Identifier
Values
[1] => [[1]] => [1] => [1,0] => 1
[2] => [[1,2]] => [1,2] => [1,0,1,0] => 3
[1,1] => [[1],[2]] => [2,1] => [1,1,0,0] => 2
[3] => [[1,2,3]] => [1,2,3] => [1,0,1,0,1,0] => 6
[2,1] => [[1,3],[2]] => [2,1,3] => [1,1,0,0,1,0] => 4
[1,1,1] => [[1],[2],[3]] => [3,2,1] => [1,1,1,0,0,0] => 3
[4] => [[1,2,3,4]] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 10
[3,1] => [[1,3,4],[2]] => [2,1,3,4] => [1,1,0,0,1,0,1,0] => 7
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => [1,1,1,0,1,0,0,0] => 5
[2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => 5
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => [1,1,1,1,0,0,0,0] => 4
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 15
[4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => 11
[3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0] => 8
[3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => 8
[2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0] => 6
[2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => 6
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0] => 5
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 21
[5,1] => [[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0] => 16
[4,2] => [[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [1,1,1,0,1,0,0,0,1,0,1,0] => 12
[4,1,1] => [[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0] => 12
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0] => 9
[3,2,1] => [[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [1,1,1,1,0,0,1,0,0,0,1,0] => 9
[3,1,1,1] => [[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => 9
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0] => 7
[2,2,1,1] => [[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0] => 7
[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 7
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => 6
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Description
Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path.
Map
reading tableau
Description
Return the RSK recording tableau of the reading word of the (standard) tableau $T$ labeled down (in English convention) each column to the shape of a partition.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.
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.