- FindStat

Your data matches 44 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000159
(load all 2 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([(0,1)],2)
 => [1]
 => 1 = 0 + 1
([(1,2)],3)
 => [1]
 => 1 = 0 + 1
([(0,2),(1,2)],3)
 => [1,1]
 => 1 = 0 + 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(2,3)],4)
 => [1]
 => 1 = 0 + 1
([(1,3),(2,3)],4)
 => [1,1]
 => 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => 1 = 0 + 1
([(0,3),(1,2)],4)
 => [1,1]
 => 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => 1 = 0 + 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => 1 = 0 + 1
([(3,4)],5)
 => [1]
 => 1 = 0 + 1
([(2,4),(3,4)],5)
 => [1,1]
 => 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => 1 = 0 + 1
([(1,4),(2,3)],5)
 => [1,1]
 => 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => 1 = 0 + 1
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => 1 = 0 + 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => 1 = 0 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 2 = 1 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => 1 = 0 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => 1 = 0 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => 2 = 1 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => 1 = 0 + 1
([(4,5)],6)
 => [1]
 => 1 = 0 + 1
([(3,5),(4,5)],6)
 => [1,1]
 => 1 = 0 + 1
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => 1 = 0 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => 1 = 0 + 1

Description

The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.

Matching statistic: St000292
(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
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([(0,1)],2)
 => [1]
 => 10 => 0
([(1,2)],3)
 => [1]
 => 10 => 0
([(0,2),(1,2)],3)
 => [1,1]
 => 110 => 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1000 => 0
([(2,3)],4)
 => [1]
 => 10 => 0
([(1,3),(2,3)],4)
 => [1,1]
 => 110 => 0
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => 1110 => 0
([(0,3),(1,2)],4)
 => [1,1]
 => 110 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => 1110 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 1000 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 10010 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 10000 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 100000 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => 1000000 => 0
([(3,4)],5)
 => [1]
 => 10 => 0
([(2,4),(3,4)],5)
 => [1,1]
 => 110 => 0
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => 1110 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => 11110 => 0
([(1,4),(2,3)],5)
 => [1,1]
 => 110 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => 1110 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => 1110 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 1000 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => 11110 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 10010 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => 10000 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => 100010 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => 1000010 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => 1000000 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 10000000 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => 11110 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 10010 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => 11000 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => 100000 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1000000 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 10000000 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => 1000010 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1000000 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => 10000010 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => 100000000 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => 10000000 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => 100000000 => 0
([(4,5)],6)
 => [1]
 => 10 => 0
([(3,5),(4,5)],6)
 => [1,1]
 => 110 => 0
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => 1110 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => 11110 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => 111110 => 0

Description

The number of ascents of a binary word.

Matching statistic: St001036
(load all 7 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([(0,1)],2)
 => [1]
 => [1,0]
 => 0
([(1,2)],3)
 => [1]
 => [1,0]
 => 0
([(0,2),(1,2)],3)
 => [1,1]
 => [1,1,0,0]
 => 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(2,3)],4)
 => [1]
 => [1,0]
 => 0
([(1,3),(2,3)],4)
 => [1,1]
 => [1,1,0,0]
 => 0
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(0,3),(1,2)],4)
 => [1,1]
 => [1,1,0,0]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 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]
 => 0
([(3,4)],5)
 => [1]
 => [1,0]
 => 0
([(2,4),(3,4)],5)
 => [1,1]
 => [1,1,0,0]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
([(1,4),(2,3)],5)
 => [1,1]
 => [1,1,0,0]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 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
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 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
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 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
([(0,4),(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
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 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]
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 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
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 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]
 => 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]
 => 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
([(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]
 => 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
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
([(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]
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
([(4,5)],6)
 => [1]
 => [1,0]
 => 0
([(3,5),(4,5)],6)
 => [1,1]
 => [1,1,0,0]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
([(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]
 => 0

Description

The number of inner corners of the parallelogram polyomino associated with the Dyck path.

