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Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
St000994: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0
[1,0,1,0]
=> [2,1] => 1
[1,1,0,0]
=> [1,2] => 0
[1,0,1,0,1,0]
=> [2,3,1] => 1
[1,0,1,1,0,0]
=> [2,1,3] => 1
[1,1,0,0,1,0]
=> [1,3,2] => 1
[1,1,0,1,0,0]
=> [3,1,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 1
Description
The number of cycle peaks and the number of cycle valleys of a permutation. A '''cycle peak''' of a permutation $\pi$ is an index $i$ such that $\pi^{-1}(i) < i > \pi(i)$. Analogously, a '''cycle valley''' is an index $i$ such that $\pi^{-1}(i) > i < \pi(i)$. Clearly, every cycle of $\pi$ contains as many peaks as valleys.
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
Mp00087: Permutations inverse first fundamental transformationPermutations
Mp00109: Permutations descent wordBinary words
St000390: Binary words ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => => ? = 0
[1,0,1,0]
=> [2,1] => [2,1] => 1 => 1
[1,1,0,0]
=> [1,2] => [1,2] => 0 => 0
[1,0,1,0,1,0]
=> [2,3,1] => [3,1,2] => 10 => 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => 10 => 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => 01 => 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,2,1] => 11 => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 00 => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,1,2,3] => 100 => 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,1,2,4] => 100 => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 101 => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,3,1,2] => 110 => 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 100 => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,2,3] => 010 => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 010 => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,2,1,3] => 110 => 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => 101 => 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,2,1,4] => 110 => 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => 001 => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => 011 => 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,3,2,1] => 111 => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 1000 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => 1000 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => 1001 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,4,1,2,3] => 1100 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => 1000 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => 1010 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 1010 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,3,1,2,4] => 1100 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [4,1,2,5,3] => 1001 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,3,1,2,5] => 1100 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 1001 => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => 1011 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,4,3,1,2] => 1110 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1000 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => 0100 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => 0100 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0101 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,4,2,3] => 0110 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 0100 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,2,1,3,4] => 1100 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,2,1,3,5] => 1100 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,1,5,2,4] => 1010 => 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,2,4,1,3] => 1010 => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => 1010 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2,1,5,4] => 1101 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,4,2,1,3] => 1110 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,1,5,4,2] => 1011 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => 1100 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => 0010 => 1
[]
=> [] => [] => ? => ? = 0
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [11,10,9,8,7,6,5,4,3,2,1] => 1111111111 => ? = 1
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [2,4,6,8,10,12,1,3,5,7,9,11] => [12,11,9,5,10,7,1,2,4,8,3,6] => 11101100010 => ? = 3
[1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [2,8,9,10,11,12,1,3,4,5,6,7] => [12,7,1,2,8,3,9,4,10,5,11,6] => 11001010101 => ? = 5
Description
The number of runs of ones in a binary word.
Mp00032: Dyck paths inverse zeta mapDyck paths
Mp00102: Dyck paths rise compositionInteger compositions
Mp00040: Integer compositions to partitionInteger partitions
St001280: Integer partitions ⟶ ℤResult quality: 98% values known / values provided: 98%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [2] => [2]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => [1,1]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => [3]
=> 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,1] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => [2,2]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => [3,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,1,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => [3,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => [5]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [4,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => [3,2]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => [4,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,2,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => [4,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [2,2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => [3,1,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => [4,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => [3,2]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [3,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> 1
[]
=> []
=> [] => ?
=> ? = 0
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 3
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1,1,1,1] => ?
=> ? = 1
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,2,1,4,1] => ?
=> ? = 3
[1,1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? => ?
=> ? = 4
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? => ?
=> ? = 4
[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 3
[1,1,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? => ?
=> ? = 3
[1,1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? => ?
=> ? = 4
[1,1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? => ?
=> ? = 4
[1,1,0,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? => ?
=> ? = 3
[1,1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? => ?
=> ? = 3
[1,1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? => ?
=> ? = 3
[1,0,1,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? => ?
=> ? = 4
[1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? => ?
=> ? = 5
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000291
Mp00032: Dyck paths inverse zeta mapDyck paths
Mp00102: Dyck paths rise compositionInteger compositions
Mp00094: Integer compositions to binary wordBinary words
St000291: Binary words ⟶ ℤResult quality: 95% values known / values provided: 95%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => 1 => 0
[1,0,1,0]
=> [1,1,0,0]
=> [2] => 10 => 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => 11 => 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => 100 => 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => 110 => 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,1] => 101 => 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => 101 => 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 111 => 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 1000 => 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => 1010 => 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => 1001 => 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => 1101 => 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => 1001 => 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => 1011 => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 1011 => 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 10000 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 10100 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 10001 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => 10010 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 11010 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => 10001 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 10110 => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 10101 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => 10001 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 11001 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 10101 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => 10011 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 11101 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => 10001 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 11001 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => 10010 => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 10010 => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 10101 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => 10011 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1,1,1,1] => 1001111111 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1,1,1] => 1001111111 => ? = 1
[]
=> []
=> [] => ? => ? = 0
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? => ? => ? = 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => ? = 1
[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] => 1111111111 => ? = 0
[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1,1,1] => 1101111111 => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1,1,1,1] => ? => ? = 1
[1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,2,1,1,1,1,1] => 1101011111 => ? = 2
[1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,2,1,1,1,1,1,1] => 1010111111 => ? = 2
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 1
[1,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,1,2,2] => 1111111010 => ? = 2
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [2,1,4,1,2] => 1011000110 => ? = 3
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,2,1,4,1] => ? => ? = 3
[1,1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? => ? => ? = 4
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,2,1,1,1,1,1,1] => 1010111111 => ? = 2
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? => ? => ? = 4
[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? => ? => ? = 3
[1,1,1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2,2,2] => 1110101010 => ? = 4
[1,1,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? => ? => ? = 3
[1,1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? => ? => ? = 4
[1,1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? => ? => ? = 4
[1,1,0,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? => ? => ? = 3
[1,1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? => ? => ? = 3
[1,1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? => ? => ? = 3
[1,0,1,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? => ? => ? = 4
[1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? => ? => ? = 5
Description
The number of descents of a binary word.
