Identifier
-
Mp00311:
Plane partitions
—to partition⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000056: Permutations ⟶ ℤ (values match St000234The number of global ascents of a permutation.)
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] => 1
[[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] => 1
[[2],[1]] => [2,1] => [1,0,1,0,1,0] => [4,1,2,3] => 1
[[1,1],[1]] => [2,1] => [1,0,1,0,1,0] => [4,1,2,3] => 1
[[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] => 1
[[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] => 1
[[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] => 1
[[3],[1]] => [3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 1
[[2,1],[1]] => [3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 1
[[1,1,1],[1]] => [3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 1
[[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] => 1
[[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] => 1
[[2,1],[1,1]] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 1
[[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] => 1
[[4],[1]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => 1
[[3,1],[1]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => 1
[[2,2],[1]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => 1
[[2,1,1],[1]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => 1
[[1,1,1,1],[1]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => 1
[[2],[2],[1],[1]] => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => 1
[[2],[2],[2]] => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 1
[[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] => 1
[[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] => 1
[[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] => 1
[[3],[3]] => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 1
[[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] => 1
[[2,1],[1,1],[1]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
[[2,1],[2,1]] => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 1
[[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] => 1
[[1,1,1],[1,1,1]] => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 1
[[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] => 1
[[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] => 1
[[3,1],[1,1]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => 1
[[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] => 1
[[2,2],[1,1]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => 1
[[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] => 1
[[2,1,1],[1,1]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => 1
[[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] => 1
[[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] => 1
[[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] => 1
[[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] => 1
[[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] => 1
[[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] => 1
[[4],[3]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 1
[[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] => 1
[[3,1],[1,1],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => 1
[[3,1],[3]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 1
[[3,1],[2,1]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 1
[[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] => 1
[[2,2],[1,1],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => 1
[[2,2],[2,1]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 1
[[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] => 1
[[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] => 1
[[2,1,1],[2,1]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 1
[[2,1,1],[1,1,1]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 1
[[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] => 1
[[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] => 1
[[3],[2],[2],[1]] => [3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => 1
[[3],[3],[1],[1]] => [3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 1
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Description
The decomposition (or block) number of a permutation.
For $\pi \in \mathcal{S}_n$, this is given by
$$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$$
This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum.
This is one plus St000234The number of global ascents of a permutation..
For $\pi \in \mathcal{S}_n$, this is given by
$$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$$
This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum.
This is one plus St000234The number of global ascents of a permutation..
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
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.
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