Processing math: 75%

Your data matches 61 different statistics following compositions of up to 3 maps.
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Matching statistic: St000913
St000913: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 1
[1,1]
=> 1
[3]
=> 1
[2,1]
=> 1
[1,1,1]
=> 1
[4]
=> 2
[3,1]
=> 1
[2,2]
=> 1
[2,1,1]
=> 1
[1,1,1,1]
=> 1
[5]
=> 4
[4,1]
=> 2
[3,2]
=> 2
[3,1,1]
=> 1
[2,2,1]
=> 1
[2,1,1,1]
=> 1
[1,1,1,1,1]
=> 1
[6]
=> 11
[5,1]
=> 4
[4,2]
=> 5
[4,1,1]
=> 2
[3,3]
=> 2
[3,2,1]
=> 2
[3,1,1,1]
=> 1
[2,2,2]
=> 1
[2,2,1,1]
=> 1
[2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 1
[7]
=> 33
[6,1]
=> 11
[5,2]
=> 12
[5,1,1]
=> 4
[4,3]
=> 10
[4,2,1]
=> 5
[4,1,1,1]
=> 2
[3,3,1]
=> 2
[3,2,2]
=> 3
[3,2,1,1]
=> 2
[3,1,1,1,1]
=> 1
[2,2,2,1]
=> 1
[2,2,1,1,1]
=> 1
[2,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> 1
[8]
=> 116
[7,1]
=> 33
[6,2]
=> 37
[6,1,1]
=> 11
[5,3]
=> 27
[5,2,1]
=> 12
Description
The number of ways to refine the partition into singletons. For example there is only one way to refine [2,2]: [2,2]>[2,1,1]>[1,1,1,1]. However, there are two ways to refine [3,2]: [3,2]>[2,2,1]>[2,1,1,1]>[1,1,1,1,1 and [3,2]>[3,1,1]>[2,1,1,1]>[1,1,1,1,1]. In other words, this is the number of saturated chains in the refinement order from the bottom element to the given partition. The sequence of values on the partitions with only one part is [[A002846]].
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00027: Dyck paths to partitionInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St001934: Integer partitions ⟶ ℤResult quality: 7% values known / values provided: 24%distinct values known / distinct values provided: 7%
Values
[1]
=> [1,0]
=> []
=> ?
=> ? = 1
[2]
=> [1,0,1,0]
=> [1]
=> []
=> ? ∊ {1,1}
[1,1]
=> [1,1,0,0]
=> []
=> ?
=> ? ∊ {1,1}
[3]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1]
=> 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? = 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 1
[2,2]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? = 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> 2
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? = 4
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [4,3,2,1]
=> ? ∊ {1,1,2,4,5,11}
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [4,3,2,1]
=> ? ∊ {1,1,2,4,5,11}
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [3,3,2,1]
=> ? ∊ {1,1,2,4,5,11}
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,2,4,5,11}
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [3,2,2,1]
=> ? ∊ {1,1,2,4,5,11}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,2,4,5,11}
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2]
=> 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [3,2,1,1]
=> 2
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> [3,2,1]
=> 2
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> [5,4,3,2,1]
=> ? ∊ {1,2,3,4,5,10,11,12,33}
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> [5,4,3,2,1]
=> ? ∊ {1,2,3,4,5,10,11,12,33}
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [3,3,2,1]
=> ? ∊ {1,2,3,4,5,10,11,12,33}
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> [4,4,3,2,1]
=> ? ∊ {1,2,3,4,5,10,11,12,33}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [2,2,2,1]
=> 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> [4,3,3,2,1]
=> ? ∊ {1,2,3,4,5,10,11,12,33}
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1]
=> 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [3,1,1,1]
=> 2
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> [4,3,2,2,1]
=> ? ∊ {1,2,3,4,5,10,11,12,33}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {1,2,3,4,5,10,11,12,33}
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> [3,2]
=> 2
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,1]
=> [4,3,2,1,1]
=> ? ∊ {1,2,3,4,5,10,11,12,33}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1]
=> [4,3,2,1]
=> ? ∊ {1,2,3,4,5,10,11,12,33}
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> [4,4,3,2,1]
=> ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,3,2,1]
=> [5,5,4,3,2,1]
=> ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [2,2,2,1]
=> 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,2,1]
=> [3,3,3,2,1]
=> ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,3,2,1]
=> [5,4,4,3,2,1]
=> ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1]
=> 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [2,1,1,1]
=> 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [2,2,2,1]
=> 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,2,2,1]
=> [4,2,2,2,1]
=> ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,3,2,1]
=> [5,4,3,3,2,1]
=> ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1]
=> 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> [3,1]
=> 2
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1,1,1]
=> [4,3,1,1,1]
=> ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,2,1]
=> [5,4,3,2,2,1]
=> ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> [3]
=> 2
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2]
=> [4,3,2]
=> ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,1]
=> [5,4,3,2,1,1]
=> ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1]
=> [5,4,3,2,1]
=> ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1]
=> [7,6,5,4,3,2,1]
=> ? ∊ {2,2,2,3,4,4,5,5,5,9,11,12,19,25,27,28,33,37,97,99,116,123,435}
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,7,6,5,4,3,2,1]
=> [7,6,5,4,3,2,1]
=> ? ∊ {2,2,2,3,4,4,5,5,5,9,11,12,19,25,27,28,33,37,97,99,116,123,435}
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,3,2,1]
=> [5,5,4,3,2,1]
=> ? ∊ {2,2,2,3,4,4,5,5,5,9,11,12,19,25,27,28,33,37,97,99,116,123,435}
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,6,6,5,4,3,2,1]
=> [6,6,5,4,3,2,1]
=> ? ∊ {2,2,2,3,4,4,5,5,5,9,11,12,19,25,27,28,33,37,97,99,116,123,435}
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,2,1]
=> [3,3,3,2,1]
=> ? ∊ {2,2,2,3,4,4,5,5,5,9,11,12,19,25,27,28,33,37,97,99,116,123,435}
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,4,4,3,2,1]
=> [4,4,4,3,2,1]
=> ? ∊ {2,2,2,3,4,4,5,5,5,9,11,12,19,25,27,28,33,37,97,99,116,123,435}
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,6,5,5,4,3,2,1]
=> [6,5,5,4,3,2,1]
=> ? ∊ {2,2,2,3,4,4,5,5,5,9,11,12,19,25,27,28,33,37,97,99,116,123,435}
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> 1
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [5,3,2,2,2,1]
=> [3,2,2,2,1]
=> ? ∊ {2,2,2,3,4,4,5,5,5,9,11,12,19,25,27,28,33,37,97,99,116,123,435}
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2,1]
=> [3,3,3,2,1]
=> ? ∊ {2,2,2,3,4,4,5,5,5,9,11,12,19,25,27,28,33,37,97,99,116,123,435}
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,3,3,2,1]
=> [5,3,3,3,2,1]
=> ? ∊ {2,2,2,3,4,4,5,5,5,9,11,12,19,25,27,28,33,37,97,99,116,123,435}
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,4,3,2,1]
=> [6,5,4,4,3,2,1]
=> ? ∊ {2,2,2,3,4,4,5,5,5,9,11,12,19,25,27,28,33,37,97,99,116,123,435}
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> [2,1]
=> 1
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 1
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,1,1]
=> [4,2,1,1,1]
=> ? ∊ {2,2,2,3,4,4,5,5,5,9,11,12,19,25,27,28,33,37,97,99,116,123,435}
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,2,2,2,1]
=> [2,2,2,2,1]
=> ? ∊ {2,2,2,3,4,4,5,5,5,9,11,12,19,25,27,28,33,37,97,99,116,123,435}
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,2,2,1]
=> [5,4,2,2,2,1]
=> ? ∊ {2,2,2,3,4,4,5,5,5,9,11,12,19,25,27,28,33,37,97,99,116,123,435}
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,3,2,1]
=> [6,5,4,3,3,2,1]
=> ? ∊ {2,2,2,3,4,4,5,5,5,9,11,12,19,25,27,28,33,37,97,99,116,123,435}
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [1,1]
=> 1
[3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [1]
=> 1
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [5,4,3]
=> [4,3]
