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Your data matches 28 different statistics following compositions of up to 3 maps.
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Matching statistic: St001726
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001726: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001726: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[2,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[3,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[4,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[5,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 3
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5
[6,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 3
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 4
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 4
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5
[7,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 3
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 4
Description
The number of visible inversions of a permutation.
A visible inversion of a permutation $\pi$ is a pair $i < j$ such that $\pi(j) \leq \min(i, \pi(i))$.
Matching statistic: St001232
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 56%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 56%
Values
[1,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[2,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[3,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[2,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[4,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[3,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[5,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[4,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 3 - 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[6,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[5,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 3 - 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 4 - 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[7,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[6,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 3 - 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 4 - 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 5 - 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 5 - 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 5 - 1
[8,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[7,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[7,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[5,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 3 - 1
[4,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[4,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 4 - 1
[3,3,3]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 3 - 1
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 4 - 1
[3,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 5 - 1
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 5 - 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ? = 6 - 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 6 - 1
[6,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 3 - 1
[5,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[5,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 4 - 1
[4,4,2]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 3 - 1
[4,3,3]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 3 - 1
[4,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 4 - 1
[4,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 5 - 1
[4,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 5 - 1
[3,3,3,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? = 4 - 1
[3,3,2,2]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 5 - 1
[3,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 5 - 1
[3,2,2,2,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ? = 6 - 1
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 6 - 1
[2,2,2,2,2]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 7 - 1
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 7 - 1
[7,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 3 - 1
[6,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[6,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 4 - 1
[5,4,2]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 3 - 1
[5,3,3]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 3 - 1
[5,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 4 - 1
[5,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 5 - 1
[5,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 5 - 1
[4,4,3]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? = 3 - 1
[4,4,2,1]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 4 - 1
[4,3,3,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? = 4 - 1
[4,3,2,2]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 5 - 1
[4,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 5 - 1
[4,2,2,2,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ? = 6 - 1
[4,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 6 - 1
[3,3,3,2]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 5 - 1
[3,3,3,1,1]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 5 - 1
[3,3,2,2,1]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 6 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001727
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 67%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 67%
Values
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => 1
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,2,4,1] => 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,2,1,3] => 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 3
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [3,2,4,5,1] => 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,4,2,1] => 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,2,3,1] => 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,1,2] => 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,2,1,3,4] => 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,1,2,3,4,5] => 4
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [3,2,4,5,6,1] => 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,4,2,5,1] => 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [4,2,3,5,1] => 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [3,4,5,1,2] => 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,2,3,1,4] => 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [4,5,1,2,3] => 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,1,2,4] => 3
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [6,2,1,3,4,5] => 4
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => ? = 5
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => [3,2,4,5,6,7,1] => ? = 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => [3,4,2,5,6,1] => 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => [4,2,3,5,6,1] => 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [3,4,5,2,1] => 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,3,2,5,1] => 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,2,3,4,1] => 3
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,3,5,1,2] => 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,5,2,1,3] => 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,3,2,1,4] => 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [6,2,3,1,4,5] => 4
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,1,2,3] => 4
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [6,3,1,2,4,5] => 4
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => [7,2,1,3,4,5,6] => ? = 5
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => [3,2,4,5,6,7,8,1] => ? = 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => [3,4,2,5,6,7,1] => ? = 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => [4,2,3,5,6,7,1] => ? = 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [3,4,5,2,6,1] => 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => [4,3,2,5,6,1] => 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => [5,2,3,4,6,1] => 3
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [3,4,5,6,1,2] => 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,3,5,2,1] => 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [4,5,2,3,1] => 3
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,3,2,4,1] => 3
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,6,1] => [6,2,3,4,1,5] => 4
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,5,3,1,2] => 3
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,3,4,1,2] => 3
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,4,2,1,3] => 4
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,6,1] => [6,3,2,1,4,5] => 4
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => [7,2,3,1,4,5,6] => ? = 5
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [5,6,1,2,3,4] => 5
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => [6,4,1,2,3,5] => 5
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => [7,3,1,2,4,5,6] => ? = 5
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [9,2,1,3,4,5,6,7,8] => [3,2,4,5,6,7,8,9,1] => ? = 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [8,3,1,2,4,5,6,7] => [3,4,2,5,6,7,8,1] => ? = 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [8,2,3,1,4,5,6,7] => [4,2,3,5,6,7,8,1] => ? = 2
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [7,4,1,2,3,5,6] => [3,4,5,2,6,7,1] => ? = 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [7,3,2,1,4,5,6] => [4,3,2,5,6,7,1] => ? = 2
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,1,5,6] => [5,2,3,4,6,7,1] => ? = 3
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,2,3,4] => [3,4,5,6,2,1] => 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1,3,5] => [4,3,5,2,6,1] => 2
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [4,5,2,3,6,1] => 3
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,6,7,1] => [7,2,3,4,1,5,6] => ? = 5
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,6,7,1] => [7,3,2,1,4,5,6] => ? = 5
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [7,4,1,2,3,5,6] => ? = 6
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,1,3,4,5,6,7,8,9] => [3,2,4,5,6,7,8,9,10,1] => ? = 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [9,3,1,2,4,5,6,7,8] => [3,4,2,5,6,7,8,9,1] => ? = 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [9,2,3,1,4,5,6,7,8] => [4,2,3,5,6,7,8,9,1] => ? = 2
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [8,4,1,2,3,5,6,7] => [3,4,5,2,6,7,8,1] => ? = 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [8,3,2,1,4,5,6,7] => [4,3,2,5,6,7,8,1] => ? = 2
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [8,2,3,4,1,5,6,7] => [5,2,3,4,6,7,8,1] => ? = 3
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [7,5,1,2,3,4,6] => [3,4,5,6,2,7,1] => ? = 1
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [7,4,2,1,3,5,6] => [4,3,5,2,6,7,1] => ? = 2
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [7,3,4,1,2,5,6] => [4,5,2,3,6,7,1] => ? = 3
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,4,1,5,6] => [5,3,2,4,6,7,1] => ? = 3
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,1,6] => [6,2,3,4,5,7,1] => ? = 4
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => [3,4,5,6,7,1,2] => ? = 1
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [6,2,3,4,5,7,1] => [7,2,3,4,5,1,6] => ? = 5
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [5,3,2,4,6,7,1] => [7,3,2,4,1,5,6] => ? = 5
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,5,2,3,6,7,1] => [7,3,4,1,2,5,6] => ? = 5
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,6,7,1] => [7,4,2,1,3,5,6] => ? = 6
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => [6,7,1,2,3,4,5] => ? = 7
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,5,1,2,3,4,6] => ? = 7
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [11,2,1,3,4,5,6,7,8,9,10] => [3,2,4,5,6,7,8,9,10,11,1] => ? = 1
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [10,3,1,2,4,5,6,7,8,9] => [3,4,2,5,6,7,8,9,10,1] => ? = 1
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,3,1,4,5,6,7,8,9] => [4,2,3,5,6,7,8,9,10,1] => ? = 2
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [9,4,1,2,3,5,6,7,8] => [3,4,5,2,6,7,8,9,1] => ? = 1
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,2,1,4,5,6,7,8] => [4,3,2,5,6,7,8,9,1] => ? = 2
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [9,2,3,4,1,5,6,7,8] => [5,2,3,4,6,7,8,9,1] => ? = 3
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [8,5,1,2,3,4,6,7] => [3,4,5,6,2,7,8,1] => ? = 1
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [8,4,2,1,3,5,6,7] => [4,3,5,2,6,7,8,1] => ? = 2
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [8,3,4,1,2,5,6,7] => [4,5,2,3,6,7,8,1] => ? = 3
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,2,4,1,5,6,7] => [5,3,2,4,6,7,8,1] => ? = 3
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [8,2,3,4,5,1,6,7] => [6,2,3,4,5,7,8,1] => ? = 4
[6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => ? = 1
[6,4,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [7,5,2,1,3,4,6] => [4,3,5,6,2,7,1] => ? = 2
[6,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [7,4,3,1,2,5,6] => [4,5,3,2,6,7,1] => ? = 3
[6,3,1,1]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [7,4,2,3,1,5,6] => [5,3,4,2,6,7,1] => ? = 3
Description
The number of invisible inversions of a permutation.
A visible inversion of a permutation $\pi$ is a pair $i < j$ such that $\pi(j) \leq \min(i, \pi(i))$. Thus, an invisible inversion satisfies $\pi(i) > \pi(j) > i$.
Matching statistic: St000319
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 78%
St000319: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 78%
Values
[1,1]
=> [2]
=> 1
[2,1]
=> [2,1]
=> 1
[1,1,1]
=> [3]
=> 2
[3,1]
=> [2,1,1]
=> 1
[2,2]
=> [2,2]
=> 1
[2,1,1]
=> [3,1]
=> 2
[1,1,1,1]
=> [4]
=> 3
[4,1]
=> [2,1,1,1]
=> 1
[3,2]
=> [2,2,1]
=> 1
[3,1,1]
=> [3,1,1]
=> 2
[2,2,1]
=> [3,2]
=> 2
[2,1,1,1]
=> [4,1]
=> 3
[1,1,1,1,1]
=> [5]
=> 4
[5,1]
=> [2,1,1,1,1]
=> 1
[4,2]
=> [2,2,1,1]
=> 1
[4,1,1]
=> [3,1,1,1]
=> 2
[3,3]
=> [2,2,2]
=> 1
[3,2,1]
=> [3,2,1]
=> 2
[3,1,1,1]
=> [4,1,1]
=> 3
[2,2,2]
=> [3,3]
=> 3
[2,2,1,1]
=> [4,2]
=> 3
[2,1,1,1,1]
=> [5,1]
=> 4
[1,1,1,1,1,1]
=> [6]
=> 5
[6,1]
=> [2,1,1,1,1,1]
=> 1
[5,2]
=> [2,2,1,1,1]
=> 1
[5,1,1]
=> [3,1,1,1,1]
=> 2
[4,3]
=> [2,2,2,1]
=> 1
[4,2,1]
=> [3,2,1,1]
=> 2
[4,1,1,1]
=> [4,1,1,1]
=> 3
[3,3,1]
=> [3,2,2]
=> 2
[3,2,2]
=> [3,3,1]
=> 3
[3,2,1,1]
=> [4,2,1]
=> 3
[3,1,1,1,1]
=> [5,1,1]
=> 4
[2,2,2,1]
=> [4,3]
=> 4
[2,2,1,1,1]
=> [5,2]
=> 4
[2,1,1,1,1,1]
=> [6,1]
=> 5
[7,1]
=> [2,1,1,1,1,1,1]
=> 1
[6,2]
=> [2,2,1,1,1,1]
=> 1
[6,1,1]
=> [3,1,1,1,1,1]
=> 2
[5,3]
=> [2,2,2,1,1]
=> 1
[5,2,1]
=> [3,2,1,1,1]
=> 2
[5,1,1,1]
=> [4,1,1,1,1]
=> 3
[4,4]
=> [2,2,2,2]
=> 1
[4,3,1]
=> [3,2,2,1]
=> 2
[4,2,2]
=> [3,3,1,1]
=> 3
[4,2,1,1]
=> [4,2,1,1]
=> 3
[4,1,1,1,1]
=> [5,1,1,1]
=> 4
[3,3,2]
=> [3,3,2]
=> 3
[3,3,1,1]
=> [4,2,2]
=> 3
[3,2,2,1]
=> [4,3,1]
=> 4
[10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> ? = 1
[9,2]
=> [2,2,1,1,1,1,1,1,1]
=> ? = 1
[9,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> ? = 2
[8,3]
=> [2,2,2,1,1,1,1,1]
=> ? = 1
[8,2,1]
=> [3,2,1,1,1,1,1,1]
=> ? = 2
[8,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> ? = 3
[7,4]
=> [2,2,2,2,1,1,1]
=> ? = 1
[7,3,1]
=> [3,2,2,1,1,1,1]
=> ? = 2
[7,2,2]
=> [3,3,1,1,1,1,1]
=> ? = 3
[7,2,1,1]
=> [4,2,1,1,1,1,1]
=> ? = 3
[7,1,1,1,1]
=> [5,1,1,1,1,1,1]
=> ? = 4
[6,5]
=> [2,2,2,2,2,1]
=> ? = 1
[6,4,1]
=> [3,2,2,2,1,1]
=> ? = 2
[6,3,2]
=> [3,3,2,1,1,1]
=> ? = 3
[6,3,1,1]
=> [4,2,2,1,1,1]
=> ? = 3
[6,2,2,1]
=> [4,3,1,1,1,1]
=> ? = 4
[6,2,1,1,1]
=> [5,2,1,1,1,1]
=> ? = 4
[6,1,1,1,1,1]
=> [6,1,1,1,1,1]
=> ? = 5
[5,5,1]
=> [3,2,2,2,2]
=> ? = 2
[5,4,2]
=> [3,3,2,2,1]
=> ? = 3
[5,4,1,1]
=> [4,2,2,2,1]
=> ? = 3
[5,3,3]
=> [3,3,3,1,1]
=> ? = 3
[5,3,2,1]
=> [4,3,2,1,1]
=> ? = 4
[5,3,1,1,1]
=> [5,2,2,1,1]
=> ? = 4
[5,2,2,2]
=> [4,4,1,1,1]
=> ? = 5
[5,2,2,1,1]
=> [5,3,1,1,1]
=> ? = 5
[5,2,1,1,1,1]
=> [6,2,1,1,1]
=> ? = 5
[4,4,3]
=> [3,3,3,2]
=> ? = 3
[4,4,2,1]
=> [4,3,2,2]
=> ? = 4
[4,4,1,1,1]
=> [5,2,2,2]
=> ? = 4
[4,3,3,1]
=> [4,3,3,1]
=> ? = 4
[4,3,2,2]
=> [4,4,2,1]
=> ? = 5
[4,3,2,1,1]
=> [5,3,2,1]
=> ? = 5
[4,3,1,1,1,1]
=> [6,2,2,1]
=> ? = 5
[4,2,2,2,1]
=> [5,4,1,1]
=> ? = 6
[4,2,2,1,1,1]
=> [6,3,1,1]
=> ? = 6
[3,3,3,2]
=> [4,4,3]
=> ? = 5
[3,3,3,1,1]
=> [5,3,3]
=> ? = 5
[3,3,2,2,1]
=> [5,4,2]
=> ? = 6
[3,3,2,1,1,1]
=> [6,3,2]
=> ? = 6
[3,2,2,2,2]
=> [5,5,1]
=> ? = 7
[3,2,2,2,1,1]
=> [6,4,1]
=> ? = 7
[2,2,2,2,2,1]
=> [6,5]
=> ? = 8
[11,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> ? = 1
[10,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> ? = 2
[9,3]
=> [2,2,2,1,1,1,1,1,1]
=> ? = 1
[9,2,1]
=> [3,2,1,1,1,1,1,1,1]
=> ? = 2
[9,1,1,1]
=> [4,1,1,1,1,1,1,1,1]
=> ? = 3
[8,4]
=> [2,2,2,2,1,1,1,1]
=> ? = 1
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 78%
St000320: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 78%
Values
[1,1]
=> [2]
=> 1
[2,1]
=> [2,1]
=> 1
[1,1,1]
=> [3]
=> 2
[3,1]
=> [2,1,1]
=> 1
[2,2]
=> [2,2]
=> 1
[2,1,1]
=> [3,1]
=> 2
[1,1,1,1]
=> [4]
=> 3
[4,1]
=> [2,1,1,1]
=> 1
[3,2]
=> [2,2,1]
=> 1
[3,1,1]
=> [3,1,1]
=> 2
[2,2,1]
=> [3,2]
=> 2
[2,1,1,1]
=> [4,1]
=> 3
[1,1,1,1,1]
=> [5]
=> 4
[5,1]
=> [2,1,1,1,1]
=> 1
[4,2]
=> [2,2,1,1]
=> 1
[4,1,1]
=> [3,1,1,1]
=> 2
[3,3]
=> [2,2,2]
=> 1
[3,2,1]
=> [3,2,1]
=> 2
[3,1,1,1]
=> [4,1,1]
=> 3
[2,2,2]
=> [3,3]
=> 3
[2,2,1,1]
=> [4,2]
=> 3
[2,1,1,1,1]
=> [5,1]
=> 4
[1,1,1,1,1,1]
=> [6]
=> 5
[6,1]
=> [2,1,1,1,1,1]
=> 1
