- FindStat

Your data matches 30 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000933
(load all 2 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000933: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of multipartitions of sizes given by an integer partition. This is, for

$\lambda = (\lambda_1,\ldots,\lambda_n)$ , this is the number of

$n$ -tuples

$(\lambda^{(1)},\ldots,\lambda^{(n)})$ of partitions

$\lambda^{(i)}$ such that

$\lambda^{(i)} \vdash \lambda_i$ .

Matching statistic: St000609
(load all 2 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000609: Set partitions ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%

Values

([],3)
 => [1,1,1]
 => [[1],[2],[3]]
 => {{1},{2},{3}}
 => 0 = 1 - 1
([],4)
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => 0 = 1 - 1
([(2,3)],4)
 => [2,1,1]
 => [[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 1 = 2 - 1
([],5)
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => {{1},{2},{3},{4},{5}}
 => 0 = 1 - 1
([(3,4)],5)
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => {{1,5},{2},{3},{4}}
 => 0 = 1 - 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => {{1,3},{2,5},{4}}
 => 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,5],[3,4]]
 => {{1,2,5},{3,4}}
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,5],[3,4]]
 => {{1,2,5},{3,4}}
 => 1 = 2 - 1
([],6)
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => {{1},{2},{3},{4},{5},{6}}
 => 0 = 1 - 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => {{1,6},{2},{3},{4},{5}}
 => 0 = 1 - 1
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => {{1,5,6},{2},{3},{4}}
 => 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => {{1,4,5,6},{2},{3}}
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => {{1,4},{2,6},{3},{5}}
 => 1 = 2 - 1
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => {{1,4,5,6},{2},{3}}
 => 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => {{1,3,6},{2,5},{4}}
 => 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => {{1,5,6},{2},{3},{4}}
 => 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => {{1,2,5,6},{3,4}}
 => 1 = 2 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => {{1,4,5,6},{2},{3}}
 => 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => {{1,4,5,6},{2},{3}}
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => {{1,2,3},{4,5,6}}
 => 2 = 3 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => {{1,4,5,6},{2},{3}}
 => 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => {{1,2},{3,4},{5,6}}
 => 3 = 4 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => {{1,2,5,6},{3,4}}
 => 1 = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => {{1,3,6},{2,5},{4}}
 => 1 = 2 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => {{1,2,5,6},{3,4}}
 => 1 = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => {{1,2,5,6},{3,4}}
 => 1 = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => {{1,2,3},{4,5,6}}
 => 2 = 3 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => {{1,2,5,6},{3,4}}
 => 1 = 2 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => {{1,4,5,6},{2},{3}}
 => 0 = 1 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => {{1,2,3},{4,5,6}}
 => 2 = 3 - 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => {{1,2,5,6},{3,4}}
 => 1 = 2 - 1
([],7)
 => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => {{1},{2},{3},{4},{5},{6},{7}}
 => 0 = 1 - 1
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => {{1,7},{2},{3},{4},{5},{6}}
 => 0 = 1 - 1
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => {{1,6,7},{2},{3},{4},{5}}
 => 0 = 1 - 1
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => {{1,5,6,7},{2},{3},{4}}
 => 0 = 1 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => {{1,4,5,6,7},{2},{3}}
 => 0 = 1 - 1
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => {{1,5},{2,7},{3},{4},{6}}
 => 1 = 2 - 1
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => {{1,5,6,7},{2},{3},{4}}
 => 0 = 1 - 1
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => {{1,4,7},{2,6},{3},{5}}
 => 1 = 2 - 1
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => {{1,6,7},{2},{3},{4},{5}}
 => 0 = 1 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => {{1,4,5,6,7},{2},{3}}
 => 0 = 1 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => {{1,3,6,7},{2,5},{4}}
 => 1 = 2 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => {{1,5,6,7},{2},{3},{4}}
 => 0 = 1 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => {{1,2,5,6,7},{3,4}}
 => 1 = 2 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => {{1,4,5,6,7},{2},{3}}
 => 0 = 1 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => {{1,5,6,7},{2},{3},{4}}
 => 0 = 1 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => [3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => {{1,3,4},{2,6,7},{5}}
 => 2 = 3 - 1
([(0,1),(0,2),(1,2),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => {{1,2,3,7,8},{4,5,6}}
 => ? = 3 - 1
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,1,1,1,1]
 => [[1,6,7,8],[2],[3],[4],[5]]
 => {{1,6,7,8},{2},{3},{4},{5}}
 => ? = 1 - 1
([(3,7),(4,7),(5,7),(6,7)],8)
 => [5,1,1,1]
 => [[1,5,6,7,8],[2],[3],[4]]
 => {{1,5,6,7,8},{2},{3},{4}}
 => ? = 1 - 1
([],8)
 => [1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => {{1},{2},{3},{4},{5},{6},{7},{8}}
 => ? = 1 - 1
([(4,7),(5,6)],8)
 => [2,2,1,1,1,1]
 => [[1,6],[2,8],[3],[4],[5],[7]]
 => {{1,6},{2,8},{3},{4},{5},{7}}
 => ? = 2 - 1
([(4,7),(5,6),(6,7)],8)
 => [4,1,1,1,1]
 => [[1,6,7,8],[2],[3],[4],[5]]
 => {{1,6,7,8},{2},{3},{4},{5}}
 => ? = 1 - 1
([(4,6),(4,7),(5,6),(5,7)],8)
 => [4,1,1,1,1]
 => [[1,6,7,8],[2],[3],[4],[5]]
 => {{1,6,7,8},{2},{3},{4},{5}}
 => ? = 1 - 1
([(2,7),(3,7),(4,6),(5,6)],8)
 => [3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => {{1,4,5},{2,7,8},{3},{6}}
 => ? = 3 - 1
([(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
 => [5,1,1,1]
 => [[1,5,6,7,8],[2],[3],[4]]
 => {{1,5,6,7,8},{2},{3},{4}}
 => ? = 1 - 1
([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
 => [3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => {{1,4,5},{2,7,8},{3},{6}}