=> 10
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> [2,1]
=> 1
[4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [2,1]
=> 1
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1]
=> [4,2,1]
=> 5
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,2,1,1,1]
=> [2,2,1,1,1]
=> 1
[3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [5,4,1,1]
=> [4,1,1]
=> 5
[3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,2,1,1,1,1]
=> [2,1,1,1,1]
=> 1
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1]
=> 1
[2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [5,4,1]
=> [4,1]
=> 5
[5,5,1]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1]
=> [3,2,1]
=> 2
[4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
Description
The number of monotone factorisations of genus zero of a permutation of given cycle type. A monotone factorisation of genus zero of a permutation πSn with cycles, including fixed points, is a tuple of r=n transpositions (a1,b1),,(ar,br) with b1br and ai<bi for all i, whose product, in this order, is π. For example, the cycle (2,3,1) has the two factorizations (2,3)(1,3) and (1,2)(2,3).
Mp00202: Integer partitions first row removalInteger partitions
Mp00095: Integer partitions to binary wordBinary words
Mp00096: Binary words Foata bijectionBinary words
St001491: Binary words ⟶ ℤResult quality: 3% values known / values provided: 23%distinct values known / distinct values provided: 3%
Values
[1]
=> []
=> => => ? = 1
[2]
=> []
=> => => ? = 1
[1,1]
=> [1]
=> 10 => 10 => 1
[3]
=> []
=> => => ? = 1
[2,1]
=> [1]
=> 10 => 10 => 1
[1,1,1]
=> [1,1]
=> 110 => 110 => 1
[4]
=> []
=> => => ? = 1
[3,1]
=> [1]
=> 10 => 10 => 1
[2,2]
=> [2]
=> 100 => 010 => 1
[2,1,1]
=> [1,1]
=> 110 => 110 => 1
[1,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 2
[5]
=> []
=> => => ? ∊ {2,4}
[4,1]
=> [1]
=> 10 => 10 => 1
[3,2]
=> [2]
=> 100 => 010 => 1
[3,1,1]
=> [1,1]
=> 110 => 110 => 1
[2,2,1]
=> [2,1]
=> 1010 => 1100 => 1
[2,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 2
[1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => ? ∊ {2,4}
[6]
=> []
=> => => ? ∊ {2,4,5,11}
[5,1]
=> [1]
=> 10 => 10 => 1
[4,2]
=> [2]
=> 100 => 010 => 1
[4,1,1]
=> [1,1]
=> 110 => 110 => 1
[3,3]
=> [3]
=> 1000 => 0010 => 1
[3,2,1]
=> [2,1]
=> 1010 => 1100 => 1
[3,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 2
[2,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[2,2,1,1]
=> [2,1,1]
=> 10110 => 11010 => ? ∊ {2,4,5,11}
[2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => ? ∊ {2,4,5,11}
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 111110 => ? ∊ {2,4,5,11}
[7]
=> []
=> => => ? ∊ {2,3,4,5,10,11,12,33}
[6,1]
=> [1]
=> 10 => 10 => 1
[5,2]
=> [2]
=> 100 => 010 => 1
[5,1,1]
=> [1,1]
=> 110 => 110 => 1
[4,3]
=> [3]
=> 1000 => 0010 => 1
[4,2,1]
=> [2,1]
=> 1010 => 1100 => 1
[4,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 2
[3,3,1]
=> [3,1]
=> 10010 => 10100 => ? ∊ {2,3,4,5,10,11,12,33}
[3,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[3,2,1,1]
=> [2,1,1]
=> 10110 => 11010 => ? ∊ {2,3,4,5,10,11,12,33}
[3,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => ? ∊ {2,3,4,5,10,11,12,33}
[2,2,2,1]
=> [2,2,1]
=> 11010 => 11100 => ? ∊ {2,3,4,5,10,11,12,33}
[2,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 110110 => ? ∊ {2,3,4,5,10,11,12,33}
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 111110 => ? ∊ {2,3,4,5,10,11,12,33}
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 1111110 => ? ∊ {2,3,4,5,10,11,12,33}
[8]
=> []
=> => => ? ∊ {1,2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[7,1]
=> [1]
=> 10 => 10 => 1
[6,2]
=> [2]
=> 100 => 010 => 1
[6,1,1]
=> [1,1]
=> 110 => 110 => 1
[5,3]
=> [3]
=> 1000 => 0010 => 1
[5,2,1]
=> [2,1]
=> 1010 => 1100 => 1
[5,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 2
[4,4]
=> [4]
=> 10000 => 00010 => ? ∊ {1,2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[4,3,1]
=> [3,1]
=> 10010 => 10100 => ? ∊ {1,2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[4,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[4,2,1,1]
=> [2,1,1]
=> 10110 => 11010 => ? ∊ {1,2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[4,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => ? ∊ {1,2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[3,3,2]
=> [3,2]
=> 10100 => 01100 => ? ∊ {1,2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[3,3,1,1]
=> [3,1,1]
=> 100110 => 101010 => ? ∊ {1,2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[3,2,2,1]
=> [2,2,1]
=> 11010 => 11100 => ? ∊ {1,2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[3,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 110110 => ? ∊ {1,2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 111110 => ? ∊ {1,2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[2,2,2,2]
=> [2,2,2]
=> 11100 => 01110 => ? ∊ {1,2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[2,2,2,1,1]
=> [2,2,1,1]
=> 110110 => 111010 => ? ∊ {1,2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> 1011110 => 1101110 => ? ∊ {1,2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 1111110 => ? ∊ {1,2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 11111110 => 11111110 => ? ∊ {1,2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[9]
=> []
=> => => ? ∊ {1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[8,1]
=> [1]
=> 10 => 10 => 1
[7,2]
=> [2]
=> 100 => 010 => 1
[7,1,1]
=> [1,1]
=> 110 => 110 => 1
[6,3]
=> [3]
=> 1000 => 0010 => 1
[6,2,1]
=> [2,1]
=> 1010 => 1100 => 1
[6,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 2
[5,4]
=> [4]
=> 10000 => 00010 => ? ∊ {1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[5,3,1]
=> [3,1]
=> 10010 => 10100 => ? ∊ {1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[5,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[5,2,1,1]
=> [2,1,1]
=> 10110 => 11010 => ? ∊ {1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[5,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => ? ∊ {1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[4,4,1]
=> [4,1]
=> 100010 => 100100 => ? ∊ {1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[4,3,2]
=> [3,2]
=> 10100 => 01100 => ? ∊ {1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[4,3,1,1]
=> [3,1,1]
=> 100110 => 101010 => ? ∊ {1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[4,2,2,1]
=> [2,2,1]
=> 11010 => 11100 => ? ∊ {1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[4,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 110110 => ? ∊ {1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 111110 => ? ∊ {1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[3,3,3]
=> [3,3]
=> 11000 => 00110 => ? ∊ {1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[3,3,2,1]
=> [3,2,1]
=> 101010 => 111000 => ? ∊ {1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[3,3,1,1,1]
=> [3,1,1,1]
=> 1001110 => 1010110 => ? ∊ {1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[3,2,2,2]
=> [2,2,2]
=> 11100 => 01110 => ? ∊ {1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[3,2,2,1,1]
=> [2,2,1,1]
=> 110110 => 111010 => ? ∊ {1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> 1011110 => 1101110 => ? ∊ {1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[9,1]
=> [1]
=> 10 => 10 => 1
[8,2]
=> [2]
=> 100 => 010 => 1
[8,1,1]
=> [1,1]
=> 110 => 110 => 1
[7,3]
=> [3]
=> 1000 => 0010 => 1
[7,2,1]
=> [2,1]
=> 1010 => 1100 => 1
[7,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 2
[6,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[10,1]
=> [1]
=> 10 => 10 => 1
[9,2]
=> [2]
=> 100 => 010 => 1
[9,1,1]
=> [1,1]
=> 110 => 110 => 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let An=K[x]/(xn). We associate to a nonempty subset S of an (n-1)-set the module MS, which is the direct sum of An-modules with indecomposable non-projective direct summands of dimension i when i is in S (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of MS. We decode the subset as a binary word so that for example the subset S={1,3} of {1,2,3} is decoded as 101.