[5,2]
=> [2,2,1,1,1]
=> 1
[5,1,1]
=> [3,1,1,1,1]
=> 2
[4,3]
=> [2,2,2,1]
=> 1
[4,2,1]
=> [3,2,1,1]
=> 2
[4,1,1,1]
=> [4,1,1,1]
=> 3
[3,3,1]
=> [3,2,2]
=> 2
[3,2,2]
=> [3,3,1]
=> 3
[3,2,1,1]
=> [4,2,1]
=> 3
[3,1,1,1,1]
=> [5,1,1]
=> 4
[2,2,2,1]
=> [4,3]
=> 4
[2,2,1,1,1]
=> [5,2]
=> 4
[2,1,1,1,1,1]
=> [6,1]
=> 5
[7,1]
=> [2,1,1,1,1,1,1]
=> 1
[6,2]
=> [2,2,1,1,1,1]
=> 1
[6,1,1]
=> [3,1,1,1,1,1]
=> 2
[5,3]
=> [2,2,2,1,1]
=> 1
[5,2,1]
=> [3,2,1,1,1]
=> 2
[5,1,1,1]
=> [4,1,1,1,1]
=> 3
[4,4]
=> [2,2,2,2]
=> 1
[4,3,1]
=> [3,2,2,1]
=> 2
[4,2,2]
=> [3,3,1,1]
=> 3
[4,2,1,1]
=> [4,2,1,1]
=> 3
[4,1,1,1,1]
=> [5,1,1,1]
=> 4
[3,3,2]
=> [3,3,2]
=> 3
[3,3,1,1]
=> [4,2,2]
=> 3
[3,2,2,1]
=> [4,3,1]
=> 4
[10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> ? = 1
[9,2]
=> [2,2,1,1,1,1,1,1,1]
=> ? = 1
[9,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> ? = 2
[8,3]
=> [2,2,2,1,1,1,1,1]
=> ? = 1
[8,2,1]
=> [3,2,1,1,1,1,1,1]
=> ? = 2
[8,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> ? = 3
[7,4]
=> [2,2,2,2,1,1,1]
=> ? = 1
[7,3,1]
=> [3,2,2,1,1,1,1]
=> ? = 2
[7,2,2]
=> [3,3,1,1,1,1,1]
=> ? = 3
[7,2,1,1]
=> [4,2,1,1,1,1,1]
=> ? = 3
[7,1,1,1,1]
=> [5,1,1,1,1,1,1]
=> ? = 4
[6,5]
=> [2,2,2,2,2,1]
=> ? = 1
[6,4,1]
=> [3,2,2,2,1,1]
=> ? = 2
[6,3,2]
=> [3,3,2,1,1,1]
=> ? = 3
[6,3,1,1]
=> [4,2,2,1,1,1]
=> ? = 3
[6,2,2,1]
=> [4,3,1,1,1,1]
=> ? = 4
[6,2,1,1,1]
=> [5,2,1,1,1,1]
=> ? = 4
[6,1,1,1,1,1]
=> [6,1,1,1,1,1]
=> ? = 5
[5,5,1]
=> [3,2,2,2,2]
=> ? = 2
[5,4,2]
=> [3,3,2,2,1]
=> ? = 3
[5,4,1,1]
=> [4,2,2,2,1]
=> ? = 3
[5,3,3]
=> [3,3,3,1,1]
=> ? = 3
[5,3,2,1]
=> [4,3,2,1,1]
=> ? = 4
[5,3,1,1,1]
=> [5,2,2,1,1]
=> ? = 4
[5,2,2,2]
=> [4,4,1,1,1]
=> ? = 5
[5,2,2,1,1]
=> [5,3,1,1,1]
=> ? = 5
[5,2,1,1,1,1]
=> [6,2,1,1,1]
=> ? = 5
[4,4,3]
=> [3,3,3,2]
=> ? = 3
[4,4,2,1]
=> [4,3,2,2]
=> ? = 4
[4,4,1,1,1]
=> [5,2,2,2]
=> ? = 4
[4,3,3,1]
=> [4,3,3,1]
=> ? = 4
[4,3,2,2]
=> [4,4,2,1]
=> ? = 5
[4,3,2,1,1]
=> [5,3,2,1]
=> ? = 5
[4,3,1,1,1,1]
=> [6,2,2,1]
=> ? = 5
[4,2,2,2,1]
=> [5,4,1,1]
=> ? = 6
[4,2,2,1,1,1]
=> [6,3,1,1]
=> ? = 6
[3,3,3,2]
=> [4,4,3]
=> ? = 5
[3,3,3,1,1]
=> [5,3,3]
=> ? = 5
[3,3,2,2,1]
=> [5,4,2]
=> ? = 6
[3,3,2,1,1,1]
=> [6,3,2]
=> ? = 6
[3,2,2,2,2]
=> [5,5,1]
=> ? = 7
[3,2,2,2,1,1]
=> [6,4,1]
=> ? = 7
[2,2,2,2,2,1]
=> [6,5]
=> ? = 8
[11,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> ? = 1
[10,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> ? = 2
[9,3]
=> [2,2,2,1,1,1,1,1,1]
=> ? = 1
[9,2,1]
=> [3,2,1,1,1,1,1,1,1]
=> ? = 2
[9,1,1,1]
=> [4,1,1,1,1,1,1,1,1]
=> ? = 3
[8,4]
=> [2,2,2,2,1,1,1,1]
=> ? = 1
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St001087
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001087: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 67%
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001087: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 67%
Values
[1,1]
=> [[1],[2]]
=> [[1,2]]
=> [1,2] => 0 = 1 - 1
[2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> [2,1,3] => 0 = 1 - 1
[1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> [1,2,3] => 1 = 2 - 1
[3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> [2,4,1,3] => 0 = 1 - 1
[2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1 = 2 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 3 - 1
[4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 0 = 1 - 1
[3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> [3,2,5,1,4] => 0 = 1 - 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1 = 2 - 1
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => 1 = 2 - 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 2 = 3 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 3 = 4 - 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 0 = 1 - 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => 0 = 1 - 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 1 = 2 - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => 0 = 1 - 1
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => 1 = 2 - 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => 2 = 3 - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => 2 = 3 - 1