 => ? = 3 - 1
([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,2,1,1]
 => [[1,4,7,8],[2,6],[3],[5]]
 => {{1,4,7,8},{2,6},{3},{5}}
 => ? = 2 - 1
([(1,3),(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,3,1]
 => [[1,3,4,8],[2,6,7],[5]]
 => {{1,3,4,8},{2,6,7},{5}}
 => ? = 3 - 1
([(1,2),(1,3),(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,3,1]
 => [[1,3,4,8],[2,6,7],[5]]
 => {{1,3,4,8},{2,6,7},{5}}
 => ? = 3 - 1
([(0,7),(1,6),(2,5),(3,4)],8)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => {{1,2},{3,4},{5,6},{7,8}}
 => ? = 8 - 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
 => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => {{1,2,7,8},{3,4},{5,6}}
 => ? = 4 - 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => {{1,2,7,8},{3,4},{5,6}}
 => ? = 4 - 1
([(0,2),(0,3),(1,2),(1,3),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 5 - 1
([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 5 - 1
([(0,6),(0,7),(1,4),(1,5),(2,4),(2,5),(3,6),(3,7),(4,5),(6,7)],8)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 5 - 1
([(2,7),(3,5),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
 => [7,1,1]
 => [[1,4,5,6,7,8,9],[2],[3]]
 => {{1,4,5,6,7,8,9},{2},{3}}
 => ? = 1 - 1
([(2,10),(3,6),(3,10),(3,13),(4,9),(4,11),(4,14),(4,15),(5,12),(5,13),(5,14),(5,15),(6,12),(6,14),(6,15),(7,8),(7,9),(7,12),(7,14),(7,15),(8,11),(8,13),(8,14),(8,15),(9,11),(9,13),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(13,14),(13,15)],16)
 => [14,1,1]
 => [[1,4,5,6,7,8,9,10,11,12,13,14,15,16],[2],[3]]
 => ?
 => ? = 1 - 1
([(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,11),(4,5),(4,7),(4,10),(5,6),(5,9),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
 => [10,1,1]
 => [[1,4,5,6,7,8,9,10,11,12],[2],[3]]
 => ?
 => ? = 1 - 1
([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => {{1,2,7,8},{3,4},{5,6}}
 => ? = 4 - 1
([(0,2),(0,3),(1,2),(1,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 5 - 1
([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4),(5,6),(5,7),(6,7)],8)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 5 - 1
([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => [6,1,1,1]
 => [[1,5,6,7,8,9],[2],[3],[4]]
 => {{1,5,6,7,8,9},{2},{3},{4}}
 => ? = 1 - 1
([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => [6,1,1,1,1]
 => [[1,6,7,8,9,10],[2],[3],[4],[5]]
 => {{1,6,7,8,9,10},{2},{3},{4},{5}}
 => ? = 1 - 1
([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
 => [6,1,1,1]
 => [[1,5,6,7,8,9],[2],[3],[4]]
 => {{1,5,6,7,8,9},{2},{3},{4}}
 => ? = 1 - 1
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
 => [9,1,1,1]
 => [[1,5,6,7,8,9,10,11,12],[2],[3],[4]]
 => ?
 => ? = 1 - 1
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
 => [11,1,1,1]
 => [[1,5,6,7,8,9,10,11,12,13,14],[2],[3],[4]]
 => ?
 => ? = 1 - 1
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
 => [11,1,1,1,1]
 => [[1,6,7,8,9,10,11,12,13,14,15],[2],[3],[4],[5]]
 => ?
 => ? = 1 - 1
([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
 => [7,1,1,1]
 => [[1,5,6,7,8,9,10],[2],[3],[4]]
 => {{1,5,6,7,8,9,10},{2},{3},{4}}
 => ? = 1 - 1
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
 => [11,1,1,1]
 => [[1,5,6,7,8,9,10,11,12,13,14],[2],[3],[4]]
 => ?
 => ? = 1 - 1
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => [13,1,1,1]
 => [[1,5,6,7,8,9,10,11,12,13,14,15,16],[2],[3],[4]]
 => ?
 => ? = 1 - 1
([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => [9,1,1]
 => [[1,4,5,6,7,8,9,10,11],[2],[3]]
 => ?
 => ? = 1 - 1
([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
 => [11,1,1]
 => [[1,4,5,6,7,8,9,10,11,12,13],[2],[3]]
 => ?
 => ? = 1 - 1
([(2,9),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
 => [8,1,1]
 => [[1,4,5,6,7,8,9,10],[2],[3]]
 => {{1,4,5,6,7,8,9,10},{2},{3}}
 => ? = 1 - 1
([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
 => [13,1,1]
 => [[1,4,5,6,7,8,9,10,11,12,13,14,15],[2],[3]]
 => ?
 => ? = 1 - 1
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
 => [8,1,1,1]
 => [[1,5,6,7,8,9,10,11],[2],[3],[4]]
 => ?
 => ? = 1 - 1
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
 => [12,1,1,1]
 => [[1,5,6,7,8,9,10,11,12,13,14,15],[2],[3],[4]]
 => ?
 => ? = 1 - 1
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
 => [10,1,1,1]
 => [[1,5,6,7,8,9,10,11,12,13],[2],[3],[4]]
 => ?
 => ? = 1 - 1
([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
 => [6,1,1,1,1,1]
 => [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
 => ?
 => ? = 1 - 1
([(2,9),(2,13),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,7),(5,8),(5,9),(5,13),(6,7),(6,10),(6,11),(6,12),(6,13),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
 => [12,1,1]
 => [[1,4,5,6,7,8,9,10,11,12,13,14],[2],[3]]
 => ?
 => ? = 1 - 1
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
 => [8,1,1,1]
 => [[1,5,6,7,8,9,10,11],[2],[3],[4]]
 => ?
 => ? = 1 - 1
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
 => [10,1,1,1]
 => [[1,5,6,7,8,9,10,11,12,13],[2],[3],[4]]
 => ?
 => ? = 1 - 1
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
 => [10,1,1,1,1]
 => [[1,6,7,8,9,10,11,12,13,14],[2],[3],[4],[5]]
 => ?
 => ? = 1 - 1
([(2,4),(2,13),(3,11),(3,12),(3,13),(4,11),(4,12),(5,8),(5,9),(5,10),(5,13),(6,8),(6,9),(6,10),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
 => [12,1,1]
 => [[1,4,5,6,7,8,9,10,11,12,13,14],[2],[3]]
 => ?
 => ? = 1 - 1
([(2,4),(2,15),(3,9),(3,15),(3,16),(4,9),(4,16),(5,11),(5,12),(5,13),(5,16),(6,11),(6,12),(6,13),(6,16),(7,10),(7,14),(7,15),(7,16),(8,11),(8,12),(8,13),(8,14),(8,16),(9,10),(9,14),(9,15),(10,11),(10,12),(10,13),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,16),(15,16)],17)
 => [15,1,1]
 => [[1,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[2],[3]]
 => ?
 => ? = 1 - 1
([(3,8),(4,6),(4,9),(5,7),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
 => [7,1,1,1]
 => [[1,5,6,7,8,9,10],[2],[3],[4]]
 => {{1,5,6,7,8,9,10},{2},{3},{4}}
 => ? = 1 - 1
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
 => [11,1,1,1]
 => [[1,5,6,7,8,9,10,11,12,13,14],[2],[3],[4]]
 => ?
 => ? = 1 - 1

