Identifier
-
Mp00080:
Set partitions
—to permutation⟶
Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤ
Values
{{1}} => [1] => [1] => [1,0] => 0
{{1,2}} => [2,1] => [1,2] => [1,0,1,0] => 1
{{1},{2}} => [1,2] => [2,1] => [1,1,0,0] => 0
{{1,2,3}} => [2,3,1] => [1,3,2] => [1,0,1,1,0,0] => 2
{{1,2},{3}} => [2,1,3] => [3,1,2] => [1,1,1,0,0,0] => 0
{{1,3},{2}} => [3,2,1] => [1,2,3] => [1,0,1,0,1,0] => 2
{{1},{2,3}} => [1,3,2] => [2,3,1] => [1,1,0,1,0,0] => 1
{{1},{2},{3}} => [1,2,3] => [3,2,1] => [1,1,1,0,0,0] => 0
{{1,2,3,4}} => [2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0] => 3
{{1,2,3},{4}} => [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0] => 0
{{1,2,4},{3}} => [2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0] => 3
{{1,2},{3,4}} => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0] => 1
{{1,2},{3},{4}} => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0] => 0
{{1,3,4},{2}} => [3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0] => 3
{{1,3},{2,4}} => [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0] => 2
{{1,3},{2},{4}} => [3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0] => 0
{{1,4},{2,3}} => [4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 3
{{1},{2,3,4}} => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0] => 2
{{1},{2,3},{4}} => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0] => 0
{{1,4},{2},{3}} => [4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0] => 3
{{1},{2,4},{3}} => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0] => 2
{{1},{2},{3,4}} => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0] => 1
{{1},{2},{3},{4}} => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0] => 0
{{1,2,3,4,5}} => [2,3,4,5,1] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => 4
{{1,2,3,4},{5}} => [2,3,4,1,5] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0] => 0
{{1,2,3,5},{4}} => [2,3,5,4,1] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0] => 4
{{1,2,3},{4,5}} => [2,3,1,5,4] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0] => 1
{{1,2,3},{4},{5}} => [2,3,1,4,5] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0] => 0
{{1,2,4,5},{3}} => [2,4,3,5,1] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0] => 4
{{1,2,4},{3,5}} => [2,4,5,1,3] => [3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0] => 2
{{1,2,4},{3},{5}} => [2,4,3,1,5] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0] => 0
{{1,2,5},{3,4}} => [2,5,4,3,1] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => 4
{{1,2},{3,4,5}} => [2,1,4,5,3] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0] => 2
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0] => 0
{{1,2,5},{3},{4}} => [2,5,3,4,1] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0] => 4
{{1,2},{3,5},{4}} => [2,1,5,4,3] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0] => 2
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => 1
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0] => 0
{{1,3,4,5},{2}} => [3,2,4,5,1] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0] => 4
{{1,3,4},{2,5}} => [3,5,4,1,2] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0] => 3
{{1,3,4},{2},{5}} => [3,2,4,1,5] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0] => 0
{{1,3,5},{2,4}} => [3,4,5,2,1] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => 4
{{1,3},{2,4,5}} => [3,4,1,5,2] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0] => 3
{{1,3},{2,4},{5}} => [3,4,1,2,5] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0] => 0
{{1,3,5},{2},{4}} => [3,2,5,4,1] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0] => 4
{{1,3},{2,5},{4}} => [3,5,1,4,2] => [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0] => 3
{{1,3},{2},{4,5}} => [3,2,1,5,4] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0] => 1
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0] => 0
{{1,4,5},{2,3}} => [4,3,2,5,1] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => 4
{{1,4},{2,3,5}} => [4,3,5,1,2] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0] => 3
{{1,4},{2,3},{5}} => [4,3,2,1,5] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0] => 0
{{1,5},{2,3,4}} => [5,3,4,2,1] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => 4
{{1},{2,3,4,5}} => [1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0] => 3
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0] => 0
{{1,5},{2,3},{4}} => [5,3,2,4,1] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0] => 4
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0] => 3
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0] => 1
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0] => 0
{{1,4,5},{2},{3}} => [4,2,3,5,1] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0] => 4
{{1,4},{2,5},{3}} => [4,5,3,1,2] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0] => 3
{{1,4},{2},{3,5}} => [4,2,5,1,3] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0] => 2
{{1,4},{2},{3},{5}} => [4,2,3,1,5] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0] => 0
{{1,5},{2,4},{3}} => [5,4,3,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 4
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0] => 3
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0] => 2
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0] => 0
{{1,5},{2},{3,4}} => [5,2,4,3,1] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => 4
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => 3
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0] => 2
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0] => 0
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0] => 4
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0] => 3
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0] => 2
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0] => 1
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0] => 0
{{1,2,3,4,5,6}} => [2,3,4,5,6,1] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
{{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [1,5,6,4,3,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => 5
{{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => [5,6,1,4,3,2] => [1,1,1,1,1,0,1,0,0,0,0,0] => 1
{{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => [6,5,1,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
{{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => [1,6,4,5,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
{{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => [4,1,6,5,3,2] => [1,1,1,1,0,0,1,1,0,0,0,0] => 2
{{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => [6,1,4,5,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [1,4,5,6,3,2] => [1,0,1,1,1,0,1,0,1,0,0,0] => 5
{{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => [4,6,5,1,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0] => 2
{{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => [6,4,5,1,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
{{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => [1,5,4,6,3,2] => [1,0,1,1,1,1,0,0,1,0,0,0] => 5
{{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => [4,5,6,1,3,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => 2
{{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => [5,6,4,1,3,2] => [1,1,1,1,1,0,1,0,0,0,0,0] => 1
{{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => [6,5,4,1,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
{{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => [1,6,5,3,4,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
{{1,2,4,5},{3,6}} => [2,4,6,5,1,3] => [3,1,5,6,4,2] => [1,1,1,0,0,1,1,0,1,0,0,0] => 3
{{1,2,4,5},{3},{6}} => [2,4,3,5,1,6] => [6,1,5,3,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => [1,3,6,5,4,2] => [1,0,1,1,0,1,1,1,0,0,0,0] => 5
{{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => [3,6,1,5,4,2] => [1,1,1,0,1,1,1,0,0,0,0,0] => 3
{{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => [6,3,1,5,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
{{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => [1,5,6,3,4,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => 5
{{1,2,4},{3,6},{5}} => [2,4,6,1,5,3] => [3,5,1,6,4,2] => [1,1,1,0,1,1,0,0,1,0,0,0] => 3
{{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => [5,6,1,3,4,2] => [1,1,1,1,1,0,1,0,0,0,0,0] => 1
{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => [6,5,1,3,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
{{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => [1,6,3,4,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
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Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
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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