Matching statistic: St001037
(load all 2 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([(0,1)],2)
 => [1]
 => [1,0]
 => 0
([(1,2)],3)
 => [1]
 => [1,0]
 => 0
([(0,2),(1,2)],3)
 => [1,1]
 => [1,1,0,0]
 => 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(2,3)],4)
 => [1]
 => [1,0]
 => 0
([(1,3),(2,3)],4)
 => [1,1]
 => [1,1,0,0]
 => 0
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(0,3),(1,2)],4)
 => [1,1]
 => [1,1,0,0]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 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]
 => 0
([(3,4)],5)
 => [1]
 => [1,0]
 => 0
([(2,4),(3,4)],5)
 => [1,1]
 => [1,1,0,0]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
([(1,4),(2,3)],5)
 => [1,1]
 => [1,1,0,0]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,0,1,0,1,0]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 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
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 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
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 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
([(0,4),(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
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 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]
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 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
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 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]
 => 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]
 => 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
([(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]
 => 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
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
([(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]
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
([(4,5)],6)
 => [1]
 => [1,0]
 => 0
([(3,5),(4,5)],6)
 => [1,1]
 => [1,1,0,0]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
([(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]
 => 0

Description

The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.

Matching statistic: St000291
(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
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([(0,1)],2)
 => [1]
 => 10 => 1 = 0 + 1
([(1,2)],3)
 => [1]
 => 10 => 1 = 0 + 1
([(0,2),(1,2)],3)
 => [1,1]
 => 110 => 1 = 0 + 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1000 => 1 = 0 + 1
([(2,3)],4)
 => [1]
 => 10 => 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 => 1 = 0 + 1
([(0,3),(1,2)],4)
 => [1,1]
 => 110 => 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => 1110 => 1 = 0 + 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 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 10000 => 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 100000 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => 1000000 => 1 = 0 + 1
([(3,4)],5)
 => [1]
 => 10 => 1 = 0 + 1
([(2,4),(3,4)],5)
 => [1,1]
 => 110 => 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => 1110 => 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => 11110 => 1 = 0 + 1
([(1,4),(2,3)],5)
 => [1,1]
 => 110 => 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => 1110 => 1 = 0 + 1
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => 1110 => 1 = 0 + 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 => 1 = 0 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 10010 => 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => 10000 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => 100010 => 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 2 = 1 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => 1000010 => 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => 1000000 => 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 10000000 => 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => 11110 => 1 = 0 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 10010 => 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => 11000 => 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => 100000 => 1 = 0 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1000000 => 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 10000000 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => 1000010 => 2 = 1 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1000000 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => 10000010 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => 100000000 => 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => 10000000 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => 100000000 => 1 = 0 + 1
([(4,5)],6)
 => [1]
 => 10 => 1 = 0 + 1
([(3,5),(4,5)],6)
 => [1,1]
 => 110 => 1 = 0 + 1
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => 1110 => 1 = 0 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => 11110 => 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => 111110 => 1 = 0 + 1

Description

The number of descents of a binary word.

Matching statistic: St000390
(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
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([(0,1)],2)
 => [1]
 => 10 => 1 = 0 + 1
([(1,2)],3)
 => [1]
 => 10 => 1 = 0 + 1
([(0,2),(1,2)],3)
 => [1,1]
 => 110 => 1 = 0 + 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1000 => 1 = 0 + 1
([(2,3)],4)
 => [1]
 => 10 => 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 => 1 = 0 + 1
([(0,3),(1,2)],4)
 => [1,1]
 => 110 => 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => 1110 => 1 = 0 + 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 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 10000 => 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 100000 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => 1000000 => 1 = 0 + 1
([(3,4)],5)
 => [1]
 => 10 => 1 = 0 + 1
([(2,4),(3,4)],5)
 => [1,1]
 => 110 => 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => 1110 => 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => 11110 => 1 = 0 + 1
([(1,4),(2,3)],5)
 => [1,1]
 => 110 => 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => 1110 => 1 = 0 + 1
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => 1110 => 1 = 0 + 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 => 1 = 0 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 10010 => 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => 10000 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => 100010 => 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 2 = 1 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => 1000010 => 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => 1000000 => 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 10000000 => 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => 11110 => 1 = 0 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 10010 => 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => 11000 => 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => 100000 => 1 = 0 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1000000 => 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 10000000 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => 1000010 => 2 = 1 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1000000 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => 10000010 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => 100000000 => 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => 10000000 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => 100000000 => 1 = 0 + 1
([(4,5)],6)
 => [1]
 => 10 => 1 = 0 + 1
([(3,5),(4,5)],6)
 => [1,1]
 => 110 => 1 = 0 + 1
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => 1110 => 1 = 0 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => 11110 => 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => 111110 => 1 = 0 + 1

Description

The number of runs of ones in a binary word.

