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Your data matches 73 different statistics following compositions of up to 3 maps.
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Matching statistic: St001440
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001440: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001440: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[3,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
[7,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[6,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[5,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
[5,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
Description
The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition.
Matching statistic: St001227
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 18%●distinct values known / distinct values provided: 6%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 18%●distinct values known / distinct values provided: 6%
Values
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 0 + 3
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 0 + 3
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 4 = 1 + 3
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 0 + 3
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 0 + 3
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 4 = 1 + 3
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 3
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 0 + 3
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 0 + 3
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 4 = 1 + 3
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 0 + 3
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 4 = 1 + 3
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 3
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 3
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 0 + 3
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 0 + 3
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 4 = 1 + 3
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 0 + 3
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 4 = 1 + 3
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 3
[2,2,2,2]
=> [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]
=> 3 = 0 + 3
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 4 = 1 + 3
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0 + 3
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 3
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 3
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 0 + 3
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 0 + 3
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 4 = 1 + 3
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 0 + 3
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 4 = 1 + 3
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 3
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 0 + 3
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 4 = 1 + 3
[3,2,2,2]
=> [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]
=> 3 = 0 + 3
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 4 = 1 + 3
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0 + 3
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 3
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 4 = 1 + 3
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 0 + 3
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0 + 3
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 3
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 3
[7,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 0 + 3
[6,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 0 + 3
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 4 = 1 + 3
[5,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[5,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 0 + 3
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 4 = 1 + 3
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 3
[4,4,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 3 = 0 + 3
[4,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 0 + 3
[4,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 4 = 1 + 3
[4,2,2,2]
=> [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]
=> 3 = 0 + 3
[4,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 4 = 1 + 3
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0 + 3
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 3
[3,3,3,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 3 = 0 + 3
[3,3,2,2]
=> [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]
=> 3 = 0 + 3
[3,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 4 = 1 + 3
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 0 + 3
[3,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 4 = 1 + 3
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 0 + 3
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0 + 3
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 3
[2,2,2,2,2]
=> [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]
=> ? = 0 + 3
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 1 + 3
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0 + 3
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0 + 3
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 3
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 3
[8,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 0 + 3
[7,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 0 + 3
[7,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 4 = 1 + 3
[6,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[6,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 0 + 3
[6,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 3
[5,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0 + 3
[5,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 3
[4,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 0 + 3
[4,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 0 + 3
[4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0 + 3
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 3
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 0 + 3
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 0 + 3
[3,2,2,2,2]
=> [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]
=> ? = 0 + 3
[3,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 1 + 3
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0 + 3
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0 + 3
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 3
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 3
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 3
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 3
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 3
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 3
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 3
[7,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 3
[6,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0 + 3
[6,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 3
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St001490
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 3% ●values known / values provided: 16%●distinct values known / distinct values provided: 3%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 3% ●values known / values provided: 16%●distinct values known / distinct values provided: 3%
Values
[1,1,1,1]
=> 1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> 1 = 0 + 1
[2,1,1,1]
=> 0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> 1 = 0 + 1
[1,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ? = 1 + 1
