- FindStat

Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001502

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001502: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. We use the code below to translate them to Dyck paths. The algebras where the statistic returns 0 are exactly the higher Auslander algebras and are of special interest. It seems like they are counted by the number of divisors function.

Matching statistic: St001491
(load all 4 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 40%

Values

([(0,2),(1,2)],3)
 => [1,1]
 => 110 => 1 = 0 + 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1000 => 1 = 0 + 1
([(1,3),(2,3)],4)
 => [1,1]
 => 110 => 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => 1110 => 2 = 1 + 1
([(0,3),(1,2)],4)
 => [1,1]
 => 110 => 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => 1110 => 2 = 1 + 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 1000 => 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 10010 => ? = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 10000 => ? = 0 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 100000 => ? = 0 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => 1000000 => ? = 0 + 1
([(2,4),(3,4)],5)
 => [1,1]
 => 110 => 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => 1110 => 2 = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => 11110 => ? = 2 + 1
([(1,4),(2,3)],5)
 => [1,1]
 => 110 => 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => 1110 => 2 = 1 + 1
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => 1110 => 2 = 1 + 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 1000 => 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => 11110 => ? = 2 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 10010 => ? = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => ? = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => 10000 => ? = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => 100010 => ? = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => ? = 0 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => ? = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => 1000010 => ? = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => 1000000 => ? = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 10000000 => ? = 0 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => 11110 => ? = 2 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 10010 => ? = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => ? = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => 11000 => ? = 3 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => 100000 => ? = 0 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1000000 => ? = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 10000000 => ? = 0 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => 1000010 => ? = 1 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1000000 => ? = 0 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => 10000010 => ? = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => 10000000 => ? = 0 + 1
([(3,5),(4,5)],6)
 => [1,1]
 => 110 => 1 = 0 + 1
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => 1110 => 2 = 1 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => 11110 => ? = 2 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => 111110 => ? = 1 + 1
([(2,5),(3,4)],6)
 => [1,1]
 => 110 => 1 = 0 + 1
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => 1110 => 2 = 1 + 1
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => 1110 => 2 = 1 + 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => 1000 => 1 = 0 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => 11110 => ? = 2 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => 11110 => ? = 2 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => 10010 => ? = 1 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => 111110 => ? = 1 + 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => 100110 => ? = 2 + 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => 1001110 => ? = 1 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => 10000 => ? = 0 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [1,1,1,1]
 => 11110 => ? = 2 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1]
 => 100010 => ? = 1 + 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => 111110 => ? = 1 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => 100000 => ? = 0 + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => 100110 => ? = 2 + 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [1,1,1,1,1]
 => 111110 => ? = 1 + 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,1,1]
 => 1000110 => ? = 2 + 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1000010 => ? = 1 + 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => 1001110 => ? = 1 + 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => 10000110 => ? = 2 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => 1000000 => ? = 0 + 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => 1000110 => ? = 2 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,1]
 => 10000010 => ? = 1 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => 10000000 => ? = 0 + 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1,1]
 => 10000110 => ? = 2 + 1
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => 1110 => 2 = 1 + 1
([(4,6),(5,6)],7)
 => [1,1]
 => 110 => 1 = 0 + 1
([(3,6),(4,6),(5,6)],7)
 => [1,1,1]
 => 1110 => 2 = 1 + 1
([(3,6),(4,5)],7)
 => [1,1]
 => 110 => 1 = 0 + 1
([(3,6),(4,5),(5,6)],7)
 => [1,1,1]
 => 1110 => 2 = 1 + 1
([(2,3),(4,6),(5,6)],7)
 => [1,1,1]
 => 1110 => 2 = 1 + 1
([(4,5),(4,6),(5,6)],7)
 => [3]
 => 1000 => 1 = 0 + 1
([(1,6),(2,5),(3,4)],7)
 => [1,1,1]
 => 1110 => 2 = 1 + 1

Description

The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let

$A_n=K[x]/(x^n)$ . We associate to a nonempty subset S of an (n-1)-set the module

$M_S$ , which is the direct sum of

$A_n$ -modules with indecomposable non-projective direct summands of dimension

$i$ when

$i$ is in

$S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of

$M_S$ . We decode the subset as a binary word so that for example the subset

$S=\{1,3 \}$ of

$\{1,2,3 \}$ is decoded as 101.