Matching statistic: St000157
Mp00032: Dyck paths inverse zeta mapDyck paths
Mp00119: Dyck paths to 321-avoiding permutation (Krattenthaler)Permutations
Mp00059: Permutations Robinson-Schensted insertion tableauStandard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 89% values known / values provided: 89%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [[1]]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [[1],[2]]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [[1,2]]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => [[1,2],[3]]
=> 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [[1,3],[2]]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [[1,2,3]]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [[1,2,3],[4]]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [[1,2,3],[4]]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [[1,3],[2,4]]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[1,2,4],[3]]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [[1,2,4],[3]]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [[1,2],[3,4]]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[1,3,4],[2]]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[1,3,4],[2]]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [[1,2,3,4],[5]]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [[1,2,3,4],[5]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [[1,3,4],[2,5]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [[1,2,3,5],[4]]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [[1,2,3,4],[5]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [[1,2,4],[3,5]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [[1,2,4],[3,5]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [[1,2,3],[4,5]]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[1,3,4],[2,5]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [[1,2,3,5],[4]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[1,3,4],[2,5]]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[1,3,5],[2,4]]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [[1,2,4,5],[3]]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [[1,2,3],[4,5]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [[1,3,5],[2,4]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [[1,2,5],[3,4]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [[1,2,3,5],[4]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [[1,2,3],[4,5]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [[1,2,3],[4,5]]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [[1,2,4],[3,5]]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [[1,2,4],[3,5]]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [[1,3,5],[2,4]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [[1,2,5],[3,4]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [2,5,7,1,3,8,4,6] => ?
=> ? = 3
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> [4,6,1,2,7,3,8,5] => ?
=> ? = 2
[1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,8,6,7] => ?
=> ? = 3
[1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,4,6,1,3,7,8,5] => ?
=> ? = 3
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,7,5,6,8] => ?
=> ? = 3
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,6,7,2,3,8,5] => ?
=> ? = 2
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,2,4,6,7,3,5,8] => ?
=> ? = 2
[1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0,1,0]
=> [3,6,1,7,2,4,5,8] => ?
=> ? = 2
[1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,6,4,5,7,8] => ?
=> ? = 2
[1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0,1,0]
=> [3,1,6,2,7,4,5,8] => ?
=> ? = 2
[1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,1,0,0,0]
=> [3,4,1,7,2,8,5,6] => ?
=> ? = 2
[1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,1,1,0,0,0]
=> [3,1,4,5,2,8,6,7] => ?
=> ? = 2
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,5,3,4,7,6,8] => ?
=> ? = 3
[1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ? = 2
[1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,2,4,6,7,8,3,5] => ?
=> ? = 2
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,6,7,2,8,4,5] => ?
=> ? = 2
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [2,5,6,1,7,3,4,8] => ?
=> ? = 2
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,5,6,2,7,3,8,4] => ?
=> ? = 1
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,6,7,4,8,5] => ?
=> ? = 2
[1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [2,3,5,6,1,7,4,8] => ?
=> ? = 2
[1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [2,5,6,1,8,3,4,7] => ?
=> ? = 3
[1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [2,1,5,6,3,8,4,7] => ?
=> ? = 3
[1,1,1,0,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [2,4,6,1,7,8,3,5] => ?
=> ? = 3
[1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,3,5,6,2,7,8,4] => ?
=> ? = 2
[1,1,1,1,0,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [2,4,5,1,6,7,3,8] => ?
=> ? = 2
[1,1,1,1,0,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,4,6,7,5,8] => ?
=> ? = 2
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2,7,8,6] => ?
=> ? = 2
[1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,3,6,7,5,8] => ?
=> ? = 3
[1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,3,5,6,4,7,8] => ?
=> ? = 2
[1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,2,4,5,3,6,7,8] => ?
=> ? = 1
[1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,4,2,6,5,7,8] => ?
=> ? = 2
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,4,6,5,7,8] => ?
=> ? = 2
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,4,2,3,5,6,7,8,9] => [[1,2,3,5,6,7,8,9],[4]]
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 1
[1,1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,6,5,7,8,9] => ?
=> ? = 2
[1,1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,1,3,5,6,4,7,8,9] => ?
=> ? = 2
[1,1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,2,6,5,7,8,9] => ?
=> ? = 2
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7,8,9] => ?