Matching statistic: St001665
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00201: Dyck paths RingelPermutations
Mp00126: Permutations cactus evacuationPermutations
St001665: Permutations ⟶ ℤResult quality: 3% values known / values provided: 21%distinct values known / distinct values provided: 3%
Values
[1]
=> [1,0,1,0]
=> [3,1,2] => [1,3,2] => 1
[2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [2,4,1,3] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [3,1,4,2] => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [2,5,1,3,4] => 1
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,4,5,2] => 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [2,6,1,3,4,5] => 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,5,4,2] => 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [2,1,4,3,5] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,2,4,3] => 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,4,5,6,2] => 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [2,7,1,3,4,5,6] => ? ∊ {2,4}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [3,4,6,1,5,2] => 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,3,5,1,4] => 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,3,2,5,4] => 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,2,5,3] => 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [4,6,1,2,5,3] => 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,4,5,6,7,2] => ? ∊ {2,4}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => [2,8,1,3,4,5,6,7] => ? ∊ {2,4,5,11}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => [3,4,7,1,5,6,2] => ? ∊ {2,4,5,11}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [2,4,6,5,1,3] => 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,4,3,6,5,2] => 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,1,5,3,4,6] => 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,6,3,2,5,4] => 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,1,4,5,3,6] => 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [5,1,6,2,4,3] => 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [4,7,1,2,5,6,3] => ? ∊ {2,4,5,11}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [3,1,4,5,6,7,8,2] => ? ∊ {2,4,5,11}
[7]
=> [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] => [2,9,1,3,4,5,6,7,8] => ? ∊ {2,3,4,5,10,11,12,33}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,1,2,7] => [3,4,8,1,5,6,7,2] => ? ∊ {2,3,4,5,10,11,12,33}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => [2,4,7,1,5,3,6] => ? ∊ {2,3,4,5,10,11,12,33}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => [3,5,1,7,4,6,2] => ? ∊ {2,3,4,5,10,11,12,33}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,3,6,1,4,5] => 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,2,4,6,5,3] => 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,4,1,2,6,5] => 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [5,1,3,6,4,2] => 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [2,6,3,5,1,4] => 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,2,6,3,5,4] => 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [3,7,1,5,2,6,4] => ? ∊ {2,3,4,5,10,11,12,33}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,2,5,6,3] => 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [4,1,6,2,5,7,3] => ? ∊ {2,3,4,5,10,11,12,33}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => [4,8,1,2,5,6,7,3] => ? ∊ {2,3,4,5,10,11,12,33}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => [3,1,4,5,6,7,8,9,2] => ? ∊ {2,3,4,5,10,11,12,33}
[8]
=> [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] => [2,10,1,3,4,5,6,7,8,9] => ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,4,5,6,7,1,2,8] => [3,4,9,1,5,6,7,8,2] => ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [2,8,4,5,6,1,3,7] => [2,4,8,1,5,6,3,7] => ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,4,5,1,8,2,7] => [3,6,1,8,4,5,7,2] => ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [2,3,7,5,1,4,6] => ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [1,2,5,7,6,4,3] => 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [4,5,1,3,7,6,2] => ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [2,1,6,3,4,5,7] => ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,3,4,6,5,2] => 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,4,1,6,3,5] => 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,2,6,4,5,3] => 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [3,7,4,1,2,6,5] => ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,1,5,6,3,4] => 2
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,5,2,6,4] => 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,6,2,3,5,4] => 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [1,2,7,6,3,5,4] => 2
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => [3,8,1,5,6,2,7,4] => ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [2,1,4,5,6,3,7] => ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [5,1,2,6,4,7,3] => ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => [4,1,7,2,5,6,8,3] => ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => [4,9,1,2,5,6,7,8,3] => ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[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]
=> [3,1,4,5,6,7,8,9,10,2] => [3,1,4,5,6,7,8,9,10,2] => ? ∊ {2,3,4,5,5,9,10,11,12,19,27,33,37,116}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,9,11,1,10] => [2,11,1,3,4,5,6,7,8,9,10] => ? ∊ {1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [10,3,4,5,6,7,8,1,2,9] => [3,4,10,1,5,6,7,8,9,2] => ? ∊ {1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [2,9,4,5,6,7,1,3,8] => [2,4,9,1,5,6,7,3,8] => ? ∊ {1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [7,3,4,5,6,1,9,2,8] => [3,7,1,9,4,5,6,8,2] => ? ∊ {1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [2,3,8,5,6,1,4,7] => [2,3,8,1,5,4,6,7] => ? ∊ {1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [8,6,4,5,1,2,3,7] => [1,4,6,8,2,7,5,3] => ? ∊ {1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [5,3,4,1,6,8,2,7] => [3,5,1,4,8,6,7,2] => ? ∊ {1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [2,3,7,1,4,5,6] => ? ∊ {1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [1,3,5,7,2,6,4] => 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => [2,5,1,7,4,3,6] => ? ∊ {1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [1,4,5,2,7,6,3] => 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [3,4,1,2,5,7,6] => ? ∊ {1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => [3,1,6,7,4,5,2] => ? ∊ {1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,3,4,6,1,5] => 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,3,4,2,6,5] => 2
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,4,2,3,6,5] => 2
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [1,4,7,2,3,6,5] => ? ∊ {1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [3,1,4,8,6,7,2,5] => [3,8,1,4,2,6,7,5] => ? ∊ {1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [2,1,3,5,4,6,7] => ? ∊ {1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,2,3,6,4] => 1
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [3,1,6,5,2,7,4] => ? ∊ {1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => [2,7,1,5,6,3,4] => ? ∊ {1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [1,5,2,7,3,6,4] => ? ∊ {1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [8,1,7,5,6,2,3,4] => [1,5,8,2,7,3,6,4] => ? ∊ {1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,9,5,6,7,8,2,4] => [3,9,1,5,6,7,2,8,4] => ? ∊ {1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [4,1,2,5,6,7,3] => ? ∊ {1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,1,4,5,2,7,8,3] => [4,1,2,6,5,7,8,3] => ? ∊ {1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [8,1,4,5,6,7,2,9,3] => [4,1,8,2,5,6,7,9,3] => ? ∊ {1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435}