Description

The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.

Matching statistic: St000589

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000589: Set partitions ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 67%

Values

([],3)
 => [1,1,1]
 => [[1],[2],[3]]
 => {{1},{2},{3}}
 => 0 = 1 - 1
([],4)
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => 0 = 1 - 1
([(2,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 1 = 2 - 1
([],5)
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => {{1},{2},{3},{4},{5}}
 => 0 = 1 - 1
([(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 0 = 1 - 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 1 = 2 - 1
([],6)
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => {{1},{2},{3},{4},{5},{6}}
 => 0 = 1 - 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => {{1,2},{3},{4},{5},{6}}
 => 0 = 1 - 1
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => {{1,2,3},{4},{5},{6}}
 => 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => {{1,2,3,4},{5},{6}}
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => {{1,2},{3,4},{5},{6}}
 => 1 = 2 - 1
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => {{1,2,3,4},{5},{6}}
 => 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => {{1,2,3},{4,5},{6}}
 => 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => {{1,2,3},{4},{5},{6}}
 => 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => {{1,2,3,4},{5,6}}
 => 1 = 2 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => {{1,2,3,4},{5},{6}}
 => 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => {{1,2,3,4},{5},{6}}
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => {{1,2,3},{4,5,6}}
 => 2 = 3 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => {{1,2,3,4},{5},{6}}
 => 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => {{1,2},{3,4},{5,6}}
 => 3 = 4 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => {{1,2,3,4},{5,6}}
 => 1 = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => {{1,2,3},{4,5},{6}}
 => 1 = 2 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => {{1,2,3,4},{5,6}}
 => 1 = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => {{1,2,3,4},{5,6}}
 => 1 = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => {{1,2,3},{4,5,6}}
 => 2 = 3 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => {{1,2,3,4},{5,6}}
 => 1 = 2 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => {{1,2,3,4},{5},{6}}
 => 0 = 1 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => {{1,2,3},{4,5,6}}
 => 2 = 3 - 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => {{1,2,3,4},{5,6}}
 => 1 = 2 - 1
([],7)
 => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => {{1},{2},{3},{4},{5},{6},{7}}
 => 0 = 1 - 1
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => {{1,2},{3},{4},{5},{6},{7}}
 => 0 = 1 - 1
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => {{1,2,3},{4},{5},{6},{7}}
 => 0 = 1 - 1
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => {{1,2,3,4},{5},{6},{7}}
 => 0 = 1 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => {{1,2,3,4,5},{6},{7}}
 => 0 = 1 - 1
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => {{1,2},{3,4},{5},{6},{7}}
 => 1 = 2 - 1
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => {{1,2,3,4},{5},{6},{7}}
 => 0 = 1 - 1
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => {{1,2,3},{4,5},{6},{7}}
 => 1 = 2 - 1
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => {{1,2,3},{4},{5},{6},{7}}
 => 0 = 1 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => [5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => {{1,2,3,4,5},{6},{7}}
 => 0 = 1 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => {{1,2,3,4},{5,6},{7}}
 => 1 = 2 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => {{1,2,3,4},{5},{6},{7}}
 => 0 = 1 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => {{1,2,3,4,5},{6,7}}
 => 1 = 2 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => {{1,2,3,4,5},{6},{7}}
 => 0 = 1 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => {{1,2,3,4},{5},{6},{7}}
 => 0 = 1 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => [3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => {{1,2,3},{4,5,6},{7}}
 => 2 = 3 - 1
([(0,1),(0,2),(1,2),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => [5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => {{1,2,3,4,5},{6,7,8}}
 => ? = 3 - 1
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8]]
 => {{1,2,3,4},{5},{6},{7},{8}}
 => ? = 1 - 1
