Your data matches 202 different statistics following compositions of up to 3 maps.
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Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00101: Dyck paths decomposition reverseDyck paths
Mp00201: Dyck paths RingelPermutations
St000534: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [2,1] => 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,3,1] => 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [3,1,2] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => 0
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 0
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [8,7,4,5,6,1,2,3] => 0
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [8,3,6,5,1,7,2,4] => 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 0
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [8,3,4,1,6,7,2,5] => 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 0
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 0
[3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => 0
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 0
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => 0
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => 0
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [6,7,8,1,2,3,4,5] => 0
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => 1
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => 0
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => 0
[5,5,3]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [8,1,2,7,6,3,4,5] => 0
[4,3,3,3]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => 0
[5,3,3,3]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [9,7,8,1,2,3,4,5,6] => 0
Description
The number of 2-rises of a permutation. A 2-rise of a permutation $\pi$ is an index $i$ such that $\pi(i)+2 = \pi(i+1)$. For 1-rises, or successions, see [[St000441]].
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
Mp00069: Permutations complementPermutations
St001086: Permutations ⟶ ℤResult quality: 90% values known / values provided: 90%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => [1] => 0
[2]
=> [1,0,1,0]
=> [2,1] => [1,2] => 0
[1,1]
=> [1,1,0,0]
=> [1,2] => [2,1] => 0
[3]
=> [1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1,3] => 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => [2,3,1] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [2,1,3,4] => 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,3,1,4] => 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,4,1] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [2,1,3,4,5] => 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,2,1,4] => 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [2,3,1,4,5] => 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,4,3,1] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [2,3,4,1,5] => 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [2,3,4,5,1] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,2,1,4,5] => 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [3,4,2,1] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,4,3,1,5] => 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [2,3,5,4,1] => 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 0
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [3,2,1,4,5,6] => 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,4,2,1,5] => 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [2,4,5,3,1] => 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [2,5,4,3,1] => 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => 0
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => [3,2,1,4,5,6,7] => 0
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => [2,4,3,1,5,6,7] => ? = 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,4,5,2,1] => 0
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,7,2,1] => [2,3,5,4,1,6,7] => ? = 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [4,3,5,2,1] => 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [4,5,3,2,1] => 0
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => 0
[3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [4,5,3,2,1,6] => 0
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [10,9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => 0
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [5,4,3,2,1,6] => 0
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,4,5,6] => [4,5,6,3,2,1] => 0
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => [5,4,3,2,1,6,7] => 0
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,2,4,5,6] => [4,6,5,3,2,1] => 1
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => [5,6,4,3,2,1] => 0
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[5,5,3]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,1,6,7] => [5,4,3,6,7,2,1] => ? = 0
[4,3,3,3]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => 0
[5,3,3,3]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,7,8,2,1] => [6,5,4,3,2,1,7,8] => 0
[6,3,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [4,5,6,7,8,9,3,2,1] => [6,5,4,3,2,1,7,8,9] => 0
[3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => [6,7,5,4,3,2,1] => ? = 0
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => 0
[5,4,4,4]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [7,6,5,4,3,2,1,8] => 0
[4,4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9] => [8,9,7,6,5,4,3,2,1] => ? = 0
[5,5,5,5]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => [7,8,6,5,4,3,2,1] => ? = 0
Description
The number of occurrences of the consecutive pattern 132 in a permutation. This is the number of occurrences of the pattern $132$, where the matched entries are all adjacent.
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00121: Dyck paths Cori-Le Borgne involutionDyck paths
Mp00031: Dyck paths to 312-avoiding permutationPermutations
St001085: Permutations ⟶ ℤResult quality: 84% values known / values provided: 84%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => 0
[2]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,2] => 0
[1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> [2,1] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => 0
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,2,1] => 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => 0
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 0
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,6,2,1,7] => ? = 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [3,4,5,2,1,6,7] => ? = 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 0
[3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => 0
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10] => 0
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [5,6,4,3,2,1] => 0
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => 0
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 0
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 1
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 0
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 0
[5,5,3]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,7,6,5,2,1] => ? = 0
[4,3,3,3]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => ? = 0
[5,3,3,3]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [6,7,8,5,4,3,2,1] => ? = 0
[6,3,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [6,7,8,9,5,4,3,2,1] => ? = 0
[3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => 0
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => 0
[5,4,4,4]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [7,8,6,5,4,3,2,1] => ? = 0
[]
=> []
=> []
=> [] => 0
[4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8,7,6,5,4,3,2,1] => 0
[4,4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => 0
[5,5,5,5]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => 0
[5,5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [10,9,8,7,6,5,4,3,2,1] => 0
[5,4,4,4,4]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [8,9,7,6,5,4,3,2,1] => ? = 0
Description
The number of occurrences of the vincular pattern |21-3 in a permutation. This is the number of occurrences of the pattern $213$, 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 smaller and the top value is strictly larger than the first entry of the permutation.