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,3,1,5,2,4,6] => [1,3,5,2,7,6,4] => 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => 1
[4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [1,3,7,4,2,6,5] => 1
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => [1,4,2,7,3,6,5] => 1
[5,4,1,1]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => [1,4,5,3,7,6,2] => 1
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => [1,2,3,5,7,4,6] => 1
Description
The number of pure excedances of a permutation. A pure excedance of a permutation π is a position i<πi such that there is no j<i with iπj<πi.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00228: Dyck paths reflect parallelogram polyominoDyck paths
Mp00201: Dyck paths RingelPermutations
St001087: Permutations ⟶ ℤResult quality: 7% values known / values provided: 18%distinct values known / distinct values provided: 7%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0 = 1 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 0 = 1 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 0 = 1 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0 = 1 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 0 = 1 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 0 = 1 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0 = 1 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0 = 1 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 2 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 0 = 1 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => ? ∊ {1,4} - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => 1 = 2 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0 = 1 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0 = 1 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0 = 1 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 1 = 2 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ? ∊ {1,4} - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [8,3,4,5,7,1,2,6] => ? ∊ {1,4,5,11} - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => ? ∊ {1,4,5,11} - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1 = 2 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 0 = 1 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => 1 = 2 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0 = 1 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 0 = 1 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 0 = 1 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 1 = 2 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => ? ∊ {1,4,5,11} - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [8,4,1,5,6,7,2,3] => ? ∊ {1,4,5,11} - 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [2,3,4,5,6,9,8,1,7] => ? ∊ {1,2,4,5,10,11,12,33} - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [2,3,4,5,8,7,1,6] => ? ∊ {1,2,4,5,10,11,12,33} - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => ? ∊ {1,2,4,5,10,11,12,33} - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => ? ∊ {1,2,4,5,10,11,12,33} - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => 1 = 2 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 0 = 1 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 0 = 1 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 0 = 1 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 1 = 2 - 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]
=> [2,6,1,3,4,5] => 2 = 3 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => ? ∊ {1,2,4,5,10,11,12,33} - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 0 = 1 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [7,3,1,5,6,2,4] => ? ∊ {1,2,4,5,10,11,12,33} - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [4,3,1,5,6,7,8,2] => ? ∊ {1,2,4,5,10,11,12,33} - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [4,3,1,5,6,7,8,9,2] => ? ∊ {1,2,4,5,10,11,12,33} - 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0]
=> [10,3,4,5,6,7,9,1,2,8] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> [8,9,4,5,6,1,2,3,7] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [2,3,8,5,7,1,4,6] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [5,3,4,1,8,7,2,6] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0 = 1 - 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 0 = 1 - 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 1 = 2 - 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => 0 = 1 - 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 1 = 2 - 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]
=> [6,3,1,2,4,5] => 0 = 1 - 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [4,3,1,8,6,7,2,5] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [6,4,1,5,2,7,8,3] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [8,9,1,5,6,7,2,3,4] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [10,4,1,5,6,7,8,9,2,3] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> [2,3,4,5,6,7,8,11,10,1,9] => ? ∊ {1,1,1,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [2,3,4,5,6,7,10,9,1,8] => ? ∊ {1,1,1,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> [9,3,4,5,6,1,8,2,7] => ? ∊ {1,1,1,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [2,9,4,5,6,8,1,3,7] => ? ∊ {1,1,1,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [2,8,4,5,7,1,3,6] => ? ∊ {1,1,1,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [8,7,5,1,2,3,4,6] => ? ∊ {1,1,1,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [8,3,4,5,1,7,2,6] => ? ∊ {1,1,1,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => ? ∊ {1,1,1,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 2 = 3 - 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,7,4,1,6,3,5] => ? ∊ {1,1,1,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ? ∊ {1,1,1,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [7,4,1,6,2,3,5] => ? ∊ {1,1,1,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => ? ∊ {1,1,1,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 0 = 1 - 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 1 = 2 - 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 0 = 1 - 1
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => ? ∊ {1,1,1,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [8,3,1,5,6,7,2,4] => ? ∊ {1,1,1,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => ? ∊ {1,1,1,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 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]
=> [6,4,1,2,3,5] => 0 = 1 - 1
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [6,3,1,5,2,7,4] => ? ∊ {1,1,1,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => ? ∊ {1,1,1,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? ∊ {1,1,1,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [8,7,1,6,2,3,4,5] => ? ∊ {1,1,1,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [8,4,1,5,6,7,2,9,3] => ? ∊ {1,1,1,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 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]
=> [5,6,1,2,3,4] => 0 = 1 - 1
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => 0 = 1 - 1
[4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => 0 = 1 - 1
[4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => 3 = 4 - 1
[5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => 3 = 4 - 1
[5,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [8,3,1,5,6,2,4,7] => 0 = 1 - 1
[4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => 0 = 1 - 1
[3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => 0 = 1 - 1
Description
The number of occurrences of the vincular pattern |12-3 in a permutation. This is the number of occurrences of the pattern 123, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive. In other words, this is the number of ascents whose bottom value is strictly larger than the first entry of the permutation.