([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(3,7),(4,7),(5,7),(6,7)],8)
 => [5,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8]]
 => {{1,2,3,4,5},{6},{7},{8}}
 => ? = 1 - 1
([],8)
 => [1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => {{1},{2},{3},{4},{5},{6},{7},{8}}
 => ? = 1 - 1
([(4,7),(5,6)],8)
 => [2,2,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8]]
 => {{1,2},{3,4},{5},{6},{7},{8}}
 => ? = 2 - 1
([(4,7),(5,6),(6,7)],8)
 => [4,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8]]
 => {{1,2,3,4},{5},{6},{7},{8}}
 => ? = 1 - 1
([(4,6),(4,7),(5,6),(5,7)],8)
 => [4,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8]]
 => {{1,2,3,4},{5},{6},{7},{8}}
 => ? = 1 - 1
([(2,7),(3,7),(4,6),(5,6)],8)
 => [3,3,1,1]
 => [[1,2,3],[4,5,6],[7],[8]]
 => {{1,2,3},{4,5,6},{7},{8}}
 => ? = 3 - 1
([(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
 => [5,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8]]
 => {{1,2,3,4,5},{6},{7},{8}}
 => ? = 1 - 1
([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
 => [3,3,1,1]
 => [[1,2,3],[4,5,6],[7],[8]]
 => {{1,2,3},{4,5,6},{7},{8}}
 => ? = 3 - 1
([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => {{1,2,3,4},{5,6},{7},{8}}
 => ? = 2 - 1
([(2,6),(2,7),(3,4),(3,5),(4,5),(4,7),(5,6),(6,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(2,3),(2,7),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(1,3),(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => {{1,2,3,4},{5,6,7},{8}}
 => ? = 3 - 1
([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(1,2),(1,3),(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => {{1,2,3,4},{5,6,7},{8}}
 => ? = 3 - 1
([(0,7),(1,6),(2,5),(3,4)],8)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => {{1,2},{3,4},{5,6},{7,8}}
 => ? = 8 - 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
 => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 4 - 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 4 - 1
([(0,2),(0,3),(1,2),(1,3),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 5 - 1
([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 5 - 1
([(0,6),(0,7),(1,4),(1,5),(2,4),(2,5),(3,6),(3,7),(4,5),(6,7)],8)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 5 - 1
([(2,7),(3,5),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
 => [7,1,1]
 => [[1,2,3,4,5,6,7],[8],[9]]
 => {{1,2,3,4,5,6,7},{8},{9}}
 => ? = 1 - 1
([(2,10),(3,6),(3,10),(3,13),(4,9),(4,11),(4,14),(4,15),(5,12),(5,13),(5,14),(5,15),(6,12),(6,14),(6,15),(7,8),(7,9),(7,12),(7,14),(7,15),(8,11),(8,13),(8,14),(8,15),(9,11),(9,13),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(13,14),(13,15)],16)
 => [14,1,1]
 => [[1,2,3,4,5,6,7,8,9,10,11,12,13,14],[15],[16]]
 => ?
 => ? = 1 - 1
([(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,11),(4,5),(4,7),(4,10),(5,6),(5,9),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
 => [10,1,1]
 => [[1,2,3,4,5,6,7,8,9,10],[11],[12]]
 => ?
 => ? = 1 - 1
([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 4 - 1
([(0,2),(0,3),(1,2),(1,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 5 - 1
([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4),(5,6),(5,7),(6,7)],8)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 5 - 1
([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => [6,1,1,1]
 => [[1,2,3,4,5,6],[7],[8],[9]]
 => {{1,2,3,4,5,6},{7},{8},{9}}
 => ? = 1 - 1
([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => [6,1,1,1,1]
 => [[1,2,3,4,5,6],[7],[8],[9],[10]]
 => {{1,2,3,4,5,6},{7},{8},{9},{10}}
 => ? = 1 - 1
([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
 => [6,1,1,1]
 => [[1,2,3,4,5,6],[7],[8],[9]]
 => {{1,2,3,4,5,6},{7},{8},{9}}
 => ? = 1 - 1
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
 => [9,1,1,1]
 => [[1,2,3,4,5,6,7,8,9],[10],[11],[12]]
 => ?
 => ? = 1 - 1
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
 => [11,1,1,1]
 => [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14]]
 => ?
 => ? = 1 - 1
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
 => [11,1,1,1,1]
 => [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14],[15]]
 => ?
 => ? = 1 - 1
([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
 => [7,1,1,1]
 => [[1,2,3,4,5,6,7],[8],[9],[10]]
 => {{1,2,3,4,5,6,7},{8},{9},{10}}
 => ? = 1 - 1
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
 => [11,1,1,1]
 => [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14]]
 => ?
 => ? = 1 - 1

Description

The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block.