[3,1,1,1]
=> 1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> 1 = 0 + 1
[2,2,1,1]
=> 0011 => [3,1,1] => [[3,3,3],[2,2]]
=> 1 = 0 + 1
[2,1,1,1,1]
=> 01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> ? = 1 + 1
[1,1,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> ? = 0 + 1
[4,1,1,1]
=> 0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> 1 = 0 + 1
[3,2,1,1]
=> 1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> 1 = 0 + 1
[3,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ? = 1 + 1
[2,2,2,1]
=> 0001 => [4,1] => [[4,4],[3]]
=> 1 = 0 + 1
[2,2,1,1,1]
=> 00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ? = 1 + 1
[2,1,1,1,1,1]
=> 011111 => [2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
=> ? = 0 + 1
[1,1,1,1,1,1,1]
=> 1111111 => [1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1],[]]
=> ? = 0 + 1
[5,1,1,1]
=> 1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> 1 = 0 + 1
[4,2,1,1]
=> 0011 => [3,1,1] => [[3,3,3],[2,2]]
=> 1 = 0 + 1
[4,1,1,1,1]
=> 01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> ? = 1 + 1
[3,3,1,1]
=> 1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> 1 = 0 + 1
[3,2,2,1]
=> 1001 => [1,3,1] => [[3,3,1],[2]]
=> 1 = 0 + 1
[3,2,1,1,1]
=> 10111 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> ? = 1 + 1
[3,1,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> ? = 0 + 1
[2,2,2,2]
=> 0000 => [5] => [[5],[]]
=> 1 = 0 + 1
[2,2,2,1,1]
=> 00011 => [4,1,1] => [[4,4,4],[3,3]]
=> ? = 1 + 1
[2,2,1,1,1,1]
=> 001111 => [3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
=> ? = 0 + 1
[2,1,1,1,1,1,1]
=> 0111111 => [2,1,1,1,1,1,1] => [[2,2,2,2,2,2,2],[1,1,1,1,1,1]]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1]
=> 11111111 => [1,1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1,1],[]]
=> ? = 0 + 1
[6,1,1,1]
=> 0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> 1 = 0 + 1
[5,2,1,1]
=> 1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> 1 = 0 + 1
[5,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ? = 1 + 1
[4,3,1,1]
=> 0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> 1 = 0 + 1
[4,2,2,1]
=> 0001 => [4,1] => [[4,4],[3]]
=> 1 = 0 + 1
[4,2,1,1,1]
=> 00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ? = 1 + 1
[4,1,1,1,1,1]
=> 011111 => [2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
=> ? = 0 + 1
[3,3,2,1]
=> 1101 => [1,1,2,1] => [[2,2,1,1],[1]]
=> 1 = 0 + 1
[3,3,1,1,1]
=> 11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ? = 1 + 1
[3,2,2,2]
=> 1000 => [1,4] => [[4,1],[]]
=> 1 = 0 + 1
[3,2,2,1,1]
=> 10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> ? = 1 + 1
[3,2,1,1,1,1]
=> 101111 => [1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
=> ? = 0 + 1
[3,1,1,1,1,1,1]
=> 1111111 => [1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1],[]]
=> ? = 0 + 1
[2,2,2,2,1]
=> 00001 => [5,1] => [[5,5],[4]]
=> ? = 1 + 1
[2,2,2,1,1,1]
=> 000111 => [4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> ? = 0 + 1
[2,2,1,1,1,1,1]
=> 0011111 => [3,1,1,1,1,1] => [[3,3,3,3,3,3],[2,2,2,2,2]]
=> ? = 0 + 1
[2,1,1,1,1,1,1,1]
=> 01111111 => [2,1,1,1,1,1,1,1] => [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1]]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,1]
=> 111111111 => [1,1,1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1,1,1],[]]
=> ? = 0 + 1
[7,1,1,1]
=> 1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> 1 = 0 + 1
[6,2,1,1]
=> 0011 => [3,1,1] => [[3,3,3],[2,2]]
=> 1 = 0 + 1
[6,1,1,1,1]
=> 01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> ? = 1 + 1
[5,3,1,1]
=> 1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> 1 = 0 + 1
[5,2,2,1]
=> 1001 => [1,3,1] => [[3,3,1],[2]]
=> 1 = 0 + 1
[5,2,1,1,1]
=> 10111 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> ? = 1 + 1
[5,1,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> ? = 0 + 1
[4,4,1,1]
=> 0011 => [3,1,1] => [[3,3,3],[2,2]]
=> 1 = 0 + 1
[4,3,2,1]
=> 0101 => [2,2,1] => [[3,3,2],[2,1]]
=> 1 = 0 + 1
[4,3,1,1,1]
=> 01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> ? = 1 + 1
[4,2,2,2]
=> 0000 => [5] => [[5],[]]
=> 1 = 0 + 1
[4,2,2,1,1]
=> 00011 => [4,1,1] => [[4,4,4],[3,3]]
=> ? = 1 + 1
[4,2,1,1,1,1]
=> 001111 => [3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
=> ? = 0 + 1
[4,1,1,1,1,1,1]
=> 0111111 => [2,1,1,1,1,1,1] => [[2,2,2,2,2,2,2],[1,1,1,1,1,1]]
=> ? = 0 + 1
[3,3,3,1]
=> 1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> 1 = 0 + 1
[3,3,2,2]
=> 1100 => [1,1,3] => [[3,1,1],[]]
=> 1 = 0 + 1
[3,3,2,1,1]
=> 11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ? = 1 + 1
[3,3,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> ? = 0 + 1
[3,2,2,2,1]
=> 10001 => [1,4,1] => [[4,4,1],[3]]
=> ? = 1 + 1
[3,2,2,1,1,1]
=> 100111 => [1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
=> ? = 0 + 1
[3,2,1,1,1,1,1]
=> 1011111 => [1,2,1,1,1,1,1] => [[2,2,2,2,2,2,1],[1,1,1,1,1]]
=> ? = 0 + 1
[3,1,1,1,1,1,1,1]
=> 11111111 => [1,1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1,1],[]]
=> ? = 0 + 1
[2,2,2,2,2]
=> 00000 => [6] => [[6],[]]
=> ? = 0 + 1
[2,2,2,2,1,1]
=> 000011 => [5,1,1] => [[5,5,5],[4,4]]
=> ? = 1 + 1
[2,2,2,1,1,1,1]
=> 0001111 => [4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]]
=> ? = 0 + 1
[2,2,1,1,1,1,1,1]
=> 00111111 => [3,1,1,1,1,1,1] => [[3,3,3,3,3,3,3],[2,2,2,2,2,2]]
=> ? = 0 + 1
[2,1,1,1,1,1,1,1,1]
=> 011111111 => [2,1,1,1,1,1,1,1,1] => [[2,2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1,1]]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,1,1]
=> 1111111111 => [1,1,1,1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1,1,1,1],[]]
=> ? = 0 + 1
[8,1,1,1]
=> 0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> 1 = 0 + 1
[7,2,1,1]
=> 1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> 1 = 0 + 1
[7,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ? = 1 + 1
[6,3,1,1]
=> 0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> 1 = 0 + 1
[6,2,2,1]
=> 0001 => [4,1] => [[4,4],[3]]
=> 1 = 0 + 1
[6,2,1,1,1]
=> 00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ? = 1 + 1
[6,1,1,1,1,1]
=> 011111 => [2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
=> ? = 0 + 1
[5,4,1,1]
=> 1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> 1 = 0 + 1
[5,3,2,1]
=> 1101 => [1,1,2,1] => [[2,2,1,1],[1]]
=> 1 = 0 + 1
[5,3,1,1,1]
=> 11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ? = 1 + 1
[5,2,2,2]
=> 1000 => [1,4] => [[4,1],[]]
=> 1 = 0 + 1
[5,2,2,1,1]
=> 10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> ? = 1 + 1
[4,4,2,1]
=> 0001 => [4,1] => [[4,4],[3]]
=> 1 = 0 + 1
[4,3,3,1]
=> 0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> 1 = 0 + 1
[4,3,2,2]
=> 0100 => [2,3] => [[4,2],[1]]
=> 1 = 0 + 1
[3,3,3,2]
=> 1110 => [1,1,1,2] => [[2,1,1,1],[]]
=> 1 = 0 + 1
[9,1,1,1]
=> 1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> 1 = 0 + 1
[8,2,1,1]
=> 0011 => [3,1,1] => [[3,3,3],[2,2]]
=> 1 = 0 + 1
[7,3,1,1]
=> 1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> 1 = 0 + 1
[7,2,2,1]
=> 1001 => [1,3,1] => [[3,3,1],[2]]
=> 1 = 0 + 1
[6,4,1,1]
=> 0011 => [3,1,1] => [[3,3,3],[2,2]]
=> 1 = 0 + 1
[6,3,2,1]
=> 0101 => [2,2,1] => [[3,3,2],[2,1]]
=> 1 = 0 + 1
[6,2,2,2]
=> 0000 => [5] => [[5],[]]
=> 1 = 0 + 1
[5,5,1,1]
=> 1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> 1 = 0 + 1
[5,4,2,1]
=> 1001 => [1,3,1] => [[3,3,1],[2]]
=> 1 = 0 + 1
[5,3,3,1]
=> 1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> 1 = 0 + 1
[5,3,2,2]
=> 1100 => [1,1,3] => [[3,1,1],[]]
=> 1 = 0 + 1
[4,4,3,1]
=> 0011 => [3,1,1] => [[3,3,3],[2,2]]
=> 1 = 0 + 1
Description
The number of connected components of a skew partition.