Matching statistic: St000308
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00201: Dyck paths RingelPermutations
Mp00064: Permutations reversePermutations
St000308: Permutations ⟶ ℤResult quality: 5% values known / values provided: 17%distinct values known / distinct values provided: 5%
Values
[1]
=> [1,0,1,0]
=> [3,1,2] => [2,1,3] => 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [3,1,4,2] => 2 = 1 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => 2 = 1 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => 2 = 1 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [3,2,1,4] => 2 = 1 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [2,5,4,1,3] => 2 = 1 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [5,1,6,4,3,2] => 2 = 1 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [4,2,1,3,5] => 3 = 2 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [3,2,4,1,5] => 2 = 1 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [2,6,5,4,1,3] => 2 = 1 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [6,1,7,5,4,3,2] => ? ∊ {2,4} + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [5,2,1,4,3,6] => 3 = 2 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => 2 = 1 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [4,2,5,1,3] => 2 = 1 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [3,5,2,1,4] => 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [3,2,5,4,1,6] => 2 = 1 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [2,7,6,5,4,1,3] => ? ∊ {2,4} + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => [7,1,8,6,5,4,3,2] => ? ∊ {1,4,5,11} + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => [6,2,1,5,4,3,7] => ? ∊ {1,4,5,11} + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [5,3,1,4,6,2] => 3 = 2 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [5,2,6,1,3,4] => 3 = 2 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [4,6,1,5,3,2] => 2 = 1 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [4,3,2,1,5] => 2 = 1 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [4,2,5,6,1,3] => 3 = 2 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [3,6,5,1,4,2] => 2 = 1 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [3,6,2,4,1,5] => 2 = 1 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [3,2,6,5,4,1,7] => ? ∊ {1,4,5,11} + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [2,8,7,6,5,4,1,3] => ? ∊ {1,4,5,11} + 1
[7]
=> [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] => [8,1,9,7,6,5,4,3,2] => ? ∊ {2,3,4,5,10,11,12,33} + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,1,2,7] => [7,2,1,6,5,4,3,8] => ? ∊ {2,3,4,5,10,11,12,33} + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => [6,3,1,5,4,7,2] => ? ∊ {2,3,4,5,10,11,12,33} + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => [6,2,7,1,4,3,5] => ? ∊ {2,3,4,5,10,11,12,33} + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [5,4,1,6,3,2] => 2 = 1 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [5,3,2,1,4,6] => 3 = 2 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [5,2,6,4,1,3] => 2 = 1 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [4,6,2,1,3,5] => 3 = 2 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [4,3,5,1,6,2] => 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,3,2,5,1,6] => 2 = 1 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [4,2,6,5,7,1,3] => ? ∊ {2,3,4,5,10,11,12,33} + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [3,6,5,2,1,4] => 2 = 1 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [3,7,2,5,4,1,6] => ? ∊ {2,3,4,5,10,11,12,33} + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => [3,2,7,6,5,4,1,8] => ? ∊ {2,3,4,5,10,11,12,33} + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => [2,9,8,7,6,5,4,1,3] => ? ∊ {2,3,4,5,10,11,12,33} + 1
[8]
=> [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] => [9,1,10,8,7,6,5,4,3,2] => ? ∊ {1,2,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,4,5,6,7,1,2,8] => [8,2,1,7,6,5,4,3,9] => ? ∊ {1,2,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [2,8,4,5,6,1,3,7] => [7,3,1,6,5,4,8,2] => ? ∊ {1,2,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,4,5,1,8,2,7] => [7,2,8,1,5,4,3,6] => ? ∊ {1,2,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [6,4,1,5,7,3,2] => ? ∊ {1,2,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [6,3,2,1,4,5,7] => 4 = 3 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [6,2,7,5,1,3,4] => ? ∊ {1,2,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [5,7,1,6,4,3,2] => ? ∊ {1,2,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [5,4,2,1,3,6] => 3 = 2 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [5,3,6,1,4,2] => 2 = 1 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [5,3,2,4,1,6] => 2 = 1 + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [5,2,6,7,4,1,3] => ? ∊ {1,2,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [4,6,3,1,5,2] => 2 = 1 + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [4,6,2,5,1,3] => 2 = 1 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [4,3,5,2,1,6] => 2 = 1 + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [4,3,2,5,6,1,7] => 3 = 2 + 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => [4,2,7,6,5,8,1,3] => ? ∊ {1,2,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [3,7,6,5,1,4,2] => ? ∊ {1,2,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [3,7,6,2,4,1,5] => ? ∊ {1,2,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => [3,8,2,6,5,4,1,7] => ? ∊ {1,2,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => [3,2,8,7,6,5,4,1,9] => ? ∊ {1,2,4,5,5,9,10,11,12,19,27,33,37,116} + 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]
=> [3,1,4,5,6,7,8,9,10,2] => [2,10,9,8,7,6,5,4,1,3] => ? ∊ {1,2,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,9,11,1,10] => [10,1,11,9,8,7,6,5,4,3,2] => ? ∊ {1,1,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [10,3,4,5,6,7,8,1,2,9] => [9,2,1,8,7,6,5,4,3,10] => ? ∊ {1,1,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [2,9,4,5,6,7,1,3,8] => [8,3,1,7,6,5,4,9,2] => ? ∊ {1,1,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [7,3,4,5,6,1,9,2,8] => [8,2,9,1,6,5,4,3,7] => ? ∊ {1,1,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [2,3,8,5,6,1,4,7] => [7,4,1,6,5,8,3,2] => ? ∊ {1,1,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [8,6,4,5,1,2,3,7] => [7,3,2,1,5,4,6,8] => ? ∊ {1,1,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [5,3,4,1,6,8,2,7] => [7,2,8,6,1,4,3,5] => ? ∊ {1,1,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [6,5,1,7,4,3,2] => ? ∊ {1,1,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [6,4,2,1,5,3,7] => 3 = 2 + 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => [6,3,7,1,4,5,2] => ? ∊ {1,1,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [6,3,2,5,1,4,7] => ? ∊ {1,1,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [6,2,7,5,4,1,3] => ? ∊ {1,1,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => [5,7,2,1,4,3,6] => ? ∊ {1,1,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [5,4,3,1,6,2] => 2 = 1 + 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [5,4,2,6,1,3] => 2 = 1 + 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [5,3,6,2,1,4] => 2 = 1 + 1
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [5,3,2,6,4,1,7] => ? ∊ {1,1,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [3,1,4,8,6,7,2,5] => [5,2,7,6,8,4,1,3] => ? ∊ {1,1,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [4,7,6,1,5,3,2] => ? ∊ {1,1,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [4,6,3,2,1,5] => 2 = 1 + 1
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [4,7,2,5,6,1,3] => ? ∊ {1,1,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => [4,3,6,5,1,7,2] => ? ∊ {1,1,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [4,3,6,2,5,1,7] => ? ∊ {1,1,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [8,1,7,5,6,2,3,4] => [4,3,2,6,5,7,1,8] => ? ∊ {1,1,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,9,5,6,7,8,2,4] => [4,2,8,7,6,5,9,1,3] => ? ∊ {1,1,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [3,7,6,5,2,1,4] => ? ∊ {1,1,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,1,4,5,2,7,8,3] => [3,8,7,2,5,4,1,6] => ? ∊ {1,1,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => [6,4,3,1,5,7,2] => 3 = 2 + 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [5,4,3,2,1,6] => 2 = 1 + 1
[5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,7,4,1,3,5,6] => [6,5,3,1,4,7,2] => 3 = 2 + 1
[5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [6,5,3,2,1,4,7] => 3 = 2 + 1
Description
The height of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices {0,1,2,,n} and root 0 in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1]. The statistic is given by the height of this tree. See also [[St000325]] for the width of this tree.