=> ? = 3
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,5,4,6,7,8,9] => [[1,2,4,6,7,8,9],[3,5]]
=> ? = 2
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7,8,9] => [[1,2,3,5,6,7,8,9],[4]]
=> ? = 1
[1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,4,7,8,9] => ?
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7,9] => [[1,2,3,4,5,6,7,9],[8]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7,8,9,10,11] => ?
=> ? = 1
[1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,2,5,4,6,7,8,9,10] => ?
=> ? = 2
[1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,3,6,7,8,9,10] => ?
=> ? = 2
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 1
[1,0,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,6,5,9,7,8,10] => ?
=> ? = 4
Description
The number of descents of a standard tableau. Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Mp00032: Dyck paths inverse zeta mapDyck paths
Mp00102: Dyck paths rise compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
St000659: Dyck paths ⟶ ℤResult quality: 71% values known / values provided: 81%distinct values known / distinct values provided: 71%
Values
[1,0]
=> [1,0]
=> [1] => [1,0]
=> ? = 0
[1,0,1,0]
=> [1,1,0,0]
=> [2] => [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[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]
=> [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,1,0,0]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,1,0,0,0]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,1,0,0]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2
[1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,1,0,0,0]
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,1,1,0,0,0,0]
=> [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3
[1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,1,0,0]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0,1,0]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2
[1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
[1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2
[1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0,1,0]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2
[1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,1,0,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,1,0,0,0]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,1,0,0,0]
=> [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,1,1,0,0,0]
=> [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,1,0,0]
=> [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,0,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[1,1,1,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3
[1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0,1,0]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2
[1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3
[1,1,1,0,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,1,0,0,0]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0]
=> [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
Description
The number of rises of length at least 2 of a Dyck path.
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
Mp00236: Permutations Clarke-Steingrimsson-Zeng inversePermutations
Mp00061: Permutations to increasing treeBinary trees
St000919: Binary trees ⟶ ℤResult quality: 71% values known / values provided: 81%distinct values known / distinct values provided: 71%
Values
[1,0]
=> [1] => [1] => [.,.]
=> ? = 0
[1,0,1,0]
=> [2,1] => [2,1] => [[.,.],.]
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => [.,[.,.]]
=> 0
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => [[[.,.],.],.]
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [[[.,.],.],[.,.]]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,3,1,2] => [[[.,.],.],[.,.]]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,1,4,3] => [[[.,.],.],[[.,.],.]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [[[.,.],.],[.,[.,.]]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,3,1,2,5] => [[[.,.],.],[.,[.,.]]]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,4,1,3,2] => [[[.,.],.],[[.,.],.]]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,1,4,3,2] => [[.,.],[[[.,.],.],.]]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,1,3,2,5] => [[.,.],[[.,.],[.,.]]]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,1,3,2,4] => [[.,.],[[.,.],[.,.]]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,7,1,8] => [7,6,5,4,3,2,1,8] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,5,6,1,8,7] => [6,5,4,3,2,1,8,7] => [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,6,8,1,7] => [8,6,5,4,3,2,1,7] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,5,6,1,7,8] => [6,5,4,3,2,1,7,8] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,5,7,8,1,6] => [8,5,4,3,2,1,7,6] => [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1,6,7,8] => [5,4,3,2,1,6,7,8] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,4,1,6,7,8,5] => [4,3,2,1,8,7,6,5] => [[[[.,.],.],.],[[[[.,.],.],.],.]]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,5,6,1,7,8,4] => [8,7,6,3,2,1,5,4] => [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,1,4,5,7,8,6] => [3,2,1,4,5,8,7,6] => [[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,4,5,6,7,1,8,3] => [8,7,2,1,6,5,4,3] => [[[[.,.],.],.],[[[[.,.],.],.],.]]
=> ? = 2
[1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [2,4,6,1,7,3,8,5] => [8,7,2,1,4,3,6,5] => [[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> ? = 3
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,4,6,8,1,3,5,7] => [8,2,1,6,4,3,5,7] => [[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> ? = 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [2,4,1,3,7,5,8,6] => [4,2,1,3,8,7,5,6] => [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> ? = 2
[1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,4,7,8,1,3,5,6] => [8,2,1,7,4,3,5,6] => [[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> ? = 2
[1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [2,5,6,8,1,3,4,7] => [8,2,1,6,3,5,4,7] => ?
=> ? = 3
[1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [2,5,1,7,3,4,8,6] => [8,7,2,1,3,5,4,6] => [[[[.,.],.],.],[.,[[.,.],[.,.]]]]
=> ? = 2
[1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [2,5,7,8,1,3,4,6] => [8,2,1,7,3,5,4,6] => [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> ? = 3
[1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [2,6,7,1,3,4,8,5] => [8,7,2,1,6,3,4,5] => [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> ? = 2
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [2,6,7,8,1,3,4,5] => [8,2,1,7,3,6,4,5] => ?
=> ? = 3
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,6,1,3,4,5,7,8] => [6,2,1,3,4,5,7,8] => ?
=> ? = 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,7,8,1,3,4,5,6] => [8,2,1,7,3,4,5,6] => [[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> ? = 2
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,7,1,3,4,5,6,8] => [7,2,1,3,4,5,6,8] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 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] => [8,2,1,3,4,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,2,6,5,8,7] => [4,3,1,2,6,5,8,7] => [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
=> ? = 3
[1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,4,2,6,8,5,7] => [4,3,1,2,8,6,5,7] => [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> ? = 2
[1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,7,5,8,6] => [4,3,1,2,8,7,5,6] => [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> ? = 2
[1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [3,4,6,8,1,2,5,7] => [8,1,6,4,3,2,5,7] => ?