Matching statistic: St000956
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000956: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 8%●distinct values known / distinct values provided: 6%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000956: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 8%●distinct values known / distinct values provided: 6%
Values
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 4 = 0 + 4
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,2,1,3,4] => 4 = 0 + 4
[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] => 5 = 1 + 4
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,1,3,2,4] => 4 = 0 + 4
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,1,2,4] => 4 = 0 + 4
[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] => 5 = 1 + 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] => ? = 0 + 4
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,1,2,4,3] => 4 = 0 + 4
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,3,2,1,4] => 4 = 0 + 4
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [6,1,3,2,4,5] => 5 = 1 + 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 = 0 + 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] => 5 = 1 + 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] => ? = 0 + 4
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [8,1,2,3,4,5,6,7] => ? = 0 + 4
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => [5,1,2,3,6,4] => 4 = 0 + 4
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,3,1,4,2] => 4 = 0 + 4
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,6,1] => [6,1,2,4,3,5] => 5 = 1 + 4
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,1,4,2,3] => 4 = 0 + 4
[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 = 0 + 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] => 5 = 1 + 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] => [7,1,3,2,4,5,6] => ? = 0 + 4
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [1,6,2,3,4,5] => 4 = 0 + 4
[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 = 1 + 4
[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] => ? = 0 + 4
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => [8,2,1,3,4,5,6,7] => ? = 0 + 4
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => [9,1,2,3,4,5,6,7,8] => ? = 0 + 4
[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] => [5,1,2,3,4,7,6] => ? = 0 + 4
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,1,5] => [5,3,1,2,6,4] => 4 = 0 + 4
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [6,1,2,3,5,4] => 5 = 1 + 4
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [5,1,4,3,2] => 4 = 0 + 4
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [5,4,1,3,2] => 4 = 0 + 4
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [5,3,2,4,6,1] => [6,3,1,4,2,5] => 5 = 1 + 4
[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] => [7,1,2,4,3,5,6] => ? = 0 + 4
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,4,3,1,2] => 4 = 0 + 4
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,6,1] => [6,1,4,2,3,5] => 5 = 1 + 4
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,1,2] => [1,6,3,2,4,5] => 4 = 0 + 4
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,6,1] => [6,4,2,1,3,5] => 5 = 1 + 4
[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] => ? = 0 + 4
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [4,2,3,5,6,7,8,1] => [8,1,3,2,4,5,6,7] => ? = 0 + 4
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [6,5,1,2,3,4] => 5 = 1 + 4
[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] => ? = 0 + 4
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,1] => [8,3,1,2,4,5,6,7] => ? = 0 + 4
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,1] => [9,2,1,3,4,5,6,7,8] => ? = 0 + 4
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [10,1,2,3,4,5,6,7,8,9] => ? = 0 + 4
[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] => [5,1,2,3,4,6,8,7] => ? = 0 + 4
[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] => [5,3,1,2,4,7,6] => ? = 0 + 4
[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] => [6,1,2,3,4,7,5] => ? = 1 + 4
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [6,4,2,3,1,5] => [5,1,4,2,6,3] => 4 = 0 + 4
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,1,5] => [5,4,1,2,6,3] => 4 = 0 + 4
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,5,1] => [6,3,1,2,5,4] => 5 = 1 + 4
[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] => [7,1,2,3,5,4,6] => ? = 0 + 4
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,1,4] => [5,1,2,6,3,4] => 4 = 0 + 4
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => 4 = 0 + 4
[4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,3,6,1] => [6,1,4,3,2,5] => 5 = 1 + 4
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,1,2] => [1,6,2,4,3,5] => 4 = 0 + 4
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,6,1] => [6,4,1,3,2,5] => 5 = 1 + 4
[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] => [7,3,1,4,2,5,6] => ? = 0 + 4
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [5,2,3,4,6,7,8,1] => [8,1,2,4,3,5,6,7] => ? = 0 + 4
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,1,3] => [5,1,6,2,3,4] => 4 = 0 + 4
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,1,2] => [1,6,4,2,3,5] => 4 = 0 + 4
[3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,5,3,2,6,1] => [6,4,3,1,2,5] => 5 = 1 + 4
[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] => [7,1,4,2,3,5,6] => ? = 0 + 4
[3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => [6,5,2,1,3,4] => 5 = 1 + 4
[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] => ? = 0 + 4
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,5,6,7,8,1] => [8,3,2,1,4,5,6,7] => ? = 0 + 4
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [4,2,3,5,6,7,8,9,1] => [9,1,3,2,4,5,6,7,8] => ? = 0 + 4
[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] => [1,7,2,3,4,5,6] => ? = 0 + 4
[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] => ? = 1 + 4
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [3,4,5,2,6,7,8,1] => [8,4,1,2,3,5,6,7] => ? = 0 + 4
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,1] => [9,3,1,2,4,5,6,7,8] => ? = 0 + 4
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => [10,2,1,3,4,5,6,7,8,9] => ? = 0 + 4
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [11,1,2,3,4,5,6,7,8,9,10] => ? = 0 + 4
[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] => [5,1,2,3,4,6,7,9,8] => ? = 0 + 4
[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] => ? => ? = 0 + 4
[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] => [6,1,2,3,4,5,8,7] => ? = 1 + 4
[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] => [5,1,4,2,3,7,6] => ? = 0 + 4
[6,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [7,3,4,2,1,5,6] => [5,4,1,2,3,7,6] => ? = 0 + 4
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,4,5,1,6] => [6,3,1,2,4,7,5] => ? = 1 + 4
[6,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,6,1] => [7,1,2,3,4,6,5] => ? = 0 + 4
[5,4,1,1]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,1,4] => [5,1,2,6,4,3] => 4 = 0 + 4
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,5] => [5,4,3,1,6,2] => 4 = 0 + 4
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,3,5,1] => [6,1,4,2,5,3] => 5 = 1 + 4
[5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,1,2] => [1,6,2,3,5,4] => 4 = 0 + 4
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,5,1] => [6,4,1,2,5,3] => 5 = 1 + 4
[5,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [6,3,2,4,5,7,1] => [7,3,1,2,5,4,6] => ? = 0 + 4
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [6,2,3,4,5,7,8,1] => [8,1,2,3,5,4,6,7] => ? = 0 + 4
[4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,6,3,2,1,4] => [5,4,1,6,2,3] => 4 = 0 + 4
[4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => [6,1,2,5,3,4] => 5 = 1 + 4
[4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,1,3] => [5,1,6,3,2,4] => 4 = 0 + 4
[4,3,2,2]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,1,2] => [1,6,4,3,2,5] => 4 = 0 + 4
[4,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [5,4,2,3,6,7,1] => [7,1,4,3,2,5,6] => ? = 0 + 4
[4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [5,3,4,2,6,7,1] => [7,4,1,3,2,5,6] => ? = 0 + 4
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,3,2,4,6,7,8,1] => [8,3,1,4,2,5,6,7] => ? = 0 + 4
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [5,2,3,4,6,7,8,9,1] => ? => ? = 0 + 4
[3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,6,7,1] => [7,4,3,1,2,5,6] => ? = 0 + 4
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [4,5,2,3,6,7,8,1] => [8,1,4,2,3,5,6,7] => ? = 0 + 4
[3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,1,2] => [1,7,3,2,4,5,6] => ? = 0 + 4
[3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,7,1] => [7,5,2,1,3,4,6] => ? = 1 + 4
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,3,5,2,6,7,8,1] => [8,4,2,1,3,5,6,7] => ? = 0 + 4
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [4,3,2,5,6,7,8,9,1] => [9,3,2,1,4,5,6,7,8] => ? = 0 + 4
Description
The maximal displacement of a permutation.
This is $\max\{ |\pi(i)-i| \mid 1 \leq i \leq n\}$ for a permutation $\pi$ of $\{1,\ldots,n\}$.
This statistic without the absolute value is the maximal drop size [[St000141]].