Matching statistic: St001715
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00201: Dyck paths RingelPermutations
Mp00062: Permutations Lehmer-code to major-code bijectionPermutations
St001715: Permutations ⟶ ℤResult quality: 5% values known / values provided: 16%distinct values known / distinct values provided: 5%
Values
[1]
=> [1,0,1,0]
=> [3,1,2] => [2,3,1] => 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [1,3,4,2] => 0 = 1 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => 0 = 1 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,2,4,5,3] => 0 = 1 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => 0 = 1 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,5,2,4] => 0 = 1 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [1,2,3,5,6,4] => 0 = 1 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,4,2,5,1] => 0 = 1 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5,1,3,2,4] => 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [4,5,2,3,1] => 0 = 1 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,4,6,2,5] => 0 = 1 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [1,2,3,4,6,7,5] => 0 = 1 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [4,2,5,3,6,1] => 1 = 2 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,3,4,5,2] => 0 = 1 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [4,1,5,2,3] => 0 = 1 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,2,3,1,4] => 1 = 2 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [5,3,6,2,4,1] => 0 = 1 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,4,5,7,2,6] => ? = 4 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => [1,2,3,4,5,7,8,6] => ? ∊ {1,4,5,11} - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => [2,5,3,6,4,7,1] => ? ∊ {1,4,5,11} - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [4,5,1,3,6,2] => 1 = 2 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [5,2,1,6,3,4] => 0 = 1 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [6,1,2,4,3,5] => 1 = 2 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => 0 = 1 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,5,6,4,2,3] => 0 = 1 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [4,1,2,6,3,5] => 0 = 1 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6,4,2,3,1,5] => 1 = 2 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [3,6,4,7,2,5,1] => ? ∊ {1,4,5,11} - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [3,1,4,5,6,8,2,7] => ? ∊ {1,4,5,11} - 1
[7]
=> [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] => [1,2,3,4,5,6,8,9,7] => ? ∊ {2,3,4,5,10,11,12,33} - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,1,2,7] => [2,3,6,4,7,5,8,1] => ? ∊ {2,3,4,5,10,11,12,33} - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => [5,3,6,1,4,7,2] => ? ∊ {2,3,4,5,10,11,12,33} - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => [2,6,3,1,7,4,5] => ? ∊ {2,3,4,5,10,11,12,33} - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [1,2,4,5,6,3] => 0 = 1 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [3,4,5,2,6,1] => 0 = 1 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,5,6,2,4] => 0 = 1 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6,3,2,4,1,5] => 1 = 2 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [5,6,1,3,4,2] => 1 = 2 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,5,6,2,3,1] => 0 = 1 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [6,1,4,7,5,2,3] => ? ∊ {2,3,4,5,10,11,12,33} - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,6,2,1,3,5] => 0 = 1 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [3,7,5,2,4,1,6] => ? ∊ {2,3,4,5,10,11,12,33} - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => [3,4,7,5,8,2,6,1] => ? ∊ {2,3,4,5,10,11,12,33} - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => [3,1,4,5,6,7,9,2,8] => ? ∊ {2,3,4,5,10,11,12,33} - 1
[8]
=> [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] => [1,2,3,4,5,6,7,9,10,8] => ? ∊ {1,1,1,1,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,4,5,6,7,1,2,8] => [2,3,4,7,5,8,6,9,1] => ? ∊ {1,1,1,1,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [2,8,4,5,6,1,3,7] => [3,6,4,7,1,5,8,2] => ? ∊ {1,1,1,1,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,4,5,1,8,2,7] => [2,3,7,4,1,8,5,6] => ? ∊ {1,1,1,1,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [5,6,1,2,4,7,3] => ? ∊ {1,1,1,1,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [4,5,6,3,2,7,1] => ? ∊ {1,1,1,1,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [4,2,1,6,7,3,5] => ? ∊ {1,1,1,1,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [7,1,2,3,5,4,6] => ? ∊ {1,1,1,1,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [3,4,2,5,6,1] => 0 = 1 - 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [5,1,2,6,3,4] => 0 = 1 - 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [4,5,2,3,6,1] => 1 = 2 - 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [1,6,7,3,5,2,4] => ? ∊ {1,1,1,1,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6,1,3,4,2,5] => 2 = 3 - 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,6,4,2,3,5] => 1 = 2 - 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [5,6,2,3,4,1] => 1 = 2 - 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [5,6,7,4,2,3,1] => ? ∊ {1,1,1,1,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => [4,7,1,5,8,6,2,3] => ? ∊ {1,1,1,1,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [4,1,2,5,7,3,6] => ? ∊ {1,1,1,1,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [5,3,7,2,1,4,6] => ? ∊ {1,1,1,1,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => [3,4,8,6,2,5,1,7] => ? ∊ {1,1,1,1,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => [3,4,5,8,6,9,2,7,1] => ? ∊ {1,1,1,1,4,5,5,9,10,11,12,19,27,33,37,116} - 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]
=> [3,1,4,5,6,7,8,9,10,2] => [3,1,4,5,6,7,8,10,2,9] => ? ∊ {1,1,1,1,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,9,11,1,10] => [1,2,3,4,5,6,7,8,10,11,9] => ? ∊ {1,1,2,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [10,3,4,5,6,7,8,1,2,9] => [2,3,4,5,8,6,9,7,10,1] => ? ∊ {1,1,2,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [2,9,4,5,6,7,1,3,8] => [3,4,7,5,8,1,6,9,2] => ? ∊ {1,1,2,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [7,3,4,5,6,1,9,2,8] => [2,3,4,8,5,1,9,6,7] => ? ∊ {1,1,2,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [2,3,8,5,6,1,4,7] => [6,4,7,1,2,5,8,3] => ? ∊ {1,1,2,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [8,6,4,5,1,2,3,7] => [5,6,3,7,4,2,8,1] => ? ∊ {1,1,2,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [5,3,4,1,6,8,2,7] => [2,5,3,1,7,8,4,6] => ? ∊ {1,1,2,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [1,2,3,5,6,7,4] => 0 = 1 - 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [4,2,5,6,3,7,1] => ? ∊ {1,1,2,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => [6,3,1,2,7,4,5] => ? ∊ {1,1,2,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [5,6,3,2,4,7,1] => ? ∊ {1,1,2,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [3,1,4,6,7,2,5] => ? ∊ {1,1,2,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => [2,7,4,3,5,1,6] => ? ∊ {1,1,2,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,3,4,5,6,2] => 0 = 1 - 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [4,1,5,6,2,3] => 0 = 1 - 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [5,6,2,1,3,4] => 0 = 1 - 1
