- FindStat

Identifier

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤ

Values

[[1,2,3]] => [1,2,3] => [1,2] => [1,0,1,0] => 2
[[1,2],[3]] => [3,1,2] => [1,2] => [1,0,1,0] => 2
[[1,2,3,4]] => [1,2,3,4] => [1,2,3] => [1,0,1,0,1,0] => 3
[[1,3,4],[2]] => [2,1,3,4] => [2,1,3] => [1,1,0,0,1,0] => 2
[[1,2,3],[4]] => [4,1,2,3] => [1,2,3] => [1,0,1,0,1,0] => 3
[[1,3],[2,4]] => [2,4,1,3] => [2,1,3] => [1,1,0,0,1,0] => 2
[[1,3],[2],[4]] => [4,2,1,3] => [2,1,3] => [1,1,0,0,1,0] => 2
[[1,2,3,4,5]] => [1,2,3,4,5] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 3
[[1,3,4,5],[2]] => [2,1,3,4,5] => [2,1,3,4] => [1,1,0,0,1,0,1,0] => 3
[[1,2,4,5],[3]] => [3,1,2,4,5] => [3,1,2,4] => [1,1,1,0,0,0,1,0] => 2
[[1,2,3,4],[5]] => [5,1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 3
[[1,3,5],[2,4]] => [2,4,1,3,5] => [2,4,1,3] => [1,1,0,1,1,0,0,0] => 2
[[1,2,5],[3,4]] => [3,4,1,2,5] => [3,4,1,2] => [1,1,1,0,1,0,0,0] => 2
[[1,3,4],[2,5]] => [2,5,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0] => 3
[[1,2,4],[3,5]] => [3,5,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0] => 2
[[1,4,5],[2],[3]] => [3,2,1,4,5] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => 2
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0] => 3
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0] => 2
[[1,4],[2,5],[3]] => [3,2,5,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => 2
[[1,3],[2,4],[5]] => [5,2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0] => 2
[[1,2],[3,4],[5]] => [5,3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0] => 2
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => 2
[[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 3
[[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => 3
[[1,2,4,5,6],[3]] => [3,1,2,4,5,6] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0] => 3
[[1,2,3,5,6],[4]] => [4,1,2,3,5,6] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 3
[[1,3,5,6],[2,4]] => [2,4,1,3,5,6] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0] => 3
[[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0] => 3
[[1,3,4,6],[2,5]] => [2,5,1,3,4,6] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0] => 2
[[1,2,4,6],[3,5]] => [3,5,1,2,4,6] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0] => 2
[[1,2,3,6],[4,5]] => [4,5,1,2,3,6] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0] => 2
[[1,3,4,5],[2,6]] => [2,6,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => 3
[[1,2,4,5],[3,6]] => [3,6,1,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0] => 3
[[1,2,3,5],[4,6]] => [4,6,1,2,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => 3
[[1,3,5,6],[2],[4]] => [4,2,1,3,5,6] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[[1,2,5,6],[3],[4]] => [4,3,1,2,5,6] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[[1,3,4,5],[2],[6]] => [6,2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => 3
[[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0] => 3
[[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[[1,3,5],[2,4,6]] => [2,4,6,1,3,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0] => 3
[[1,2,5],[3,4,6]] => [3,4,6,1,2,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0] => 3
[[1,3,4],[2,5,6]] => [2,5,6,1,3,4] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0] => 2
[[1,2,4],[3,5,6]] => [3,5,6,1,2,4] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0] => 2
[[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0] => 2
[[1,4,6],[2,5],[3]] => [3,2,5,1,4,6] => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0] => 2
[[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0] => 2
[[1,2,6],[3,5],[4]] => [4,3,5,1,2,6] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0] => 2
[[1,4,5],[2,6],[3]] => [3,2,6,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => 3
[[1,3,5],[2,6],[4]] => [4,2,6,1,3,5] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[[1,2,5],[3,6],[4]] => [4,3,6,1,2,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[[1,3,5],[2,4],[6]] => [6,2,4,1,3,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0] => 3
[[1,2,5],[3,4],[6]] => [6,3,4,1,2,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0] => 3
[[1,3,4],[2,5],[6]] => [6,2,5,1,3,4] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0] => 2
[[1,2,4],[3,5],[6]] => [6,3,5,1,2,4] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0] => 2
[[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0] => 2
[[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[[1,4,5],[2],[3],[6]] => [6,3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => 3
[[1,3,5],[2],[4],[6]] => [6,4,2,1,3,5] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[[1,2,5],[3],[4],[6]] => [6,4,3,1,2,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[[1,4],[2,5],[3,6]] => [3,6,2,5,1,4] => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0] => 2
[[1,3],[2,5],[4,6]] => [4,6,2,5,1,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0] => 2
[[1,2],[3,5],[4,6]] => [4,6,3,5,1,2] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0] => 2
[[1,5],[2,6],[3],[4]] => [4,3,2,6,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[[1,4],[2,5],[3],[6]] => [6,3,2,5,1,4] => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0] => 2
[[1,3],[2,5],[4],[6]] => [6,4,2,5,1,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0] => 2
[[1,2],[3,5],[4],[6]] => [6,4,3,5,1,2] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0] => 2
[[1,5],[2],[3],[4],[6]] => [6,4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2

search for individual values

searching the database for the individual values of this statistic

Description

The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ .

Map

restriction

Description

The permutation obtained by removing the largest letter.
This map is undefined for the empty permutation.

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.