Matching statistic: St000612

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000612: Set partitions ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 67%

Values

([],3)
 => [1,1,1]
 => [[1],[2],[3]]
 => {{1},{2},{3}}
 => 0 = 1 - 1
([],4)
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => 0 = 1 - 1
([(2,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 1 = 2 - 1
([],5)
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => {{1},{2},{3},{4},{5}}
 => 0 = 1 - 1
([(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 0 = 1 - 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 1 = 2 - 1
([],6)
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => {{1},{2},{3},{4},{5},{6}}
 => 0 = 1 - 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => {{1,2},{3},{4},{5},{6}}
 => 0 = 1 - 1
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => {{1,2,3},{4},{5},{6}}
 => 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => {{1,2,3,4},{5},{6}}
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => {{1,2},{3,4},{5},{6}}
 => 1 = 2 - 1
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => {{1,2,3,4},{5},{6}}
 => 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => {{1,2,3},{4,5},{6}}
 => 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => {{1,2,3},{4},{5},{6}}
 => 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => {{1,2,3,4},{5,6}}
 => 1 = 2 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => {{1,2,3,4},{5},{6}}
 => 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => {{1,2,3,4},{5},{6}}
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => {{1,2,3},{4,5,6}}
 => 2 = 3 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => {{1,2,3,4},{5},{6}}
 => 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => {{1,2},{3,4},{5,6}}
 => 3 = 4 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => {{1,2,3,4},{5,6}}
 => 1 = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => {{1,2,3},{4,5},{6}}
 => 1 = 2 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => {{1,2,3,4},{5,6}}
 => 1 = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => {{1,2,3,4},{5,6}}
 => 1 = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => {{1,2,3},{4,5,6}}
 => 2 = 3 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => {{1,2,3,4},{5,6}}
 => 1 = 2 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => {{1,2,3,4},{5},{6}}
 => 0 = 1 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => {{1,2,3},{4,5,6}}
 => 2 = 3 - 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => {{1,2,3,4},{5,6}}
 => 1 = 2 - 1
([],7)
 => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => {{1},{2},{3},{4},{5},{6},{7}}
 => 0 = 1 - 1
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => {{1,2},{3},{4},{5},{6},{7}}
 => 0 = 1 - 1
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => {{1,2,3},{4},{5},{6},{7}}
 => 0 = 1 - 1
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => {{1,2,3,4},{5},{6},{7}}
 => 0 = 1 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => {{1,2,3,4,5},{6},{7}}
 => 0 = 1 - 1
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => {{1,2},{3,4},{5},{6},{7}}
 => 1 = 2 - 1
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => {{1,2,3,4},{5},{6},{7}}
 => 0 = 1 - 1
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => {{1,2,3},{4,5},{6},{7}}
 => 1 = 2 - 1
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => {{1,2,3},{4},{5},{6},{7}}
 => 0 = 1 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => [5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => {{1,2,3,4,5},{6},{7}}
 => 0 = 1 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => {{1,2,3,4},{5,6},{7}}
 => 1 = 2 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => {{1,2,3,4},{5},{6},{7}}
 => 0 = 1 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => {{1,2,3,4,5},{6,7}}
 => 1 = 2 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => {{1,2,3,4,5},{6},{7}}
 => 0 = 1 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => {{1,2,3,4},{5},{6},{7}}
 => 0 = 1 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => [3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => {{1,2,3},{4,5,6},{7}}
 => 2 = 3 - 1
([(0,1),(0,2),(1,2),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => [5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => {{1,2,3,4,5},{6,7,8}}
 => ? = 3 - 1
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8]]
 => {{1,2,3,4},{5},{6},{7},{8}}
 => ? = 1 - 1
([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(3,7),(4,7),(5,7),(6,7)],8)
 => [5,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8]]
 => {{1,2,3,4,5},{6},{7},{8}}
 => ? = 1 - 1
([],8)
 => [1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => {{1},{2},{3},{4},{5},{6},{7},{8}}
 => ? = 1 - 1
([(4,7),(5,6)],8)
 => [2,2,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8]]
 => {{1,2},{3,4},{5},{6},{7},{8}}
 => ? = 2 - 1
([(4,7),(5,6),(6,7)],8)
 => [4,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8]]
 => {{1,2,3,4},{5},{6},{7},{8}}
 => ? = 1 - 1
([(4,6),(4,7),(5,6),(5,7)],8)
 => [4,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8]]
 => {{1,2,3,4},{5},{6},{7},{8}}
 => ? = 1 - 1
([(2,7),(3,7),(4,6),(5,6)],8)
 => [3,3,1,1]
 => [[1,2,3],[4,5,6],[7],[8]]
 => {{1,2,3},{4,5,6},{7},{8}}
 => ? = 3 - 1
([(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
 => [5,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8]]
 => {{1,2,3,4,5},{6},{7},{8}}
 => ? = 1 - 1
([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
 => [3,3,1,1]
 => [[1,2,3],[4,5,6],[7],[8]]
 => {{1,2,3},{4,5,6},{7},{8}}
 => ? = 3 - 1
([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => {{1,2,3,4},{5,6},{7},{8}}
 => ? = 2 - 1
([(2,6),(2,7),(3,4),(3,5),(4,5),(4,7),(5,6),(6,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(2,3),(2,7),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(1,3),(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => {{1,2,3,4},{5,6,7},{8}}
 => ? = 3 - 1
([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => [6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1 - 1
([(1,2),(1,3),(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => {{1,2,3,4},{5,6,7},{8}}
 => ? = 3 - 1
([(0,7),(1,6),(2,5),(3,4)],8)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => {{1,2},{3,4},{5,6},{7,8}}
 => ? = 8 - 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
 => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 4 - 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 4 - 1
([(0,2),(0,3),(1,2),(1,3),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 5 - 1
([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 5 - 1
([(0,6),(0,7),(1,4),(1,5),(2,4),(2,5),(3,6),(3,7),(4,5),(6,7)],8)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 5 - 1
([(2,7),(3,5),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
 => [7,1,1]
 => [[1,2,3,4,5,6,7],[8],[9]]
 => {{1,2,3,4,5,6,7},{8},{9}}
 => ? = 1 - 1
([(2,10),(3,6),(3,10),(3,13),(4,9),(4,11),(4,14),(4,15),(5,12),(5,13),(5,14),(5,15),(6,12),(6,14),(6,15),(7,8),(7,9),(7,12),(7,14),(7,15),(8,11),(8,13),(8,14),(8,15),(9,11),(9,13),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(13,14),(13,15)],16)
 => [14,1,1]
 => [[1,2,3,4,5,6,7,8,9,10,11,12,13,14],[15],[16]]
 => ?
 => ? = 1 - 1
([(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,11),(4,5),(4,7),(4,10),(5,6),(5,9),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
 => [10,1,1]
 => [[1,2,3,4,5,6,7,8,9,10],[11],[12]]
 => ?
 => ? = 1 - 1
([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 4 - 1
([(0,2),(0,3),(1,2),(1,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 5 - 1
([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4),(5,6),(5,7),(6,7)],8)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 5 - 1
([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => [6,1,1,1]
 => [[1,2,3,4,5,6],[7],[8],[9]]
 => {{1,2,3,4,5,6},{7},{8},{9}}
 => ? = 1 - 1
([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => [6,1,1,1,1]
 => [[1,2,3,4,5,6],[7],[8],[9],[10]]
 => {{1,2,3,4,5,6},{7},{8},{9},{10}}
 => ? = 1 - 1
([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
 => [6,1,1,1]
 => [[1,2,3,4,5,6],[7],[8],[9]]
 => {{1,2,3,4,5,6},{7},{8},{9}}
 => ? = 1 - 1
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
 => [9,1,1,1]
 => [[1,2,3,4,5,6,7,8,9],[10],[11],[12]]
 => ?
 => ? = 1 - 1
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
 => [11,1,1,1]
 => [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14]]
 => ?
 => ? = 1 - 1
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
 => [11,1,1,1,1]
 => [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14],[15]]
 => ?
 => ? = 1 - 1
([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
 => [7,1,1,1]
 => [[1,2,3,4,5,6,7],[8],[9],[10]]
 => {{1,2,3,4,5,6,7},{8},{9},{10}}
 => ? = 1 - 1
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
 => [11,1,1,1]
 => [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14]]
 => ?
 => ? = 1 - 1

