Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000222
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000222: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [2,1] => 0
[2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3,1,2] => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => 2
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => 2
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,4,6,1,2,5] => 2
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [4,1,5,6,2,3] => 2
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,6,1,5] => 3
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 3
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => 3
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => 3
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 0
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => 3
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => 3
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 3
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,4,5,6,2] => 3
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,4,6,1,3,5] => 3
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,4,1,6,2,5] => 3
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 3
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 3
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 3
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,1,5,6,2,4] => 3
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => 2
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,1,5,2,6,3] => 3
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [2,3,4,1,6,5] => 4
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [4,6,1,2,3,5] => 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,3,4,6,2,5] => 4
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [2,3,5,1,4,6] => 4
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => 4
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 4
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 4
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [3,1,4,5,2,6] => 4
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => 4
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 4
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => 4
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [5,1,6,2,3,4] => 1
Description
The number of alignments in the permutation.
Matching statistic: St000698
(load all 4 compositions to match this statistic)
Mapping
St000698: Integer partitions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[1]
 => ? = 0
[2]
 => 1
[1,1]
 => 1
[3]
 => 1
[2,1]
 => 0
[1,1,1]
 => 1
[4]
 => 2
[3,1]
 => 2
[2,2]
 => 2
[2,1,1]
 => 2
[1,1,1,1]
 => 2
[5]
 => 2
[4,1]
 => 1
[3,2]
 => 2
[3,1,1]
 => 2
[2,2,1]
 => 2
[2,1,1,1]
 => 1
[1,1,1,1,1]
 => 2
[5,1]
 => 3
[4,2]
 => 3
[4,1,1]
 => 3
[3,3]
 => 3
[3,2,1]
 => 0
[3,1,1,1]
 => 3
[2,2,2]
 => 3
[2,2,1,1]
 => 3
[2,1,1,1,1]
 => 3
[5,2]
 => 3
[5,1,1]
 => 3
[4,3]
 => 2
[4,2,1]
 => 3
[4,1,1,1]
 => 2
[3,3,1]
 => 3
[3,2,2]
 => 3
[3,2,1,1]
 => 3
[3,1,1,1,1]
 => 3
[2,2,2,1]
 => 2
[2,2,1,1,1]
 => 3
[5,3]
 => 4
[5,2,1]
 => 1
[5,1,1,1]
 => 4
[4,4]
 => 4
[4,3,1]
 => 4
[4,2,2]
 => 4
[4,2,1,1]
 => 4
[4,1,1,1,1]
 => 4
[3,3,2]
 => 4
[3,3,1,1]
 => 4
[3,2,2,1]
 => 4
[3,2,1,1,1]
 => 1
[2,2,2,2]
 => 4
[5,4,3,1]
 => ? = 6
[5,4,2,2]
 => ? = 6
[5,4,2,1,1]
 => ? = 6
[5,3,3,2]
 => ? = 6
[5,3,3,1,1]
 => ? = 6
[5,3,2,2,1]
 => ? = 6
[4,4,3,2]
 => ? = 6
[4,4,3,1,1]
 => ? = 6
[4,4,2,2,1]
 => ? = 6
[4,3,3,2,1]
 => ? = 6
[5,4,3,2]
 => ? = 4
[5,4,3,1,1]
 => ? = 4
[5,4,2,2,1]
 => ? = 4
[5,3,3,2,1]
 => ? = 4
[4,4,3,2,1]
 => ? = 4
[5,4,3,2,1]
 => ? = 0
Description
The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$. This statistic counts the $2$-rim hooks that are removed in this process to obtain a $2$-core.
Matching statistic: St000789
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000789: Perfect matchings ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 43%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => 1 = 0 + 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => 2 = 1 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => 2 = 1 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => 2 = 1 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => 1 = 0 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [(1,10),(2,3),(4,9),(5,8),(6,7)]
 => 2 = 1 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [(1,12),(2,9),(3,8),(4,7),(5,6),(10,11)]
 => ? = 2 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => 3 = 2 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [(1,10),(2,5),(3,4),(6,9),(7,8)]
 => 3 = 2 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [(1,10),(2,3),(4,9),(5,6),(7,8)]
 => 3 = 2 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [(1,12),(2,3),(4,11),(5,10),(6,9),(7,8)]
 => ? = 2 + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [(1,14),(2,11),(3,10),(4,9),(5,8),(6,7),(12,13)]
 => ? = 2 + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [(1,12),(2,9),(3,8),(4,5),(6,7),(10,11)]
 => ? = 1 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => 3 = 2 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [(1,10),(2,3),(4,7),(5,6),(8,9)]
 => 3 = 2 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8)]
 => 3 = 2 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [(1,12),(2,3),(4,11),(5,10),(6,7),(8,9)]
 => ? = 1 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,14),(2,3),(4,13),(5,12),(6,11),(7,10),(8,9)]
 => ? = 2 + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [(1,14),(2,11),(3,10),(4,9),(5,6),(7,8),(12,13)]
 => ? = 3 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [(1,12),(2,9),(3,6),(4,5),(7,8),(10,11)]