=> ? = 2
[1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [3,4,7,8,1,2,5,6] => [8,1,7,4,3,2,5,6] => ?
=> ? = 2
[1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,6,4,8,7] => [6,5,3,1,2,4,8,7] => [[[[.,.],.],.],[.,[.,[[.,.],.]]]]
=> ? = 2
[1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [3,1,5,6,2,8,4,7] => [8,6,3,1,2,5,4,7] => [[[[.,.],.],.],[.,[[.,.],[.,.]]]]
=> ? = 2
[1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [3,1,5,2,7,4,8,6] => [8,7,5,3,1,2,4,6] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1
[1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [3,1,6,2,7,4,8,5] => [8,7,3,1,2,4,6,5] => [[[[.,.],.],.],[.,[.,[[.,.],.]]]]
=> ? = 2
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,6,7,8,1,2,4,5] => [8,1,7,3,2,6,4,5] => ?
=> ? = 3
[1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0]
=> [4,1,5,2,6,3,8,7] => [6,5,1,2,4,3,8,7] => [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
=> ? = 3
[1,1,1,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> [4,1,5,2,7,3,8,6] => [8,7,5,1,2,4,3,6] => [[[[.,.],.],.],[.,[[.,.],[.,.]]]]
=> ? = 2
[1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0]
=> [4,1,5,2,7,8,3,6] => [8,5,1,2,4,3,7,6] => [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
=> ? = 3
[1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
=> [4,5,6,8,1,2,3,7] => [8,1,6,2,5,4,3,7] => ?
=> ? = 3
[1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [4,5,6,1,2,3,7,8] => [6,1,5,2,4,3,7,8] => ?
=> ? = 3
[1,1,1,0,1,1,0,0,1,0,0,1,0,0,1,0]
=> [4,1,6,2,7,3,8,5] => [8,7,1,2,4,3,6,5] => [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
=> ? = 3
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [1,5,2,6,3,7,4,8] => [1,7,6,2,3,5,4,8] => ?
=> ? = 2
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [1,5,2,6,3,8,4,7] => [1,8,6,2,3,5,4,7] => ?
=> ? = 2
[1,1,1,1,0,0,1,0,0,1,1,1,0,0,0,0]
=> [1,5,2,6,3,4,7,8] => [1,6,2,3,5,4,7,8] => ?
=> ? = 2
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [1,5,2,7,3,8,4,6] => [1,8,2,3,5,4,7,6] => ?
=> ? = 3
[1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [5,1,6,2,7,3,8,4] => [8,7,1,2,6,5,3,4] => [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> ? = 2
[1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [5,7,1,2,3,4,6,8] => [7,1,5,2,3,4,6,8] => ?
=> ? = 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [6,7,1,2,3,4,8,5] => [8,7,1,6,2,3,4,5] => [[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> ? = 2
[1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [1,7,2,8,3,4,5,6] => [1,8,2,3,7,4,5,6] => ?
=> ? = 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,8,6] => [8,7,1,2,3,4,5,6] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 1
Description
The number of maximal left branches of a binary tree. A maximal left branch of a binary tree is an inclusion wise maximal path which consists of left edges only. This statistic records the number of distinct maximal left branches in the tree.
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
Mp00239: Permutations CorteelPermutations
Mp00236: Permutations Clarke-Steingrimsson-Zeng inversePermutations
St000884: Permutations ⟶ ℤResult quality: 76% values known / values provided: 76%distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => [2,3,1] => 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => [3,1,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [2,4,1,3] => 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => [3,4,1,2] => 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [3,2,4,1] => 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [2,3,4,1,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => [2,3,5,1,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => [2,4,5,1,3] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => [2,4,3,5,1] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [2,4,1,3,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [2,5,1,3,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [1,3,5,2,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => [3,4,5,1,2] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => [3,4,1,2,5] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => [3,2,4,5,1] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => [3,4,2,5,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => [3,2,4,1,5] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => [3,5,1,2,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => [3,2,5,1,4] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,6,7,1,5] => [7,2,3,4,6,5,1] => [2,3,4,6,5,7,1] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => [7,2,3,6,5,4,1] => [2,3,5,6,4,7,1] => ? = 2
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,5,7,1,3,4,6] => [7,2,5,3,4,1,6] => [2,5,3,4,7,1,6] => ? = 2
