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Your data matches 22 different statistics following compositions of up to 3 maps.
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Matching statistic: St001440
St001440: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 0
[1,1]
=> 1
[3]
=> 0
[2,1]
=> 1
[1,1,1]
=> 0
[4]
=> 0
[3,1]
=> 1
[2,2]
=> 0
[2,1,1]
=> 1
[1,1,1,1]
=> 0
[5]
=> 0
[4,1]
=> 1
[3,2]
=> 1
[3,1,1]
=> 1
[2,2,1]
=> 1
[2,1,1,1]
=> 1
[1,1,1,1,1]
=> 0
[6]
=> 0
[5,1]
=> 1
[4,2]
=> 1
[4,1,1]
=> 2
[3,3]
=> 1
[3,2,1]
=> 3
[3,1,1,1]
=> 1
[2,2,2]
=> 0
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 0
[7]
=> 0
[6,1]
=> 1
[5,2]
=> 2
[5,1,1]
=> 2
[4,3]
=> 2
[4,2,1]
=> 5
[4,1,1,1]
=> 3
[3,3,1]
=> 3
[3,2,2]
=> 3
[3,2,1,1]
=> 5
[3,1,1,1,1]
=> 2
[2,2,2,1]
=> 2
[2,2,1,1,1]
=> 2
[2,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> 0
[8]
=> 0
[7,1]
=> 1
[6,2]
=> 2
[6,1,1]
=> 3
[5,3]
=> 4
[5,2,1]
=> 8
Description
The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition.
Matching statistic: St000764
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
St000764: Integer compositions ⟶ ℤResult quality: 6% ●values known / values provided: 16%●distinct values known / distinct values provided: 6%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
St000764: Integer compositions ⟶ ℤResult quality: 6% ●values known / values provided: 16%●distinct values known / distinct values provided: 6%
Values
[1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> [2,2] => 1 = 0 + 1
[2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [3,3] => 1 = 0 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [2,3,1] => 2 = 1 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [4,4] => 1 = 0 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [2,2,2] => 1 = 0 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [2,4,2] => 2 = 1 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> [5,5] => 1 = 0 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [3,2,3] => 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [3,4,1] => 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [2,3,2,1] => 2 = 1 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> [2,5,3] => ? = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> [6,6] => 1 = 0 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> [4,2,4] => ? ∊ {1,1,1} + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [3,3,2] => 1 = 0 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [2,3,3] => 2 = 1 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [2,2,3,1] => 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> [2,4,2,2] => ? ∊ {1,1,1} + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> [2,6,4] => ? ∊ {1,1,1} + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12,14]]
=> [7,7] => ? ∊ {0,0,1,1,1,1,1,2,2,3} + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> [5,2,5] => ? ∊ {0,0,1,1,1,1,1,2,2,3} + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> [4,3,3] => ? ∊ {0,0,1,1,1,1,1,2,2,3} + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> [3,3,4] => ? ∊ {0,0,1,1,1,1,1,2,2,3} + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> [4,5,1] => ? ∊ {0,0,1,1,1,1,1,2,2,3} + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [2,2,2,2] => 1 = 0 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> [2,4,3,1] => ? ∊ {0,0,1,1,1,1,1,2,2,3} + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> [3,5,2] => ? ∊ {0,0,1,1,1,1,1,2,2,3} + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> [2,3,3,2] => ? ∊ {0,0,1,1,1,1,1,2,2,3} + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> [2,5,2,3] => ? ∊ {0,0,1,1,1,1,1,2,2,3} + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> [2,7,5] => ? ∊ {0,0,1,1,1,1,1,2,2,3} + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,7,15],[8,9,10,11,12,13,14,16]]
=> [8,8] => ? ∊ {0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,7,13],[6,8,9,10,11,12,14]]
=> [6,2,6] => ? ∊ {0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[1,2,3,4,7,11],[5,6,8,9,10,12]]
=> [5,3,4] => ? ∊ {0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[1,2,3,5,6,11],[4,7,8,9,10,12]]
=> [4,3,5] => ? ∊ {0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[1,2,3,7,9],[4,5,6,8,10]]
=> [4,4,2] => ? ∊ {0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> [3,2,2,3] => 1 = 0 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> [2,4,4] => ? ∊ {0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> [3,2,4,1] => ? ∊ {0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> [3,4,2,1] => ? ∊ {0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> [2,3,2,2,1] => ? ∊ {0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[1,3,4,5,6,9],[2,7,8,10,11,12]]
=> [2,5,3,2] => ? ∊ {0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> [2,2,4,2] => ? ∊ {0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,3,4,5,7,8],[2,6,9,10,11,12]]
=> [2,4,3,3] => ? ∊ {0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[1,3,4,5,6,7,9],[2,8,10,11,12,13,14]]
=> [2,6,2,4] => ? ∊ {0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,9],[2,10,11,12,13,14,15,16]]
=> [2,8,6] => ? ∊ {0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,7,8,17],[9,10,11,12,13,14,15,16,18]]
=> [9,9] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,8,15],[7,9,10,11,12,13,14,16]]
=> [7,2,7] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,5,8,13],[6,7,9,10,11,12,14]]
=> [6,3,5] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,6,7,13],[5,8,9,10,11,12,14]]
=> [5,3,6] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[1,2,3,4,8,11],[5,6,7,9,10,12]]
=> [5,4,3] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,7,11],[4,6,8,9,10,12]]
=> [4,2,2,4] => 1 = 0 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[1,2,4,5,6,11],[3,7,8,9,10,12]]
=> [3,4,5] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[1,2,3,4,9,10],[5,6,7,8,11,12]]
=> [5,6,1] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[1,2,4,7,9],[3,5,6,8,10]]
=> [3,2,3,2] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[1,2,5,6,9],[3,4,7,8,10]]
=> [3,4,3] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[1,3,4,6,9],[2,5,7,8,10]]
=> [2,3,2,3] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[1,3,4,5,6,10],[2,7,8,9,11,12]]
=> [2,5,4,1] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,7,8],[3,4,6,9,10]]