Matching statistic: St000653
(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
St000653: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 7%●distinct values known / distinct values provided: 6%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000653: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 7%●distinct values known / distinct values provided: 6%
Values
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4 = 0 + 4
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 4 = 0 + 4
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5 = 1 + 4
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 4 = 0 + 4
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 4 = 0 + 4
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 5 = 1 + 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] => ? = 0 + 4
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 4 = 0 + 4
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 4 = 0 + 4
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => 5 = 1 + 4
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 4 = 0 + 4
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => 5 = 1 + 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] => ? = 0 + 4
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 0 + 4
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => 4 = 0 + 4
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 4 = 0 + 4
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,6,1] => 5 = 1 + 4
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 4 = 0 + 4
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 4 = 0 + 4
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,6,1] => 5 = 1 + 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] => ? = 0 + 4
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => 4 = 0 + 4
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => 5 = 1 + 4
[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] => ? = 0 + 4
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? = 0 + 4
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? = 0 + 4
[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] => ? = 0 + 4
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,1,5] => 4 = 0 + 4
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => 5 = 1 + 4
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 4 = 0 + 4
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 4 = 0 + 4
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [5,3,2,4,6,1] => 5 = 1 + 4
[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] => ? = 0 + 4
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 4 = 0 + 4
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,6,1] => 5 = 1 + 4
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,1,2] => 4 = 0 + 4
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,6,1] => 5 = 1 + 4
[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] => ? = 0 + 4
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [4,2,3,5,6,7,8,1] => ? = 0 + 4
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => 5 = 1 + 4
[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] => ? = 0 + 4
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,1] => ? = 0 + 4
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,1] => ? = 0 + 4
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => ? = 0 + 4
[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] => ? = 0 + 4
[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] => ? = 0 + 4
[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] => ? = 1 + 4
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [6,4,2,3,1,5] => 4 = 0 + 4
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,1,5] => 4 = 0 + 4
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,5,1] => 5 = 1 + 4
[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] => ? = 0 + 4
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,1,4] => 4 = 0 + 4
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 4 = 0 + 4
[4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,3,6,1] => 5 = 1 + 4
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,1,2] => 4 = 0 + 4
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,6,1] => 5 = 1 + 4
[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] => ? = 0 + 4
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [5,2,3,4,6,7,8,1] => ? = 0 + 4
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,1,3] => 4 = 0 + 4
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,1,2] => 4 = 0 + 4
[3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,5,3,2,6,1] => 5 = 1 + 4
[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] => ? = 0 + 4
[3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => 5 = 1 + 4
[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] => ? = 0 + 4
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,5,6,7,8,1] => ? = 0 + 4
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [4,2,3,5,6,7,8,9,1] => ? = 0 + 4
[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] => ? = 0 + 4
[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] => ? = 1 + 4
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [3,4,5,2,6,7,8,1] => ? = 0 + 4
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,1] => ? = 0 + 4
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => ? = 0 + 4
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => ? = 0 + 4
[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] => ? = 0 + 4
[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] => ? = 0 + 4
[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] => ? = 1 + 4
[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] => ? = 0 + 4
[6,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [7,3,4,2,1,5,6] => ? = 0 + 4
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,4,5,1,6] => ? = 1 + 4
[6,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,6,1] => ? = 0 + 4
[5,4,1,1]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,1,4] => 4 = 0 + 4
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,5] => 4 = 0 + 4
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,3,5,1] => 5 = 1 + 4
[5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,1,2] => 4 = 0 + 4
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,5,1] => 5 = 1 + 4
[5,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [6,3,2,4,5,7,1] => ? = 0 + 4
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [6,2,3,4,5,7,8,1] => ? = 0 + 4
[4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,6,3,2,1,4] => 4 = 0 + 4
[4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => 5 = 1 + 4
[4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,1,3] => 4 = 0 + 4
[4,3,2,2]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,1,2] => 4 = 0 + 4
[4,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [5,4,2,3,6,7,1] => ? = 0 + 4
[4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [5,3,4,2,6,7,1] => ? = 0 + 4
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,3,2,4,6,7,8,1] => ? = 0 + 4
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [5,2,3,4,6,7,8,9,1] => ? = 0 + 4
[3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,6,7,1] => ? = 0 + 4
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [4,5,2,3,6,7,8,1] => ? = 0 + 4
[3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,1,2] => ? = 0 + 4
[3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,7,1] => ? = 1 + 4
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,3,5,2,6,7,8,1] => ? = 0 + 4
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [4,3,2,5,6,7,8,9,1] => ? = 0 + 4
Description
The last descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the largest index $0 \leq i < n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(0) = n+1$.
Matching statistic: St001480
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001480: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 7%●distinct values known / distinct values provided: 6%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001480: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 7%●distinct values known / distinct values provided: 6%
Values
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 0 + 4
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 4 = 0 + 4
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5 = 1 + 4
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4 = 0 + 4
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4 = 0 + 4
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 5 = 1 + 4
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 4
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4 = 0 + 4
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 0 + 4
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 5 = 1 + 4
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4 = 0 + 4
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 5 = 1 + 4
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 4
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 4
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 4 = 0 + 4
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4 = 0 + 4
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 5 = 1 + 4
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4 = 0 + 4
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4 = 0 + 4
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 5 = 1 + 4
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 0 + 4
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 4 = 0 + 4
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 5 = 1 + 4
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 4
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 4
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 4
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 4
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> 4 = 0 + 4
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 5 = 1 + 4
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 0 + 4
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 0 + 4
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 5 = 1 + 4
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 0 + 4
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 0 + 4
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 5 = 1 + 4
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 4 = 0 + 4
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 5 = 1 + 4
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 4
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 0 + 4
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 5 = 1 + 4
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 0 + 4
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 4
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 4
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 4
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 4
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 0 + 4
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 4
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 4 = 0 + 4
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> 4 = 0 + 4
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 5 = 1 + 4
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 4
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 4 = 0 + 4
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 0 + 4
[4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 5 = 1 + 4
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 5 = 1 + 4
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 0 + 4
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 0 + 4
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 4 = 0 + 4
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 4 = 0 + 4
[3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 5 = 1 + 4
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 0 + 4
[3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 5 = 1 + 4
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 0 + 4
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 4
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 0 + 4
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 0 + 4
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 1 + 4
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 0 + 4
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 4
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 4
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 4
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 4
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 0 + 4
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 4
[6,3,1,1]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 0 + 4
[6,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 0 + 4
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 1 + 4
[6,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 4
[5,4,1,1]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> 4 = 0 + 4
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4 = 0 + 4
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 5 = 1 + 4
[5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 4 = 0 + 4
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 5 = 1 + 4
[5,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 0 + 4
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 4
[4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> 4 = 0 + 4
[4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 5 = 1 + 4
[4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 4 = 0 + 4
[4,3,2,2]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 4 = 0 + 4
[4,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 0 + 4
[4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 0 + 4
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 0 + 4
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 0 + 4
[3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 4
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 0 + 4
[3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 0 + 4
[3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 1 + 4
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 0 + 4
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 4
Description
The number of simple summands of the module J^2/J^3. Here J is the Jacobson radical of the Nakayama algebra algebra corresponding to the Dyck path.
