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Your data matches 37 different statistics following compositions of up to 3 maps.
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Matching statistic: St001083
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001083: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001083: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [2,1] => 0
[2]
=> [1,0,1,0]
=> [1,0,1,0]
=> [3,1,2] => 0
[1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> [2,3,1] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 0
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,3,4,1,2,5,6] => 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => 2
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => 0
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,9,1,2,3,4,5,6,7] => 0
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [6,7,8,1,2,3,4,5] => 0
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,1,2,3,4,5,6,9,7] => 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => 0
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,9,1,2,3,4,5,6,7,8] => 0
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [9,7,8,1,2,3,4,5,6] => 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [9,1,2,3,4,5,6,7,10,8] => 1
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => 0
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [10,9,8,1,2,3,4,5,6,7] => 0
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [8,1,2,7,6,3,4,5] => 2
Description
The number of boxed occurrences of 132 in a permutation.
This is the number of occurrences of the pattern $132$ such that any entry between the three matched entries is either larger than the largest matched entry or smaller than the smallest matched entry.
Matching statistic: St001960
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 67%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0]
=> [1,0]
=> [2,1] => 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,3,1] => 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [3,1,2] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => ? = 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => ? = 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => ? = 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ? = 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => ? = 0
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ? = 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ? = 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => ? = 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ? = 0
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => ? = 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 2
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [8,3,4,5,6,7,1,2] => ? = 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,9,1,8] => ? = 0
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,8,1,6,7] => ? = 0
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => ? = 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => ? = 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ? = 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [9,3,4,5,6,7,8,1,2] => ? = 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,10,1,9] => ? = 0
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,6,9,1,7,8] => ? = 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? = 0
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [10,3,4,5,6,7,8,9,1,2] => ? = 1
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,6,7,10,1,8,9] => ? = 0
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [2,3,8,7,6,1,4,5] => ? = 2
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => ? = 1
[7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [2,3,4,5,6,10,1,7,8,9] => ? = 0
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [2,3,8,1,4,5,6,7] => ? = 0
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 1
[4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [8,4,1,2,3,7,5,6] => ? = 2
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 0
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => ? = 0
[]
=> []
=> []
=> [1] => ? = 0
[4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => ? = 0
[5,5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => ? = 0
[5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => ? = 0
Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Matching statistic: St000805
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000805: Integer compositions ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 67%
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000805: Integer compositions ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0]
=> 10 => [1,1] => 1 = 0 + 1
[2]
=> [1,0,1,0]
=> 1010 => [1,1,1,1] => 1 = 0 + 1
[1,1]
=> [1,1,0,0]
=> 1100 => [2,2] => 1 = 0 + 1
[3]
=> [1,0,1,0,1,0]
=> 101010 => [1,1,1,1,1,1] => 1 = 0 + 1
[2,1]
=> [1,0,1,1,0,0]
=> 101100 => [1,1,2,2] => 1 = 0 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => [2,1,1,2] => 2 = 1 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => [1,1,1,1,1,1,1,1] => 1 = 0 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => [1,1,1,1,2,2] => 1 = 0 + 1
