Identifier
-
Mp00080:
Set partitions
—to permutation⟶
Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤ (values match St000053The number of valleys of the Dyck path., St001068Number of torsionless simple modules in the corresponding Nakayama algebra., St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.)
Values
{{1}} => [1] => [1] => [1,0] => 1
{{1,2}} => [2,1] => [2,1] => [1,1,0,0] => 1
{{1},{2}} => [1,2] => [1,2] => [1,0,1,0] => 2
{{1,2,3}} => [2,3,1] => [3,2,1] => [1,1,1,0,0,0] => 1
{{1,2},{3}} => [2,1,3] => [2,1,3] => [1,1,0,0,1,0] => 2
{{1,3},{2}} => [3,2,1] => [2,3,1] => [1,1,0,1,0,0] => 2
{{1},{2,3}} => [1,3,2] => [1,3,2] => [1,0,1,1,0,0] => 2
{{1},{2},{3}} => [1,2,3] => [1,2,3] => [1,0,1,0,1,0] => 3
{{1,2,3,4}} => [2,3,4,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0] => 1
{{1,2,3},{4}} => [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => 2
{{1,2,4},{3}} => [2,4,3,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0] => 2
{{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0] => 2
{{1,2},{3},{4}} => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0] => 3
{{1,3,4},{2}} => [3,2,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0] => 2
{{1,3},{2,4}} => [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0] => 1
{{1,3},{2},{4}} => [3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0] => 3
{{1,4},{2,3}} => [4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0] => 2
{{1},{2,3,4}} => [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0] => 2
{{1},{2,3},{4}} => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0] => 3
{{1,4},{2},{3}} => [4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0] => 3
{{1},{2,4},{3}} => [1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0] => 3
{{1},{2},{3,4}} => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0] => 3
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 4
{{1,2,3,4,5}} => [2,3,4,5,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0] => 1
{{1,2,3,4},{5}} => [2,3,4,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
{{1,2,3,5},{4}} => [2,3,5,4,1] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0] => 2
{{1,2,3},{4,5}} => [2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0] => 2
{{1,2,3},{4},{5}} => [2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => 3
{{1,2,4,5},{3}} => [2,4,3,5,1] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0] => 2
{{1,2,4},{3,5}} => [2,4,5,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0] => 1
{{1,2,4},{3},{5}} => [2,4,3,1,5] => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0] => 3
{{1,2,5},{3,4}} => [2,5,4,3,1] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0] => 2
{{1,2},{3,4,5}} => [2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0] => 2
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0] => 3
{{1,2,5},{3},{4}} => [2,5,3,4,1] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0] => 3
{{1,2},{3,5},{4}} => [2,1,5,4,3] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0] => 3
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0] => 3
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => 4
{{1,3,4,5},{2}} => [3,2,4,5,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0] => 2
{{1,3,4},{2,5}} => [3,5,4,1,2] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0] => 2
{{1,3,4},{2},{5}} => [3,2,4,1,5] => [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0] => 3
{{1,3,5},{2,4}} => [3,4,5,2,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0] => 1
{{1,3},{2,4,5}} => [3,4,1,5,2] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0] => 1
{{1,3},{2,4},{5}} => [3,4,1,2,5] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0] => 2
{{1,3,5},{2},{4}} => [3,2,5,4,1] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0] => 3
{{1,3},{2,5},{4}} => [3,5,1,4,2] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0] => 2
{{1,3},{2},{4,5}} => [3,2,1,5,4] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0] => 3
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => 4
{{1,4,5},{2,3}} => [4,3,2,5,1] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0] => 2
{{1,4},{2,3,5}} => [4,3,5,1,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0] => 1
{{1,4},{2,3},{5}} => [4,3,2,1,5] => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0] => 3
{{1,5},{2,3,4}} => [5,3,4,2,1] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0] => 2
{{1},{2,3,4,5}} => [1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 2
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0] => 3
{{1,5},{2,3},{4}} => [5,3,2,4,1] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0] => 3
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0] => 3
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0] => 3
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => 4
{{1,4,5},{2},{3}} => [4,2,3,5,1] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0] => 3
{{1,4},{2,5},{3}} => [4,5,3,1,2] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => 2
{{1,4},{2},{3,5}} => [4,2,5,1,3] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0] => 2
{{1,4},{2},{3},{5}} => [4,2,3,1,5] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0] => 4
{{1,5},{2,4},{3}} => [5,4,3,2,1] => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0] => 3
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0] => 3
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0] => 2
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => 4
{{1,5},{2},{3,4}} => [5,2,4,3,1] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0] => 3
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0] => 3
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => 3
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => 4
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => 4
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => 4
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => 4
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 4
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 5
{{1,2,3,4,5,6}} => [2,3,4,5,6,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
{{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [5,6,4,3,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2
{{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
{{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => 3
{{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => [4,6,5,3,2,1] => [1,1,1,1,0,1,1,0,0,0,0,0] => 2
{{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => [6,3,2,1,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
{{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => [4,5,3,2,1,6] => [1,1,1,1,0,1,0,0,0,0,1,0] => 3
{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [5,4,6,3,2,1] => [1,1,1,1,1,0,0,1,0,0,0,0] => 2
{{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => [3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0] => 2
{{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => [3,2,1,5,4,6] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
{{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => [4,5,6,3,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0] => 3
{{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => [3,2,1,5,6,4] => [1,1,1,0,0,0,1,1,0,1,0,0] => 3
{{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => [3,2,1,4,6,5] => [1,1,1,0,0,0,1,0,1,1,0,0] => 3
{{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => [3,2,1,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0] => 4
{{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => [3,6,5,4,2,1] => [1,1,1,0,1,1,1,0,0,0,0,0] => 2
{{1,2,4,5},{3,6}} => [2,4,6,5,1,3] => [5,2,1,6,4,3] => [1,1,1,1,1,0,0,0,1,0,0,0] => 2
{{1,2,4,5},{3},{6}} => [2,4,3,5,1,6] => [3,5,4,2,1,6] => [1,1,1,0,1,1,0,0,0,0,1,0] => 3
{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
{{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => [6,5,2,1,4,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
{{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => [5,2,1,4,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
{{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => [3,5,6,4,2,1] => [1,1,1,0,1,1,0,1,0,0,0,0] => 3
{{1,2,4},{3,6},{5}} => [2,4,6,1,5,3] => [5,6,2,1,4,3] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2
{{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => [3,4,2,1,6,5] => [1,1,1,0,1,0,0,0,1,1,0,0] => 3
{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => [3,4,2,1,5,6] => [1,1,1,0,1,0,0,0,1,0,1,0] => 4
{{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => [4,3,6,5,2,1] => [1,1,1,1,0,0,1,1,0,0,0,0] => 2
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Description
The number of peaks of a Dyck path.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
Clarke-Steingrimsson-Zeng inverse
Description
The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map $\Phi$ in [1, sec.3].
This is the inverse of the map $\Phi$ in [1, sec.3].
Map
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
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