=> [3,3,3,1] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> [2,3,4,1] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> [2,2,3,2,1] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [[1,3,4,5,7,9],[2,6,8,10,11,12]]
=> [2,4,2,2,2] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [[1,3,4,5,6,7,10],[2,8,9,11,12,13,14]]
=> [2,6,3,3] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[1,2,5,6,7,8],[3,4,9,10,11,12]]
=> [3,6,3] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [[1,3,4,6,7,8],[2,5,9,10,11,12]]
=> [2,3,4,3] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,8,9],[2,7,10,11,12,13,14]]
=> [2,5,3,4] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,10],[2,9,11,12,13,14,15,16]]
=> [2,7,2,5] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 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]
=> [[1,3,4,5,6,7,8,9,10],[2,11,12,13,14,15,16,17,18]]
=> [2,9,7] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,7,8,9,19],[10,11,12,13,14,15,16,17,18,20]]
=> [10,10] => ? ∊ {0,1,3,3,3,3,4,5,5,5,5,6,6,8,9,9,12,12,13,13,18,18,19,19,21,21,24,24} + 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> [3,3,2,2] => 1 = 0 + 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> [2,2,3,3] => 2 = 1 + 1
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [[1,2,3,6,8,11],[4,5,7,9,10,12]]
=> [4,3,2,3] => 1 = 0 + 1
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [[1,2,4,5,8,11],[3,6,7,9,10,12]]
=> [3,3,3,3] => 1 = 0 + 1
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,6,7,11],[3,5,8,9,10,12]]
=> [3,2,3,4] => 2 = 1 + 1
Description
The number of strong records in an integer composition.
A strong record is an element $a_i$ such that $a_i > a_j$ for all $j < i$. In particular, the first part of a composition is a strong record.
Theorem 1.1 of [1] provides the generating function for compositions with parts in a given set according to the sum of the parts, the number of parts and the number of strong records.
Matching statistic: St001487
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 8% ●values known / values provided: 14%●distinct values known / distinct values provided: 8%
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 8% ●values known / values provided: 14%●distinct values known / distinct values provided: 8%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [[2],[]]
=> 1 = 0 + 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> 1 = 0 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 2 = 1 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? = 0 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> 1 = 0 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 2 = 1 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? ∊ {0,0,0} + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? ∊ {0,0,0} + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 2 = 1 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ? ∊ {0,0,0} + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4],[]]
=> ? ∊ {0,1,1} + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> ? ∊ {0,1,1} + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 1 = 0 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> 2 = 1 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> 2 = 1 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2],[1,1]]
=> ? ∊ {0,1,1} + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[4,4,4,4],[]]
=> ? ∊ {0,0,1,1,1,1,2,3} + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3],[]]
=> ? ∊ {0,0,1,1,1,1,2,3} + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? ∊ {0,0,1,1,1,1,2,3} + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ? ∊ {0,0,1,1,1,1,2,3} + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? ∊ {0,0,1,1,1,1,2,3} + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> 1 = 0 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 3 = 2 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? ∊ {0,0,1,1,1,1,2,3} + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> 2 = 1 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [[3,3,3,2],[1,1,1]]
=> ? ∊ {0,0,1,1,1,1,2,3} + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[4,4,4,2],[1,1,1]]
=> ? ∊ {0,0,1,1,1,1,2,3} + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[5,5,5,5],[]]
=> ? ∊ {0,0,2,2,2,2,2,2,3,3,3,5,5} + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[5,5,5],[]]
=> ? ∊ {0,0,2,2,2,2,2,2,3,3,3,5,5} + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [[4,4,3],[]]
=> ? ∊ {0,0,2,2,2,2,2,2,3,3,3,5,5} + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [[4,4,4],[1]]
=> ? ∊ {0,0,2,2,2,2,2,2,3,3,3,5,5} + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ? ∊ {0,0,2,2,2,2,2,2,3,3,3,5,5} + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> ? ∊ {0,0,2,2,2,2,2,2,3,3,3,5,5} + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> ? ∊ {0,0,2,2,2,2,2,2,3,3,3,5,5} + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? ∊ {0,0,2,2,2,2,2,2,3,3,3,5,5} + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ? ∊ {0,0,2,2,2,2,2,2,3,3,3,5,5} + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> 2 = 1 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [[4,3,2],[1,1]]
=> ? ∊ {0,0,2,2,2,2,2,2,3,3,3,5,5} + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> 2 = 1 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> ? ∊ {0,0,2,2,2,2,2,2,3,3,3,5,5} + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [[5,5,2],[1,1]]
=> ? ∊ {0,0,2,2,2,2,2,2,3,3,3,5,5} + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[5,5,5,2],[1,1,1]]
=> ? ∊ {0,0,2,2,2,2,2,2,3,3,3,5,5} + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [[5,5,5,5,5],[]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [[4,4,4,4,4],[]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [[4,4,4,4],[1]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [[4,4,4,3],[]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [[5,5],[]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [[3,3,3,2],[]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> 2 = 1 + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [[5,2],[1]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [[4,4,4,2],[2,1,1]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1,1]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [[4,4,3,2],[1,1,1]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[4,4,4,4,2],[1,1,1,1]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 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]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[5,5,5,5,2],[1,1,1,1]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [[6,6,6,6,6],[]]
