- FindStat

Identifier

Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001067: Dyck paths ⟶ ℤ (values match St000932The number of occurrences of the pattern UDU in a Dyck path., St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 625 entries. <<<

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$F_{1} = 1$

$F_{2} = 1 + q$

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

$F_{4} = 4 + 6\ q + 3\ q^{2} + q^{3}$

$F_{5} = 9 + 16\ q + 12\ q^{2} + 4\ q^{3} + q^{4}$

$F_{6} = 21 + 45\ q + 40\ q^{2} + 20\ q^{3} + 5\ q^{4} + q^{5}$

$F_{7} = 51 + 126\ q + 135\ q^{2} + 80\ q^{3} + 30\ q^{4} + 6\ q^{5} + q^{6}$

Description

The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.

Map

inverse zeta map

Description

The inverse zeta map on Dyck paths.
See its inverse, the zeta map Mp00030zeta map, for the definition and details.