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => 2 = 3 - 1
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => 3 = 4 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 4 = 5 - 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => 0 = 1 - 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [[1,6],[2,7],[3],[4],[5]]
=> [5,4,3,2,7,1,6] => ? = 1 - 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => 1 = 2 - 1
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [[1,5],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5] => ? = 1 - 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => ? = 2 - 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => 2 = 3 - 1
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => ? = 2 - 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[1,4,6],[2,5,7],[3]]
=> [3,2,5,7,1,4,6] => ? = 3 - 1
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,4,6,7],[2,5],[3]]
=> [3,2,5,1,4,6,7] => ? = 3 - 1
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => 3 = 4 - 1
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => ? = 4 - 1
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [[1,3,5,6,7],[2,4]]
=> [2,4,1,3,5,6,7] => 3 = 4 - 1
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => 4 = 5 - 1
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => 0 = 1 - 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [[1,7],[2,8],[3],[4],[5],[6]]
=> [6,5,4,3,2,8,1,7] => ? = 1 - 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => 1 = 2 - 1
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [[1,6],[2,7],[3,8],[4],[5]]
=> [5,4,3,8,2,7,1,6] => ? = 1 - 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [[1,6,8],[2,7],[3],[4],[5]]
=> [5,4,3,2,7,1,6,8] => ? = 2 - 1
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => 2 = 3 - 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8]]
=> [4,8,3,7,2,6,1,5] => ? = 1 - 1
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [[1,5,8],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5,8] => ? = 2 - 1
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [[1,5,7],[2,6,8],[3],[4]]
=> [4,3,2,6,8,1,5,7] => ? = 3 - 1
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [[1,5,7,8],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7,8] => ? = 3 - 1
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => 3 = 4 - 1
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [[1,4,7],[2,5,8],[3,6]]
=> [3,6,2,5,8,1,4,7] => ? = 3 - 1
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [[1,4,7,8],[2,5],[3,6]]
=> [3,6,2,5,1,4,7,8] => ? = 3 - 1
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [[1,4,6,8],[2,5,7],[3]]
=> [3,2,5,7,1,4,6,8] => ? = 4 - 1
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [[1,4,6,7,8],[2,5],[3]]
=> [3,2,5,1,4,6,7,8] => ? = 4 - 1
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => 4 = 5 - 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7] => 4 = 5 - 1
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [[1,3,5,7,8],[2,4,6]]
=> [2,4,6,1,3,5,7,8] => ? = 5 - 1
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [[1,3,5,6,7,8],[2,4]]
=> [2,4,1,3,5,6,7,8] => ? = 5 - 1
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1,9] => 0 = 1 - 1
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> [[1,8],[2,9],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,9,1,8] => ? = 1 - 1
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8,9] => 1 = 2 - 1
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [[1,7],[2,8],[3,9],[4],[5],[6]]
=> [6,5,4,3,9,2,8,1,7] => ? = 1 - 1
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [[1,7,9],[2,8],[3],[4],[5],[6]]
=> [6,5,4,3,2,8,1,7,9] => ? = 2 - 1
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9] => 2 = 3 - 1
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [[1,6],[2,7],[3,8],[4,9],[5]]
=> [5,4,9,3,8,2,7,1,6] => ? = 1 - 1
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [[1,6,9],[2,7],[3,8],[4],[5]]
=> [5,4,3,8,2,7,1,6,9] => ? = 2 - 1
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [[1,6,8],[2,7,9],[3],[4],[5]]
=> [5,4,3,2,7,9,1,6,8] => ? = 3 - 1
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> [[1,6,8,9],[2,7],[3],[4],[5]]
