- FindStat

Identifier

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤ

Values

[[1],[2]] => [2,1] => [1,2] => [1,0,1,0] => 2

[[1],[3]] => [2,1] => [1,2] => [1,0,1,0] => 2

[[2],[3]] => [2,1] => [1,2] => [1,0,1,0] => 2

[[1,1,2]] => [1,2,3] => [2,3,1] => [1,1,0,1,0,0] => 2

[[1,2,2]] => [1,2,3] => [2,3,1] => [1,1,0,1,0,0] => 2

[[2,2,2]] => [1,2,3] => [2,3,1] => [1,1,0,1,0,0] => 2

[[1],[4]] => [2,1] => [1,2] => [1,0,1,0] => 2

[[2],[4]] => [2,1] => [1,2] => [1,0,1,0] => 2

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

[[1,1,3]] => [1,2,3] => [2,3,1] => [1,1,0,1,0,0] => 2

[[1,2,3]] => [1,2,3] => [2,3,1] => [1,1,0,1,0,0] => 2

[[1,3,3]] => [1,2,3] => [2,3,1] => [1,1,0,1,0,0] => 2

[[2,2,3]] => [1,2,3] => [2,3,1] => [1,1,0,1,0,0] => 2

[[2,3,3]] => [1,2,3] => [2,3,1] => [1,1,0,1,0,0] => 2

[[3,3,3]] => [1,2,3] => [2,3,1] => [1,1,0,1,0,0] => 2

[[1],[2],[3]] => [3,2,1] => [1,3,2] => [1,0,1,1,0,0] => 2

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

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

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

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

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

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

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

[[1],[5]] => [2,1] => [1,2] => [1,0,1,0] => 2

[[2],[5]] => [2,1] => [1,2] => [1,0,1,0] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1],[6]] => [2,1] => [1,2] => [1,0,1,0] => 2

[[2],[6]] => [2,1] => [1,2] => [1,0,1,0] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{(5, 2)} = 2\ q^{2} + 4\ q^{3} + 5\ q^{4}$

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

reading word permutation

Description

Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottommost 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.

Map

Kreweras complement

Description

Sends the permutation

$\pi \in \mathfrak{S}_n$ to the permutation

$\pi^{-1}c$ where

$c = (1,\ldots,n)$ is the long cycle.