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [2,6,1,7,3,4,5] => [7,2,1,6,4,5,3] => [2,6,4,5,7,1,3] => ? = 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,6,7,1,3,4,5] => [7,2,6,3,4,5,1] => [2,6,3,4,5,7,1] => ? = 2
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,1,2,5,6,7] => [4,3,2,1,5,6,7] => [3,2,4,1,5,6,7] => ? = 2
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [3,5,6,1,2,4,7] => [6,5,3,2,4,1,7] => [3,5,2,4,6,1,7] => ? = 2
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [3,6,1,2,4,5,7] => [6,3,2,1,4,5,7] => [3,2,6,1,4,5,7] => ? = 2
[1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,4,2,6,7,3,5] => [1,7,2,4,6,5,3] => [1,4,6,5,7,2,3] => ? = 2
[1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [4,5,1,6,2,3,7] => [6,5,2,4,3,1,7] => [4,5,2,3,6,1,7] => ? = 2
[1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [4,5,6,1,2,3,7] => [6,5,4,3,2,1,7] => [4,3,5,2,6,1,7] => ? = 3
[1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [4,1,6,2,3,5,7] => [6,1,4,3,2,5,7] => [4,3,6,1,2,5,7] => ? = 2
[1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [4,6,1,2,3,7,5] => [7,4,2,3,1,6,5] => [4,2,3,6,7,1,5] => ? = 2
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [4,6,1,2,3,5,7] => [6,4,2,3,1,5,7] => [4,2,3,6,1,5,7] => ? = 2
[1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,5,6,2,7,3,4] => [1,7,6,3,5,4,2] => [1,5,6,3,4,7,2] => ? = 2
[1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,5,6,7,2,3,4] => [1,7,6,5,4,3,2] => [1,5,4,6,3,7,2] => ? = 3
[1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,5,2,7,3,4,6] => [1,7,2,5,4,3,6] => [1,5,4,7,2,3,6] => ? = 2
[1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,5,7,2,3,4,6] => [1,7,5,3,4,2,6] => [1,5,3,4,7,2,6] => ? = 2
[1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [5,1,2,6,3,4,7] => [6,1,2,5,4,3,7] => [5,4,6,1,2,3,7] => ? = 2
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [5,1,6,2,3,7,4] => [7,1,5,3,4,6,2] => [5,3,4,6,7,1,2] => ? = 2
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [5,1,6,2,3,4,7] => [6,1,5,3,4,2,7] => [5,3,4,6,1,2,7] => ? = 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,1,2,3,4] => [7,6,5,3,4,2,1] => [5,3,4,6,2,7,1] => ? = 3
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [5,6,1,2,3,4,7] => [6,5,2,3,4,1,7] => [5,2,3,4,6,1,7] => ? = 2
[1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,7,4,6] => [7,1,2,3,5,4,6] => [5,7,1,2,3,4,6] => ? = 1
[1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,6,2,3,7,4,5] => [1,7,2,3,6,5,4] => [1,6,5,7,2,3,4] => ? = 2
[1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,6,2,7,3,4,5] => [1,7,2,6,4,5,3] => [1,6,4,5,7,2,3] => ? = 2
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,6,7,2,3,4,5] => [1,7,6,3,4,5,2] => [1,6,3,4,5,7,2] => ? = 2
[1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,1,2,3,7,4,5] => [7,1,2,3,6,5,4] => [6,5,7,1,2,3,4] => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,5,7,8,1,6] => [8,2,3,4,5,7,6,1] => [2,3,4,5,7,6,8,1] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,5,6,1,7,8,4] => [8,2,3,5,4,6,7,1] => [2,3,5,4,6,7,8,1] => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,4,5,6,7,1,8,3] => [8,2,6,4,5,3,7,1] => [2,4,5,6,3,7,8,1] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [2,4,6,1,7,3,8,5] => [8,2,4,3,6,5,7,1] => [2,4,3,6,5,7,8,1] => ? = 3
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,4,6,8,1,3,5,7] => [8,2,6,4,3,5,1,7] => [2,4,6,3,5,8,1,7] => ? = 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [2,4,1,3,7,5,8,6] => [4,2,1,3,8,5,7,6] => [2,4,1,3,7,8,5,6] => ? = 2
[1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,4,7,8,1,3,5,6] => [8,2,7,4,3,5,6,1] => ? => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [2,1,5,3,7,4,8,6] => [2,1,8,3,5,4,7,6] => [2,1,5,7,8,3,4,6] => ? = 2
[1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [2,5,6,8,1,3,4,7] => [8,2,6,5,4,3,1,7] => [2,5,4,6,3,8,1,7] => ? = 3
[1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [2,5,1,7,3,4,8,6] => [8,2,1,5,4,3,7,6] => [2,5,4,7,8,1,3,6] => ? = 2
[1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [2,5,7,8,1,3,4,6] => [8,2,7,5,4,3,6,1] => [2,5,4,7,3,6,8,1] => ? = 3
[1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [2,6,7,1,3,4,8,5] => [8,2,6,3,4,5,7,1] => [2,6,3,4,5,7,8,1] => ? = 2
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [2,6,7,8,1,3,4,5] => [8,2,7,6,4,5,3,1] => [2,6,4,5,7,3,8,1] => ? = 3
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,7,8,1,3,4,5,6] => [8,2,7,3,4,5,6,1] => [2,7,3,4,5,6,8,1] => ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,3,5,6,2,7,4,8] => [1,7,3,5,4,6,2,8] => [1,3,5,4,6,7,2,8] => ? = 2
[1,1,0,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,5,8,2,4,6,7] => [1,8,3,5,4,2,6,7] => ? => ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,3,2,6,4,7,5,8] => [1,3,2,7,4,6,5,8] => [1,3,2,6,7,4,5,8] => ? = 2