=> ? ∊ {0,0,1,3,3,3,3,4,5,5,5,5,6,6,8,9,9,12,12,13,13,18,18,19,19,21,21,24,24} + 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 2 = 1 + 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> 1 = 0 + 1
Description
The number of inner corners of a skew partition.
Matching statistic: St000689
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000689: Dyck paths ⟶ ℤResult quality: 8% ●values known / values provided: 11%●distinct values known / distinct values provided: 8%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000689: Dyck paths ⟶ ℤResult quality: 8% ●values known / values provided: 11%●distinct values known / distinct values provided: 8%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,1,0,0]
=> 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1}
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1}
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1}
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1}
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1}
[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]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,2,3}
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,2,3}
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,2,3}
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,2,3}
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,2,3}
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,2,3}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,1,1,1,1,1,2,3}
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,1,2,3}
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [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]
=> ? ∊ {0,0,1,1,1,1,1,2,3}
[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]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [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]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[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]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
Description
The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid.
The correspondence between LNakayama algebras and Dyck paths is explained in [[St000684]]. A module $M$ is $n$-rigid, if $\operatorname{Ext}^i(M,M)=0$ for $1\leq i\leq n$.
This statistic gives the maximal $n$ such that the minimal generator-cogenerator module $A \oplus D(A)$ of the LNakayama algebra $A$ corresponding to a Dyck path is $n$-rigid.
An application is to check for maximal $n$-orthogonal objects in the module category in the sense of [2].
Matching statistic: St000768
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
St000768: Integer compositions ⟶ ℤResult quality: 6% ●values known / values provided: 11%●distinct values known / distinct values provided: 6%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
St000768: Integer compositions ⟶ ℤResult quality: 6% ●values known / values provided: 11%●distinct values known / distinct values provided: 6%
Values
[1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> [3,1] => 0
[2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [2,3,1] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [4,2] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [3,4,1] => 1
[2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [3,2,1] => 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [5,3] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> [4,5,1] => ? ∊ {0,0}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [2,2,3,1] => 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [2,4,2] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [4,2,2] => 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> [6,4] => ? ∊ {0,0}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> [5,6,1] => ? ∊ {1,1,1,1}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> [3,2,4,1] => ? ∊ {1,1,1,1}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [2,3,2,1] => 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [4,3,1] => 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [3,3,2] => 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> [5,2,3] => ? ∊ {1,1,1,1}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> [7,5] => ? ∊ {1,1,1,1}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12,14]]
=> [6,7,1] => ? ∊ {0,0,1,1,1,1,1,2,2,3}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> [4,2,5,1] => ? ∊ {0,0,1,1,1,1,1,2,2,3}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> [3,3,3,1] => ? ∊ {0,0,1,1,1,1,1,2,2,3}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> [2,3,4,1] => ? ∊ {0,0,1,1,1,1,1,2,2,3}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> [3,5,2] => ? ∊ {0,0,1,1,1,1,1,2,2,3}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [3,2,2,1] => 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> [5,3,2] => ? ∊ {0,0,1,1,1,1,1,2,2,3}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> [2,5,3] => ? ∊ {0,0,1,1,1,1,1,2,2,3}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> [4,3,3] => ? ∊ {0,0,1,1,1,1,1,2,2,3}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> [6,2,4] => ? ∊ {0,0,1,1,1,1,1,2,2,3}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> [8,6] => ? ∊ {0,0,1,1,1,1,1,2,2,3}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,7,15],[8,9,10,11,12,13,14,16]]
=> [7,8,1] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,7,13],[6,8,9,10,11,12,14]]
=> [5,2,6,1] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[1,2,3,4,7,11],[5,6,8,9,10,12]]
=> [4,3,4,1] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[1,2,3,5,6,11],[4,7,8,9,10,12]]
=> [3,3,5,1] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[1,2,3,7,9],[4,5,6,8,10]]
=> [3,4,2,1] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> [2,2,2,3,1] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> [5,4,1] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> [2,2,4,2] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> [2,4,2,2] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> [4,2,2,2] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[1,3,4,5,6,9],[2,7,8,10,11,12]]
=> [6,3,3] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> [3,4,3] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,3,4,5,7,8],[2,6,9,10,11,12]]
=> [5,3,4] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[1,3,4,5,6,7,9],[2,8,10,11,12,13,14]]