[2,2]
=> [1,1,1,0,0,0]
=> 111000 => [3,3] => 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => [1,1,2,1,1,2] => 2 = 1 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => [2,1,1,1,1,2] => 2 = 1 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => [1,1,1,1,1,1,1,1,1,1] => ? = 0 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => [1,1,1,1,1,1,2,2] => ? = 0 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => [1,1,3,3] => 1 = 0 + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => [1,1,1,1,2,1,1,2] => ? = 1 + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => [3,2,1,2] => 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => [1,1,2,1,1,1,1,2] => ? = 1 + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => [2,1,1,1,1,1,1,2] => ? = 1 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => [1,1,1,1,1,1,1,1,1,1,1,1] => ? = 0 + 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => [1,1,1,1,1,1,1,1,2,2] => ? = 0 + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => [1,1,1,1,3,3] => ? = 0 + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => [3,1,1,3] => 2 = 1 + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => [1,1,3,2,1,2] => ? = 0 + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => [4,4] => 1 = 0 + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => [3,2,1,1,1,2] => ? = 1 + 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 110101010100 => [2,1,1,1,1,1,1,1,1,2] => ? = 1 + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 10101010101010 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1] => ? = 0 + 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => [1,1,1,1,1,1,1,1,1,1,2,2] => ? = 0 + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => [1,1,3,1,1,3] => ? = 1 + 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => [3,1,1,2,1,2] => ? = 2 + 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => [1,1,4,4] => ? = 0 + 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 101110010100 => [1,1,3,2,1,1,1,2] => ? = 0 + 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => [4,3,1,2] => ? = 2 + 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 11010101010100 => [2,1,1,1,1,1,1,1,1,1,1,2] => ? = 1 + 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1010101010101010 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => ? = 0 + 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1010101010101100 => [1,1,1,1,1,1,1,1,1,1,1,1,2,2] => ? = 0 + 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 10101010111000 => [1,1,1,1,1,1,1,1,3,3] => ? = 0 + 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => [3,1,1,1,1,3] => ? = 1 + 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 101011110000 => [1,1,1,1,4,4] => ? = 0 + 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => [3,1,2,4] => ? = 1 + 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => [4,1,1,4] => ? = 1 + 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1101010101010100 => [2,1,1,1,1,1,1,1,1,1,1,1,1,2] => ? = 1 + 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010101010 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => ? = 0 + 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101010101100 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2] => ? = 0 + 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 1010101010111000 => [1,1,1,1,1,1,1,1,1,1,3,3] => ? = 0 + 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => [5,5] => ? = 0 + 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 110101010101010100 => [2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2] => ? = 1 + 1
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 10101010101010101010 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => ? = 0 + 1
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 101010101010111000 => [1,1,1,1,1,1,1,1,1,1,1,1,3,3] => ? = 0 + 1
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> 10101110101000 => [1,1,1,1,3,1,1,1,1,3] => ? = 2 + 1
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 111010101000 => [3,1,1,1,1,1,1,3] => ? = 1 + 1
[7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 101010101011110000 => ? => ? = 0 + 1
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 10101111100000 => [1,1,1,1,5,5] => ? = 0 + 1
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 111110010000 => [5,2,1,4] => ? = 1 + 1
[4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> 11101110000100 => [3,1,3,4,1,2] => ? = 2 + 1
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 111111000000 => [6,6] => ? = 0 + 1
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => [7,7] => ? = 0 + 1
[]
=> []
=> => [] => ? = 0 + 1
[4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> 1111111100000000 => [8,8] => ? = 0 + 1
[5,5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> 11111111110000000000 => [10,10] => ? = 0 + 1
[5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> 111111111000000000 => [9,9] => ? = 0 + 1
Description
The number of peaks of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of contiguous subsequences consisting of an up step, a sequence of horizontal steps, and a down step.