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,3,2,6,4,8,5,7] => [1,3,2,8,4,6,5,7] => [1,3,2,6,8,4,5,7] => ? = 2
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,3,2,7,4,8,5,6] => [1,3,2,8,4,7,6,5] => ? => ? = 3
[1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,4,2,6,8,5,7] => [4,1,3,2,8,6,5,7] => [3,4,1,2,6,8,5,7] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,8,1,2] => [8,7,3,4,5,6,2,1] => [3,4,5,6,7,2,8,1] => ? = 2
[1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [3,4,6,8,1,2,5,7] => [8,6,3,4,2,5,1,7] => ? => ? = 2
Description
The number of isolated descents of a permutation. A descent $i$ is isolated if neither $i+1$ nor $i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
Mp00240: Permutations weak exceedance partitionSet partitions
Mp00080: Set partitions to permutationPermutations
St000703: Permutations ⟶ ℤResult quality: 67% values known / values provided: 67%distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1] => {{1}}
=> [1] => 0
[1,0,1,0]
=> [2,1] => {{1,2}}
=> [2,1] => 1
[1,1,0,0]
=> [1,2] => {{1},{2}}
=> [1,2] => 0
[1,0,1,0,1,0]
=> [2,3,1] => {{1,2,3}}
=> [2,3,1] => 1
[1,0,1,1,0,0]
=> [2,1,3] => {{1,2},{3}}
=> [2,1,3] => 1
[1,1,0,0,1,0]
=> [1,3,2] => {{1},{2,3}}
=> [1,3,2] => 1
[1,1,0,1,0,0]
=> [3,1,2] => {{1,3},{2}}
=> [3,2,1] => 1
[1,1,1,0,0,0]
=> [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> [2,3,4,1] => 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> [2,3,1,4] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> [2,1,4,3] => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => {{1,2,4},{3}}
=> [2,4,3,1] => 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> [1,3,4,2] => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => {{1,3,4},{2}}
=> [3,2,4,1] => 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => {{1,3},{2,4}}
=> [3,4,1,2] => 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => {{1,3},{2},{4}}
=> [3,2,1,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => {{1},{2,4},{3}}
=> [1,4,3,2] => 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => {{1,4},{2},{3}}
=> [4,2,3,1] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> [2,3,4,1,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> [2,3,1,5,4] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => {{1,2,3,5},{4}}
=> [2,3,5,4,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> [2,1,4,5,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => {{1,2,4,5},{3}}
=> [2,4,3,5,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> [2,4,5,1,3] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => {{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> [2,1,5,4,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => {{1,2,5},{3},{4}}
=> [2,5,3,4,1] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => {{1,3,4,5},{2}}
=> [3,2,4,5,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => {{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => {{1,3},{2,4,5}}
=> [3,4,1,5,2] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => {{1,3,5},{2,4}}
=> [3,4,5,2,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> [3,4,1,2,5] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> [3,2,1,5,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => {{1,3,5},{2},{4}}
=> [3,2,5,4,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,7,1,6] => {{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5] => {{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,7,1,5,6] => {{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => {{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => ? = 2
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,4,1,3,5,6,7] => {{1,2,4},{3},{5},{6},{7}}
=> [2,4,3,1,5,6,7] => ? = 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,5,7,1,3,4,6] => {{1,2,5},{3,7},{4},{6}}
=> [2,5,7,4,1,6,3] => ? = 2
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,5,1,3,4,6,7] => {{1,2,5},{3},{4},{6},{7}}
=> [2,5,3,4,1,6,7] => ? = 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,1,6,3,4,5,7] => {{1,2},{3,6},{4},{5},{7}}
=> [2,1,6,4,5,3,7] => ? = 2
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [2,6,1,7,3,4,5] => {{1,2,6},{3},{4,7},{5}}
=> [2,6,3,7,5,1,4] => ? = 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,6,7,1,3,4,5] => {{1,2,6},{3,7},{4},{5}}
=> [2,6,7,4,5,1,3] => ? = 2
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,6,1,3,4,5,7] => {{1,2,6},{3},{4},{5},{7}}
=> [2,6,3,4,5,1,7] => ? = 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,1,3,4,7,5,6] => {{1,2},{3},{4},{5,7},{6}}
=> [2,1,3,4,7,6,5] => ? = 2
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,1,3,7,4,5,6] => {{1,2},{3},{4,7},{5},{6}}