=> [7,2,5] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,9],[2,10,11,12,13,14,15,16]]
=> [9,7] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,7,8,17],[9,10,11,12,13,14,15,16,18]]
=> [8,9,1] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,8,15],[7,9,10,11,12,13,14,16]]
=> [6,2,7,1] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,5,8,13],[6,7,9,10,11,12,14]]
=> [5,3,5,1] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,6,7,13],[5,8,9,10,11,12,14]]
=> [4,3,6,1] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[1,2,3,4,8,11],[5,6,7,9,10,12]]
=> [4,4,3,1] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,7,11],[4,6,8,9,10,12]]
=> [3,2,2,4,1] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[1,2,4,5,6,11],[3,7,8,9,10,12]]
=> [2,4,5,1] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[1,2,3,4,9,10],[5,6,7,8,11,12]]
=> [4,6,2] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[1,2,4,7,9],[3,5,6,8,10]]
=> [2,2,3,2,1] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[1,2,5,6,9],[3,4,7,8,10]]
=> [2,4,3,1] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[1,3,4,6,9],[2,5,7,8,10]]
=> [4,2,3,1] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[1,3,4,5,6,10],[2,7,8,9,11,12]]
=> [6,4,2] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,7,8],[3,4,6,9,10]]
=> [2,3,3,2] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> [4,4,2] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> [3,3,2,2] => 0
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [[1,3,4,5,7,9],[2,6,8,10,11,12]]
=> [5,2,2,3] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [[1,3,4,5,6,7,10],[2,8,9,11,12,13,14]]
=> [7,3,4] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[1,2,5,6,7,8],[3,4,9,10,11,12]]
=> [2,6,4] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [[1,3,4,6,7,8],[2,5,9,10,11,12]]
=> [4,4,4] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,8,9],[2,7,10,11,12,13,14]]
=> [6,3,5] => ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [[1,3,4,6,7,9],[2,5,8,10,11,12]]
=> [4,3,2,3] => 0
Description
The number of peaks in an integer composition.
A peak is an ascent followed by a descent, i.e., a subsequence $c_{i-1} c_i c_{i+1}$ with $c_i > \max(c_{i-1}, c_{i+1})$.
Matching statistic: St001113
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001113: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 11%●distinct values known / distinct values provided: 6%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001113: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 11%●distinct values known / distinct values provided: 6%
Values
[1]
=> 10 => [1,2] => [1,0,1,1,0,0]
=> 0
[2]
=> 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[1,1]
=> 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[3]
=> 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0
[2,1]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,1,1]
=> 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[4]
=> 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[3,1]
=> 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 0
[2,2]
=> 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[2,1,1]
=> 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 0
[1,1,1,1]
=> 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[5]
=> 100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1}
[4,1]
=> 100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1}
[3,2]
=> 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 0
[3,1,1]
=> 100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1}
[2,2,1]
=> 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1}
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1}
[6]
=> 1000000 => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,1,1,1,2,2,3}
[5,1]
=> 1000010 => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? ∊ {0,0,0,1,1,1,2,2,3}
[4,2]
=> 100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? ∊ {0,0,0,1,1,1,2,2,3}
[4,1,1]
=> 1000110 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,1,1,1,2,2,3}
[3,3]
=> 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[3,2,1]
=> 101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,0,0,1,1,1,2,2,3}
[3,1,1,1]
=> 1001110 => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,1,1,1,2,2,3}
[2,2,2]
=> 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[2,2,1,1]
=> 110110 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,1,1,1,2,2,3}
[2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,1,1,1,2,2,3}
[1,1,1,1,1,1]
=> 1111110 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,1,1,1,2,2,3}
[7]
=> 10000000 => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[6,1]
=> 10000010 => [1,6,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[5,2]
=> 1000100 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[5,1,1]
=> 10000110 => [1,5,1,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[4,3]
=> 101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[4,2,1]
=> 1001010 => [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[4,1,1,1]
=> 10001110 => [1,4,1,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[3,3,1]
=> 110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[3,2,2]
=> 101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[3,2,1,1]
=> 1010110 => [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[2,2,2,1]
=> 111010 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[1,1,1,1,1,1,1]
=> 11111110 => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[8]
=> 100000000 => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[7,1]
=> 100000010 => [1,7,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[6,2]
=> 10000100 => [1,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[6,1,1]
=> 100000110 => [1,6,1,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[5,3]
=> 1001000 => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[5,2,1]
=> 10001010 => [1,4,2,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[5,1,1,1]
=> 100001110 => [1,5,1,1,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,4]
=> 110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,3,1]
=> 1010010 => [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,2,2]
=> 1001100 => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,2,1,1]
=> 10010110 => [1,3,2,1,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,1,1,1,1]
=> 100011110 => [1,4,1,1,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,3,2]
=> 110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,3,1,1]