Matching statistic: St001291
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 7%●distinct values known / distinct values provided: 6%
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 7%●distinct values known / distinct values provided: 6%
Values
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 0 + 5
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5 = 0 + 5
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6 = 1 + 5
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5 = 0 + 5
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 5 = 0 + 5
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 6 = 1 + 5
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0 + 5
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5 = 0 + 5
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 5 = 0 + 5
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 6 = 1 + 5
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 5 = 0 + 5
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 6 = 1 + 5
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 0 + 5
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 5
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 5 = 0 + 5
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 5 = 0 + 5
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 6 = 1 + 5
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 5 = 0 + 5
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 6 = 1 + 5
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 0 + 5
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 5 = 0 + 5
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 6 = 1 + 5
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 0 + 5
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 0 + 5
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 5
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 0 + 5
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 5 = 0 + 5
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 6 = 1 + 5
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 5 = 0 + 5
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 5 = 0 + 5
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 6 = 1 + 5
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 0 + 5
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 5 = 0 + 5
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 6 = 1 + 5
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> 5 = 0 + 5
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 6 = 1 + 5
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 0 + 5
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 0 + 5
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 6 = 1 + 5
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 0 + 5
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 0 + 5
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 5
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 5
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 0 + 5
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 0 + 5
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 5
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 5 = 0 + 5
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 5 = 0 + 5
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 6 = 1 + 5
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0 + 5
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 5 = 0 + 5
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 0 + 5
[4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 6 = 1 + 5
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 5 = 0 + 5
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 6 = 1 + 5
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> ? = 0 + 5
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0 + 5
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> 5 = 0 + 5
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> 5 = 0 + 5
[3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 6 = 1 + 5
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> ? = 0 + 5
[3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 6 = 1 + 5
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 5
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 0 + 5
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 5
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 0 + 5
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 1 + 5
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 0 + 5
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> ? = 0 + 5
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 5
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 5
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 0 + 5
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 0 + 5
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 1 + 5
[6,3,1,1]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 0 + 5
[6,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 0 + 5
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 + 5
[6,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0 + 5
[5,4,1,1]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 5 = 0 + 5
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 5 = 0 + 5
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 6 = 1 + 5
[5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5 = 0 + 5
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 6 = 1 + 5
[5,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0 + 5
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0 + 5
[4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 5 = 0 + 5
[4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 6 = 1 + 5
[4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 5 = 0 + 5
[4,3,2,2]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 5 = 0 + 5
[4,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> ? = 0 + 5
[4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 5
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 0 + 5
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 5
[3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 0 + 5
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> ? = 0 + 5
[3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> ? = 0 + 5
[3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 1 + 5
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> ? = 0 + 5
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 0 + 5
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
Matching statistic: St000354
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 7%●distinct values known / distinct values provided: 6%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 7%●distinct values known / distinct values provided: 6%
Values
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => 3 = 0 + 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => 3 = 0 + 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,6,5,4,3,1] => 4 = 1 + 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => 3 = 0 + 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,5,4,1,3] => 3 = 0 + 3
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,4,5,1,6] => [2,6,5,4,1,3] => 4 = 1 + 3
[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] => [2,7,6,5,4,3,1] => ? = 0 + 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => 3 = 0 + 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => 3 = 0 + 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,4,1,5,6] => [2,6,5,1,4,3] => 4 = 1 + 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,5,4,1,3] => 3 = 0 + 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => [2,6,5,4,1,3] => 4 = 1 + 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => [2,7,6,5,4,1,3] => ? = 0 + 3
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [2,8,7,6,5,4,3,1] => ? = 0 + 3
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [3,1,2,4,5,6] => [3,1,6,5,4,2] => 3 = 0 + 3
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => 3 = 0 + 3
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => [2,6,1,5,4,3] => 4 = 1 + 3
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [2,5,1,4,3] => 3 = 0 + 3
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [2,5,1,4,3] => 3 = 0 + 3
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,4,1,6,5] => [2,6,5,1,4,3] => 4 = 1 + 3
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,3,4,5,1,6,7] => [2,7,6,5,1,4,3] => ? = 0 + 3
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [3,6,5,4,1,2] => 3 = 0 + 3
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => [2,6,5,4,1,3] => 4 = 1 + 3
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,4,5,7,1,6] => [2,7,6,5,4,1,3] => ? = 0 + 3
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1,8] => [2,8,7,6,5,4,1,3] => ? = 0 + 3
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => [2,9,8,7,6,5,4,3,1] => ? = 0 + 3
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [4,1,2,3,5,6,7] => [4,1,7,6,5,3,2] => ? = 0 + 3
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [3,1,2,4,6,5] => [3,1,6,5,4,2] => 3 = 0 + 3
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,5,6] => [2,1,6,5,4,3] => 4 = 1 + 3
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => 3 = 0 + 3
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 3 = 0 + 3
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,4,6,5] => [2,6,1,5,4,3] => 4 = 1 + 3
