- FindStat

Identifier

Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001244: Dyck paths ⟶ ℤ (values match St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

([(3,4)],5) => [1] => [1,0,1,0] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(4,5)],6) => [1] => [1,0,1,0] => 1

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

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

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

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

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

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

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

([(3,4),(3,5),(4,5)],6) => [3] => [1,1,1,0,0,0,1,0] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(5,6)],7) => [1] => [1,0,1,0] => 1

([(4,6),(5,6)],7) => [2] => [1,1,0,0,1,0] => 1

([(3,6),(4,6),(5,6)],7) => [3] => [1,1,1,0,0,0,1,0] => 1

([(2,6),(3,6),(4,6),(5,6)],7) => [4] => [1,1,1,1,0,0,0,0,1,0] => 1

([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1

([(3,6),(4,5)],7) => [1,1] => [1,0,1,1,0,0] => 1

([(3,6),(4,5),(5,6)],7) => [3] => [1,1,1,0,0,0,1,0] => 1

([(2,3),(4,6),(5,6)],7) => [2,1] => [1,0,1,0,1,0] => 1

([(4,5),(4,6),(5,6)],7) => [3] => [1,1,1,0,0,0,1,0] => 1

([(2,6),(3,6),(4,5),(5,6)],7) => [4] => [1,1,1,1,0,0,0,0,1,0] => 1

([(1,2),(3,6),(4,6),(5,6)],7) => [3,1] => [1,1,0,1,0,0,1,0] => 1

([(3,6),(4,5),(4,6),(5,6)],7) => [4] => [1,1,1,1,0,0,0,0,1,0] => 1

([(1,6),(2,6),(3,6),(4,5),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1

([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => 1

([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1

([(3,5),(3,6),(4,5),(4,6)],7) => [4] => [1,1,1,1,0,0,0,0,1,0] => 1

([(1,6),(2,6),(3,5),(4,5)],7) => [2,2] => [1,1,0,0,1,1,0,0] => 1

([(2,6),(3,4),(3,5),(4,6),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1

([(1,6),(2,6),(3,4),(4,5),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1

([(0,6),(1,6),(2,6),(3,5),(4,5)],7) => [3,2] => [1,1,0,0,1,0,1,0] => 1

([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1

([(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1

([(1,6),(2,6),(3,5),(4,5),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1

([(1,6),(2,5),(3,4)],7) => [1,1,1] => [1,0,1,1,1,0,0,0] => 1

([(2,6),(3,5),(4,5),(4,6)],7) => [4] => [1,1,1,1,0,0,0,0,1,0] => 1

([(1,2),(3,6),(4,5),(5,6)],7) => [3,1] => [1,1,0,1,0,0,1,0] => 1

([(0,3),(1,2),(4,6),(5,6)],7) => [2,1,1] => [1,0,1,1,0,1,0,0] => 1

([(2,3),(4,5),(4,6),(5,6)],7) => [3,1] => [1,1,0,1,0,0,1,0] => 1

([(1,6),(2,5),(3,4),(4,6),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1

([(0,1),(2,6),(3,6),(4,5),(5,6)],7) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => 1

([(2,5),(3,4),(3,6),(4,6),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1

([(1,2),(3,6),(4,5),(4,6),(5,6)],7) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => 1

([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => 1

([(2,5),(2,6),(3,4),(3,6),(4,5)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1

([(1,6),(2,5),(3,4),(3,5),(4,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1

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searching the database for the individual values of this statistic

Description

The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path.
For the projective dimension see St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. and for 1-regularity see St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra.. After applying the inverse zeta map Mp00032inverse zeta map, this statistic matches the number of rises of length at least 2 St000659The number of rises of length at least 2 of a Dyck path..

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.

Map

to edge-partition of connected components

Description

Sends a graph to the partition recording the number of edges in its connected components.