=> [5,4,3,2,7,1,6,8,9] => ? = 3 - 1
[5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => 3 = 4 - 1
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [[1,5,9],[2,6],[3,7],[4,8]]
=> [4,8,3,7,2,6,1,5,9] => ? = 2 - 1
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [[1,5,8],[2,6,9],[3,7],[4]]
=> [4,3,7,2,6,9,1,5,8] => ? = 3 - 1
[4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [[1,5,8,9],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5,8,9] => ? = 3 - 1
[4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [[1,5,7,9],[2,6,8],[3],[4]]
=> [4,3,2,6,8,1,5,7,9] => ? = 4 - 1
[4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [[1,5,7,8,9],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7,8,9] => ? = 4 - 1
[4,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => 4 = 5 - 1
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9]]
=> [3,6,9,2,5,8,1,4,7] => ? = 3 - 1
[3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [[1,4,7,9],[2,5,8],[3,6]]
=> [3,6,2,5,8,1,4,7,9] => ? = 4 - 1
[3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [[1,4,7,8,9],[2,5],[3,6]]
=> [3,6,2,5,1,4,7,8,9] => ? = 4 - 1
[3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [[1,4,6,8],[2,5,7,9],[3]]
=> [3,2,5,7,9,1,4,6,8] => ? = 5 - 1
[3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [[1,4,6,8,9],[2,5,7],[3]]
=> [3,2,5,7,1,4,6,8,9] => ? = 5 - 1
[3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [[1,4,6,7,8,9],[2,5],[3]]
=> [3,2,5,1,4,6,7,8,9] => ? = 5 - 1
[2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [[1,3,5,7,9],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7,9] => ? = 6 - 1
[2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [[1,3,5,7,8,9],[2,4,6]]
=> [2,4,6,1,3,5,7,8,9] => ? = 6 - 1
[9,1]
=> [[1,2,3,4,5,6,7,8,9],[10]]
=> [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,2,1,10] => 0 = 1 - 1
[8,2]
=> [[1,2,3,4,5,6,7,8],[9,10]]
=> [[1,9],[2,10],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,10,1,9] => ? = 1 - 1
[8,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10]]
=> [[1,9,10],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1,9,10] => 1 = 2 - 1
[7,3]
=> [[1,2,3,4,5,6,7],[8,9,10]]
=> [[1,8],[2,9],[3,10],[4],[5],[6],[7]]
=> [7,6,5,4,3,10,2,9,1,8] => ? = 1 - 1
[7,2,1]
=> [[1,2,3,4,5,6,7],[8,9],[10]]
=> [[1,8,10],[2,9],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,9,1,8,10] => ? = 2 - 1
[7,1,1,1]
=> [[1,2,3,4,5,6,7],[8],[9],[10]]
=> [[1,8,9,10],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8,9,10] => 2 = 3 - 1
[6,4]
=> [[1,2,3,4,5,6],[7,8,9,10]]
=> [[1,7],[2,8],[3,9],[4,10],[5],[6]]
=> [6,5,4,10,3,9,2,8,1,7] => ? = 1 - 1
[6,3,1]
=> [[1,2,3,4,5,6],[7,8,9],[10]]
=> [[1,7,10],[2,8],[3,9],[4],[5],[6]]
=> [6,5,4,3,9,2,8,1,7,10] => ? = 2 - 1
[6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> [[1,7,9],[2,8,10],[3],[4],[5],[6]]
=> [6,5,4,3,2,8,10,1,7,9] => ? = 3 - 1
[6,2,1,1]
=> [[1,2,3,4,5,6],[7,8],[9],[10]]
=> [[1,7,9,10],[2,8],[3],[4],[5],[6]]
=> [6,5,4,3,2,8,1,7,9,10] => ? = 3 - 1
[6,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10]]
=> [[1,7,8,9,10],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9,10] => 3 = 4 - 1
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [[1,6],[2,7],[3,8],[4,9],[5,10]]
=> [5,10,4,9,3,8,2,7,1,6] => ? = 1 - 1
[5,4,1]
=> [[1,2,3,4,5],[6,7,8,9],[10]]
=> [[1,6,10],[2,7],[3,8],[4,9],[5]]
=> [5,4,9,3,8,2,7,1,6,10] => ? = 2 - 1
[5,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10]]
=> [[1,6,9],[2,7,10],[3,8],[4],[5]]
=> [5,4,3,8,2,7,10,1,6,9] => ? = 3 - 1
[5,1,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9],[10]]
=> [[1,6,7,8,9,10],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9,10] => 4 = 5 - 1
[2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> [2,4,6,8,10,1,3,5,7,9] => 6 = 7 - 1
Description
The number of occurrences of the vincular pattern |12-3 in a permutation.
This is the number of occurrences of the pattern $123$, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive.
In other words, this is the number of ascents whose bottom value is strictly larger than the first entry of the permutation.