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
St000664: Permutations ⟶ ℤResult quality: 66% values known / values provided: 66%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => 0
[2]
=> [1,0,1,0]
=> [2,1] => 0
[1,1]
=> [1,1,0,0]
=> [1,2] => 0
[3]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => ? = 0
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1] => ? = 0
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 0
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => ? = 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 0
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,7,2,1] => ? = 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 0
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4,3,2,1] => ? = 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 0
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [10,9,8,7,6,5,4,3,2,1] => ? = 0
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,4,5,6] => 0
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 0
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,2,4,5,6] => 1
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 0
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0
[5,5,3]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,1,6,7] => ? = 0
[4,3,3,3]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 0
[5,3,3,3]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,7,8,2,1] => ? = 0
[6,3,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [4,5,6,7,8,9,3,2,1] => ? = 0
[3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? = 0
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => 0
[5,4,4,4]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 0
[]
=> []
=> [] => ? = 0
[4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[4,4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9] => ? = 0
[5,5,5,5]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => ? = 0
[5,5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8,9,10] => ? = 0
[5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8,9] => ? = 0
[5,4,4,4,4]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? = 0
Description
The number of right ropes of a permutation. Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, [[St000441]], and a right rope is a large ascent after a raft of $\pi$. See Definition 3.10 and Example 3.11 in [1].
Matching statistic: St000125
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00201: Dyck paths RingelPermutations
Mp00072: Permutations binary search tree: left to rightBinary trees
St000125: Binary trees ⟶ ℤResult quality: 66% values known / values provided: 66%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [2,1] => [[.,.],.]
=> 0
[2]
=> [1,0,1,0]
=> [3,1,2] => [[.,[.,.]],.]
=> 0
[1,1]
=> [1,1,0,0]
=> [2,3,1] => [[.,.],[.,.]]
=> 0
[3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => [[[.,[.,.]],.],.]
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [[.,[.,[.,.]]],[.,.]]
=> 0
[2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [[.,[[.,[.,.]],.]],.]
=> 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],.]
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [[.,[.,[.,[.,.]]]],[.,.]]
=> 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [[.,[.,[[.,[.,.]],.]]],.]
=> 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [[.,.],[[[.,.],.],.]]
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [[.,[[.,[.,[.,.]]],.]],.]
=> 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [[.,[.,[.,[.,.]]]],[.,.]]
=> 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [[.,[.,[.,.]]],[.,[.,.]]]
=> 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [[.,[.,.]],[[[.,.],.],.]]
=> 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [[.,.],[[[.,[.,.]],.],.]]
=> 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> ? = 0
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [[.,[[.,[.,.]],[.,.]]],.]
=> 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [[[.,[.,.]],[[.,.],.]],.]
=> 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [[.,[.,.]],[.,[.,[.,.]]]]
=> 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [[.,.],[.,[[[.,.],.],.]]]
=> 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> ? = 0
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> ? = 0
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,1,2,3,8,7,4,6] => [[.,[.,[.,[.,.]]]],[[[.,.],.],.]]
=> ? = 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [[[[.,[.,[.,.]]],.],.],.]
=> 0
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,1,2,8,7,3,5,6] => [[.,[.,[.,.]]],[[[.,[.,.]],.],.]]
=> ? = 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [[[.,[.,.]],[.,[.,.]]],.]
=> 0
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]
=> ? = 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [[.,.],[.,[.,[.,[.,.]]]]]
=> 0
[3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [[.,[[.,[.,.]],[.,[.,.]]]],.]
=> 0
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]
=> ? = 0
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> 0
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> 0
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,1,2,5,6,7,8,3] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> ? = 0
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => [[.,.],[[[.,.],[.,[.,.]]],.]]