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [5,3,6,7,2,4,1] => ? ∊ {1,1,2,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [3,1,4,8,6,7,2,5] => [7,1,5,8,3,6,2,4] => ? ∊ {1,1,2,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [5,1,2,3,7,4,6] => ? ∊ {1,1,2,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6,2,3,4,1,5] => 2 = 3 - 1
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [7,1,5,4,2,3,6] => ? ∊ {1,1,2,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => [6,4,7,1,3,5,2] => ? ∊ {1,1,2,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [6,7,4,2,3,5,1] => ? ∊ {1,1,2,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [8,1,7,5,6,2,3,4] => [6,7,4,8,5,2,3,1] => ? ∊ {1,1,2,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,9,5,6,7,8,2,4] => [4,5,8,1,6,9,7,2,3] => ? ∊ {1,1,2,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [4,5,1,7,2,3,6] => ? ∊ {1,1,2,2,2,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [2,3,4,5,6,1] => 0 = 1 - 1
[5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => [1,2,4,5,6,7,3] => 0 = 1 - 1
Description
The number of non-records in a permutation. A record in a permutation π is a value π(j) which is a left-to-right minimum, a left-to-right maximum, a right-to-left minimum, or a right-to-left maximum. For example, in the permutation π=[1,4,3,2,5], the values 1 is a left-to-right minimum, 1,4,5 are left-to-right maxima, 5,2,1 are right-to-left minima and 5 is a right-to-left maximum. Hence, 3 is the unique non-record. Permutations without non-records are called square [1].
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00201: Dyck paths RingelPermutations
St001390: Permutations ⟶ ℤResult quality: 3% values known / values provided: 15%distinct values known / distinct values provided: 3%
Values
[1]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 1 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 2 = 1 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 2 = 1 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 2 = 1 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 3 = 2 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 2 = 1 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 2 = 1 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {2,4} + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 3 = 2 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 1 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2 = 1 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 2 = 1 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? ∊ {2,4} + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? ∊ {2,4,5,11} + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {2,4,5,11} + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => 3 = 2 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 3 = 2 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 2 = 1 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 2 = 1 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 2 = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 2 = 1 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 2 = 1 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {2,4,5,11} + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? ∊ {2,4,5,11} + 1
[7]
=> [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] => ? ∊ {2,3,4,5,10,11,12,33} + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,1,2,7] => ? ∊ {2,3,4,5,10,11,12,33} + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? ∊ {2,3,4,5,10,11,12,33} + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? ∊ {2,3,4,5,10,11,12,33} + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 2 = 1 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 3 = 2 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 2 = 1 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 3 = 2 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 2 = 1 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? ∊ {2,3,4,5,10,11,12,33} + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 2 = 1 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? ∊ {2,3,4,5,10,11,12,33} + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => ? ∊ {2,3,4,5,10,11,12,33} + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => ? ∊ {2,3,4,5,10,11,12,33} + 1
[8]
=> [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] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,4,5,6,7,1,2,8] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [2,8,4,5,6,1,3,7] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,4,5,1,8,2,7] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => 3 = 2 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 2 = 1 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => 2 = 1 + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 2 = 1 + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 2 = 1 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 2 = 1 + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} + 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]
=> [3,1,4,5,6,7,8,9,10,2] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,9,11,1,10] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [10,3,4,5,6,7,8,1,2,9] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [2,9,4,5,6,7,1,3,8] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [7,3,4,5,6,1,9,2,8] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [2,3,8,5,6,1,4,7] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [8,6,4,5,1,2,3,7] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [5,3,4,1,6,8,2,7] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 2 = 1 + 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 2 = 1 + 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 2 = 1 + 1
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [3,1,4,8,6,7,2,5] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 2 = 1 + 1
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [8,1,7,5,6,2,3,4] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} + 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 2 = 1 + 1
Description
The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. For a given permutation π, this is the index of the row containing π1(1) of the recording tableau of π (obtained by [[Mp00070]]).