Matching statistic: St001189
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 56%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 56%
Values
[1,1]
=> [[1],[2]]
=> [2,1] => [1,1,0,0]
=> 1
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 2
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 3
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 4
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,1,1,0,1,0,0,0,1,0,1,0]
=> 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [1,1,1,1,0,0,1,0,0,0,1,0]
=> 2
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 3
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> 2
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> 2
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 3
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> 2
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> 3
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> 3
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> 4
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 4
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 4
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 5
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 1
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 2
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 1
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> ? = 2
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? = 3
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> ? = 3
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> ? = 3
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> ? = 3
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 4
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 4
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 5
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 5
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 5
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 5
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [2,1,3,4,5,6,7,8,9] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [3,4,1,2,5,6,7,8,9] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [3,2,1,4,5,6,7,8,9] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 2
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 3
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8,9] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 3
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => [1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 2
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 3
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5,9] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0,1,0]
=> ? = 3
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0,1,0]
=> ? = 4
[4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8,9] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0,1,0]
=> ? = 4
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 5
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? = 3
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6] => [1,1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0,0]
=> ? = 4
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6] => [1,1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0,0]
=> ? = 4
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 5
[3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> ? = 5
[3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6,9] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0]
=> ? = 5
[2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3] => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> ? = 6
[2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> [8,6,4,3,9,2,7,1,5] => [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> ? = 6
[9,1]
=> [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,1,3,4,5,6,7,8,9,10] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> [3,4,1,2,5,6,7,8,9,10] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[8,1,1]
=> [[1,4,5,6,7,8,9,10],[2],[3]]
=> [3,2,1,4,5,6,7,8,9,10] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[7,3]
=> [[1,2,3,7,8,9,10],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9,10] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[7,2,1]
=> [[1,3,6,7,8,9,10],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9,10] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[7,1,1,1]
=> [[1,5,6,7,8,9,10],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9,10] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3
Description
The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000366
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 56%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 56%
Values
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 0 = 1 - 1
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => [1,3,2] => 0 = 1 - 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => 0 = 1 - 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => 0 = 1 - 1
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => 1 = 2 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 2 = 3 - 1
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,5,4] => 0 = 1 - 1
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,4,2,5,3] => 0 = 1 - 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => 1 = 2 - 1
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [3,1,5,4,2] => 1 = 2 - 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => 2 = 3 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 3 = 4 - 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => 0 = 1 - 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [1,2,5,3,6,4] => 0 = 1 - 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [1,2,3,6,5,4] => 1 = 2 - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => 0 = 1 - 1
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [1,4,2,6,5,3] => 1 = 2 - 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [1,2,6,5,4,3] => 2 = 3 - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [5,3,1,6,4,2] => 2 = 3 - 1
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [3,1,6,5,4,2] => 2 = 3 - 1
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [1,6,5,4,3,2] => 3 = 4 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => 4 = 5 - 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 0 = 1 - 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [1,2,3,6,4,7,5] => 0 = 1 - 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => [1,2,3,4,7,6,5] => 1 = 2 - 1
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [1,5,2,6,3,7,4] => ? = 1 - 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => [1,2,5,3,7,6,4] => 1 = 2 - 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => 2 = 3 - 1
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => [4,1,5,2,7,6,3] => ? = 2 - 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [1,6,4,2,7,5,3] => ? = 3 - 1
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => [1,4,2,7,6,5,3] => 2 = 3 - 1
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => [1,2,7,6,5,4,3] => 3 = 4 - 1
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => [5,3,1,7,6,4,2] => ? = 4 - 1
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => [3,1,7,6,5,4,2] => ? = 4 - 1
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => [1,7,6,5,4,3,2] => 4 = 5 - 1
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,8,7] => 0 = 1 - 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => [1,2,3,4,7,5,8,6] => ? = 1 - 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => [1,2,3,4,5,8,7,6] => ? = 2 - 1
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => [1,2,6,3,7,4,8,5] => ? = 1 - 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => [1,2,3,6,4,8,7,5] => ? = 2 - 1
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [8,7,6,1,2,3,4,5] => [1,2,3,4,8,7,6,5] => ? = 3 - 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,8,4] => 0 = 1 - 1
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => [1,5,2,6,3,8,7,4] => ? = 2 - 1
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => [1,2,7,5,3,8,6,4] => ? = 3 - 1
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => [1,2,5,3,8,7,6,4] => ? = 3 - 1
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [8,7,6,5,1,2,3,4] => [1,2,3,8,7,6,5,4] => ? = 4 - 1
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => [4,1,7,5,2,8,6,3] => ? = 3 - 1
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => [4,1,5,2,8,7,6,3] => ? = 3 - 1
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => [1,6,4,2,8,7,5,3] => ? = 4 - 1
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => [1,4,2,8,7,6,5,3] => ? = 4 - 1
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,1,2,3] => [1,2,8,7,6,5,4,3] => ? = 5 - 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [7,5,3,1,8,6,4,2] => 4 = 5 - 1
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => [5,3,1,8,7,6,4,2] => ? = 5 - 1
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => [3,1,8,7,6,5,4,2] => ? = 5 - 1
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> [9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,9,8] => 0 = 1 - 1
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> [8,9,1,2,3,4,5,6,7] => [1,2,3,4,5,8,6,9,7] => ? = 1 - 1
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [9,8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,9,8,7] => ? = 2 - 1
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [7,8,9,1,2,3,4,5,6] => [1,2,3,7,4,8,5,9,6] => ? = 1 - 1
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [9,7,8,1,2,3,4,5,6] => [1,2,3,4,7,5,9,8,6] => ? = 2 - 1
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> [9,8,7,1,2,3,4,5,6] => [1,2,3,4,5,9,8,7,6] => ? = 3 - 1
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [6,7,8,9,1,2,3,4,5] => [1,6,2,7,3,8,4,9,5] => ? = 1 - 1
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [9,6,7,8,1,2,3,4,5] => [1,2,6,3,7,4,9,8,5] => ? = 2 - 1
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [8,9,6,7,1,2,3,4,5] => [1,2,3,8,6,4,9,7,5] => ? = 3 - 1
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> [9,8,6,7,1,2,3,4,5] => [1,2,3,6,4,9,8,7,5] => ? = 3 - 1
[5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [9,8,7,6,1,2,3,4,5] => [1,2,3,4,9,8,7,6,5] => ? = 4 - 1
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [9,5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,9,8,4] => ? = 2 - 1
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [8,9,5,6,7,1,2,3,4] => [1,5,2,8,6,3,9,7,4] => ? = 3 - 1
[4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [9,8,5,6,7,1,2,3,4] => [1,5,2,6,3,9,8,7,4] => ? = 3 - 1
[4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,1,2,3,4] => [1,2,7,5,3,9,8,6,4] => ? = 4 - 1
[4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,1,2,3,4] => [1,2,5,3,9,8,7,6,4] => ? = 4 - 1
[4,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,1,2,3,4] => [1,2,3,9,8,7,6,5,4] => ? = 5 - 1
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [7,4,1,8,5,2,9,6,3] => ? = 3 - 1
[3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [9,7,8,4,5,6,1,2,3] => [4,1,7,5,2,9,8,6,3] => ? = 4 - 1
[3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [9,8,7,4,5,6,1,2,3] => [4,1,5,2,9,8,7,6,3] => ? = 4 - 1
[3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [8,9,6,7,4,5,1,2,3] => [1,8,6,4,2,9,7,5,3] => ? = 5 - 1
[3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [9,8,6,7,4,5,1,2,3] => [1,6,4,2,9,8,7,5,3] => ? = 5 - 1
[3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [9,8,7,6,4,5,1,2,3] => [1,4,2,9,8,7,6,5,3] => ? = 5 - 1
[2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,3,4,1,2] => [7,5,3,1,9,8,6,4,2] => ? = 6 - 1
[2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,3,4,1,2] => [5,3,1,9,8,7,6,4,2] => ? = 6 - 1
[9,1]
=> [[1,2,3,4,5,6,7,8,9],[10]]
=> [10,1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,10,9] => 0 = 1 - 1
[8,2]
=> [[1,2,3,4,5,6,7,8],[9,10]]
=> [9,10,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,9,7,10,8] => ? = 1 - 1
[8,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10]]
=> [10,9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,10,9,8] => ? = 2 - 1
[7,3]
=> [[1,2,3,4,5,6,7],[8,9,10]]
=> [8,9,10,1,2,3,4,5,6,7] => [1,2,3,4,8,5,9,6,10,7] => ? = 1 - 1
[7,2,1]
=> [[1,2,3,4,5,6,7],[8,9],[10]]
=> [10,8,9,1,2,3,4,5,6,7] => [1,2,3,4,5,8,6,10,9,7] => ? = 2 - 1
[7,1,1,1]
=> [[1,2,3,4,5,6,7],[8],[9],[10]]
=> [10,9,8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,10,9,8,7] => ? = 3 - 1
Description
The number of double descents of a permutation.