Matching statistic: St000318
(load all 2 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

([(0,1)],2)
 => [1]
 => 2 = 0 + 2
([(1,2)],3)
 => [1]
 => 2 = 0 + 2
([(0,2),(1,2)],3)
 => [1,1]
 => 2 = 0 + 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 2 = 0 + 2
([(2,3)],4)
 => [1]
 => 2 = 0 + 2
([(1,3),(2,3)],4)
 => [1,1]
 => 2 = 0 + 2
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => 2 = 0 + 2
([(0,3),(1,2)],4)
 => [1,1]
 => 2 = 0 + 2
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => 2 = 0 + 2
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 2 = 0 + 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 3 = 1 + 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 2 = 0 + 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 2 = 0 + 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => 2 = 0 + 2
([(3,4)],5)
 => [1]
 => 2 = 0 + 2
([(2,4),(3,4)],5)
 => [1,1]
 => 2 = 0 + 2
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => 2 = 0 + 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => 2 = 0 + 2
([(1,4),(2,3)],5)
 => [1,1]
 => 2 = 0 + 2
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => 2 = 0 + 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => 2 = 0 + 2
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 2 = 0 + 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => 2 = 0 + 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 3 = 1 + 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 3 = 1 + 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => 2 = 0 + 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => 3 = 1 + 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 2 = 0 + 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 3 = 1 + 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => 3 = 1 + 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => 2 = 0 + 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 2 = 0 + 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => 2 = 0 + 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 3 = 1 + 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => 3 = 1 + 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => 2 = 0 + 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => 2 = 0 + 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => 2 = 0 + 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 2 = 0 + 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => 3 = 1 + 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 2 = 0 + 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => 3 = 1 + 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => 2 = 0 + 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => 2 = 0 + 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => 2 = 0 + 2
([(4,5)],6)
 => [1]
 => 2 = 0 + 2
([(3,5),(4,5)],6)
 => [1,1]
 => 2 = 0 + 2
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => 2 = 0 + 2
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => 2 = 0 + 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => 2 = 0 + 2
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => [6,5]
 => ? = 1 + 2
([(0,4),(0,5),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,5]
 => ? = 1 + 2
([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,6]
 => ? = 0 + 2

Description

The number of addable cells of the Ferrers diagram of an integer partition.

Matching statistic: St001124
(load all 2 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%

Values

([(0,1)],2)
 => [1]
 => ? = 0
([(1,2)],3)
 => [1]
 => ? = 0
([(0,2),(1,2)],3)
 => [1,1]
 => 0
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 0
([(2,3)],4)
 => [1]
 => ? = 0
([(1,3),(2,3)],4)
 => [1,1]
 => 0
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => 0
([(0,3),(1,2)],4)
 => [1,1]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => 0
([(3,4)],5)
 => [1]
 => ? = 0
([(2,4),(3,4)],5)
 => [1,1]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => 0
([(1,4),(2,3)],5)
 => [1,1]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => 0
([(4,5)],6)
 => [1]
 => ? = 0
([(3,5),(4,5)],6)
 => [1,1]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => 0
([(2,5),(3,4)],6)
 => [1,1]
 => 0
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => 0
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [3]
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => 0
([(5,6)],7)
 => [1]
 => ? = 0
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => [6,5]
 => ? = 1
([(0,4),(0,5),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,5]
 => ? = 1
([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,6]
 => ? = 0

Description

The multiplicity of the standard representation in the Kronecker square corresponding to a partition. The Kronecker coefficient is the multiplicity

$g_{\mu,\nu}^\lambda$ of the Specht module

$S^\lambda$ in

$S^\mu\otimes S^\nu$ :

$S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda$ This statistic records the Kronecker coefficient

$g_{\lambda,\lambda}^{(n-1)1}$ , for

$\lambda\vdash n > 1$ . For

$n\leq1$ the statistic is undefined. It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.