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
St001520: Permutations ⟶ ℤResult quality: 5% values known / values provided: 15%distinct values known / distinct values provided: 5%
Values
[1]
=> [1,0,1,0]
=> [2,1] => 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 0 = 1 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 0 = 1 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 0 = 1 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0 = 1 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 0 = 1 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 1 = 2 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 0 = 1 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 0 = 1 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 0 = 1 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => ? ∊ {2,4} - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => 0 = 1 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0 = 1 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 0 = 1 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0 = 1 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 1 = 2 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => ? ∊ {2,4} - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,7,6] => ? ∊ {2,4,5,11} - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,6,4] => ? ∊ {2,4,5,11} - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => 1 = 2 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 0 = 1 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 0 = 1 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 1 = 2 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 0 = 1 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 0 = 1 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 0 = 1 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,6,1,3,4,5] => ? ∊ {2,4,5,11} - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? ∊ {2,4,5,11} - 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,8,7] => ? ∊ {1,2,4,5,10,11,12,33} - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,7,5] => ? ∊ {1,2,4,5,10,11,12,33} - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,5,2,3,6,4] => ? ∊ {1,2,4,5,10,11,12,33} - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [4,1,2,3,6,5] => ? ∊ {1,2,4,5,10,11,12,33} - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 0 = 1 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 1 = 2 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 0 = 1 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => 1 = 2 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 0 = 1 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 2 = 3 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,6,3,4,5] => ? ∊ {1,2,4,5,10,11,12,33} - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 0 = 1 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,5,1,3,4,6] => ? ∊ {1,2,4,5,10,11,12,33} - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,1,3,4,5,6] => ? ∊ {1,2,4,5,10,11,12,33} - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => ? ∊ {1,2,4,5,10,11,12,33} - 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,9,8] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,8,6] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,6,2,3,4,7,5] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,7,6] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,2,5,3,6,4] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [4,5,1,2,6,3] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [3,1,2,4,6,5] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 0 = 1 - 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 0 = 1 - 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 1 = 2 - 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,3,6,4,5] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 0 = 1 - 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 1 = 2 - 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,5,6,1,3,4] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,1,7,3,4,5,6] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,1,3,5,6] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,6,1,3,4,5,7] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 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]
=> [2,1,3,4,5,6,7,8,9] => ? ∊ {1,1,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,8,10,9] => ? ∊ {1,1,1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,9,7] => ? ∊ {1,1,1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,7,2,3,4,5,8,6] => ? ∊ {1,1,1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,8,7] => ? ∊ {1,1,1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,2,6,3,4,7,5] => ? ∊ {1,1,1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [5,6,1,2,3,7,4] => ? ∊ {1,1,1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [4,1,2,3,5,7,6] => ? ∊ {1,1,1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => ? ∊ {1,1,1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [4,1,5,2,6,3] => ? ∊ {1,1,1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,4,2,3,6,5] => ? ∊ {1,1,1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [3,5,1,2,6,4] => ? ∊ {1,1,1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,6,5] => ? ∊ {1,1,1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [4,1,2,5,3,6] => ? ∊ {1,1,1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 1 = 2 - 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 0 = 1 - 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 0 = 1 - 1
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,5,1,6,3,4] => ? ∊ {1,1,1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,1,3,7,4,5,6] => ? ∊ {1,1,1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => ? ∊ {1,1,1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 1 = 2 - 1
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,5,3,4,6] => ? ∊ {1,1,1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,3,6,2,4,5] => ? ∊ {1,1,1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,6,1,3,5] => ? ∊ {1,1,1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,6,7,1,3,4,5] => ? ∊ {1,1,1,1,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1 = 2 - 1
Description
The number of strict 3-descents. A '''strict 3-descent''' of a permutation π of {1,2,,n} is a pair (i,i+3) with i+3n and π(i)>π(i+3).
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00201: Dyck paths RingelPermutations
St001728: Permutations ⟶ ℤResult quality: 3% values known / values provided: 15%distinct values known / distinct values provided: 3%
Values
[1]
=> [1,0,1,0]
=> [3,1,2] => 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 0 = 1 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0 = 1 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0 = 1 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0 = 1 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0 = 1 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 0 = 1 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1 = 2 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0 = 1 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0 = 1 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 0 = 1 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {2,4} - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1 = 2 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0 = 1 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0 = 1 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0 = 1 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 0 = 1 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? ∊ {2,4} - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? ∊ {1,4,5,11} - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {1,4,5,11} - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => 1 = 2 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 1 = 2 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 0 = 1 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0 = 1 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 1 = 2 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 0 = 1 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 0 = 1 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {1,4,5,11} - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? ∊ {1,4,5,11} - 1
[7]
=> [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] => ? ∊ {2,3,4,5,10,11,12,33} - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,1,2,7] => ? ∊ {2,3,4,5,10,11,12,33} - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? ∊ {2,3,4,5,10,11,12,33} - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? ∊ {2,3,4,5,10,11,12,33} - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 0 = 1 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 1 = 2 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 0 = 1 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 1 = 2 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 0 = 1 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 0 = 1 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? ∊ {2,3,4,5,10,11,12,33} - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 0 = 1 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? ∊ {2,3,4,5,10,11,12,33} - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => ? ∊ {2,3,4,5,10,11,12,33} - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => ? ∊ {2,3,4,5,10,11,12,33} - 1
[8]
=> [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] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,4,5,6,7,1,2,8] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [2,8,4,5,6,1,3,7] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,4,5,1,8,2,7] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => 1 = 2 - 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 0 = 1 - 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => 0 = 1 - 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 0 = 1 - 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 0 = 1 - 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0 = 1 - 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 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]
=> [3,1,4,5,6,7,8,9,10,2] => ? ∊ {1,2,2,3,4,5,5,9,10,11,12,19,27,33,37,116} - 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,9,11,1,10] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [10,3,4,5,6,7,8,1,2,9] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [2,9,4,5,6,7,1,3,8] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [7,3,4,5,6,1,9,2,8] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [2,3,8,5,6,1,4,7] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [8,6,4,5,1,2,3,7] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [5,3,4,1,6,8,2,7] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 0 = 1 - 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 0 = 1 - 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 0 = 1 - 1
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [3,1,4,8,6,7,2,5] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 0 = 1 - 1
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [8,1,7,5,6,2,3,4] => ? ∊ {1,1,2,2,2,3,4,4,5,5,5,9,10,11,12,19,25,27,28,33,37,97,99,116,123,435} - 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 0 = 1 - 1
Description
The number of invisible descents of a permutation. A visible descent of a permutation π is a position i such that π(i+1)min. Thus, an invisible descent satisfies \pi(i) > \pi(i+1) > i.
The following 51 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000353The number of inner valleys of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001162The minimum jump of a permutation. St001205The number of non-simple indecomposable projective-injective modules of the algebra eAe in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. 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). St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001737The number of descents of type 2 in a permutation. St000092The number of outer peaks of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length 3. St000664The number of right ropes of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001095The number of non-isomorphic posets with precisely one further covering relation. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001201The grade of the simple module S_0 in the special CNakayama algebra corresponding to the Dyck path. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001741The largest integer such that all patterns of this size are contained in the permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001846The number of elements which do not have a complement in the lattice. St001722The number of minimal chains with small intervals between a binary word and the top element. St001964The interval resolution global dimension of a poset. St000782The indicator function of whether a given perfect matching is an L & P matching. St000804The number of occurrences of the vincular pattern |123 in a permutation. St001490The number of connected components of a skew partition. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St000326The position of the first one in a binary word after appending a 1 at the end. St001487The number of inner corners of a skew partition. St001198The 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. St001206The 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. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000629The defect of a binary word. St001330The hat guessing number of a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000456The monochromatic index of a connected graph. St001896The number of right descents of a signed permutations.