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Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St000870
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [[4,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 3
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2],[3,1]]
=> [3,1]
=> [1]
=> 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [[4,3,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [[3,3,3,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 3
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [[4,3,3],[2,2]]
=> [2,2]
=> [2]
=> 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [[4,4,3],[3,2]]
=> [3,2]
=> [2]
=> 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [[4,4,3],[2,2]]
=> [2,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [[4,4,4],[3,3]]
=> [3,3]
=> [3]
=> 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [[4,4,4],[3,2]]
=> [3,2]
=> [2]
=> 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [[4,4,4],[2,2]]
=> [2,2]
=> [2]
=> 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 3
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [[4,3,3],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [[4,4,3],[3,1]]
=> [3,1]
=> [1]
=> 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [[4,4,3],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 2
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [[4,3,3],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [[4,4,3],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4],[3,1]]
=> [3,1]
=> [1]
=> 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [[4,4,4],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [[4,4,4],[1,1]]
=> [1,1]
=> [1]
=> 1
Description
The product of the hook lengths of the diagonal cells in an integer partition.
For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells $(i,i)$ of a partition.
Matching statistic: St001006
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001006: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 14%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001006: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 14%
Values
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [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
[1,1,0,0,1,0,1,0,1,0]
=> [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]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [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
[1,1,0,0,1,1,0,0,1,0]
=> [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
[1,1,0,0,1,1,1,0,0,0]
=> [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
[1,1,0,1,0,0,1,0,1,0]
=> [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]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [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
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [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
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> ? = 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 3
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [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
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> ? = 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [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]
=> 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 3
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [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
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [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]
=> ? = 2
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [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
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [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]
=> 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [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
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [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
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [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
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [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
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 2
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> ? = 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [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
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [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
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [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
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [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]
=> ? = 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [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
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 2
[1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [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
[1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [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]
=> 2
[1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [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
[1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [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
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [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
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [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]
=> 2
[1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [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
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [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