=> 1
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> 0
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> 0
[5,5,3]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [6,5,4,1,2,7,8,3] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> ? = 0
[4,3,3,3]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[5,3,3,3]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [4,1,2,5,6,7,8,9,3] => [[.,[.,[.,.]]],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[6,3,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [5,1,2,3,6,7,8,9,10,4] => [[.,[.,[.,[.,.]]]],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [8,3,4,5,6,7,1,2] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> ? = 0
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 0
[5,4,4,4]
=> [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] => [[.,[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 0
[]
=> []
=> [1] => [.,.]
=> 0
[4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 0
[4,4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [10,3,4,5,6,7,8,9,1,2] => [[[.,[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> ? = 0
[5,5,5,5]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [9,3,4,5,6,7,8,1,2] => [[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]],.]
=> ? = 0
[5,5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ? = 0
[5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0
[5,4,4,4,4]
=> [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] => [[.,[.,.]],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 0
Description
The number of occurrences of the contiguous pattern {{{[.,[[[.,.],.],.]]}}} in a binary tree. [[oeis:A005773]] counts binary trees avoiding this pattern.
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00031: Dyck paths to 312-avoiding permutationPermutations
Mp00149: Permutations Lehmer code rotationPermutations
St001513: Permutations ⟶ ℤResult quality: 66% values known / values provided: 66%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => [1] => 0
[2]
=> [1,0,1,0]
=> [1,2] => [2,1] => 0
[1,1]
=> [1,1,0,0]
=> [2,1] => [1,2] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => [2,3,1] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [2,1,3] => 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [2,3,4,1] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,3,1,4] => 0
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,4,1,2] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [2,3,4,5,1] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,3,4,1,5] => 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,3,4] => 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,3,5,1,4] => 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,3,1,2] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,4,5,1,3] => 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [3,4,5,1,2] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [2,3,4,5,6,1] => 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [2,3,1,4,5] => 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,1,2,3] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [2,5,4,1,3] => 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [4,3,5,1,2] => 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [2,3,4,1,5,6] => 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [2,5,1,3,4] => 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => [4,5,3,1,2] => 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,3,4,5] => 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [5,4,3,1,2] => 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => [2,3,4,5,6,7,8,1] => ? = 0
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [2,3,4,5,1,6,7] => ? = 0
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [2,3,4,7,6,1,5] => ? = 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => [4,5,1,2,3] => 0
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [2,3,6,5,7,1,4] => ? = 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => [4,1,2,3,5] => 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => [5,1,2,3,4] => 0
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => [2,3,4,5,6,7,8,9,1] => ? = 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0
[3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => [2,6,1,3,4,5] => 0
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10] => [2,3,4,5,6,7,8,9,10,1] => ? = 0
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [2,1,3,4,5,6] => 0
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,5,6,3,2,1] => [5,6,1,2,3,4] => 0
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [2,3,1,4,5,6,7] => ? = 0
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,4,6,3,2,1] => [6,5,1,2,3,4] => 1
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [5,6,4,3,2,1] => [6,1,2,3,4,5] => 0
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0
[5,5,3]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,7,6,5,2,1] => [4,5,1,2,3,6,7] => ? = 0
[4,3,3,3]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => [2,1,3,4,5,6,7] => ? = 0
[5,3,3,3]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,8,7,6,5,4,3] => [2,3,1,4,5,6,7,8] => ? = 0
[6,3,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,9,8,7,6,5,4] => [2,3,4,1,5,6,7,8,9] => ? = 0
[3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => [7,1,2,3,4,5,6] => ? = 0
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 0
[5,4,4,4]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8] => ? = 0
[]
=> []
=> [] => [] => ? = 0
[4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => ? = 0
[4,4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [8,9,7,6,5,4,3,2,1] => [9,1,2,3,4,5,6,7,8] => ? = 0
[5,5,5,5]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [7,8,6,5,4,3,2,1] => [8,1,2,3,4,5,6,7] => ? = 0
[5,5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [10,9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => ? = 0
[5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => ? = 0
[5,4,4,4,4]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9] => ? = 0
Description
The number of nested exceedences of a permutation. For a permutation $\pi$, this is the number of pairs $i,j$ such that $i < j < \pi(j) < \pi(i)$. For exceedences, see [[St000155]].