=> 1100110 => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,2,2,1]
=> 1011010 => [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,2,1,1,1]
=> 10101110 => [1,2,2,1,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,1,1,1,1,1]
=> 100111110 => [1,3,1,1,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[2,2,2,2]
=> 111100 => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[2,2,2,1,1]
=> 1110110 => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[2,2,1,1,1,1]
=> 11011110 => [1,1,2,1,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[2,1,1,1,1,1,1]
=> 101111110 => [1,2,1,1,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
Description
Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra.
Matching statistic: St001730
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 6% ●values known / values provided: 11%●distinct values known / distinct values provided: 6%
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 6% ●values known / values provided: 11%●distinct values known / distinct values provided: 6%
Values
[1]
=> [1,0]
=> 10 => 11 => 0
[2]
=> [1,0,1,0]
=> 1010 => 1111 => 0
[1,1]
=> [1,1,0,0]
=> 1100 => 1001 => 1
[3]
=> [1,0,1,0,1,0]
=> 101010 => 111111 => 0
[2,1]
=> [1,0,1,1,0,0]
=> 101100 => 111001 => 0
[1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 100001 => 1
[4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 11111111 => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 11111001 => 0
[2,2]
=> [1,1,1,0,0,0]
=> 111000 => 101101 => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 11100001 => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 10000001 => 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 1111111111 => ? ∊ {0,1,1,1,1}
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 1111111001 => ? ∊ {0,1,1,1,1}
[3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 11101101 => 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 1111100001 => ? ∊ {0,1,1,1,1}
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 10110001 => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 1110000001 => ? ∊ {0,1,1,1,1}
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 1000000001 => ? ∊ {0,1,1,1,1}
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => 111111111111 => ? ∊ {0,0,1,1,1,1,2,2,3}
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => 111111111001 => ? ∊ {0,0,1,1,1,1,2,2,3}
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 1111101101 => ? ∊ {0,0,1,1,1,1,2,2,3}
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => 111111100001 => ? ∊ {0,0,1,1,1,1,2,2,3}
[3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 10111101 => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 1110110001 => ? ∊ {0,0,1,1,1,1,2,2,3}
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => 111110000001 => ? ∊ {0,0,1,1,1,1,2,2,3}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 10100101 => 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => 1011000001 => ? ∊ {0,0,1,1,1,1,2,2,3}
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 101101010100 => 111000000001 => ? ∊ {0,0,1,1,1,1,2,2,3}
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 110101010100 => 100000000001 => ? ∊ {0,0,1,1,1,1,2,2,3}
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 10101010101010 => 11111111111111 => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => 11111111111001 => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 101010111000 => 111111101101 => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 10101010110100 => 11111111100001 => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 1110111101 => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 101011100100 => 111110110001 => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 10101011010100 => 11111110000001 => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => 1011110001 => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 1110100101 => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 101110010100 => 111011000001 => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 10101101010100 => 11111000000001 => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => 1010010001 => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 111001010100 => 101100000001 => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 10110101010100 => 11100000000001 => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 11010101010100 => 10000000000001 => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1010101010101010 => 1111111111111111 => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1010101010101100 => 1111111111111001 => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 10101010111000 => 11111111101101 => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 1010101010110100 => 1111111111100001 => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 101011101000 => 111110111101 => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> 10101011100100 => 11111110110001 => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 1010101011010100 => 1111111110000001 => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => 1011111101 => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 101110100100 => 111011110001 => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 101011110000 => 111110100101 => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> 10101110010100 => 11111011000001 => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 1010101101010100 => 1111111000000001 => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => 1011100101 => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 111010010100 => 101111000001 => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 101111000100 => 111010010001 => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> 10111001010100 => 11101100000001 => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 1010110101010100 => 1111100000000001 => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => 1010000101 => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 111100010100 => 101001000001 => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 11100101010100 => 10110000000001 => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1011010101010100 => 1110000000000001 => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
Description
The number of times the path corresponding to a binary word crosses the base line.
Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.
Matching statistic: St001964
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 11%●distinct values known / distinct values provided: 6%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 11%●distinct values known / distinct values provided: 6%
Values
[1]
=> [1,0]
=> [2,1] => ([],2)
=> 0
[2]
=> [1,0,1,0]
=> [3,1,2] => ([(1,2)],3)
=> ? ∊ {0,1}
[1,1]
=> [1,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> ? ∊ {0,1}
[3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => ([(2,3)],4)
=> ? = 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 1
[2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
=> ? = 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ([(0,5),(1,3),(3,4),(4,2),(4,5)],6)
=> ? ∊ {1,1,1}
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ([(1,5),(4,3),(5,2),(5,4)],6)
=> ? ∊ {1,1,1}
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> ? ∊ {1,1,1}
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? ∊ {0,0,1,1,1,1,1,2,2,3}
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ([(0,6),(1,4),(3,5),(4,3),(5,2),(5,6)],7)
=> ? ∊ {0,0,1,1,1,1,1,2,2,3}
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,4),(4,2),(4,5),(5,3)],6)
=> ? ∊ {0,0,1,1,1,1,1,2,2,3}
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ([(1,5),(4,3),(5,6),(6,2),(6,4)],7)
=> ? ∊ {0,0,1,1,1,1,1,2,2,3}
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,3}
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ([(0,2),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3)],6)
=> ? ∊ {0,0,1,1,1,1,1,2,2,3}
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ([(1,6),(4,5),(5,3),(6,2),(6,4)],7)
=> ? ∊ {0,0,1,1,1,1,1,2,2,3}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ([(0,5),(1,2),(1,3),(1,5),(5,4)],6)
=> ? ∊ {0,0,1,1,1,1,1,2,2,3}
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ([(0,6),(1,5),(1,6),(3,4),(4,2),(5,3)],7)
=> ? ∊ {0,0,1,1,1,1,1,2,2,3}
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? ∊ {0,0,1,1,1,1,1,2,2,3}
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => ([(1,7),(3,4),(4,6),(5,3),(6,2),(7,5)],8)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => ([(0,7),(1,5),(3,4),(4,6),(5,3),(6,2),(6,7)],8)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => ([(0,6),(1,4),(4,5),(5,2),(5,6),(6,3)],7)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,1,2,3,4,7,5,6] => ([(1,6),(4,7),(5,2),(6,4),(7,3),(7,5)],8)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ([(1,4),(1,5),(4,3),(5,2)],6)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => ([(0,3),(1,4),(1,5),(1,6),(2,6),(3,2),(3,4),(3,5)],7)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [8,1,2,3,7,4,5,6] => ([(1,6),(4,5),(5,2),(6,7),(7,3),(7,4)],8)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => ([(1,4),(2,3),(2,5),(4,5)],6)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,3),(1,5),(4,2),(5,4)],6)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => ([(0,3),(0,5),(0,6),(1,4),(1,5),(1,6),(3,4),(4,2)],7)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => ([(0,7),(1,6),(3,5),(4,3),(5,2),(6,4),(6,7)],8)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => ([(0,5),(1,4),(4,2),(4,3),(4,5)],6)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => ([(0,6),(1,4),(1,6),(4,3),(5,2),(6,5)],7)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [8,1,7,2,3,4,5,6] => ([(1,3),(1,7),(4,6),(5,4),(6,2),(7,5)],8)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => ([(2,7),(4,6),(5,4),(6,3),(7,5)],8)
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[8]
=> [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] => ([(1,8),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],9)
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[7,1]
=> [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] => ([(0,8),(1,6),(3,5),(4,3),(5,7),(6,4),(7,2),(7,8)],9)
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => ([(0,7),(1,5),(4,6),(5,4),(6,2),(6,7),(7,3)],8)
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [9,1,2,3,4,5,8,6,7] => ([(1,7),(4,5),(5,8),(6,2),(7,4),(8,3),(8,6)],9)
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => ([(1,6),(4,3),(5,2),(6,4),(6,5)],7)
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,1,2,3,8,7,4,6] => ([(0,5),(0,6),(0,7),(1,3),(2,7),(3,4),(4,2),(4,5),(4,6)],8)
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [9,1,2,3,4,8,5,6,7] => ([(1,7),(4,8),(5,6),(6,2),(7,4),(8,3),(8,5)],9)
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => ([(3,4),(4,5)],6)
=> 0
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => ([(1,4),(1,5),(3,6),(4,3),(5,2),(5,6)],7)
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ([(0,6),(1,4),(4,3),(4,6),(5,2),(6,5)],7)
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,1,2,8,7,3,5,6] => ([(0,5),(0,6),(0,7),(1,4),(3,7),(4,3),(4,5),(4,6),(7,2)],8)
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [8,1,2,3,9,4,5,6,7] => ([(0,8),(1,6),(3,5),(4,3),(5,2),(6,7),(7,4),(7,8)],9)
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => ([(0,5),(1,4),(2,3),(2,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => ([(0,5),(1,5),(1,6),(2,3),(3,6),(6,4)],7)
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ([(0,6),(1,4),(1,6),(4,5),(6,2),(6,3),(6,5)],7)
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,7,8,2,4,5,6] => ([(0,6),(0,7),(1,4),(1,7),(4,6),(5,3),(6,5),(7,2)],8)
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [9,1,2,8,3,4,5,6,7] => ([(1,8),(4,5),(5,7),(6,4),(7,2),(8,3),(8,6)],9)
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => ([(1,3),(2,4),(4,5)],6)
=> 0
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => ([(0,6),(1,5),(5,2),(5,3),(5,6),(6,4)],7)
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [2,8,7,1,3,4,5,6] => ([(0,7),(1,3),(1,4),(1,7),(5,2),(6,5),(7,6)],8)