Description
Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000788
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000788: Perfect matchings ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 7%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000788: Perfect matchings ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 7%
Values
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [(1,2),(3,14),(4,9),(5,6),(7,8),(10,11),(12,13)]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> ? = 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]
=> ? = 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [(1,2),(3,18),(4,15),(5,6),(7,8),(9,14),(10,11),(12,13),(16,17)]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [(1,2),(3,18),(4,13),(5,6),(7,12),(8,11),(9,10),(14,15),(16,17)]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [(1,16),(2,11),(3,4),(5,10),(6,9),(7,8),(12,13),(14,15)]
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,18),(6,13),(7,12),(8,11),(9,10),(14,15),(16,17)]
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [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)]
=> ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [(1,2),(3,18),(4,13),(5,6),(7,8),(9,12),(10,11),(14,15),(16,17)]
=> ? = 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,18),(8,13),(9,12),(10,11),(14,15),(16,17)]
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [(1,2),(3,14),(4,9),(5,8),(6,7),(10,11),(12,13)]
=> ? = 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [(1,2),(3,18),(4,11),(5,6),(7,8),(9,10),(12,13),(14,15),(16,17)]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,6),(7,14),(8,11),(9,10),(12,13)]
=> ? = 3
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [(1,14),(2,13),(3,4),(5,12),(6,9),(7,8),(10,11)]
=> ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [(1,2),(3,4),(5,16),(6,15),(7,14),(8,11),(9,10),(12,13)]
=> ? = 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,9),(7,8),(10,11)]
=> ? = 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,7),(5,6),(8,9)]
=> ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,6),(7,8),(9,14),(10,11),(12,13)]
=> ? = 3
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [(1,14),(2,13),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [(1,2),(3,4),(5,16),(6,15),(7,8),(9,14),(10,11),(12,13)]
=> ? = 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [(1,12),(2,11),(3,4),(5,10),(6,7),(8,9)]
=> ? = 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,4),(5,6),(7,16),(8,15),(9,14),(10,11),(12,13)]
=> ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,12),(8,9),(10,11)]
=> ? = 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [(1,2),(3,16),(4,13),(5,6),(7,12),(8,11),(9,10),(14,15)]
=> ? = 3
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [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)]
=> ? = 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [(1,2),(3,4),(5,16),(6,13),(7,12),(8,11),(9,10),(14,15)]
=> ? = 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1]
=> [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
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [(1,2),(3,16),(4,13),(5,6),(7,8),(9,12),(10,11),(14,15)]
=> ? = 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,16),(8,13),(9,12),(10,11),(14,15)]
=> ? = 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [(1,2),(3,12),(4,9),(5,8),(6,7),(10,11)]
=> ? = 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [(1,2),(3,16),(4,11),(5,6),(7,8),(9,10),(12,13),(14,15)]
=> ? = 3
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [(1,14),(2,9),(3,4),(5,6),(7,8),(10,11),(12,13)]
=> ? = 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,16),(6,11),(7,8),(9,10),(12,13),(14,15)]
=> ? = 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [(1,12),(2,7),(3,4),(5,6),(8,9),(10,11)]
=> ? = 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,16),(8,11),(9,10),(12,13),(14,15)]
=> ? = 2
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,14),(6,9),(7,8),(10,11),(12,13)]
=> ? = 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
=> ? = 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,16),(10,11),(12,13),(14,15)]
=> ? = 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,6),(7,12),(8,11),(9,10)]
=> ? = 2
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [(1,12),(2,11),(3,4),(5,10),(6,9),(7,8)]
=> ? = 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ? = 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> 1
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,6),(7,8),(9,12),(10,11)]
=> ? = 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,6),(7,14),(8,13),(9,12),(10,11)]
=> ? = 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [(1,2),(3,14),(4,11),(5,6),(7,8),(9,10),(12,13)]
=> ? = 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)]
=> ? = 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> [1,0,1,1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ?
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> 1
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> 1
[1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> 1
[1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 1
[1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 1
[1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1
Description
The number of nesting-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 nestings 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 nesting-similar, it is enough to check that $M$ and $N$ have the same number of nestings, and that the distribution of nestings agrees for their direct children.
[thm 3.5, 1], see also [2], gives the number of equivalence classes of nesting-similar matchings with $n$ arcs as $$2\cdot 4^{n-1} - \frac{3n-1}{2n+2}\binom{2n}{n}.$$ [prop 3.6, 1] has further interpretations of this number.
Matching statistic: St001043
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001043: Perfect matchings ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 7%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001043: Perfect matchings ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 7%
Values
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [(1,2),(3,14),(4,9),(5,6),(7,8),(10,11),(12,13)]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> ? = 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]
=> ? = 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [(1,2),(3,18),(4,15),(5,6),(7,8),(9,14),(10,11),(12,13),(16,17)]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [(1,2),(3,18),(4,13),(5,6),(7,12),(8,11),(9,10),(14,15),(16,17)]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [(1,16),(2,11),(3,4),(5,10),(6,9),(7,8),(12,13),(14,15)]
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,18),(6,13),(7,12),(8,11),(9,10),(14,15),(16,17)]
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [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)]
=> ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [(1,2),(3,18),(4,13),(5,6),(7,8),(9,12),(10,11),(14,15),(16,17)]
=> ? = 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,18),(8,13),(9,12),(10,11),(14,15),(16,17)]
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [(1,2),(3,14),(4,9),(5,8),(6,7),(10,11),(12,13)]
=> ? = 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [(1,2),(3,18),(4,11),(5,6),(7,8),(9,10),(12,13),(14,15),(16,17)]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,6),(7,14),(8,11),(9,10),(12,13)]
=> ? = 3
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [(1,14),(2,13),(3,4),(5,12),(6,9),(7,8),(10,11)]
=> ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [(1,2),(3,4),(5,16),(6,15),(7,14),(8,11),(9,10),(12,13)]
=> ? = 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,9),(7,8),(10,11)]
=> ? = 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,7),(5,6),(8,9)]
=> ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,6),(7,8),(9,14),(10,11),(12,13)]
=> ? = 3
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [(1,14),(2,13),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [(1,2),(3,4),(5,16),(6,15),(7,8),(9,14),(10,11),(12,13)]
=> ? = 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [(1,12),(2,11),(3,4),(5,10),(6,7),(8,9)]
=> ? = 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,4),(5,6),(7,16),(8,15),(9,14),(10,11),(12,13)]
=> ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,12),(8,9),(10,11)]
=> ? = 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [(1,2),(3,16),(4,13),(5,6),(7,12),(8,11),(9,10),(14,15)]
=> ? = 3
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [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)]
=> ? = 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [(1,2),(3,4),(5,16),(6,13),(7,12),(8,11),(9,10),(14,15)]
=> ? = 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1]
=> [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
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [(1,2),(3,16),(4,13),(5,6),(7,8),(9,12),(10,11),(14,15)]
=> ? = 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,16),(8,13),(9,12),(10,11),(14,15)]
=> ? = 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [(1,2),(3,12),(4,9),(5,8),(6,7),(10,11)]
=> ? = 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [(1,2),(3,16),(4,11),(5,6),(7,8),(9,10),(12,13),(14,15)]
=> ? = 3
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [(1,14),(2,9),(3,4),(5,6),(7,8),(10,11),(12,13)]
=> ? = 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,16),(6,11),(7,8),(9,10),(12,13),(14,15)]
=> ? = 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [(1,12),(2,7),(3,4),(5,6),(8,9),(10,11)]
=> ? = 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,16),(8,11),(9,10),(12,13),(14,15)]
=> ? = 2
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,14),(6,9),(7,8),(10,11),(12,13)]
=> ? = 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
=> ? = 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,16),(10,11),(12,13),(14,15)]
=> ? = 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,6),(7,12),(8,11),(9,10)]
=> ? = 2
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [(1,12),(2,11),(3,4),(5,10),(6,9),(7,8)]
=> ? = 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ? = 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> 1
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,6),(7,8),(9,12),(10,11)]
=> ? = 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,6),(7,14),(8,13),(9,12),(10,11)]
=> ? = 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [(1,2),(3,14),(4,11),(5,6),(7,8),(9,10),(12,13)]
=> ? = 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)]
=> ? = 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> [1,0,1,1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ?
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> 1
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> 1
[1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> 1
[1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 1
[1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 1
[1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1
Description
The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching.
The bijection between perfect matchings of $\{1,\dots,2n\}$ and trees with $n+1$ leaves is described in Example 5.2.6 of [1].
Matching statistic: St000787
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000787: Perfect matchings ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 7%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000787: Perfect matchings ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 7%
Values
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [(1,2),(3,14),(4,9),(5,6),(7,8),(10,11),(12,13)]
=> ? = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]
=> ? = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [(1,2),(3,18),(4,15),(5,6),(7,8),(9,14),(10,11),(12,13),(16,17)]
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [(1,2),(3,18),(4,13),(5,6),(7,12),(8,11),(9,10),(14,15),(16,17)]
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [(1,16),(2,11),(3,4),(5,10),(6,9),(7,8),(12,13),(14,15)]
=> ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,18),(6,13),(7,12),(8,11),(9,10),(14,15),(16,17)]
=> ? = 1 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [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)]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [(1,2),(3,18),(4,13),(5,6),(7,8),(9,12),(10,11),(14,15),(16,17)]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,18),(8,13),(9,12),(10,11),(14,15),(16,17)]
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [(1,2),(3,14),(4,9),(5,8),(6,7),(10,11),(12,13)]
=> ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [(1,2),(3,18),(4,11),(5,6),(7,8),(9,10),(12,13),(14,15),(16,17)]
=> ? = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,6),(7,14),(8,11),(9,10),(12,13)]
=> ? = 3 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [(1,14),(2,13),(3,4),(5,12),(6,9),(7,8),(10,11)]
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [(1,2),(3,4),(5,16),(6,15),(7,14),(8,11),(9,10),(12,13)]
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,9),(7,8),(10,11)]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,7),(5,6),(8,9)]
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,6),(7,8),(9,14),(10,11),(12,13)]
=> ? = 3 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [(1,14),(2,13),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> ? = 1 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [(1,2),(3,4),(5,16),(6,15),(7,8),(9,14),(10,11),(12,13)]
=> ? = 1 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [(1,12),(2,11),(3,4),(5,10),(6,7),(8,9)]
=> ? = 1 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,4),(5,6),(7,16),(8,15),(9,14),(10,11),(12,13)]
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,12),(8,9),(10,11)]
=> ? = 1 - 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> ? = 2 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [(1,2),(3,16),(4,13),(5,6),(7,12),(8,11),(9,10),(14,15)]
=> ? = 3 - 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [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)]
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [(1,2),(3,4),(5,16),(6,13),(7,12),(8,11),(9,10),(14,15)]
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1]
=> [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
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [(1,2),(3,16),(4,13),(5,6),(7,8),(9,12),(10,11),(14,15)]
=> ? = 3 - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,16),(8,13),(9,12),(10,11),(14,15)]
=> ? = 2 - 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [(1,2),(3,12),(4,9),(5,8),(6,7),(10,11)]
=> ? = 2 - 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [(1,2),(3,16),(4,11),(5,6),(7,8),(9,10),(12,13),(14,15)]
=> ? = 3 - 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [(1,14),(2,9),(3,4),(5,6),(7,8),(10,11),(12,13)]
=> ? = 1 - 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,16),(6,11),(7,8),(9,10),(12,13),(14,15)]
=> ? = 1 - 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [(1,12),(2,7),(3,4),(5,6),(8,9),(10,11)]
=> ? = 1 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,16),(8,11),(9,10),(12,13),(14,15)]
=> ? = 2 - 1
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,14),(6,9),(7,8),(10,11),(12,13)]
=> ? = 1 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
=> ? = 2 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,16),(10,11),(12,13),(14,15)]
=> ? = 1 - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> 0 = 1 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,6),(7,12),(8,11),(9,10)]
=> ? = 2 - 1
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [(1,12),(2,11),(3,4),(5,10),(6,9),(7,8)]
=> ? = 1 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ? = 1 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,6),(7,8),(9,12),(10,11)]
=> ? = 2 - 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,6),(7,14),(8,13),(9,12),(10,11)]
=> ? = 1 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [(1,2),(3,14),(4,11),(5,6),(7,8),(9,10),(12,13)]
=> ? = 1 - 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)]
=> ? = 2 - 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> 0 = 1 - 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> [1,0,1,1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 1 - 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> 0 = 1 - 1
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> 0 = 1 - 1
[1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> 0 = 1 - 1
[1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> 0 = 1 - 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 0 = 1 - 1
[1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> 0 = 1 - 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 0 = 1 - 1
[1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 0 = 1 - 1
Description
The number of flips required to make a perfect matching noncrossing.
A crossing in a perfect matching is a pair of arcs $\{a,b\}$ and $\{c,d\}$ such that $a < c < b < d$. Replacing any such pair by either $\{a,c\}$ and $\{b,d\}$ or by $\{a,d\}$, $\{b,c\}$ produces a perfect matching with fewer crossings.
This statistic is the minimal number of such flips required to turn a given matching into a noncrossing matching.
Matching statistic: St001133
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001133: Perfect matchings ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 7%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001133: Perfect matchings ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 7%
Values
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [(1,2),(3,14),(4,9),(5,6),(7,8),(10,11),(12,13)]
=> ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> ? = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]
=> ? = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [(1,2),(3,18),(4,15),(5,6),(7,8),(9,14),(10,11),(12,13),(16,17)]
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [(1,2),(3,18),(4,13),(5,6),(7,12),(8,11),(9,10),(14,15),(16,17)]
=> ? = 2 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [(1,16),(2,11),(3,4),(5,10),(6,9),(7,8),(12,13),(14,15)]
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,18),(6,13),(7,12),(8,11),(9,10),(14,15),(16,17)]
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [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)]
=> ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [(1,2),(3,18),(4,13),(5,6),(7,8),(9,12),(10,11),(14,15),(16,17)]
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,18),(8,13),(9,12),(10,11),(14,15),(16,17)]
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [(1,2),(3,14),(4,9),(5,8),(6,7),(10,11),(12,13)]
=> ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [(1,2),(3,18),(4,11),(5,6),(7,8),(9,10),(12,13),(14,15),(16,17)]
=> ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,6),(7,14),(8,11),(9,10),(12,13)]
=> ? = 3 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [(1,14),(2,13),(3,4),(5,12),(6,9),(7,8),(10,11)]
=> ? = 2 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [(1,2),(3,4),(5,16),(6,15),(7,14),(8,11),(9,10),(12,13)]
=> ? = 2 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,9),(7,8),(10,11)]
=> ? = 1 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,7),(5,6),(8,9)]
=> ? = 2 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,6),(7,8),(9,14),(10,11),(12,13)]
=> ? = 3 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [(1,14),(2,13),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [(1,2),(3,4),(5,16),(6,15),(7,8),(9,14),(10,11),(12,13)]
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [(1,12),(2,11),(3,4),(5,10),(6,7),(8,9)]
=> ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,4),(5,6),(7,16),(8,15),(9,14),(10,11),(12,13)]
=> ? = 2 + 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,12),(8,9),(10,11)]
=> ? = 1 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> ? = 2 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [(1,2),(3,16),(4,13),(5,6),(7,12),(8,11),(9,10),(14,15)]
=> ? = 3 + 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [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)]
=> ? = 2 + 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [(1,2),(3,4),(5,16),(6,13),(7,12),(8,11),(9,10),(14,15)]
=> ? = 2 + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1]
=> [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
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [(1,2),(3,16),(4,13),(5,6),(7,8),(9,12),(10,11),(14,15)]
=> ? = 3 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,16),(8,13),(9,12),(10,11),(14,15)]
=> ? = 2 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [(1,2),(3,12),(4,9),(5,8),(6,7),(10,11)]
=> ? = 2 + 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [(1,2),(3,16),(4,11),(5,6),(7,8),(9,10),(12,13),(14,15)]
=> ? = 3 + 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [(1,14),(2,9),(3,4),(5,6),(7,8),(10,11),(12,13)]
=> ? = 1 + 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,16),(6,11),(7,8),(9,10),(12,13),(14,15)]
=> ? = 1 + 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [(1,12),(2,7),(3,4),(5,6),(8,9),(10,11)]
=> ? = 1 + 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,16),(8,11),(9,10),(12,13),(14,15)]
=> ? = 2 + 1
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,14),(6,9),(7,8),(10,11),(12,13)]
=> ? = 1 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
=> ? = 2 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,16),(10,11),(12,13),(14,15)]
=> ? = 1 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> 2 = 1 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,6),(7,12),(8,11),(9,10)]
=> ? = 2 + 1
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [(1,12),(2,11),(3,4),(5,10),(6,9),(7,8)]
=> ? = 1 + 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ? = 1 + 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,6),(7,8),(9,12),(10,11)]
=> ? = 2 + 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,6),(7,14),(8,13),(9,12),(10,11)]
=> ? = 1 + 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [(1,2),(3,14),(4,11),(5,6),(7,8),(9,10),(12,13)]
=> ? = 1 + 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)]
=> ? = 2 + 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> 2 = 1 + 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> [1,0,1,1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 1 + 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> 2 = 1 + 1
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> 2 = 1 + 1
[1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> 2 = 1 + 1
[1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> 2 = 1 + 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 2 = 1 + 1
[1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> 2 = 1 + 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 2 = 1 + 1
Description
The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching.
The bijection between perfect matchings of $\{1,\dots,2n\}$ and trees with $n+1$ leaves is described in Example 5.2.6 of [1].
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