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00119: Dyck paths to 321-avoiding permutation (Krattenthaler)Permutations
Mp00149: Permutations Lehmer code rotationPermutations
St001728: Permutations ⟶ ℤResult quality: 66% values known / values provided: 66%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => [1] => 0
[2]
=> [1,0,1,0]
=> [1,2] => [2,1] => 0
[1,1]
=> [1,1,0,0]
=> [2,1] => [1,2] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => [2,3,1] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [2,1,3] => 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [2,3,4,1] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,3,1,4] => 0
[2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => [1,3,2] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,4,1,2] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [2,3,4,5,1] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,3,4,1,5] => 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,4,3] => 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,3,5,1,4] => 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,4,5,1,3] => 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [3,4,5,1,2] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [2,3,4,5,6,1] => 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,3,1,5,4] => 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [4,1,3,2] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,5,3,1,4] => 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [1,3,4,2] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [4,2,5,1,3] => 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => [2,3,4,1,6,5] => 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,5,1,4,3] => 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [4,5,2,1,3] => 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,4,5,3] => 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [5,2,3,1,4] => 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => [2,3,4,5,6,7,8,1] => ? = 0
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,5,6] => [2,3,4,5,1,7,6] => ? = 0
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,4,7,5] => [2,3,4,7,5,1,6] => ? = 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => [4,5,1,3,2] => 0
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,3,6,7,4] => [2,3,6,4,7,1,5] => ? = 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [4,1,3,5,2] => 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [5,1,3,4,2] => 0
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => [2,3,4,5,6,7,8,9,1] => ? = 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [1,3,4,5,2] => 0
[3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,2,3,4] => [2,6,1,4,5,3] => 0
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10] => [2,3,4,5,6,7,8,9,10,1] => ? = 0
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => [2,1,4,5,6,3] => 0
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,5,6,1,2,3] => [5,6,1,3,4,2] => 0
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,3,4,5,6] => [2,3,1,5,6,7,4] => ? = 0
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,1,6,2,3,4] => [6,2,1,4,5,3] => 1
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [5,6,1,2,3,4] => [6,1,3,4,5,2] => 0
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,1,2,3,4,5] => [1,3,4,5,6,2] => 0
[5,5,3]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,7,1,2,5,6] => [4,5,1,3,6,7,2] => ? = 0
[4,3,3,3]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,2,3,4,5,6] => [2,1,4,5,6,7,3] => ? = 0
[5,3,3,3]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,8,3,4,5,6,7] => [2,3,1,5,6,7,8,4] => ? = 0
[6,3,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,9,4,5,6,7,8] => [2,3,4,1,6,7,8,9,5] => ? = 0
[3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,1,2,3,4,5] => [7,1,3,4,5,6,2] => ? = 0
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,1,2,3,4,5,6] => [1,3,4,5,6,7,2] => 0
[5,4,4,4]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,2,3,4,5,6,7] => [2,1,4,5,6,7,8,3] => ? = 0
[]
=> []
=> [] => [] => ? = 0
[4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7] => [1,3,4,5,6,7,8,2] => ? = 0
[4,4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [8,9,1,2,3,4,5,6,7] => [9,1,3,4,5,6,7,8,2] => ? = 0
[5,5,5,5]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [7,8,1,2,3,4,5,6] => [8,1,3,4,5,6,7,2] => ? = 0
[5,5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [10,1,2,3,4,5,6,7,8,9] => [1,3,4,5,6,7,8,9,10,2] => ? = 0
[5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [9,1,2,3,4,5,6,7,8] => [1,3,4,5,6,7,8,9,2] => ? = 0
[5,4,4,4,4]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,2,3,4,5,6,7,8] => [2,1,4,5,6,7,8,9,3] => ? = 0
Description
The number of invisible descents of a permutation. A visible descent of a permutation $\pi$ is a position $i$ such that $\pi(i+1) \leq \min(i, \pi(i))$. Thus, an invisible descent satisfies $\pi(i) > \pi(i+1) > i$.
