- FindStat

Identifier

Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤ (values match St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1117 entries. <<<

search for individual values

searching the database for the individual values of this statistic

Description

The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.

Map

lower permutation

Description

The lower bound in the Grassmann interval corresponding to the decorated permutation.
Let $I$ be the anti-exceedance set of a decorated permutation $w$. Let $v$ be the $k$-Grassmannian permutation determined by $v[k] = w^{-1}(I)$ and let $u$ be the permutation satisfying $u = wv$. Then $[u, v]$ is the Grassmann interval corresponding to $w$.
This map returns $u$.

Map

complement

Description

Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$

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.