Description

The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block.

Matching statistic: St001500

Mapping

Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001500: Dyck paths ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 67%

Values

([],3)
 => []
 => []
 => []
 => ? = 1
([],4)
 => []
 => []
 => []
 => ? = 1
([(2,3)],4)
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1
([(0,3),(1,2)],4)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
([],5)
 => []
 => []
 => []
 => ? = 1
([(3,4)],5)
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1
([(2,4),(3,4)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
([(1,4),(2,3)],5)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
([(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]
 => 2
([],6)
 => []
 => []
 => []
 => ? = 1
([(4,5)],6)
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1
([(3,5),(4,5)],6)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
([(2,5),(3,4)],6)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
([(2,5),(3,4),(4,5)],6)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 4
([(0,1),(2,5),(3,4),(4,5)],6)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [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]
 => 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [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]
 => 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [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]
 => 2
([],7)
 => []
 => []
 => []
 => ? = 1
([(5,6)],7)
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1
([(4,6),(5,6)],7)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
([(3,6),(4,6),(5,6)],7)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
([(3,6),(4,5)],7)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
([(3,6),(4,5),(5,6)],7)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
([(2,3),(4,6),(5,6)],7)
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
([(4,5),(4,6),(5,6)],7)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 3
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [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]
 => 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [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]
 => 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [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]
 => 1
([(1,6),(2,5),(3,4)],7)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 4
([(2,6),(3,5),(4,5),(4,6)],7)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [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]
 => ? = 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [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]
 => ? = 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2
([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2
([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 2
([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2
([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 2
([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [9,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 2
([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? = 1
([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 2
([(0,1),(0,2),(1,2),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1
([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1
([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1
([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1
([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1
([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1
([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1
([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1
([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1
([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1
([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1
([(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1
([],8)
 => ?
 => ?
 => ?
 => ? = 1
([(4,7),(5,6)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(4,7),(5,6),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1
([(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 1
([(2,7),(3,7),(4,6),(5,6)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 1
([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 1
([(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1
([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
 => ?
 => ?
 => ?
 => ? = 1
([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 2
([(2,6),(2,7),(3,4),(3,5),(4,5),(4,7),(5,6),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1
([(2,3),(2,7),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1
([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 1
([(1,3),(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 3
([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 1

Description

The global dimension of magnitude 1 Nakayama algebras. We use the code below to translate them to Dyck paths.