Matching statistic: St000386
(load all 2 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%

Values

([(0,1)],2)
 => [1]
 => [1,0]
 => [1,0]
 => 0
([(1,2)],3)
 => [1]
 => [1,0]
 => [1,0]
 => 0
([(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
([(2,3)],4)
 => [1]
 => [1,0]
 => [1,0]
 => 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]
 => [1,0,1,0,1,0]
 => 0
([(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]
 => [1,0,1,0,1,0]
 => 0
([(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]
 => [1,1,1,0,0,0,1,0]
 => 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
([(3,4)],5)
 => [1]
 => [1,0]
 => [1,0]
 => 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]
 => [1,0,1,0,1,0]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
([(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]
 => [1,0,1,0,1,0]
 => 0
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 0
([(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]
 => [1,0,1,0,1,0,1,0]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 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]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
([(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]
 => [1,1,1,1,0,0,0,0,1,0]
 => 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]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
([(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]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 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]
 => [1,0,1,0,1,0,1,0]
 => 0
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 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]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
([(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]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 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]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => 0
([(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
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => 0
([(4,5)],6)
 => [1]
 => [1,0]
 => [1,0]
 => 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]
 => [1,0,1,0,1,0]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
([(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]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,4),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,5),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,3),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,2),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(0,6),(1,4),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,4),(2,3),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,4),(1,4),(1,6),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(0,6),(1,5),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,4),(1,2),(1,6),(2,5),(3,5),(3,6),(4,6),(5,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,4),(1,3),(2,5),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(0,6),(1,2),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,3,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => ? = 2
([(0,6),(1,5),(2,3),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,3,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => ? = 2
([(0,3),(1,4),(1,5),(2,4),(2,6),(3,5),(4,6),(5,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,2),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(0,3),(1,2),(1,5),(2,4),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,5),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,3),(1,3),(1,4),(2,5),(2,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5)],7)
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
([(0,5),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => [6,3,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => ? = 2
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => [6,3,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => ? = 2

Description

The number of factors DDU in a Dyck path.

Matching statistic: St000053
(load all 4 compositions to match this statistic)

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%

Values

([(0,1)],2)
 => [1]
 => [1,0,1,0]
 => 1 = 0 + 1
([(1,2)],3)
 => [1]
 => [1,0,1,0]
 => 1 = 0 + 1
([(0,2),(1,2)],3)
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
([(2,3)],4)
 => [1]
 => [1,0,1,0]
 => 1 = 0 + 1
([(1,3),(2,3)],4)
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
([(0,3),(1,2)],4)
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 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 = 0 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1 = 0 + 1
([(3,4)],5)
 => [1]
 => [1,0,1,0]
 => 1 = 0 + 1
([(2,4),(3,4)],5)
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
([(1,4),(2,3)],5)
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 1 + 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 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 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 = 0 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 2 = 1 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(4,5)],6)
 => [1]
 => [1,0,1,0]
 => 1 = 0 + 1
([(3,5),(4,5)],6)
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
([(2,5),(3,4)],6)
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 1 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 1 + 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 1 + 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 1 + 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 1 + 1
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 1 + 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 1 + 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 1 + 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 1 + 1
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 1 + 1
([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(0,6),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 1 + 1
([(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 1 + 1
([(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 1 + 1
([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 1 + 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 1 + 1
([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 1 + 1
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 1 + 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
([(1,2),(1,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1

Description

The number of valleys of the Dyck path.

The following 34 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000024The number of double up and double down steps of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001487The number of inner corners of a skew partition. St000068The number of minimal elements in a poset. St001846The number of elements which do not have a complement in the lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001720The minimal length of a chain of small intervals in a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St000181The number of connected components of the Hasse diagram for the poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000907The number of maximal antichains of minimal length in a poset.