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,4,1,5,6,7] => [2,7,6,1,5,4,3] => ? = 0 + 3
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [2,5,1,4,3] => 3 = 0 + 3
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,3,6,1,4,5] => [2,6,5,1,4,3] => 4 = 1 + 3
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,4,5,1,6,2] => [3,6,5,1,4,2] => 3 = 0 + 3
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,5,1,6,4] => [2,6,5,1,4,3] => 4 = 1 + 3
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,3,4,5,1,7,6] => [2,7,6,5,1,4,3] => ? = 0 + 3
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,7,8] => [2,8,7,6,5,1,4,3] => ? = 0 + 3
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => [2,6,5,4,1,3] => 4 = 1 + 3
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,6,7,1,5] => [2,7,6,5,4,1,3] => ? = 0 + 3
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,8,1,7] => [2,8,7,6,5,4,1,3] => ? = 0 + 3
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1,9] => [2,9,8,7,6,5,4,1,3] => ? = 0 + 3
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [2,10,9,8,7,6,5,4,3,1] => ? = 0 + 3
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,7,8] => [5,1,8,7,6,4,3,2] => ? = 0 + 3
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [4,1,2,3,5,7,6] => [4,1,7,6,5,3,2] => ? = 0 + 3
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [3,1,2,4,5,6,7] => [3,1,7,6,5,4,2] => ? = 1 + 3
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [3,1,2,6,4,5] => [3,1,6,5,4,2] => 3 = 0 + 3
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,6,4] => [3,1,6,5,4,2] => 3 = 0 + 3
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,3,4,6,5] => [2,1,6,5,4,3] => 4 = 1 + 3
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,1,4,5,6,7] => [2,7,1,6,5,4,3] => ? = 0 + 3
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [3,6,1,2,4,5] => [3,6,1,5,4,2] => 3 = 0 + 3
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => 3 = 0 + 3
[4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5] => [2,6,1,5,4,3] => 4 = 1 + 3
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [3,4,1,5,6,2] => [3,6,1,5,4,2] => 3 = 0 + 3
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,1,5,6,4] => [2,6,1,5,4,3] => 4 = 1 + 3
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [2,3,4,1,5,7,6] => [2,7,6,1,5,4,3] => ? = 0 + 3
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,3,4,5,1,6,7,8] => [2,8,7,6,1,5,4,3] => ? = 0 + 3
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,5,6,1,2,4] => [3,6,5,1,4,2] => 3 = 0 + 3
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,4,5,1,2,6] => [3,6,5,1,4,2] => 3 = 0 + 3
[3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => [2,6,5,1,4,3] => 4 = 1 + 3
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,3,4,7,1,5,6] => [2,7,6,5,1,4,3] => ? = 0 + 3
[3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,5,1,6,3] => [2,6,5,1,4,3] => 4 = 1 + 3
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,3,4,6,1,7,5] => [2,7,6,5,1,4,3] => ? = 0 + 3
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,8,7] => [2,8,7,6,5,1,4,3] => ? = 0 + 3
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1,8,9] => [2,9,8,7,6,5,1,4,3] => ? = 0 + 3
[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] => [3,7,6,5,4,1,2] => ? = 0 + 3
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,5,6,7,1,4] => [2,7,6,5,4,1,3] => ? = 1 + 3
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,7,8,1,6] => [2,8,7,6,5,4,1,3] => ? = 0 + 3
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,9,1,8] => [2,9,8,7,6,5,4,1,3] => ? = 0 + 3
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1,10] => [2,10,9,8,7,6,5,4,1,3] => ? = 0 + 3
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [2,11,10,9,8,7,6,5,4,3,1] => ? = 0 + 3
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7,8,9] => [6,1,9,8,7,5,4,3,2] => ? = 0 + 3
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,8,7] => [5,1,8,7,6,4,3,2] => ? = 0 + 3
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [4,1,2,3,5,6,7,8] => [4,1,8,7,6,5,3,2] => ? = 1 + 3
[6,3,1,1]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [4,1,2,3,7,5,6] => [4,1,7,6,5,3,2] => ? = 0 + 3
[6,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [4,1,2,3,6,7,5] => [4,1,7,6,5,3,2] => ? = 0 + 3
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [3,1,2,4,5,7,6] => [3,1,7,6,5,4,2] => ? = 1 + 3
[6,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,6,7] => [2,1,7,6,5,4,3] => ? = 0 + 3
[5,4,1,1]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => 3 = 0 + 3
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4,6] => [3,1,6,5,4,2] => 3 = 0 + 3
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,3,6,4,5] => [2,1,6,5,4,3] => 4 = 1 + 3
[5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,4,5,6,2] => [3,1,6,5,4,2] => 3 = 0 + 3
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,3,5,6,4] => [2,1,6,5,4,3] => 4 = 1 + 3
[5,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [2,3,1,4,5,7,6] => [2,7,1,6,5,4,3] => ? = 0 + 3
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [2,3,4,1,5,6,7,8] => [2,8,7,1,6,5,4,3] => ? = 0 + 3
[4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4,6] => [3,6,1,5,4,2] => 3 = 0 + 3
[4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,6,1,3,4,5] => [2,6,1,5,4,3] => 4 = 1 + 3
[4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,5,1,6,2,4] => [3,6,1,5,4,2] => 3 = 0 + 3
[4,3,2,2]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [3,4,1,5,2,6] => [3,6,1,5,4,2] => 3 = 0 + 3
[4,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,3,4,1,7,5,6] => [2,7,6,1,5,4,3] => ? = 0 + 3
[4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [2,3,4,1,6,7,5] => [2,7,6,1,5,4,3] => ? = 0 + 3
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [2,3,4,5,1,6,8,7] => [2,8,7,6,1,5,4,3] => ? = 0 + 3
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,7,8,9] => [2,9,8,7,6,1,5,4,3] => ? = 0 + 3
[3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5,7] => [2,7,6,5,1,4,3] => ? = 0 + 3
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,3,4,5,8,1,6,7] => [2,8,7,6,5,1,4,3] => ? = 0 + 3
[3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [3,4,5,6,1,7,2] => [3,7,6,5,1,4,2] => ? = 0 + 3
[3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,5,6,1,7,4] => [2,7,6,5,1,4,3] => ? = 1 + 3
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,3,4,5,7,1,8,6] => [2,8,7,6,5,1,4,3] => ? = 0 + 3
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1,9,8] => [2,9,8,7,6,5,1,4,3] => ? = 0 + 3
Description
The number of recoils of a permutation.
A '''recoil''', or '''inverse descent''' of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$.
In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.
Matching statistic: St000702
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 7%●distinct values known / distinct values provided: 6%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 7%●distinct values known / distinct values provided: 6%
Values
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => 3 = 0 + 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => 3 = 0 + 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [1,6,5,4,3,2] => 4 = 1 + 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => 3 = 0 + 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => 3 = 0 + 3
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => [1,6,5,4,3,2] => 4 = 1 + 3
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 0 + 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => 3 = 0 + 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 3 = 0 + 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => [1,6,5,4,3,2] => 4 = 1 + 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => 3 = 0 + 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => [1,6,5,4,3,2] => 4 = 1 + 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 0 + 3
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ? = 0 + 3
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,4,3,1,6] => [2,6,5,4,1,3] => 3 = 0 + 3
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => 3 = 0 + 3
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => [1,6,5,4,3,2] => 4 = 1 + 3
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => 3 = 0 + 3
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => 3 = 0 + 3
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,4,5,6,3,2] => [1,6,5,4,3,2] => 4 = 1 + 3
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => [1,7,6,5,4,3,2] => ? = 0 + 3
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => [2,1,6,5,4,3] => 3 = 0 + 3
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,5,4,2] => [1,6,5,4,3,2] => 4 = 1 + 3
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => [1,7,6,5,4,3,2] => ? = 0 + 3
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ? = 0 + 3
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => ? = 0 + 3
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [3,6,5,4,2,1,7] => [3,7,6,5,2,1,4] => ? = 0 + 3
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,4,5,3,1,6] => [2,6,5,4,1,3] => 3 = 0 + 3
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => [1,6,5,4,3,2] => 4 = 1 + 3
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => 3 = 0 + 3
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => 3 = 0 + 3
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,4,5,3,6,2] => [1,6,5,4,3,2] => 4 = 1 + 3