=> [2,1,3,7,5,6,4] => ? = 2
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,1,7,3,4,5,6] => {{1,2},{3,7},{4},{5},{6}}
=> [2,1,7,4,5,6,3] => ? = 2
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,1,3,4,5,6] => {{1,2,7},{3},{4},{5},{6}}
=> [2,7,3,4,5,6,1] => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,1,2] => {{1,3,5,7},{2,4,6}}
=> [3,4,5,6,7,2,1] => ? = 2
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [3,6,1,2,4,5,7] => {{1,3},{2,6},{4},{5},{7}}
=> [3,6,1,4,5,2,7] => ? = 2
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [3,1,2,4,5,7,6] => {{1,3},{2},{4},{5},{6,7}}
=> [3,2,1,4,5,7,6] => ? = 2
[1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,4,7,2,3,5,6] => {{1},{2,4},{3,7},{5},{6}}
=> [1,4,7,2,5,6,3] => ? = 2
[1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [4,1,6,2,3,5,7] => {{1,4},{2},{3,6},{5},{7}}
=> [4,2,6,1,5,3,7] => ? = 2
[1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [4,6,1,2,3,7,5] => {{1,4},{2,6,7},{3},{5}}
=> [4,6,3,1,5,7,2] => ? = 2
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [4,6,1,2,3,5,7] => {{1,4},{2,6},{3},{5},{7}}
=> [4,6,3,1,5,2,7] => ? = 2
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [4,1,2,3,5,7,6] => {{1,4},{2},{3},{5},{6,7}}
=> [4,2,3,1,5,7,6] => ? = 2
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [4,1,2,3,5,6,7] => {{1,4},{2},{3},{5},{6},{7}}
=> [4,2,3,1,5,6,7] => ? = 1
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,7,6] => {{1},{2,5},{3},{4},{6,7}}
=> [1,5,3,4,2,7,6] => ? = 2
[1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,5,2,7,3,4,6] => {{1},{2,5},{3},{4,7},{6}}
=> [1,5,3,7,2,6,4] => ? = 2
[1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,5,7,2,3,4,6] => {{1},{2,5},{3,7},{4},{6}}
=> [1,5,7,4,2,6,3] => ? = 2
[1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [5,1,2,3,6,4,7] => {{1,5,6},{2},{3},{4},{7}}
=> [5,2,3,4,6,1,7] => ? = 1
[1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [5,1,2,6,3,4,7] => {{1,5},{2},{3},{4,6},{7}}
=> [5,2,3,6,1,4,7] => ? = 2
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [5,1,6,2,3,7,4] => {{1,5},{2},{3,6,7},{4}}
=> [5,2,6,4,1,7,3] => ? = 2
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [5,1,6,2,3,4,7] => {{1,5},{2},{3,6},{4},{7}}
=> [5,2,6,4,1,3,7] => ? = 2
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [5,6,1,2,3,7,4] => {{1,5},{2,6,7},{3},{4}}
=> [5,6,3,4,1,7,2] => ? = 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,1,2,3,4] => {{1,5},{2,6},{3,7},{4}}
=> [5,6,7,4,1,2,3] => ? = 3
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,7,6] => {{1,5},{2},{3},{4},{6,7}}
=> [5,2,3,4,1,7,6] => ? = 2
[1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,7,4,6] => {{1,5,7},{2},{3},{4},{6}}
=> [5,2,3,4,7,6,1] => ? = 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [5,1,2,3,4,6,7] => {{1,5},{2},{3},{4},{6},{7}}
=> [5,2,3,4,1,6,7] => ? = 1
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,6,2,3,4,7,5] => {{1},{2,6,7},{3},{4},{5}}
=> [1,6,3,4,5,7,2] => ? = 1
[1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,6,2,3,7,4,5] => {{1},{2,6},{3},{4},{5,7}}
=> [1,6,3,4,7,2,5] => ? = 2
[1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,6,2,7,3,4,5] => {{1},{2,6},{3},{4,7},{5}}
=> [1,6,3,7,5,2,4] => ? = 2
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,6,7,2,3,4,5] => {{1},{2,6},{3,7},{4},{5}}
=> [1,6,7,4,5,2,3] => ? = 2
[1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,1,2,3,7,4,5] => {{1,6},{2},{3},{4},{5,7}}
=> [6,2,3,4,7,1,5] => ? = 2
[1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [6,1,2,7,3,4,5] => {{1,6},{2},{3},{4,7},{5}}
=> [6,2,3,7,5,1,4] => ? = 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [6,7,1,2,3,4,5] => {{1,6},{2,7},{3},{4},{5}}
=> [6,7,3,4,5,1,2] => ? = 2
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [6,1,2,3,4,5,7] => {{1,6},{2},{3},{4},{5},{7}}
=> [6,2,3,4,5,1,7] => ? = 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,7,2,3,4,5,6] => {{1},{2,7},{3},{4},{5},{6}}
=> [1,7,3,4,5,6,2] => ? = 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [7,1,2,3,4,5,6] => {{1,7},{2},{3},{4},{5},{6}}
=> [7,2,3,4,5,6,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,6,8,1,7] => {{1,2,3,4,5,6,8},{7}}
=> [2,3,4,5,6,8,7,1] => ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,4,3,8,5,6,7] => {{1,2},{3,4},{5,8},{6},{7}}
=> ? => ? = 3
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,4,6,8,1,3,5,7] => {{1,2,4,8},{3,6},{5},{7}}
=> [2,4,6,8,5,3,7,1] => ? = 2
[1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,4,7,8,1,3,5,6] => {{1,2,4,8},{3,7},{5},{6}}
=> [2,4,7,8,5,6,3,1] => ? = 2
Description
The number of deficiencies of a permutation. This is defined as $$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$$ The number of exceedances is [[St000155]].
Mp00229: Dyck paths Delest-ViennotDyck paths
Mp00023: Dyck paths to non-crossing permutationPermutations
Mp00239: Permutations CorteelPermutations
St000374: Permutations ⟶ ℤResult quality: 66% values known / values provided: 66%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [2,1] => 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1
[1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,2,4,1] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,3,1] => 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,4,1,2] => 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,4,1] => 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,2,3,5,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,2,5,4,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,2,5,1,3] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,2,4,1,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,2,4,5,1] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,3,5,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [2,5,3,4,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [2,4,3,1,5] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [3,5,1,4,2] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [3,5,4,1,2] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [3,4,1,2,5] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [2,3,1,5,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [2,4,3,5,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [3,4,1,5,2] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [2,3,1,4,5] => 1
[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]
=> [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,1,7] => [6,2,3,4,5,1,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,4,5,1,7,6] => [5,2,3,4,1,7,6] => ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,5,1,6,7] => [5,2,3,4,1,6,7] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,4,6,5,7,1] => [5,2,3,4,7,6,1] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,7,6,5,1] => [6,2,3,4,7,1,5] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [2,3,4,1,5,7,6] => [4,2,3,1,5,7,6] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,4,1,5,6,7] => [4,2,3,1,5,6,7] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,7,5,6,4,1] => [5,2,3,7,6,1,4] => ? = 2
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [2,4,3,1,5,6,7] => [3,2,4,1,5,6,7] => ? = 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,7,5,4,3,6,1] => [4,2,5,6,1,7,3] => ? = 2
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [2,5,3,4,1,6,7] => [3,2,4,5,1,6,7] => ? = 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,6,4,5,3,7] => [2,1,4,5,6,3,7] => ? = 2
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,7,3,6,5,4,1] => [3,2,5,6,7,1,4] => ? = 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,6,4,5,3,1] => [4,2,5,6,7,1,3] => ? = 2
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [2,6,3,4,5,1,7] => [3,2,4,5,6,1,7] => ? = 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [2,1,3,4,7,6,5] => [2,1,3,4,6,7,5] => ? = 2
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [2,1,3,7,5,6,4] => [2,1,3,5,6,7,4] => ? = 2
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [2,1,7,4,5,6,3] => [2,1,4,5,6,7,3] => ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => [1,7,3,4,5,6,2] => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,6,2,7] => [1,6,3,4,5,2,7] => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,3,4,5,2,7,6] => [1,5,3,4,2,7,6] => ? = 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,4,5,7,6,2] => [1,6,3,4,5,7,2] => ? = 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,3,4,5,2,6,7] => [1,5,3,4,2,6,7] => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,6,5,7,2] => [1,5,3,4,7,6,2] => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,4,7,6,5,2] => [1,6,3,4,7,2,5] => ? = 2
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,3,4,6,5,2,7] => [1,5,3,4,6,2,7] => ? = 1
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,4,7,5,6,2] => [1,5,3,4,6,7,2] => ? = 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,6,5,4,7,2] => [1,5,3,7,2,6,4] => ? = 2
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,7,5,6,4,2] => [1,5,3,7,6,2,4] => ? = 2
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,3,6,5,4,2,7] => [1,5,3,6,2,4,7] => ? = 2
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,7,5,4,6,2] => [1,5,3,6,2,7,4] => ? = 2
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,7,6,5,4,2] => [1,5,3,6,7,2,4] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [7,3,4,5,6,2,1] => [3,7,4,5,6,1,2] => ? = 2
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,5,4,2,1,7] => [3,5,4,6,1,2,7] => ? = 2
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [6,3,2,4,5,1,7] => [3,4,1,5,6,2,7] => ? = 2
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [3,2,1,4,5,7,6] => [2,3,1,4,5,7,6] => ? = 2
[1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => [1,4,7,2,5,6,3] => ? = 2
[1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,6,4,5,3,7,2] => [1,4,7,5,2,6,3] => ? = 2
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,7,4,5,6,3,2] => [1,4,7,5,6,2,3] => ? = 2
[1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,7,4,6,5,3,2] => [1,4,6,5,7,2,3] => ? = 2
[1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,4,3,5,2,1,7] => [3,4,6,5,1,2,7] => ? = 2
[1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [6,2,4,3,5,1,7] => [2,4,5,1,6,3,7] => ? = 2
[1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [6,4,3,2,5,7,1] => [3,4,5,1,7,6,2] => ? = 2
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [6,4,3,2,5,1,7] => [3,4,5,1,6,2,7] => ? = 2
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [5,2,3,4,6,7,1] => [2,3,4,7,5,6,1] => ? = 1
[1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [5,2,3,4,6,1,7] => [2,3,4,6,5,1,7] => ? = 1
[1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,5,4,1,7] => [2,3,5,6,1,4,7] => ? = 2
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [6,2,5,4,3,7,1] => [2,4,5,7,1,6,3] => ? = 2
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [6,2,5,4,3,1,7] => [2,4,5,6,1,3,7] => ? = 2
Description
The number of exclusive right-to-left minima of a permutation. This is the number of right-to-left minima that are not left-to-right maxima. This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3. Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$. See also [[St000213]] and [[St000119]].
The following 53 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000251The number of nonsingleton blocks of a set partition. St000211The rank of the set partition. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000035The number of left outer peaks of a permutation. St000386The number of factors DDU in a Dyck path. St000996The number of exclusive left-to-right maxima of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000834The number of right outer peaks of a permutation. St000354The number of recoils of a permutation. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000201The number of leaf nodes in a binary tree. St000702The number of weak deficiencies of a permutation. St000021The number of descents of a permutation. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001298The number of repeated entries in the Lehmer code of a permutation. St000325The width of the tree associated to a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000647The number of big descents of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000353The number of inner valleys of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000710The number of big deficiencies of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001728The number of invisible descents of a permutation. St000711The number of big exceedences of a permutation. St000092The number of outer peaks of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001330The hat guessing number of a graph. St000871The number of very big ascents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St000256The number of parts from which one can substract 2 and still get an integer partition. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001597The Frobenius rank of a skew partition. St000023The number of inner peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St001905The number of preferred parking spots in a parking function less than the index of the car. St001960The number of descents of a permutation minus one if its first entry is not one. St001487The number of inner corners of a skew partition. St000028The number of stack-sorts needed to sort a permutation. St000862The number of parts of the shifted shape of a permutation. St001624The breadth of a lattice.