Matching statistic: St001732
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00201: Dyck paths RingelPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St001732: Dyck paths ⟶ ℤResult quality: 66% values known / values provided: 66%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[2]
=> [1,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1]
=> [1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 1 = 0 + 1
[3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1 = 0 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1 = 0 + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 0 + 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,1,2,3,8,7,4,6] => [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,1,2,8,7,3,5,6] => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1 = 0 + 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1 = 0 + 1
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1 = 0 + 1
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,1,2,5,6,7,8,3] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> 2 = 1 + 1
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1 = 0 + 1
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[5,5,3]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [6,5,4,1,2,7,8,3] => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 0 + 1
[4,3,3,3]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[5,3,3,3]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [4,1,2,5,6,7,8,9,3] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[6,3,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [5,1,2,3,6,7,8,9,10,4] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [8,3,4,5,6,7,1,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[5,4,4,4]
=> [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] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[]
=> []
=> [1] => [1,0]
=> 1 = 0 + 1
[4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[4,4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [10,3,4,5,6,7,8,9,1,2] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[5,5,5,5]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [9,3,4,5,6,7,8,1,2] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[5,5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[5,4,4,4,4]
=> [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] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
Description
The number of peaks visible from the left. This is, the number of left-to-right maxima of the heights of the peaks of a Dyck path.
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00142: Dyck paths promotionDyck paths
St001493: Dyck paths ⟶ ℤResult quality: 64% values known / values provided: 64%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 0 + 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[5,5,3]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
[4,3,3,3]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[5,3,3,3]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[6,3,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0 + 1
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[5,4,4,4]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[]
=> []
=> []
=> ? = 0 + 1
[4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[4,4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[5,5,5,5]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0 + 1
[5,5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[5,4,4,4,4]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
Description
The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra.
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00030: Dyck paths zeta mapDyck paths
Mp00120: Dyck paths Lalanne-Kreweras involutionDyck paths
St001221: Dyck paths ⟶ ℤResult quality: 64% values known / values provided: 64%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 0
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 0
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 0
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[5,5,3]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 0
[4,3,3,3]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[5,3,3,3]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[6,3,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[5,4,4,4]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[]
=> []
=> []
=> []
=> ? = 0
[4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[4,4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[5,5,5,5]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[5,5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 0
[5,5,5,5,5]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[5,4,4,4,4]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
Description
The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module.
The following 192 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St000659The number of rises of length at least 2 of a Dyck path. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St000989The number of final rises of a permutation. St001394The genus of a permutation. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St000359The number of occurrences of the pattern 23-1. St001095The number of non-isomorphic posets with precisely one further covering relation. St000570The Edelman-Greene number of a permutation. 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 $K[x]/(x^n)$. St001960The number of descents of a permutation minus one if its first entry is not one. St000648The number of 2-excedences of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St000358The number of occurrences of the pattern 31-2. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000732The number of double deficiencies of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000367The number of simsun double descents of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000779The tier of a permutation. St001868The number of alignments of type NE of a signed permutation. St000454The largest eigenvalue of a graph if it is integral. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001811The Castelnuovo-Mumford regularity of a permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St000455The second largest eigenvalue of a graph if it is integral. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001621The number of atoms of a lattice. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001867The number of alignments of type EN of a signed permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001896The number of right descents of a signed permutations. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St000417The size of the automorphism group of the ordered tree. St001058The breadth of the ordered tree. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000068The number of minimal elements in a poset. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation. St001410The minimal entry of a semistandard tableau. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. 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. St000295The length of the border of a binary word. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001884The number of borders of a binary word. St001889The size of the connectivity set of a signed permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000039The number of crossings of a permutation. St000091The descent variation of a composition. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000462The major index minus the number of excedences of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000496The rcs statistic of a set partition. St000516The number of stretching pairs of a permutation. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000623The number of occurrences of the pattern 52341 in a permutation. St000666The number of right tethers of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000962The 3-shifted major index of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001381The fertility of a permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001537The number of cyclic crossings of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St001810The number of fixed points of a permutation smaller than its largest moved point. St001847The number of occurrences of the pattern 1432 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St000021The number of descents of a permutation. St000056The decomposition (or block) number of a permutation. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000253The crossing number of a set partition. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000654The first descent of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000729The minimal arc length of a set partition. St000864The number of circled entries of the shifted recording tableau of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001461The number of topologically connected components of the chord diagram of a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001806The upper middle entry of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001928The number of non-overlapping descents in a permutation. St000084The number of subtrees. St000105The number of blocks in the set partition. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000504The cardinality of the first block of a set partition. St000542The number of left-to-right-minima of a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000823The number of unsplittable factors of the set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set 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 $eA$. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001624The breadth of a lattice. St000879The number of long braid edges in the graph of braid moves of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau.