[[1,1,1],[1,1,1]] => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 0
[[4],[1],[1]] => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => 1
[[4],[2]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => 2
[[3,1],[1],[1]] => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => 1
[[3,1],[2]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => 2
[[3,1],[1,1]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => 2
[[2,2],[1],[1]] => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => 1
[[2,2],[2]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => 2
[[2,2],[1,1]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => 2
[[2,1,1],[1],[1]] => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => 1
[[2,1,1],[2]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => 2
[[2,1,1],[1,1]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => 2
[[1,1,1,1],[1],[1]] => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => 1
[[1,1,1,1],[1,1]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => 2
[[2],[2],[2],[1]] => [2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 1
[[1,1],[1,1],[1,1],[1]] => [2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 1
[[3],[2],[1],[1]] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => 2
[[3],[2],[2]] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => 1
[[3],[3],[1]] => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => 1
[[2,1],[2],[1],[1]] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => 2
[[2,1],[2],[2]] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => 1
[[2,1],[1,1],[1],[1]] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => 2
[[2,1],[1,1],[1,1]] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => 1
[[2,1],[2,1],[1]] => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => 1
[[1,1,1],[1,1],[1],[1]] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => 2
[[1,1,1],[1,1],[1,1]] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => 1
[[1,1,1],[1,1,1],[1]] => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => 1
[[4],[1],[1],[1]] => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 1
[[4],[2],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => 3
[[4],[3]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 2
[[3,1],[1],[1],[1]] => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 1
[[3,1],[2],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => 3
[[3,1],[1,1],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => 3
[[3,1],[3]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 2
[[3,1],[2,1]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 2
[[2,2],[1],[1],[1]] => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 1
[[2,2],[2],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => 3
[[2,2],[1,1],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => 3
[[2,2],[2,1]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 2
[[2,1,1],[1],[1],[1]] => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 1
[[2,1,1],[2],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => 3
[[2,1,1],[1,1],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => 3
[[2,1,1],[2,1]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 2
[[2,1,1],[1,1,1]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 2
[[1,1,1,1],[1],[1],[1]] => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 1
[[1,1,1,1],[1,1],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => 3
[[1,1,1,1],[1,1,1]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 2
[[3],[2],[2],[1]] => [3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => 2
[[3],[3],[1],[1]] => [3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 0
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Description
The number of ascent tops in the permutation such that all smaller elements appear before.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
to partition
Description
The underlying integer partition of a plane partition.
This is the partition whose parts are the sums of the individual rows of the plane partition.
This is the partition whose parts are the sums of the individual rows of the plane partition.
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