- FindStat

Your data matches 8 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000819
(load all 22 compositions to match this statistic)

Mapping

St000819: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The propagating number of a perfect matching. In a perfect matching of

$\{1,\dots,2n\}$ , this is the number of pairs

$(i,j)$ with

$i\leq n < j$ .

Matching statistic: St000835

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000835: Integer partitions ⟶ ℤResult quality: 70% ●values known / values provided: 87%●distinct values known / distinct values provided: 70%

Values

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

Description

The minimal difference in size when partitioning the integer partition into two subpartitions. This is the optimal value of the optimisation version of the partition problem [1].

Matching statistic: St000992

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000992: Integer partitions ⟶ ℤResult quality: 70% ●values known / values provided: 87%●distinct values known / distinct values provided: 70%

Values

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

Description

The alternating sum of the parts of an integer partition. For a partition

$\lambda = (\lambda_1,\ldots,\lambda_k)$ , this is

$\lambda_1 - \lambda_2 + \cdots \pm \lambda_k$ .

Matching statistic: St001055

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001055: Integer partitions ⟶ ℤResult quality: 70% ●values known / values provided: 87%●distinct values known / distinct values provided: 70%

Values

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

Description

The Grundy value for the game of removing cells of a row in an integer partition. Two players alternately remove any positive number of cells in a row of the Ferrers diagram of an integer partition, such that the result is still a Ferrers diagram. The player facing the empty partition looses.

Matching statistic: St000714

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000714: Integer partitions ⟶ ℤResult quality: 70% ●values known / values provided: 87%●distinct values known / distinct values provided: 70%

Values

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

Description

The number of semistandard Young tableau of given shape, with entries at most 2. This is also the dimension of the corresponding irreducible representation of

$GL_2$ .

Matching statistic: St001232

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 60%

Values

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

Description

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

Matching statistic: St001879
(load all 2 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00047: Ordered trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 50%

Values

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

Description

The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.

Matching statistic: St001880
(load all 2 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00047: Ordered trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 50%

Values

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

Description

The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.