Matching statistic: St000534
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000534: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 67%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000534: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0]
=> [(1,2)]
=> [2,1] => 0
[2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 0
[1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => 0
[3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => ? = 0
[2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => ? = 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => ? = 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => ? = 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,7,9,6,10,8,5] => ? = 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? = 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => ? = 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [3,5,2,7,4,9,6,10,8,1] => ? = 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [2,1,4,3,6,5,8,7,11,12,10,9] => ? = 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => ? = 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => ? = 0
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => ? = 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [4,5,7,3,2,9,6,10,8,1] => ? = 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> [3,5,2,7,4,9,6,11,8,12,10,1] => ? = 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13] => ? = 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,14),(12,13)]
=> [2,1,4,3,6,5,8,7,10,9,13,14,12,11] => ? = 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => ? = 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,2,10,8,1] => ? = 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => ? = 0
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
=> [2,1,6,7,9,5,4,11,8,12,10,3] => ? = 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [5,6,7,9,4,3,2,10,8,1] => ? = 2
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
=> [3,5,2,7,4,9,6,11,8,13,10,14,12,1] => ? = 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15] => ? = 0
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,15,16,14,13] => ? = 0
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,14),(10,13),(11,12)]
=> [2,1,4,3,6,5,8,7,12,13,14,11,10,9] => ? = 0
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,2,1] => ? = 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,4,3,9,10,11,12,8,7,6,5] => ? = 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [4,7,8,3,9,10,6,5,2,1] => ? = 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [5,7,8,9,4,10,6,3,2,1] => ? = 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,16),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15)]
=> [3,5,2,7,4,9,6,11,8,13,10,15,12,16,14,1] => ? = 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15,18,17] => ? = 0
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,18),(16,17)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13,17,18,16,15] => ? = 0
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,16),(12,15),(13,14)]
=> [2,1,4,3,6,5,8,7,10,9,14,15,16,13,12,11] => ? = 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [6,7,8,9,10,5,4,3,2,1] => ? = 0
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,18),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17)]
=> [3,5,2,7,4,9,6,11,8,13,10,15,12,17,14,18,16,1] => ? = 1
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18),(19,20)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15,18,17,20,19] => ? = 0
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,18),(14,17),(15,16)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,16,17,18,15,14,13] => ? = 0
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,8),(9,10),(11,12)]
=> [2,1,4,3,8,10,12,7,13,9,14,11,6,5] => ? = 2
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)]
=> [4,6,8,3,10,5,11,7,12,9,2,1] => ? = 1
[7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,18),(12,17),(13,16),(14,15)]
=> ? => ? = 0
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> [2,1,4,3,10,11,12,13,14,9,8,7,6,5] => ? = 0
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,7),(5,6),(8,9)]
=> [6,7,9,10,11,5,4,12,8,3,2,1] => ? = 1
[4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [(1,14),(2,11),(3,4),(5,10),(6,9),(7,8),(12,13)]
=> [4,8,9,3,10,11,13,7,6,5,2,14,12,1] => ? = 2
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
=> [7,8,9,10,11,12,6,5,4,3,2,1] => ? = 0
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [8,9,10,11,12,13,14,7,6,5,4,3,2,1] => ? = 0
[]
=> []
=> []
=> ? => ? = 0
[4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> [9,10,11,12,13,14,15,16,8,7,6,5,4,3,2,1] => ? = 0
[5,5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11)]
=> [11,12,13,14,15,16,17,18,19,20,10,9,8,7,6,5,4,3,2,1] => ? = 0
Description
The number of 2-rises of a permutation.
A 2-rise of a permutation $\pi$ is an index $i$ such that $\pi(i)+2 = \pi(i+1)$.
For 1-rises, or successions, see [[St000441]].
Matching statistic: St000451
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 67%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0]
=> [(1,2)]
=> [2,1] => 2 = 0 + 2
[2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 2 = 0 + 2