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,6,5,4,7,3,2] => [1,7,6,5,4,3,2] => ? = 0 + 3
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => 3 = 0 + 3
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,4,3,6,5,2] => [1,6,5,4,3,2] => 4 = 1 + 3
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,1,5,6,4,3] => [2,1,6,5,4,3] => 3 = 0 + 3
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,3,5,6,4,2] => [1,6,5,4,3,2] => 4 = 1 + 3
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,5,6,7,4,3,2] => [1,7,6,5,4,3,2] => ? = 0 + 3
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,7,6,8,5,4,3,2] => [1,8,7,6,5,4,3,2] => ? = 0 + 3
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => [1,6,5,4,3,2] => 4 = 1 + 3
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => [1,7,6,5,4,3,2] => ? = 0 + 3
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,6,8,7,5,4,3,2] => [1,8,7,6,5,4,3,2] => ? = 0 + 3
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => ? = 0 + 3
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,9,8,7,6,5,4,3,2] => [1,10,9,8,7,6,5,4,3,2] => ? = 0 + 3
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [4,7,6,5,3,2,1,8] => [4,8,7,6,3,2,1,5] => ? = 0 + 3
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [3,5,6,4,2,1,7] => [3,7,6,5,2,1,4] => ? = 0 + 3
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,6,5,4,3,1,7] => [2,7,6,5,4,1,3] => ? = 1 + 3
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,4,3,5,1,6] => [2,6,5,4,1,3] => 3 = 0 + 3
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,5,4,1,6] => [2,6,5,4,1,3] => 3 = 0 + 3
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,5,3,2,6] => [1,6,5,4,3,2] => 4 = 1 + 3
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,5,4,3,7,2] => [1,7,6,5,4,3,2] => ? = 0 + 3
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,3,1,6,5] => [2,6,5,1,4,3] => 3 = 0 + 3
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => 3 = 0 + 3
[4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => [1,6,5,4,3,2] => 4 = 1 + 3
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,5,4,6,3] => [2,1,6,5,4,3] => 3 = 0 + 3
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,4,6,2] => [1,6,5,4,3,2] => 4 = 1 + 3
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,5,6,4,7,3,2] => [1,7,6,5,4,3,2] => ? = 0 + 3
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,7,6,5,8,4,3,2] => [1,8,7,6,5,4,3,2] => ? = 0 + 3
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,1,6,5,4] => [2,6,1,5,4,3] => 3 = 0 + 3
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,5,3] => [2,1,6,5,4,3] => 3 = 0 + 3
[3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,5,2] => [1,6,5,4,3,2] => 4 = 1 + 3
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,5,4,7,6,3,2] => [1,7,6,5,4,3,2] => ? = 0 + 3
[3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,4,3] => [1,6,5,4,3,2] => 4 = 1 + 3
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => [1,7,6,5,4,3,2] => ? = 0 + 3
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,6,7,8,5,4,3,2] => [1,8,7,6,5,4,3,2] => ? = 0 + 3
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,7,9,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => ? = 0 + 3
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,6,5,4,3] => [2,1,7,6,5,4,3] => ? = 0 + 3
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,7,6,5,4,2] => [1,7,6,5,4,3,2] => ? = 1 + 3
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,5,8,7,6,4,3,2] => [1,8,7,6,5,4,3,2] => ? = 0 + 3
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,7,9,8,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => ? = 0 + 3
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9,10,8,7,6,5,4,3,2] => [1,10,9,8,7,6,5,4,3,2] => ? = 0 + 3
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,7,6,5,4,3,2] => [1,11,10,9,8,7,6,5,4,3,2] => ? = 0 + 3
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [5,8,7,6,4,3,2,1,9] => ? => ? = 0 + 3
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [4,6,7,5,3,2,1,8] => [4,8,7,6,3,2,1,5] => ? = 0 + 3
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [3,7,6,5,4,2,1,8] => ? => ? = 1 + 3
[6,3,1,1]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [3,5,4,6,2,1,7] => [3,7,6,5,2,1,4] => ? = 0 + 3
[6,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [3,4,6,5,2,1,7] => [3,7,6,5,2,1,4] => ? = 0 + 3
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [2,5,6,4,3,1,7] => [2,7,6,5,4,1,3] => ? = 1 + 3
[6,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,5,4,3,2,7] => [1,7,6,5,4,3,2] => ? = 0 + 3
[5,4,1,1]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,3,1,5,6] => [2,6,5,1,4,3] => 3 = 0 + 3
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,1,6] => [2,6,5,4,1,3] => 3 = 0 + 3
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,3,5,2,6] => [1,6,5,4,3,2] => 4 = 1 + 3
[5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,4,3,6] => [2,1,6,5,4,3] => 3 = 0 + 3
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,4,2,6] => [1,6,5,4,3,2] => 4 = 1 + 3
[5,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,4,3,7,2] => [1,7,6,5,4,3,2] => ? = 0 + 3
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,7,6,5,4,8,3,2] => [1,8,7,6,5,4,3,2] => ? = 0 + 3
[4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5] => [2,6,5,1,4,3] => 3 = 0 + 3
[4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => [1,6,5,4,3,2] => 4 = 1 + 3
[4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,3,1,5,6,4] => [2,6,1,5,4,3] => 3 = 0 + 3
[4,3,2,2]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,1,4,5,6,3] => [2,1,6,5,4,3] => 3 = 0 + 3
[4,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,5,4,6,7,3,2] => [1,7,6,5,4,3,2] => ? = 0 + 3
[4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,4,6,5,7,3,2] => [1,7,6,5,4,3,2] => ? = 0 + 3
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,6,7,5,8,4,3,2] => [1,8,7,6,5,4,3,2] => ? = 0 + 3
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,8,7,6,9,5,4,3,2] => ? => ? = 0 + 3
[3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,4,5,7,6,3,2] => [1,7,6,5,4,3,2] => ? = 0 + 3
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,6,5,8,7,4,3,2] => [1,8,7,6,5,4,3,2] => ? = 0 + 3
[3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,6,7,5,4,3] => [2,1,7,6,5,4,3] => ? = 0 + 3
[3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,6,7,5,4,2] => [1,7,6,5,4,3,2] => ? = 1 + 3
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,5,7,8,6,4,3,2] => ? => ? = 0 + 3
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,7,8,9,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => ? = 0 + 3
Description
The number of weak deficiencies of a permutation.
This is defined as
$$\operatorname{wdec}(\sigma)=\#\{i:\sigma(i) \leq i\}.$$
The number of weak exceedances is [[St000213]], the number of deficiencies is [[St000703]].
Matching statistic: St001566
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001566: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 7%●distinct values known / distinct values provided: 6%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001566: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 7%●distinct values known / distinct values provided: 6%
Values
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => 3 = 0 + 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => 3 = 0 + 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,6,5,4,3,1] => 4 = 1 + 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => 3 = 0 + 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,5,4,1,3] => 3 = 0 + 3
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,4,5,1,6] => [2,6,5,4,1,3] => 4 = 1 + 3
[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] => [2,7,6,5,4,3,1] => ? = 0 + 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => 3 = 0 + 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => 3 = 0 + 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,4,1,5,6] => [2,6,5,1,4,3] => 4 = 1 + 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,5,4,1,3] => 3 = 0 + 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => [2,6,5,4,1,3] => 4 = 1 + 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => [2,7,6,5,4,1,3] => ? = 0 + 3
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [2,8,7,6,5,4,3,1] => ? = 0 + 3
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [3,1,2,4,5,6] => [3,1,6,5,4,2] => 3 = 0 + 3
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => 3 = 0 + 3
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => [2,6,1,5,4,3] => 4 = 1 + 3
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [2,5,1,4,3] => 3 = 0 + 3
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [2,5,1,4,3] => 3 = 0 + 3
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,4,1,6,5] => [2,6,5,1,4,3] => 4 = 1 + 3
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,3,4,5,1,6,7] => [2,7,6,5,1,4,3] => ? = 0 + 3
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [3,6,5,4,1,2] => 3 = 0 + 3
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => [2,6,5,4,1,3] => 4 = 1 + 3
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,4,5,7,1,6] => [2,7,6,5,4,1,3] => ? = 0 + 3
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1,8] => [2,8,7,6,5,4,1,3] => ? = 0 + 3
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => [2,9,8,7,6,5,4,3,1] => ? = 0 + 3
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [4,1,2,3,5,6,7] => [4,1,7,6,5,3,2] => ? = 0 + 3