Matching statistic: St001232

Mapping

Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 50%

Values

([],3)
 => []
 => []
 => []
 => ? = 1 - 1
([],4)
 => []
 => []
 => []
 => ? = 1 - 1
([(2,3)],4)
 => [1]
 => [1,0]
 => [1,0]
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
([],5)
 => []
 => []
 => []
 => ? = 1 - 1
([(3,4)],5)
 => [1]
 => [1,0]
 => [1,0]
 => 0 = 1 - 1
([(2,4),(3,4)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
([(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 = 2 - 1
([],6)
 => []
 => []
 => []
 => ? = 1 - 1
([(4,5)],6)
 => [1]
 => [1,0]
 => [1,0]
 => 0 = 1 - 1
([(3,5),(4,5)],6)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
([(2,5),(3,4),(4,5)],6)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 4 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [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 = 2 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [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 = 1 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 3 - 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [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 = 2 - 1
([],7)
 => []
 => []
 => []
 => ? = 1 - 1
([(5,6)],7)
 => [1]
 => [1,0]
 => [1,0]
 => 0 = 1 - 1
([(4,6),(5,6)],7)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
([(3,6),(4,6),(5,6)],7)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
([(3,6),(4,5)],7)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
([(3,6),(4,5),(5,6)],7)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
([(2,3),(4,6),(5,6)],7)
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([(4,5),(4,6),(5,6)],7)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [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 = 1 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [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 = 1 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [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 = 1 - 1
([(1,6),(2,5),(3,4)],7)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 4 - 1
([(0,3),(1,2),(4,6),(5,6)],7)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 4 - 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 3 - 1
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [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]
 => ? = 1 - 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [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]
 => ? = 1 - 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 1 - 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 4 - 1
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 3 - 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 3 - 1
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 3 - 1
([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [9,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 3 - 1
([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => ? = 3 - 1
([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? = 1 - 1
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 3 - 1
([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 2 - 1
([(0,1),(0,2),(1,2),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 3 - 1
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1 - 1
([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1 - 1
([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1 - 1
([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1 - 1
([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1 - 1
([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1 - 1
([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1 - 1
([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1 - 1
([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1 - 1
([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1 - 1
([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1 - 1
([(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1 - 1
([],8)
 => ?
 => ?
 => ?
 => ? = 1 - 1
([(4,7),(5,6)],8)
 => ?
 => ?
 => ?
 => ? = 2 - 1
([(4,7),(5,6),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1 - 1
([(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 1 - 1
([(2,7),(3,7),(4,6),(5,6)],8)
 => ?
 => ?
 => ?
 => ? = 3 - 1
([(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 1 - 1
([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ?
 => ?
 => ?
 => ? = 1 - 1
([(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
 => ?
 => ?
 => ?
 => ? = 1 - 1

Description

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

Matching statistic: St001613
(load all 3 compositions to match this statistic)

Mapping

Mp00243: Graphs —weak duplicate order⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001613: Lattices ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 17%

Values

([],3)
 => ([],1)
 => ([(0,1)],2)
 => 1
([],4)
 => ([],1)
 => ([(0,1)],2)
 => 1
([(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(0,3),(1,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2
([],5)
 => ([],1)
 => ([(0,1)],2)
 => 1
([(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(1,4),(2,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2
([(0,1),(2,4),(3,4)],5)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2
([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ?
 => ? = 2
([],6)
 => ([],1)
 => ([(0,1)],2)
 => 1
([(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(2,5),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(2,5),(3,4)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2
([(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 1
([(1,2),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2
([(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1
([(0,5),(1,4),(2,3)],6)
 => ([],6)
 => ?
 => ? = 4
([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => ?
 => ? = 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => ([],5)
 => ?
 => ? = 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],5)
 => ?
 => ? = 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([],6)
 => ?
 => ? = 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],6)
 => ?
 => ? = 2
([],7)
 => ([],1)
 => ([(0,1)],2)
 => 1
([(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(3,6),(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(3,6),(4,5)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2
([(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 1
([(2,3),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2
([(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2
([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 3
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 3
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,9),(3,8),(4,10),(4,11),(5,7),(5,10),(6,7),(6,11),(7,13),(8,12),(9,12),(10,3),(10,13),(11,2),(11,13),(12,1),(13,8),(13,9)],14)
 => ? = 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1
([(1,6),(2,5),(3,4)],7)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ?
 => ? = 4
([(2,6),(3,5),(4,5),(4,6)],7)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ?
 => ? = 1
([(1,2),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ?
 => ? = 2
([(0,3),(1,2),(4,6),(5,6)],7)
 => ([],6)
 => ?
 => ? = 4
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 2
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 2
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ?
 => ? = 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ?
 => ? = 2
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,5),(4,5)],6)
 => ?
 => ? = 2
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 1
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
 => ? = 1
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
 => ? = 1
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
 => ? = 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 3
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 3
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1

Description

The binary logarithm of the size of the center of a lattice. An element of a lattice is central if it is neutral and has a complement. The subposet induced by central elements is a Boolean lattice.

Matching statistic: St001719
(load all 3 compositions to match this statistic)

Mapping

Mp00243: Graphs —weak duplicate order⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 17%

Values

([],3)
 => ([],1)
 => ([(0,1)],2)
 => 1
([],4)
 => ([],1)
 => ([(0,1)],2)
 => 1
([(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(0,3),(1,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2
([],5)
 => ([],1)
 => ([(0,1)],2)
 => 1
([(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(1,4),(2,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2
([(0,1),(2,4),(3,4)],5)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2
([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ?
 => ? = 2
([],6)
 => ([],1)
 => ([(0,1)],2)
 => 1
([(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(2,5),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(2,5),(3,4)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2
([(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 1
([(1,2),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2
([(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1
([(0,5),(1,4),(2,3)],6)
 => ([],6)
 => ?
 => ? = 4
([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => ?
 => ? = 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => ([],5)
 => ?
 => ? = 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],5)
 => ?
 => ? = 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([],6)
 => ?
 => ? = 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],6)
 => ?
 => ? = 2
([],7)
 => ([],1)
 => ([(0,1)],2)
 => 1
([(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(3,6),(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(3,6),(4,5)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2
([(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 1
([(2,3),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2
([(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2
([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 3
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 3
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,9),(3,8),(4,10),(4,11),(5,7),(5,10),(6,7),(6,11),(7,13),(8,12),(9,12),(10,3),(10,13),(11,2),(11,13),(12,1),(13,8),(13,9)],14)
 => ? = 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1
([(1,6),(2,5),(3,4)],7)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ?
 => ? = 4
([(2,6),(3,5),(4,5),(4,6)],7)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ?
 => ? = 1
([(1,2),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ?
 => ? = 2
([(0,3),(1,2),(4,6),(5,6)],7)
 => ([],6)
 => ?
 => ? = 4
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 2
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 2
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ?
 => ? = 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ?
 => ? = 2
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,5),(4,5)],6)
 => ?
 => ? = 2
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 1
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
 => ? = 1
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
 => ? = 1
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
 => ? = 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 3
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 3
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1

Description

The number of shortest chains of small intervals from the bottom to the top in a lattice. An interval

$[a, b]$ in a lattice is small if

$b$ is a join of elements covering

$a$ .