[1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => 2 = 0 + 2
[3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 2 = 0 + 2
[2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 2 = 0 + 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 3 = 1 + 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => ? = 0 + 2
[2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => 2 = 0 + 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => ? = 1 + 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 2 = 0 + 2
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => ? = 0 + 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => ? = 0 + 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,7,9,6,10,8,5] => ? = 1 + 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? = 1 + 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => ? = 1 + 2
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [3,5,2,7,4,9,6,10,8,1] => ? = 1 + 2
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 2 = 0 + 2
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [2,1,4,3,6,5,8,7,11,12,10,9] => ? = 0 + 2
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => ? = 0 + 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 1 + 2
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => ? = 0 + 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => ? = 0 + 2
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [4,5,7,3,2,9,6,10,8,1] => ? = 1 + 2
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> [3,5,2,7,4,9,6,11,8,12,10,1] => ? = 1 + 2
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13] => ? = 0 + 2
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,14),(12,13)]
=> [2,1,4,3,6,5,8,7,10,9,13,14,12,11] => ? = 0 + 2
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => ? = 1 + 2
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,2,10,8,1] => ? = 2 + 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => ? = 0 + 2
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
=> [2,1,6,7,9,5,4,11,8,12,10,3] => ? = 0 + 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [5,6,7,9,4,3,2,10,8,1] => ? = 2 + 2
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
=> [3,5,2,7,4,9,6,11,8,13,10,14,12,1] => ? = 1 + 2
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15] => ? = 0 + 2
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,15,16,14,13] => ? = 0 + 2
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,14),(10,13),(11,12)]
=> [2,1,4,3,6,5,8,7,12,13,14,11,10,9] => ? = 0 + 2
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,2,1] => ? = 1 + 2
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,4,3,9,10,11,12,8,7,6,5] => ? = 0 + 2
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [4,7,8,3,9,10,6,5,2,1] => ? = 1 + 2
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [5,7,8,9,4,10,6,3,2,1] => ? = 1 + 2
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,16),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15)]
=> [3,5,2,7,4,9,6,11,8,13,10,15,12,16,14,1] => ? = 1 + 2
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15,18,17] => ? = 0 + 2
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,18),(16,17)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13,17,18,16,15] => ? = 0 + 2
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,16),(12,15),(13,14)]
=> [2,1,4,3,6,5,8,7,10,9,14,15,16,13,12,11] => ? = 0 + 2
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [6,7,8,9,10,5,4,3,2,1] => ? = 0 + 2
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,18),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17)]
=> [3,5,2,7,4,9,6,11,8,13,10,15,12,17,14,18,16,1] => ? = 1 + 2
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18),(19,20)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15,18,17,20,19] => ? = 0 + 2
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,18),(14,17),(15,16)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,16,17,18,15,14,13] => ? = 0 + 2
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,8),(9,10),(11,12)]
=> [2,1,4,3,8,10,12,7,13,9,14,11,6,5] => ? = 2 + 2
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)]
=> [4,6,8,3,10,5,11,7,12,9,2,1] => ? = 1 + 2
[7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,18),(12,17),(13,16),(14,15)]
=> ? => ? = 0 + 2
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> [2,1,4,3,10,11,12,13,14,9,8,7,6,5] => ? = 0 + 2
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,7),(5,6),(8,9)]
=> [6,7,9,10,11,5,4,12,8,3,2,1] => ? = 1 + 2
[4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [(1,14),(2,11),(3,4),(5,10),(6,9),(7,8),(12,13)]
=> [4,8,9,3,10,11,13,7,6,5,2,14,12,1] => ? = 2 + 2
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
=> [7,8,9,10,11,12,6,5,4,3,2,1] => ? = 0 + 2
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [8,9,10,11,12,13,14,7,6,5,4,3,2,1] => ? = 0 + 2
[]
=> []
=> []
=> ? => ? = 0 + 2
[4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> [9,10,11,12,13,14,15,16,8,7,6,5,4,3,2,1] => ? = 0 + 2
[5,5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11)]
=> [11,12,13,14,15,16,17,18,19,20,10,9,8,7,6,5,4,3,2,1] => ? = 0 + 2
Description
The length of the longest pattern of the form k 1 2...(k-1).
Matching statistic: St000842
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000842: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 67%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000842: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0]
=> [(1,2)]
=> [2,1] => 2 = 0 + 2
[2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 2 = 0 + 2
[1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => 2 = 0 + 2
[3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 2 = 0 + 2
[2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 2 = 0 + 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 3 = 1 + 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 0 + 2
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => ? = 0 + 2
[2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => 2 = 0 + 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => ? = 1 + 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1 + 2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 2 = 0 + 2
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => ? = 0 + 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => ? = 0 + 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,7,9,6,10,8,5] => ? = 1 + 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? = 1 + 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => ? = 1 + 2
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [3,5,2,7,4,9,6,10,8,1] => ? = 1 + 2
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 2 = 0 + 2
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [2,1,4,3,6,5,8,7,11,12,10,9] => ? = 0 + 2
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => ? = 0 + 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 1 + 2
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => ? = 0 + 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => ? = 0 + 2
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [4,5,7,3,2,9,6,10,8,1] => ? = 1 + 2
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> [3,5,2,7,4,9,6,11,8,12,10,1] => ? = 1 + 2
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13] => ? = 0 + 2
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,14),(12,13)]
=> [2,1,4,3,6,5,8,7,10,9,13,14,12,11] => ? = 0 + 2
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => ? = 1 + 2
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,2,10,8,1] => ? = 2 + 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => ? = 0 + 2
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
=> [2,1,6,7,9,5,4,11,8,12,10,3] => ? = 0 + 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [5,6,7,9,4,3,2,10,8,1] => ? = 2 + 2
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
=> [3,5,2,7,4,9,6,11,8,13,10,14,12,1] => ? = 1 + 2
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15] => ? = 0 + 2
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,15,16,14,13] => ? = 0 + 2
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,14),(10,13),(11,12)]
=> [2,1,4,3,6,5,8,7,12,13,14,11,10,9] => ? = 0 + 2
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,2,1] => ? = 1 + 2
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,4,3,9,10,11,12,8,7,6,5] => ? = 0 + 2
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [4,7,8,3,9,10,6,5,2,1] => ? = 1 + 2
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [5,7,8,9,4,10,6,3,2,1] => ? = 1 + 2
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,16),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15)]
=> [3,5,2,7,4,9,6,11,8,13,10,15,12,16,14,1] => ? = 1 + 2
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15,18,17] => ? = 0 + 2
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,18),(16,17)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13,17,18,16,15] => ? = 0 + 2
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,16),(12,15),(13,14)]
=> [2,1,4,3,6,5,8,7,10,9,14,15,16,13,12,11] => ? = 0 + 2
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [6,7,8,9,10,5,4,3,2,1] => ? = 0 + 2
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,18),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17)]
=> [3,5,2,7,4,9,6,11,8,13,10,15,12,17,14,18,16,1] => ? = 1 + 2
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18),(19,20)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15,18,17,20,19] => ? = 0 + 2
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,18),(14,17),(15,16)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,16,17,18,15,14,13] => ? = 0 + 2
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,8),(9,10),(11,12)]
=> [2,1,4,3,8,10,12,7,13,9,14,11,6,5] => ? = 2 + 2
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)]
=> [4,6,8,3,10,5,11,7,12,9,2,1] => ? = 1 + 2
[7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,18),(12,17),(13,16),(14,15)]
=> ? => ? = 0 + 2
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> [2,1,4,3,10,11,12,13,14,9,8,7,6,5] => ? = 0 + 2
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,7),(5,6),(8,9)]
=> [6,7,9,10,11,5,4,12,8,3,2,1] => ? = 1 + 2
[4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [(1,14),(2,11),(3,4),(5,10),(6,9),(7,8),(12,13)]
=> [4,8,9,3,10,11,13,7,6,5,2,14,12,1] => ? = 2 + 2
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
=> [7,8,9,10,11,12,6,5,4,3,2,1] => ? = 0 + 2
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [8,9,10,11,12,13,14,7,6,5,4,3,2,1] => ? = 0 + 2
[]
=> []
=> []
=> ? => ? = 0 + 2
[4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> [9,10,11,12,13,14,15,16,8,7,6,5,4,3,2,1] => ? = 0 + 2
[5,5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11)]
=> [11,12,13,14,15,16,17,18,19,20,10,9,8,7,6,5,4,3,2,1] => ? = 0 + 2
Description
The breadth of a permutation.
According to [1, Def.1.6], this is the minimal Manhattan distance between two ones in the permutation matrix of $\pi$: $$\min\{|i-j|+|\pi(i)-\pi(j)|: i\neq j\}.$$
According to [1, Def.1.3], a permutation $\pi$ is $k$-prolific, if the set of permutations obtained from $\pi$ by deleting any $k$ elements and standardising has maximal cardinality, i.e., $\binom{n}{k}$.
By [1, Thm.2.22], a permutation is $k$-prolific if and only if its breath is at least $k+2$.
By [1, Cor.4.3], the smallest permutations that are $k$-prolific have size $\lceil k^2+2k+1\rceil$, and by [1, Thm.4.4], there are $k$-prolific permutations of any size larger than this.
According to [2] the proportion of $k$-prolific permutations in the set of all permutations is asymptotically equal to $\exp(-k^2-k)$.
Matching statistic: St001198
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 67%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 67%
Values
[1]
=> [[1]]
=> [1] => [1,0]
=> ? = 0 + 2
[2]
=> [[1,2]]
=> [1,2] => [1,0,1,0]
=> 2 = 0 + 2
[1,1]
=> [[1],[2]]
=> [2,1] => [1,1,0,0]
=> ? = 0 + 2
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> ? = 0 + 2
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> ? = 1 + 2
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2 = 0 + 2
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? = 1 + 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 + 2
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2 = 0 + 2
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 0 + 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> 3 = 1 + 2
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 0 + 2
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 2
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 2
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 2
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 1 + 2
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 + 2
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0 + 2
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 2
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 + 2
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 2
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 2
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 2
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0 + 2
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 1 + 2
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0 + 2
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 + 2
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 + 2
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 + 2
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 2
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> [9,1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 2
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> [8,9,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 2
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? = 0 + 2
[1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 + 2
[10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [1,2,3,4,5,6,7,8,9,10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 2
[8,2]
=> [[1,2,3,4,5,6,7,8],[9,10]]
=> [9,10,1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 2
[6,4]
=> [[1,2,3,4,5,6],[7,8,9,10]]
=> [7,8,9,10,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? = 2 + 2
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,7,8,9,10,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 1 + 2
[7,2,2]
=> [[1,2,3,4,5,6,7],[8,9],[10,11]]
=> ? => ?
=> ? = 0 + 2
[5,3,3]
=> [[1,2,3,4,5],[6,7,8],[9,10,11]]
=> [9,10,11,6,7,8,1,2,3,4,5] => ?
=> ? = 0 + 2
[3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [10,11,7,8,9,4,5,6,1,2,3] => ?
=> ? = 1 + 2
[4,4,3,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12]]
=> [12,9,10,11,5,6,7,8,1,2,3,4] => ?
=> ? = 2 + 2
[3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? => ?
=> ? = 0 + 2
[4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> ? => ?
=> ? = 0 + 2
[]
=> []
=> [] => []
=> ? = 0 + 2
[4,4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16],[17,18,19,20]]
=> ? => ?
=> ? = 0 + 2
[5,5,5,5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15],[16,17,18,19,20],[21,22,23,24,25],[26,27,28,29,30]]
=> ? => ?
=> ? = 0 + 2
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001206
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 67%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 67%
Values
[1]
=> [[1]]
=> [1] => [1,0]
=> ? = 0 + 2
[2]
=> [[1,2]]
=> [1,2] => [1,0,1,0]
=> 2 = 0 + 2
[1,1]
=> [[1],[2]]
=> [2,1] => [1,1,0,0]
=> ? = 0 + 2
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> ? = 0 + 2
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> ? = 1 + 2
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2 = 0 + 2
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? = 1 + 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 + 2
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2 = 0 + 2
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 0 + 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> 3 = 1 + 2
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 0 + 2
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 2
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 2
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 2
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 1 + 2
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 + 2
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0 + 2
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 2
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 + 2
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 2
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 2
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 2
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0 + 2
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 1 + 2
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0 + 2
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 + 2
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 + 2
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 + 2
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 2
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> [9,1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 2
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> [8,9,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 2
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? = 0 + 2
[1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 + 2
[10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [1,2,3,4,5,6,7,8,9,10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 2
[8,2]
=> [[1,2,3,4,5,6,7,8],[9,10]]
=> [9,10,1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 2
[6,4]
=> [[1,2,3,4,5,6],[7,8,9,10]]
=> [7,8,9,10,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? = 2 + 2
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,7,8,9,10,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 1 + 2
[7,2,2]
=> [[1,2,3,4,5,6,7],[8,9],[10,11]]
=> ? => ?
=> ? = 0 + 2
[5,3,3]
=> [[1,2,3,4,5],[6,7,8],[9,10,11]]
=> [9,10,11,6,7,8,1,2,3,4,5] => ?
=> ? = 0 + 2
[3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [10,11,7,8,9,4,5,6,1,2,3] => ?