A double descent of a permutation $\pi$ is a position $i$ such that $\pi(i) > \pi(i+1) > \pi(i+2)$.
Matching statistic: St000371
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 56%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 56%
Values
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 0 = 1 - 1
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => [1,3,2] => 0 = 1 - 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => 0 = 1 - 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => 0 = 1 - 1
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => 1 = 2 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 2 = 3 - 1
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,5,4] => 0 = 1 - 1
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,4,2,5,3] => 0 = 1 - 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => 1 = 2 - 1
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [3,1,5,4,2] => 1 = 2 - 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => 2 = 3 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 3 = 4 - 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => 0 = 1 - 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [1,2,5,3,6,4] => 0 = 1 - 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [1,2,3,6,5,4] => 1 = 2 - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => 0 = 1 - 1
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [1,4,2,6,5,3] => 1 = 2 - 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [1,2,6,5,4,3] => 2 = 3 - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [5,3,1,6,4,2] => 2 = 3 - 1
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [3,1,6,5,4,2] => 2 = 3 - 1
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [1,6,5,4,3,2] => 3 = 4 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => 4 = 5 - 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 0 = 1 - 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [1,2,3,6,4,7,5] => 0 = 1 - 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => [1,2,3,4,7,6,5] => 1 = 2 - 1
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [1,5,2,6,3,7,4] => ? = 1 - 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => [1,2,5,3,7,6,4] => 1 = 2 - 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => 2 = 3 - 1
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => [4,1,5,2,7,6,3] => ? = 2 - 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [1,6,4,2,7,5,3] => ? = 3 - 1
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => [1,4,2,7,6,5,3] => 2 = 3 - 1
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => [1,2,7,6,5,4,3] => 3 = 4 - 1
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => [5,3,1,7,6,4,2] => ? = 4 - 1
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => [3,1,7,6,5,4,2] => ? = 4 - 1
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => [1,7,6,5,4,3,2] => 4 = 5 - 1
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,8,7] => 0 = 1 - 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => [1,2,3,4,7,5,8,6] => ? = 1 - 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => [1,2,3,4,5,8,7,6] => ? = 2 - 1
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => [1,2,6,3,7,4,8,5] => ? = 1 - 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => [1,2,3,6,4,8,7,5] => ? = 2 - 1
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [8,7,6,1,2,3,4,5] => [1,2,3,4,8,7,6,5] => ? = 3 - 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,8,4] => 0 = 1 - 1
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => [1,5,2,6,3,8,7,4] => ? = 2 - 1
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => [1,2,7,5,3,8,6,4] => ? = 3 - 1
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => [1,2,5,3,8,7,6,4] => ? = 3 - 1
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [8,7,6,5,1,2,3,4] => [1,2,3,8,7,6,5,4] => ? = 4 - 1
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => [4,1,7,5,2,8,6,3] => ? = 3 - 1
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => [4,1,5,2,8,7,6,3] => ? = 3 - 1
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => [1,6,4,2,8,7,5,3] => ? = 4 - 1
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => [1,4,2,8,7,6,5,3] => ? = 4 - 1
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,1,2,3] => [1,2,8,7,6,5,4,3] => ? = 5 - 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [7,5,3,1,8,6,4,2] => 4 = 5 - 1
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => [5,3,1,8,7,6,4,2] => ? = 5 - 1
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => [3,1,8,7,6,5,4,2] => ? = 5 - 1
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> [9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,9,8] => 0 = 1 - 1
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> [8,9,1,2,3,4,5,6,7] => [1,2,3,4,5,8,6,9,7] => ? = 1 - 1
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [9,8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,9,8,7] => ? = 2 - 1
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [7,8,9,1,2,3,4,5,6] => [1,2,3,7,4,8,5,9,6] => ? = 1 - 1
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [9,7,8,1,2,3,4,5,6] => [1,2,3,4,7,5,9,8,6] => ? = 2 - 1
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> [9,8,7,1,2,3,4,5,6] => [1,2,3,4,5,9,8,7,6] => ? = 3 - 1
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [6,7,8,9,1,2,3,4,5] => [1,6,2,7,3,8,4,9,5] => ? = 1 - 1
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [9,6,7,8,1,2,3,4,5] => [1,2,6,3,7,4,9,8,5] => ? = 2 - 1
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [8,9,6,7,1,2,3,4,5] => [1,2,3,8,6,4,9,7,5] => ? = 3 - 1
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> [9,8,6,7,1,2,3,4,5] => [1,2,3,6,4,9,8,7,5] => ? = 3 - 1
[5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [9,8,7,6,1,2,3,4,5] => [1,2,3,4,9,8,7,6,5] => ? = 4 - 1
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [9,5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,9,8,4] => ? = 2 - 1
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [8,9,5,6,7,1,2,3,4] => [1,5,2,8,6,3,9,7,4] => ? = 3 - 1
[4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [9,8,5,6,7,1,2,3,4] => [1,5,2,6,3,9,8,7,4] => ? = 3 - 1
[4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,1,2,3,4] => [1,2,7,5,3,9,8,6,4] => ? = 4 - 1
[4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,1,2,3,4] => [1,2,5,3,9,8,7,6,4] => ? = 4 - 1
[4,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,1,2,3,4] => [1,2,3,9,8,7,6,5,4] => ? = 5 - 1
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [7,4,1,8,5,2,9,6,3] => ? = 3 - 1
[3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [9,7,8,4,5,6,1,2,3] => [4,1,7,5,2,9,8,6,3] => ? = 4 - 1
[3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [9,8,7,4,5,6,1,2,3] => [4,1,5,2,9,8,7,6,3] => ? = 4 - 1
[3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [8,9,6,7,4,5,1,2,3] => [1,8,6,4,2,9,7,5,3] => ? = 5 - 1
[3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [9,8,6,7,4,5,1,2,3] => [1,6,4,2,9,8,7,5,3] => ? = 5 - 1
[3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [9,8,7,6,4,5,1,2,3] => [1,4,2,9,8,7,6,5,3] => ? = 5 - 1
[2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,3,4,1,2] => [7,5,3,1,9,8,6,4,2] => ? = 6 - 1
[2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,3,4,1,2] => [5,3,1,9,8,7,6,4,2] => ? = 6 - 1
[9,1]
=> [[1,2,3,4,5,6,7,8,9],[10]]
=> [10,1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,10,9] => 0 = 1 - 1
[8,2]
=> [[1,2,3,4,5,6,7,8],[9,10]]
=> [9,10,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,9,7,10,8] => ? = 1 - 1
[8,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10]]
=> [10,9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,10,9,8] => ? = 2 - 1
[7,3]
=> [[1,2,3,4,5,6,7],[8,9,10]]
=> [8,9,10,1,2,3,4,5,6,7] => [1,2,3,4,8,5,9,6,10,7] => ? = 1 - 1
[7,2,1]
=> [[1,2,3,4,5,6,7],[8,9],[10]]
=> [10,8,9,1,2,3,4,5,6,7] => [1,2,3,4,5,8,6,10,9,7] => ? = 2 - 1
[7,1,1,1]
=> [[1,2,3,4,5,6,7],[8],[9],[10]]
=> [10,9,8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,10,9,8,7] => ? = 3 - 1
Description
The number of mid points of decreasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima.