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> 0
Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Matching statistic: St000326
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 6% ●values known / values provided: 11%●distinct values known / distinct values provided: 6%
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 6% ●values known / values provided: 11%●distinct values known / distinct values provided: 6%
Values
[1]
=> [1,0]
=> 10 => 10 => 1 = 0 + 1
[2]
=> [1,0,1,0]
=> 1010 => 0110 => 2 = 1 + 1
[1,1]
=> [1,1,0,0]
=> 1100 => 1010 => 1 = 0 + 1
[3]
=> [1,0,1,0,1,0]
=> 101010 => 100110 => 1 = 0 + 1
[2,1]
=> [1,0,1,1,0,0]
=> 101100 => 110010 => 1 = 0 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 011010 => 2 = 1 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 01100110 => 2 = 1 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 01110010 => 2 = 1 + 1
[2,2]
=> [1,1,1,0,0,0]
=> 111000 => 101010 => 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 10110010 => 1 = 0 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 10011010 => 1 = 0 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? => ? ∊ {0,1,1,1,1} + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? => ? ∊ {0,1,1,1,1} + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 11010010 => 1 = 0 + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? => ? ∊ {0,1,1,1,1} + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 01011010 => 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? => ? ∊ {0,1,1,1,1} + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => ? => ? ∊ {0,1,1,1,1} + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => ? => ? ∊ {0,0,1,1,1,1,2,2,3} + 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? => ? ∊ {0,0,1,1,1,1,2,2,3} + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? => ? ∊ {0,0,1,1,1,1,2,2,3} + 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? => ? ∊ {0,0,1,1,1,1,2,2,3} + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 01101010 => 2 = 1 + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? => ? ∊ {0,0,1,1,1,1,2,2,3} + 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => ? => ? ∊ {0,0,1,1,1,1,2,2,3} + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 10101010 => 1 = 0 + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => ? => ? ∊ {0,0,1,1,1,1,2,2,3} + 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 101101010100 => ? => ? ∊ {0,0,1,1,1,1,2,2,3} + 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 110101010100 => ? => ? ∊ {0,0,1,1,1,1,2,2,3} + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 10101010101010 => ? => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => ? => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 101010111000 => ? => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 10101010110100 => ? => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 101011100100 => ? => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 10101011010100 => ? => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => ? => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 101110010100 => ? => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 10101101010100 => ? => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => ? => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 111001010100 => ? => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 10110101010100 => ? => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 11010101010100 => ? => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5} + 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1010101010101010 => ? => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1010101010101100 => ? => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 10101010111000 => ? => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 1010101010110100 => ? => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 101011101000 => ? => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> 10101011100100 => ? => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 1010101011010100 => ? => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => ? => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 101110100100 => ? => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 101011110000 => ? => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> 10101110010100 => ? => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 1010101101010100 => ? => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => ? => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 111010010100 => ? => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 101111000100 => ? => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> 10111001010100 => ? => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 1010110101010100 => ? => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => ? => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 111100010100 => ? => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 11100101010100 => ? => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1011010101010100 => ? => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12} + 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000761
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
St000761: Integer compositions ⟶ ℤResult quality: 6% ●values known / values provided: 9%●distinct values known / distinct values provided: 6%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
St000761: Integer compositions ⟶ ℤResult quality: 6% ●values known / values provided: 9%●distinct values known / distinct values provided: 6%
Values
[1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> [3,1] => 0
[2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [2,3,1] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [4,2] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [3,4,1] => 1