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [3,1,2,4,6,5] => [3,1,6,5,4,2] => 3 = 0 + 3
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,5,6] => [2,1,6,5,4,3] => 4 = 1 + 3
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => 3 = 0 + 3
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 3 = 0 + 3
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,4,6,5] => [2,6,1,5,4,3] => 4 = 1 + 3
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,4,1,5,6,7] => [2,7,6,1,5,4,3] => ? = 0 + 3
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [2,5,1,4,3] => 3 = 0 + 3
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,3,6,1,4,5] => [2,6,5,1,4,3] => 4 = 1 + 3
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,4,5,1,6,2] => [3,6,5,1,4,2] => 3 = 0 + 3
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,5,1,6,4] => [2,6,5,1,4,3] => 4 = 1 + 3
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,3,4,5,1,7,6] => [2,7,6,5,1,4,3] => ? = 0 + 3
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,7,8] => [2,8,7,6,5,1,4,3] => ? = 0 + 3
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => [2,6,5,4,1,3] => 4 = 1 + 3
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,6,7,1,5] => [2,7,6,5,4,1,3] => ? = 0 + 3
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,8,1,7] => [2,8,7,6,5,4,1,3] => ? = 0 + 3
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1,9] => [2,9,8,7,6,5,4,1,3] => ? = 0 + 3
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [2,10,9,8,7,6,5,4,3,1] => ? = 0 + 3
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,7,8] => [5,1,8,7,6,4,3,2] => ? = 0 + 3
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [4,1,2,3,5,7,6] => [4,1,7,6,5,3,2] => ? = 0 + 3
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [3,1,2,4,5,6,7] => [3,1,7,6,5,4,2] => ? = 1 + 3
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [3,1,2,6,4,5] => [3,1,6,5,4,2] => 3 = 0 + 3
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,6,4] => [3,1,6,5,4,2] => 3 = 0 + 3
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,3,4,6,5] => [2,1,6,5,4,3] => 4 = 1 + 3
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,1,4,5,6,7] => [2,7,1,6,5,4,3] => ? = 0 + 3
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [3,6,1,2,4,5] => [3,6,1,5,4,2] => 3 = 0 + 3
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => 3 = 0 + 3
[4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5] => [2,6,1,5,4,3] => 4 = 1 + 3
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [3,4,1,5,6,2] => [3,6,1,5,4,2] => 3 = 0 + 3
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,1,5,6,4] => [2,6,1,5,4,3] => 4 = 1 + 3
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [2,3,4,1,5,7,6] => [2,7,6,1,5,4,3] => ? = 0 + 3
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,3,4,5,1,6,7,8] => [2,8,7,6,1,5,4,3] => ? = 0 + 3
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,5,6,1,2,4] => [3,6,5,1,4,2] => 3 = 0 + 3
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,4,5,1,2,6] => [3,6,5,1,4,2] => 3 = 0 + 3
[3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => [2,6,5,1,4,3] => 4 = 1 + 3
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,3,4,7,1,5,6] => [2,7,6,5,1,4,3] => ? = 0 + 3
[3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,5,1,6,3] => [2,6,5,1,4,3] => 4 = 1 + 3
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,3,4,6,1,7,5] => [2,7,6,5,1,4,3] => ? = 0 + 3
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,8,7] => [2,8,7,6,5,1,4,3] => ? = 0 + 3
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1,8,9] => [2,9,8,7,6,5,1,4,3] => ? = 0 + 3
[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] => [3,7,6,5,4,1,2] => ? = 0 + 3
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,5,6,7,1,4] => [2,7,6,5,4,1,3] => ? = 1 + 3
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,7,8,1,6] => [2,8,7,6,5,4,1,3] => ? = 0 + 3
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,9,1,8] => [2,9,8,7,6,5,4,1,3] => ? = 0 + 3
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1,10] => [2,10,9,8,7,6,5,4,1,3] => ? = 0 + 3
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [2,11,10,9,8,7,6,5,4,3,1] => ? = 0 + 3
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7,8,9] => [6,1,9,8,7,5,4,3,2] => ? = 0 + 3
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,8,7] => [5,1,8,7,6,4,3,2] => ? = 0 + 3
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [4,1,2,3,5,6,7,8] => [4,1,8,7,6,5,3,2] => ? = 1 + 3
[6,3,1,1]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [4,1,2,3,7,5,6] => [4,1,7,6,5,3,2] => ? = 0 + 3
[6,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [4,1,2,3,6,7,5] => [4,1,7,6,5,3,2] => ? = 0 + 3
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [3,1,2,4,5,7,6] => [3,1,7,6,5,4,2] => ? = 1 + 3
[6,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,6,7] => [2,1,7,6,5,4,3] => ? = 0 + 3
[5,4,1,1]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => 3 = 0 + 3
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4,6] => [3,1,6,5,4,2] => 3 = 0 + 3
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,3,6,4,5] => [2,1,6,5,4,3] => 4 = 1 + 3
[5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,4,5,6,2] => [3,1,6,5,4,2] => 3 = 0 + 3
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,3,5,6,4] => [2,1,6,5,4,3] => 4 = 1 + 3
[5,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [2,3,1,4,5,7,6] => [2,7,1,6,5,4,3] => ? = 0 + 3
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [2,3,4,1,5,6,7,8] => [2,8,7,1,6,5,4,3] => ? = 0 + 3
[4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4,6] => [3,6,1,5,4,2] => 3 = 0 + 3
[4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,6,1,3,4,5] => [2,6,1,5,4,3] => 4 = 1 + 3
[4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,5,1,6,2,4] => [3,6,1,5,4,2] => 3 = 0 + 3
[4,3,2,2]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [3,4,1,5,2,6] => [3,6,1,5,4,2] => 3 = 0 + 3
[4,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,3,4,1,7,5,6] => [2,7,6,1,5,4,3] => ? = 0 + 3
[4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [2,3,4,1,6,7,5] => [2,7,6,1,5,4,3] => ? = 0 + 3
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [2,3,4,5,1,6,8,7] => [2,8,7,6,1,5,4,3] => ? = 0 + 3
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,7,8,9] => [2,9,8,7,6,1,5,4,3] => ? = 0 + 3
[3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5,7] => [2,7,6,5,1,4,3] => ? = 0 + 3
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,3,4,5,8,1,6,7] => [2,8,7,6,5,1,4,3] => ? = 0 + 3
[3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [3,4,5,6,1,7,2] => [3,7,6,5,1,4,2] => ? = 0 + 3
[3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,5,6,1,7,4] => [2,7,6,5,1,4,3] => ? = 1 + 3
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,3,4,5,7,1,8,6] => [2,8,7,6,5,1,4,3] => ? = 0 + 3
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1,9,8] => [2,9,8,7,6,5,1,4,3] => ? = 0 + 3
Description
The length of the longest arithmetic progression in a permutation.
For a permutation $\pi$ of length $n$, this is the biggest $k$ such that there exist $1 \leq i_1 < \dots < i_k \leq n$ with
$$\pi(i_2) - \pi(i_1) = \pi(i_3) - \pi(i_2) = \dots = \pi(i_k) - \pi(i_{k-1}).$$
The following 63 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000051The size of the left subtree of a binary tree. St000204The number of internal nodes of a binary tree. St000304The load of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000356The number of occurrences of the pattern 13-2. St000863The length of the first row of the shifted shape of a permutation. St000957The number of Bruhat lower covers of a permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St000235The number of indices that are not cyclical small weak excedances. St000240The number of indices that are not small excedances. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000692Babson and Steingrímsson's statistic of a permutation. St000508Eigenvalues of the random-to-random operator acting on a simple module. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001557The number of inversions of the second entry of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000744The length of the path to the largest entry in a standard Young tableau. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000044The number of vertices of the unicellular map given by a perfect matching. St000017The number of inversions of a standard tableau. St001721The degree of a binary word. St000016The number of attacking pairs of a standard tableau. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000924The number of topologically connected components of a perfect matching. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001958The degree of the polynomial interpolating the values of a permutation. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001513The number of nested exceedences of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001667The maximal size of a pair of weak twins for a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St000731The number of double exceedences of a permutation. St000787The number of flips required to make a perfect matching noncrossing. St000788The number of nesting-similar perfect matchings of a perfect matching. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
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