This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence.
See also [[St000119]].
Matching statistic: St001336
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001336: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 56%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001336: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 56%
Values
[1,1]
=> [[1],[2]]
=> [2,1] => ([(0,1)],2)
=> 0 = 1 - 1
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => ([(1,2)],3)
=> 0 = 1 - 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => ([(2,3)],4)
=> 0 = 1 - 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 0 = 1 - 1
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 1 - 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => ([(4,5)],6)
=> 0 = 1 - 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 0 = 1 - 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ([(5,6)],7)
=> 0 = 1 - 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => ([(3,5),(3,6),(4,5),(4,6)],7)
=> 0 = 1 - 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ([(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 0 = 1 - 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(4,5),(5,6)],7)
=> 1 = 2 - 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 2 = 3 - 1
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => ([(0,1),(0,6),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => ([(6,7)],8)
=> ? = 1 - 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ([(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 1 - 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => ([(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 1 - 1
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => ([(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 2 - 1
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ? = 1 - 1
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => ([(1,6),(1,7),(2,4),(2,5),(2,7),(3,4),(3,5),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 2 - 1
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => ([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3 - 1
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => ([(2,3),(2,7),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => ([(0,1),(0,6),(0,7),(1,4),(1,5),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3 - 1
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => ([(0,2),(0,3),(0,7),(1,2),(1,3),(1,7),(2,6),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => ([(1,2),(1,4),(1,5),(1,7),(2,3),(2,5),(2,6),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => ([(1,2),(1,7),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 5 - 1
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => ([(0,1),(0,7),(1,6),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 - 1
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [2,1,3,4,5,6,7,8,9] => ([(7,8)],9)
=> ? = 1 - 1
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [3,4,1,2,5,6,7,8,9] => ([(5,7),(5,8),(6,7),(6,8)],9)
=> ? = 1 - 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [3,2,1,4,5,6,7,8,9] => ([(6,7),(6,8),(7,8)],9)
=> ? = 2 - 1
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => ([(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ? = 1 - 1
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9] => ([(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 2 - 1
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => ([(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => ([(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8)],9)
=> ? = 1 - 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => ([(2,7),(2,8),(3,5),(3,6),(3,8),(4,5),(4,6),(4,8),(5,7),(6,7),(7,8)],9)
=> ? = 2 - 1
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => ([(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
=> ? = 3 - 1
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8,9] => ([(3,4),(3,8),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => ([(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 - 1
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => ([(0,7),(0,8),(1,4),(1,5),(1,6),(1,8),(2,4),(2,5),(2,6),(2,8),(3,4),(3,5),(3,6),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ? = 2 - 1
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9] => ([(1,2),(1,7),(1,8),(2,5),(2,6),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
=> ? = 3 - 1
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5,9] => ([(1,3),(1,4),(1,8),(2,3),(2,4),(2,8),(3,7),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9] => ([(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 - 1
[4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8,9] => ([(2,3),(2,8),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 - 1
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9] => ([(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ? = 3 - 1
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6] => ([(0,1),(0,6),(0,8),(1,5),(1,7),(2,3),(2,5),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 4 - 1
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6] => ([(0,2),(0,3),(0,8),(1,2),(1,3),(1,8),(2,7),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 - 1
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
=> ? = 5 - 1
[3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9] => ([(1,2),(1,3),(1,7),(1,8),(2,3),(2,6),(2,8),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6,9] => ([(1,2),(1,8),(2,7),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(0,8),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 6 - 1
[2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> [8,6,4,3,9,2,7,1,5] => ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,6),(1,8),(2,5),(2,7),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 6 - 1
[9,1]
=> [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,1,3,4,5,6,7,8,9,10] => ([(8,9)],10)
=> ? = 1 - 1
[8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> [3,4,1,2,5,6,7,8,9,10] => ([(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 1 - 1
[8,1,1]
=> [[1,4,5,6,7,8,9,10],[2],[3]]
=> [3,2,1,4,5,6,7,8,9,10] => ([(7,8),(7,9),(8,9)],10)
=> ? = 2 - 1
[7,3]
=> [[1,2,3,7,8,9,10],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9,10] => ([(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9)],10)
=> ? = 1 - 1
Description
The minimal number of vertices in a graph whose complement is triangle-free.
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000651The maximal size of a rise in a permutation. St000356The number of occurrences of the pattern 13-2. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St000836The number of descents of distance 2 of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000653The last descent of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000652The maximal difference between successive positions of a permutation. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000837The number of ascents of distance 2 of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St000454The largest eigenvalue of a graph if it is integral. St001812The biclique partition number of a graph. St001330The hat guessing number of a graph.
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