- FindStat

Identifier

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001215: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 873 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 4\ q + q^{2}$

$F_{4} = 1 + 13\ q + 9\ q^{2} + q^{3}$

$F_{5} = 1 + 46\ q + 56\ q^{2} + 16\ q^{3} + q^{4}$

$F_{6} = 1 + 199\ q + 334\ q^{2} + 160\ q^{3} + 25\ q^{4} + q^{5}$

Description

Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. Then the statistic gives the vector space dimension of the second Ext-group between X and the regular module.
For the first 196 values, the statistic also gives the number of indecomposable non-projective modules

$X$ such that

$\tau(X)$ has codominant dimension equal to one and projective dimension equal to one.

Map

Lalanne-Kreweras involution

Description

The Lalanne-Kreweras involution on Dyck paths.
Label the upsteps from left to right and record the labels on the first up step of each double rise. Do the same for the downsteps. Then form the Dyck path whose ascent lengths and descent lengths are the consecutives differences of the labels.

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.