=> ? = 1 + 2
[4,4,3,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12]]
=> [12,9,10,11,5,6,7,8,1,2,3,4] => ?
=> ? = 2 + 2
[3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? => ?
=> ? = 0 + 2
[4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> ? => ?
=> ? = 0 + 2
[]
=> []
=> [] => []
=> ? = 0 + 2
[4,4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16],[17,18,19,20]]
=> ? => ?
=> ? = 0 + 2
[5,5,5,5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15],[16,17,18,19,20],[21,22,23,24,25],[26,27,28,29,30]]
=> ? => ?
=> ? = 0 + 2
Description
The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
Matching statistic: St000408
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000408: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 67%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000408: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0]
=> [(1,2)]
=> [2,1] => 0
[2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 0
[1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => 0
[3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => ? = 0
[2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => ? = 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => ? = 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => ? = 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,7,9,6,10,8,5] => ? = 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? = 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => ? = 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [3,5,2,7,4,9,6,10,8,1] => ? = 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => ? = 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [2,1,4,3,6,5,8,7,11,12,10,9] => ? = 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => ? = 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => ? = 0
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => ? = 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [4,5,7,3,2,9,6,10,8,1] => ? = 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> [3,5,2,7,4,9,6,11,8,12,10,1] => ? = 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13] => ? = 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,14),(12,13)]
=> [2,1,4,3,6,5,8,7,10,9,13,14,12,11] => ? = 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => ? = 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,2,10,8,1] => ? = 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => ? = 0
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
=> [2,1,6,7,9,5,4,11,8,12,10,3] => ? = 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [5,6,7,9,4,3,2,10,8,1] => ? = 2
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
=> [3,5,2,7,4,9,6,11,8,13,10,14,12,1] => ? = 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15] => ? = 0
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,15,16,14,13] => ? = 0
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,14),(10,13),(11,12)]
=> [2,1,4,3,6,5,8,7,12,13,14,11,10,9] => ? = 0
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,2,1] => ? = 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,4,3,9,10,11,12,8,7,6,5] => ? = 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [4,7,8,3,9,10,6,5,2,1] => ? = 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [5,7,8,9,4,10,6,3,2,1] => ? = 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,16),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15)]
=> [3,5,2,7,4,9,6,11,8,13,10,15,12,16,14,1] => ? = 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15,18,17] => ? = 0
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,18),(16,17)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13,17,18,16,15] => ? = 0
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,16),(12,15),(13,14)]
=> [2,1,4,3,6,5,8,7,10,9,14,15,16,13,12,11] => ? = 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [6,7,8,9,10,5,4,3,2,1] => ? = 0
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,18),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17)]
=> [3,5,2,7,4,9,6,11,8,13,10,15,12,17,14,18,16,1] => ? = 1
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18),(19,20)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15,18,17,20,19] => ? = 0
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,18),(14,17),(15,16)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,16,17,18,15,14,13] => ? = 0
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,8),(9,10),(11,12)]
=> [2,1,4,3,8,10,12,7,13,9,14,11,6,5] => ? = 2
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)]
=> [4,6,8,3,10,5,11,7,12,9,2,1] => ? = 1
[7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,18),(12,17),(13,16),(14,15)]
=> ? => ? = 0
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> [2,1,4,3,10,11,12,13,14,9,8,7,6,5] => ? = 0
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,7),(5,6),(8,9)]
=> [6,7,9,10,11,5,4,12,8,3,2,1] => ? = 1
[4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [(1,14),(2,11),(3,4),(5,10),(6,9),(7,8),(12,13)]
=> [4,8,9,3,10,11,13,7,6,5,2,14,12,1] => ? = 2
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
=> [7,8,9,10,11,12,6,5,4,3,2,1] => ? = 0
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [8,9,10,11,12,13,14,7,6,5,4,3,2,1] => ? = 0
[]
=> []
=> []
=> ? => ? = 0
[4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> [9,10,11,12,13,14,15,16,8,7,6,5,4,3,2,1] => ? = 0
Description
The number of occurrences of the pattern 4231 in a permutation.