 => ? = 3 + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [(1,12),(2,9),(3,4),(5,8),(6,7),(10,11)]
 => ? = 3 + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [(1,12),(2,7),(3,6),(4,5),(8,11),(9,10)]
 => ? = 3 + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => 1 = 0 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [(1,12),(2,3),(4,11),(5,8),(6,7),(9,10)]
 => ? = 3 + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [(1,12),(2,5),(3,4),(6,11),(7,10),(8,9)]
 => ? = 3 + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [(1,12),(2,3),(4,11),(5,6),(7,10),(8,9)]
 => ? = 3 + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [(1,14),(2,3),(4,13),(5,12),(6,11),(7,8),(9,10)]
 => ? = 3 + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => [(1,14),(2,11),(3,10),(4,7),(5,6),(8,9),(12,13)]
 => ? = 3 + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
 => [(1,14),(2,11),(3,10),(4,5),(6,9),(7,8),(12,13)]
 => ? = 3 + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [(1,12),(2,7),(3,6),(4,5),(8,9),(10,11)]
 => ? = 2 + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [(1,12),(2,9),(3,4),(5,6),(7,8),(10,11)]
 => ? = 3 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [(1,12),(2,3),(4,9),(5,8),(6,7),(10,11)]
 => ? = 2 + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [(1,12),(2,7),(3,4),(5,6),(8,11),(9,10)]
 => ? = 3 + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [(1,12),(2,5),(3,4),(6,11),(7,8),(9,10)]
 => ? = 3 + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [(1,12),(2,3),(4,11),(5,6),(7,8),(9,10)]
 => ? = 3 + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => [(1,14),(2,3),(4,13),(5,12),(6,9),(7,8),(10,11)]
 => ? = 3 + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [(1,12),(2,3),(4,5),(6,11),(7,10),(8,9)]
 => ? = 2 + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [(1,14),(2,3),(4,13),(5,12),(6,7),(8,11),(9,10)]
 => ? = 3 + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
 => [(1,14),(2,11),(3,8),(4,7),(5,6),(9,10),(12,13)]
 => ? = 4 + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [(1,14),(2,11),(3,10),(4,5),(6,7),(8,9),(12,13)]
 => ? = 1 + 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => [(1,14),(2,11),(3,4),(5,10),(6,9),(7,8),(12,13)]
 => ? = 4 + 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => [(1,14),(2,9),(3,8),(4,7),(5,6),(10,13),(11,12)]
 => ? = 4 + 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [(1,12),(2,7),(3,4),(5,6),(8,9),(10,11)]
 => ? = 4 + 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [(1,12),(2,5),(3,4),(6,9),(7,8),(10,11)]
 => ? = 4 + 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [(1,12),(2,3),(4,9),(5,6),(7,8),(10,11)]
 => ? = 4 + 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [(1,14),(2,3),(4,13),(5,10),(6,9),(7,8),(11,12)]
 => ? = 4 + 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [(1,12),(2,5),(3,4),(6,7),(8,11),(9,10)]
 => ? = 4 + 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [(1,12),(2,3),(4,7),(5,6),(8,11),(9,10)]
 => ? = 4 + 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [(1,12),(2,3),(4,5),(6,11),(7,8),(9,10)]
 => ? = 4 + 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [(1,14),(2,3),(4,13),(5,12),(6,7),(8,9),(10,11)]
 => ? = 1 + 1
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [(1,14),(2,5),(3,4),(6,13),(7,12),(8,11),(9,10)]
 => ? = 4 + 1
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [(1,14),(2,3),(4,13),(5,6),(7,12),(8,11),(9,10)]
 => ? = 4 + 1
[5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [(1,14),(2,9),(3,8),(4,7),(5,6),(10,11),(12,13)]
 => ? = 4 + 1
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
 => [(1,14),(2,11),(3,8),(4,5),(6,7),(9,10),(12,13)]
 => ? = 4 + 1
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
 => [(1,14),(2,11),(3,6),(4,5),(7,10),(8,9),(12,13)]
 => ? = 4 + 1
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => [(1,14),(2,11),(3,4),(5,10),(6,7),(8,9),(12,13)]
 => ? = 4 + 1
[5,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => [(1,14),(2,3),(4,11),(5,10),(6,9),(7,8),(12,13)]
 => ? = 4 + 1
[4,4,1]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
 => [(1,14),(2,9),(3,8),(4,5),(6,7),(10,13),(11,12)]
 => ? = 4 + 1
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
 => ? = 3 + 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [(1,12),(2,3),(4,7),(5,6),(8,9),(10,11)]
 => ? = 3 + 1
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [(1,12),(2,3),(4,5),(6,9),(7,8),(10,11)]
 => ? = 3 + 1
[4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [(1,14),(2,3),(4,13),(5,10),(6,7),(8,9),(11,12)]
 => ? = 4 + 1
[3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [(1,14),(2,7),(3,6),(4,5),(8,13),(9,12),(10,11)]
 => ? = 4 + 1
Description
The number of crossing-similar perfect matchings of a perfect matching. Consider the infinite tree $T$ defined in [1] as follows. $T$ has the perfect matchings on $\{1,\dots,2n\}$ on level $n$, with children obtained by inserting an arc with opener $1$. For example, the matching $[(1,2)]$ has the three children $[(1,2),(3,4)]$, $[(1,3),(2,4)]$ and $[(1,4),(2,3)]$. Two perfect matchings $M$ and $N$ on $\{1,\dots,2n\}$ are nesting-similar, if the distribution of the number of crossings agrees on all levels of the subtrees of $T$ rooted at $M$ and $N$. [thm 1.2, 1] shows that to find out whether $M$ and $N$ are crossing-similar, it is enough to check that $M$ and $N$ have the same number of crossings, and that the distribution of crossings agrees for their direct children. [thm 3.3, 1], see also [2], gives the number of equivalence classes of crossing-similar matchings with $n$ arcs as $$2^{n-2}\left(\binom{n}{2}+2\right).$$