Matching statistic: St001881
(load all 3 compositions to match this statistic)

Mapping

Mp00243: Graphs —weak duplicate order⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001881: Lattices ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 17%

Values

([],3)
 => ([],1)
 => ([(0,1)],2)
 => 1
([],4)
 => ([],1)
 => ([(0,1)],2)
 => 1
([(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(0,3),(1,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2
([],5)
 => ([],1)
 => ([(0,1)],2)
 => 1
([(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(1,4),(2,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2
([(0,1),(2,4),(3,4)],5)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2
([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ?
 => ? = 2
([],6)
 => ([],1)
 => ([(0,1)],2)
 => 1
([(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(2,5),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(2,5),(3,4)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2
([(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 1
([(1,2),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2
([(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1
([(0,5),(1,4),(2,3)],6)
 => ([],6)
 => ?
 => ? = 4
([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => ?
 => ? = 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => ([],5)
 => ?
 => ? = 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],5)
 => ?
 => ? = 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([],6)
 => ?
 => ? = 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],6)
 => ?
 => ? = 2
([],7)
 => ([],1)
 => ([(0,1)],2)
 => 1
([(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(3,6),(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(3,6),(4,5)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2
([(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 1
([(2,3),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2
([(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2
([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 3
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 3
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,9),(3,8),(4,10),(4,11),(5,7),(5,10),(6,7),(6,11),(7,13),(8,12),(9,12),(10,3),(10,13),(11,2),(11,13),(12,1),(13,8),(13,9)],14)
 => ? = 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1
([(1,6),(2,5),(3,4)],7)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ?
 => ? = 4
([(2,6),(3,5),(4,5),(4,6)],7)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ?
 => ? = 1
([(1,2),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ?
 => ? = 2
([(0,3),(1,2),(4,6),(5,6)],7)
 => ([],6)
 => ?
 => ? = 4
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 2
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 2
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ?
 => ? = 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ?
 => ? = 2
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,5),(4,5)],6)
 => ?
 => ? = 2
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 1
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
 => ? = 1
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
 => ? = 1
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
 => ? = 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 3
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 3
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1

Description

The number of factors of a lattice as a Cartesian product of lattices. Since the cardinality of a lattice is the product of the cardinalities of its factors, this statistic is one whenever the cardinality of the lattice is prime.

Matching statistic: St001845
(load all 3 compositions to match this statistic)

Mapping

Mp00243: Graphs —weak duplicate order⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001845: Lattices ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 17%

Values

([],3)
 => ([],1)
 => ([(0,1)],2)
 => 0 = 1 - 1
([],4)
 => ([],1)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2 - 1
([],5)
 => ([],1)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0 = 1 - 1
([(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2 - 1
([(0,1),(2,4),(3,4)],5)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ?
 => ? = 2 - 1
([],6)
 => ([],1)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0 = 1 - 1
([(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2 - 1
([(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 1 - 1
([(1,2),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2 - 1
([(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 3 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
 => ([],6)
 => ?
 => ? = 4 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => ?
 => ? = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => ([],5)
 => ?
 => ? = 3 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],5)
 => ?
 => ? = 2 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 1 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([],6)
 => ?
 => ? = 3 - 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],6)
 => ?
 => ? = 2 - 1
([],7)
 => ([],1)
 => ([(0,1)],2)
 => 0 = 1 - 1
([(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0 = 1 - 1
([(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0 = 1 - 1
([(3,6),(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0 = 1 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0 = 1 - 1
([(3,6),(4,5)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2 - 1
([(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 1 - 1
([(2,3),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2 - 1
([(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 0 = 1 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 1 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 1 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 1 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0 = 1 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 3 - 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 1 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 3 - 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 0 = 1 - 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,9),(3,8),(4,10),(4,11),(5,7),(5,10),(6,7),(6,11),(7,13),(8,12),(9,12),(10,3),(10,13),(11,2),(11,13),(12,1),(13,8),(13,9)],14)
 => ? = 1 - 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 1 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0 = 1 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 0 = 1 - 1
([(1,6),(2,5),(3,4)],7)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ?
 => ? = 4 - 1
([(2,6),(3,5),(4,5),(4,6)],7)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ?
 => ? = 1 - 1
([(1,2),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ?
 => ? = 2 - 1
([(0,3),(1,2),(4,6),(5,6)],7)
 => ([],6)
 => ?
 => ? = 4 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 2 - 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ?
 => ? = 1 - 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ?
 => ? = 2 - 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,5),(4,5)],6)
 => ?
 => ? = 2 - 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 1 - 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 1 - 1
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
 => ? = 1 - 1
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
 => ? = 1 - 1
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
 => ? = 1 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2 - 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 3 - 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 3 - 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 0 = 1 - 1

Description

The number of join irreducibles minus the rank of a lattice. A lattice is join-extremal, if this statistic is

$0$ .

The following 20 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000567The sum of the products of all pairs of parts. St000929The constant term of the character polynomial of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St001271The competition number of a graph.