Identifier
-
Mp00311:
Plane partitions
—to partition⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001208: Permutations ⟶ ℤ
Values
[[1]] => [1] => [1,0,1,0] => [2,1] => 1
[[1],[1]] => [1,1] => [1,0,1,1,0,0] => [2,3,1] => 1
[[2]] => [2] => [1,1,0,0,1,0] => [3,1,2] => 1
[[1,1]] => [2] => [1,1,0,0,1,0] => [3,1,2] => 1
[[1],[1],[1]] => [1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1
[[2],[1]] => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 1
[[1,1],[1]] => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 1
[[3]] => [3] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 1
[[2,1]] => [3] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 1
[[1,1,1]] => [3] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 1
[[1],[1],[1],[1]] => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[[2],[1],[1]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => 1
[[2],[2]] => [2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 1
[[1,1],[1],[1]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => 1
[[1,1],[1,1]] => [2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 1
[[3],[1]] => [3,1] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => 1
[[2,1],[1]] => [3,1] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => 1
[[1,1,1],[1]] => [3,1] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => 1
[[4]] => [4] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 1
[[3,1]] => [4] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 1
[[2,2]] => [4] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 1
[[2,1,1]] => [4] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 1
[[1,1,1,1]] => [4] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 1
[[2],[1],[1],[1]] => [2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => 1
[[2],[2],[1]] => [2,2,1] => [1,0,1,0,1,1,0,0] => [3,4,2,1] => 1
[[1,1],[1],[1],[1]] => [2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => 1
[[1,1],[1,1],[1]] => [2,2,1] => [1,0,1,0,1,1,0,0] => [3,4,2,1] => 1
[[3],[1],[1]] => [3,1,1] => [1,0,1,1,0,0,1,0] => [4,2,3,1] => 1
[[3],[2]] => [3,2] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => 1
[[2,1],[1],[1]] => [3,1,1] => [1,0,1,1,0,0,1,0] => [4,2,3,1] => 1
[[2,1],[2]] => [3,2] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => 1
[[2,1],[1,1]] => [3,2] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => 1
[[1,1,1],[1],[1]] => [3,1,1] => [1,0,1,1,0,0,1,0] => [4,2,3,1] => 1
[[1,1,1],[1,1]] => [3,2] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => 1
[[4],[1]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => 1
[[3,1],[1]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => 1
[[2,2],[1]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => 1
[[2,1,1],[1]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => 1
[[1,1,1,1],[1]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => 1
[[2],[2],[1],[1]] => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => 1
[[2],[2],[2]] => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 1
[[1,1],[1,1],[1],[1]] => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => 1
[[1,1],[1,1],[1,1]] => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 1
[[3],[1],[1],[1]] => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => 1
[[3],[2],[1]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => 1
[[3],[3]] => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 1
[[2,1],[1],[1],[1]] => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => 1
[[2,1],[2],[1]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => 1
[[2,1],[1,1],[1]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => 1
[[2,1],[2,1]] => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 1
[[1,1,1],[1],[1],[1]] => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => 1
[[1,1,1],[1,1],[1]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => 1
[[1,1,1],[1,1,1]] => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 1
[[4],[1],[1]] => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => 1
[[4],[2]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => 1
[[3,1],[1],[1]] => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => 1
[[3,1],[2]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => 1
[[3,1],[1,1]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => 1
[[2,2],[1],[1]] => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => 1
[[2,2],[2]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => 1
[[2,2],[1,1]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => 1
[[2,1,1],[1],[1]] => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => 1
[[2,1,1],[2]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => 1
[[2,1,1],[1,1]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => 1
[[1,1,1,1],[1],[1]] => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => 1
[[1,1,1,1],[1,1]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => 1
[[2],[2],[2],[1]] => [2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => 1
[[1,1],[1,1],[1,1],[1]] => [2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => 1
[[3],[2],[1],[1]] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => 1
[[3],[2],[2]] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => 1
[[3],[3],[1]] => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => 1
[[2,1],[2],[1],[1]] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => 1
[[2,1],[2],[2]] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => 1
[[2,1],[1,1],[1],[1]] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => 1
[[2,1],[1,1],[1,1]] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => 1
[[2,1],[2,1],[1]] => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => 1
[[1,1,1],[1,1],[1],[1]] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => 1
[[1,1,1],[1,1],[1,1]] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => 1
[[1,1,1],[1,1,1],[1]] => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => 1
[[4],[1],[1],[1]] => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 1
[[4],[2],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => 1
[[4],[3]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 1
[[3,1],[1],[1],[1]] => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 1
[[3,1],[2],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => 1
[[3,1],[1,1],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => 1
[[3,1],[3]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 1
[[3,1],[2,1]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 1
[[2,2],[1],[1],[1]] => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 1
[[2,2],[2],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => 1
[[2,2],[1,1],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => 1
[[2,2],[2,1]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 1
[[2,1,1],[1],[1],[1]] => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 1
[[2,1,1],[2],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => 1
[[2,1,1],[1,1],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => 1
[[2,1,1],[2,1]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 1
[[2,1,1],[1,1,1]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 1
[[1,1,1,1],[1],[1],[1]] => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 1
[[1,1,1,1],[1,1],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => 1
[[1,1,1,1],[1,1,1]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 1
[[3],[2],[2],[1]] => [3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => 1
[[3],[3],[1],[1]] => [3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => 1
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Description
The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$.
Map
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].
This bijection is defined in [1, Section 2].
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.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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