[2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [3,2,1] => 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [5,3] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> [4,5,1] => ? ∊ {0,0}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [2,2,3,1] => 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [2,4,2] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [4,2,2] => 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> [6,4] => ? ∊ {0,0}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> [5,6,1] => ? ∊ {1,1,1,1}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> [3,2,4,1] => ? ∊ {1,1,1,1}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [2,3,2,1] => 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [4,3,1] => 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [3,3,2] => 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> [5,2,3] => ? ∊ {1,1,1,1}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> [7,5] => ? ∊ {1,1,1,1}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12,14]]
=> [6,7,1] => ? ∊ {0,0,1,1,1,1,1,2,2,3}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> [4,2,5,1] => ? ∊ {0,0,1,1,1,1,1,2,2,3}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> [3,3,3,1] => ? ∊ {0,0,1,1,1,1,1,2,2,3}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> [2,3,4,1] => ? ∊ {0,0,1,1,1,1,1,2,2,3}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> [3,5,2] => ? ∊ {0,0,1,1,1,1,1,2,2,3}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [3,2,2,1] => 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> [5,3,2] => ? ∊ {0,0,1,1,1,1,1,2,2,3}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> [2,5,3] => ? ∊ {0,0,1,1,1,1,1,2,2,3}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> [4,3,3] => ? ∊ {0,0,1,1,1,1,1,2,2,3}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> [6,2,4] => ? ∊ {0,0,1,1,1,1,1,2,2,3}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> [8,6] => ? ∊ {0,0,1,1,1,1,1,2,2,3}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,7,15],[8,9,10,11,12,13,14,16]]
=> [7,8,1] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,7,13],[6,8,9,10,11,12,14]]
=> [5,2,6,1] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[1,2,3,4,7,11],[5,6,8,9,10,12]]
=> [4,3,4,1] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[1,2,3,5,6,11],[4,7,8,9,10,12]]
=> [3,3,5,1] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[1,2,3,7,9],[4,5,6,8,10]]
=> [3,4,2,1] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> [2,2,2,3,1] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> [5,4,1] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> [2,2,4,2] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> [2,4,2,2] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> [4,2,2,2] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[1,3,4,5,6,9],[2,7,8,10,11,12]]
=> [6,3,3] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> [3,4,3] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,3,4,5,7,8],[2,6,9,10,11,12]]
=> [5,3,4] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[1,3,4,5,6,7,9],[2,8,10,11,12,13,14]]
=> [7,2,5] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,9],[2,10,11,12,13,14,15,16]]
=> [9,7] => ? ∊ {0,0,1,1,2,2,2,2,2,2,3,3,3,5,5}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,7,8,17],[9,10,11,12,13,14,15,16,18]]
=> [8,9,1] => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,8,15],[7,9,10,11,12,13,14,16]]
=> [6,2,7,1] => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,5,8,13],[6,7,9,10,11,12,14]]
=> [5,3,5,1] => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,6,7,13],[5,8,9,10,11,12,14]]
=> [4,3,6,1] => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[1,2,3,4,8,11],[5,6,7,9,10,12]]
=> [4,4,3,1] => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,7,11],[4,6,8,9,10,12]]
=> [3,2,2,4,1] => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[1,2,4,5,6,11],[3,7,8,9,10,12]]
=> [2,4,5,1] => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[1,2,3,4,9,10],[5,6,7,8,11,12]]
=> [4,6,2] => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[1,2,4,7,9],[3,5,6,8,10]]
=> [2,2,3,2,1] => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[1,2,5,6,9],[3,4,7,8,10]]
=> [2,4,3,1] => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[1,3,4,6,9],[2,5,7,8,10]]
=> [4,2,3,1] => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[1,3,4,5,6,10],[2,7,8,9,11,12]]
=> [6,4,2] => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,7,8],[3,4,6,9,10]]
=> [2,3,3,2] => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> [4,4,2] => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> [3,3,2,2] => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [[1,3,4,5,7,9],[2,6,8,10,11,12]]
=> [5,2,2,3] => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [[1,3,4,5,6,7,10],[2,8,9,11,12,13,14]]
=> [7,3,4] => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[1,2,5,6,7,8],[3,4,9,10,11,12]]
=> [2,6,4] => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [[1,3,4,6,7,8],[2,5,9,10,11,12]]
=> [4,4,4] => ? ∊ {0,0,1,1,1,1,2,2,3,3,4,4,4,4,6,6,6,8,8,9,9,12}
Description
The number of ascents in an integer composition.
A composition has an ascent, or rise, at position $i$ if $a_i < a_{i+1}$.
The following 12 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000807The sum of the heights of the valleys of the associated bargraph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001520The number of strict 3-descents. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000805The number of peaks of the associated bargraph. 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 $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.
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