It is a necessary condition that a permutation $\pi$ avoids this pattern for the Schubert variety associated to $\pi$ to be smooth [2].
Matching statistic: St000440
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000440: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 67%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000440: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0]
=> [(1,2)]
=> [2,1] => 0
[2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 0
[1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => 0
[3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => ? = 0
[2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => ? = 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => ? = 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => ? = 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,7,9,6,10,8,5] => ? = 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? = 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => ? = 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [3,5,2,7,4,9,6,10,8,1] => ? = 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => ? = 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [2,1,4,3,6,5,8,7,11,12,10,9] => ? = 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => ? = 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => ? = 0
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => ? = 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [4,5,7,3,2,9,6,10,8,1] => ? = 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> [3,5,2,7,4,9,6,11,8,12,10,1] => ? = 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13] => ? = 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,14),(12,13)]
=> [2,1,4,3,6,5,8,7,10,9,13,14,12,11] => ? = 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => ? = 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,2,10,8,1] => ? = 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => ? = 0
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
=> [2,1,6,7,9,5,4,11,8,12,10,3] => ? = 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [5,6,7,9,4,3,2,10,8,1] => ? = 2
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
=> [3,5,2,7,4,9,6,11,8,13,10,14,12,1] => ? = 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15] => ? = 0
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,15,16,14,13] => ? = 0
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,14),(10,13),(11,12)]
=> [2,1,4,3,6,5,8,7,12,13,14,11,10,9] => ? = 0
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,2,1] => ? = 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,4,3,9,10,11,12,8,7,6,5] => ? = 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [4,7,8,3,9,10,6,5,2,1] => ? = 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [5,7,8,9,4,10,6,3,2,1] => ? = 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,16),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15)]
=> [3,5,2,7,4,9,6,11,8,13,10,15,12,16,14,1] => ? = 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15,18,17] => ? = 0
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,18),(16,17)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13,17,18,16,15] => ? = 0
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,16),(12,15),(13,14)]
=> [2,1,4,3,6,5,8,7,10,9,14,15,16,13,12,11] => ? = 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [6,7,8,9,10,5,4,3,2,1] => ? = 0
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,18),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(14,15),(16,17)]
=> [3,5,2,7,4,9,6,11,8,13,10,15,12,17,14,18,16,1] => ? = 1
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18),(19,20)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15,18,17,20,19] => ? = 0
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,18),(14,17),(15,16)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,16,17,18,15,14,13] => ? = 0
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,8),(9,10),(11,12)]
=> [2,1,4,3,8,10,12,7,13,9,14,11,6,5] => ? = 2
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)]
=> [4,6,8,3,10,5,11,7,12,9,2,1] => ? = 1
[7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,18),(12,17),(13,16),(14,15)]
=> ? => ? = 0
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> [2,1,4,3,10,11,12,13,14,9,8,7,6,5] => ? = 0
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,7),(5,6),(8,9)]
=> [6,7,9,10,11,5,4,12,8,3,2,1] => ? = 1
[4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [(1,14),(2,11),(3,4),(5,10),(6,9),(7,8),(12,13)]
=> [4,8,9,3,10,11,13,7,6,5,2,14,12,1] => ? = 2
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
=> [7,8,9,10,11,12,6,5,4,3,2,1] => ? = 0
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [8,9,10,11,12,13,14,7,6,5,4,3,2,1] => ? = 0
[]
=> []
=> []
=> ? => ? = 0
[4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> [9,10,11,12,13,14,15,16,8,7,6,5,4,3,2,1] => ? = 0
Description
The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation.
There is a bijection between permutations avoiding these two pattern and Schröder paths [1,2].
The following 27 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001866The nesting alignments of a signed permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000217The number of occurrences of the pattern 312 in a permutation. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000664The number of right ropes of a permutation. St000779The tier of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001705The number of occurrences of the pattern 2413 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001490The number of connected components of a skew partition. St000022The number of fixed points of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000546The number of global descents of a permutation. St000731The number of double exceedences of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000862The number of parts of the shifted shape of a permutation.
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