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Mp00051: Ordered trees to Dyck pathDyck paths
St001226: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> 2
[[],[]]
=> [1,0,1,0]
=> 2
[[[]]]
=> [1,1,0,0]
=> 3
[[],[],[]]
=> [1,0,1,0,1,0]
=> 2
[[],[[]]]
=> [1,0,1,1,0,0]
=> 3
[[[]],[]]
=> [1,1,0,0,1,0]
=> 3
[[[],[]]]
=> [1,1,0,1,0,0]
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> 4
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> 2
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 3
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> 3
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 4
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> 3
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> 4
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> 4
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> 3
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> 2
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> 5
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 4
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 4
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 5
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 5
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
Description
The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. That is the number of i such that $Ext_A^1(J,e_i J)=0$.
Mp00051: Ordered trees to Dyck pathDyck paths
Mp00120: Dyck paths Lalanne-Kreweras involutionDyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> 1 = 2 - 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 3 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 4 - 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
Description
The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.
Mp00051: Ordered trees to Dyck pathDyck paths
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
St000056: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1] => 1 = 2 - 1
[[],[]]
=> [1,0,1,0]
=> [2,1] => 1 = 2 - 1
[[[]]]
=> [1,1,0,0]
=> [1,2] => 2 = 3 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1 = 2 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [2,1,3] => 2 = 3 - 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,3,2] => 2 = 3 - 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [3,1,2] => 1 = 2 - 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,2,3] => 3 = 4 - 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 1 = 2 - 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2 = 3 - 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1 = 2 - 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 3 = 4 - 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 2 = 3 - 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 3 = 4 - 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 1 = 2 - 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 3 = 4 - 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 1 = 2 - 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 2 = 3 - 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2 = 3 - 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 1 = 2 - 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4 = 5 - 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 2 = 3 - 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 2 = 3 - 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 1 = 2 - 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 3 = 4 - 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 2 = 3 - 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 3 = 4 - 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 1 = 2 - 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 3 = 4 - 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 1 = 2 - 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 2 = 3 - 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 2 = 3 - 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 1 = 2 - 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 4 = 5 - 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 2 = 3 - 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 3 = 4 - 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 3 = 4 - 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 2 = 3 - 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 4 = 5 - 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 1 = 2 - 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 3 = 4 - 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => 2 = 3 - 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 4 = 5 - 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 1 = 2 - 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 2 = 3 - 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => 2 = 3 - 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => 1 = 2 - 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 4 = 5 - 1
Description
The decomposition (or block) number of a permutation. For $\pi \in \mathcal{S}_n$, this is given by $$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$$ This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum. This is one plus [[St000234]].
Mp00051: Ordered trees to Dyck pathDyck paths
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
St001461: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1] => 1 = 2 - 1
[[],[]]
=> [1,0,1,0]
=> [2,1] => 1 = 2 - 1
[[[]]]
=> [1,1,0,0]
=> [1,2] => 2 = 3 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1 = 2 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [2,1,3] => 2 = 3 - 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,3,2] => 2 = 3 - 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [3,1,2] => 1 = 2 - 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,2,3] => 3 = 4 - 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 1 = 2 - 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2 = 3 - 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1 = 2 - 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 3 = 4 - 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 2 = 3 - 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 3 = 4 - 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 1 = 2 - 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 3 = 4 - 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 1 = 2 - 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 2 = 3 - 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2 = 3 - 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 1 = 2 - 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4 = 5 - 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 2 = 3 - 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 2 = 3 - 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 1 = 2 - 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 3 = 4 - 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 2 = 3 - 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 3 = 4 - 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 1 = 2 - 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 3 = 4 - 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 1 = 2 - 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 2 = 3 - 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 2 = 3 - 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 1 = 2 - 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 4 = 5 - 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 2 = 3 - 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 3 = 4 - 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 3 = 4 - 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 2 = 3 - 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 4 = 5 - 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 1 = 2 - 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 3 = 4 - 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => 2 = 3 - 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 4 = 5 - 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 1 = 2 - 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 2 = 3 - 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => 2 = 3 - 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => 1 = 2 - 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 4 = 5 - 1
Description
The number of topologically connected components of the chord diagram of a permutation. The chord diagram of a permutation $\pi\in\mathfrak S_n$ is obtained by placing labels $1,\dots,n$ in cyclic order on a cycle and drawing a (straight) arc from $i$ to $\pi(i)$ for every label $i$. This statistic records the number of topologically connected components in the chord diagram. In particular, if two arcs cross, all four labels connected by the two arcs are in the same component. The permutation $\pi\in\mathfrak S_n$ stabilizes an interval $I=\{a,a+1,\dots,b\}$ if $\pi(I)=I$. It is stabilized-interval-free, if the only interval $\pi$ stablizes is $\{1,\dots,n\}$. Thus, this statistic is $1$ if $\pi$ is stabilized-interval-free.
Mp00051: Ordered trees to Dyck pathDyck paths
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
St000234: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1] => 0 = 2 - 2
[[],[]]
=> [1,0,1,0]
=> [2,1] => 0 = 2 - 2
[[[]]]
=> [1,1,0,0]
=> [1,2] => 1 = 3 - 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> [2,3,1] => 0 = 2 - 2
[[],[[]]]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1 = 3 - 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,3,2] => 1 = 3 - 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [3,1,2] => 0 = 2 - 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,2,3] => 2 = 4 - 2
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 0 = 2 - 2
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1 = 3 - 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 1 = 3 - 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 0 = 2 - 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 2 = 4 - 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1 = 3 - 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 2 = 4 - 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 0 = 2 - 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 2 = 4 - 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 0 = 2 - 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 1 = 3 - 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1 = 3 - 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 0 = 2 - 2
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 3 = 5 - 2
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 0 = 2 - 2
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 1 = 3 - 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 1 = 3 - 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 0 = 2 - 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 2 = 4 - 2
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 1 = 3 - 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 2 = 4 - 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 0 = 2 - 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 2 = 4 - 2
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 0 = 2 - 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 1 = 3 - 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 1 = 3 - 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 0 = 2 - 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 3 = 5 - 2
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 1 = 3 - 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 2 = 4 - 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2 = 4 - 2
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 1 = 3 - 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 3 = 5 - 2
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 0 = 2 - 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 2 = 4 - 2
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => 1 = 3 - 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 3 = 5 - 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 0 = 2 - 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 1 = 3 - 2
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => 1 = 3 - 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => 0 = 2 - 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 3 = 5 - 2
Description
The number of global ascents of a permutation. The global ascents are the integers $i$ such that $$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$$ Equivalently, by the pigeonhole principle, $$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$$ For $n > 1$ it can also be described as an occurrence of the mesh pattern $$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$ or equivalently $$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$ see [3]. According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives [[oeis:A003319]].
Mp00051: Ordered trees to Dyck pathDyck paths
Mp00120: Dyck paths Lalanne-Kreweras involutionDyck paths
Mp00101: Dyck paths decomposition reverseDyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> [1,0]
=> 2
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 3
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 3
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 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]
=> 2
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 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]
=> 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]
=> 2
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
Description
The position of the first down step of a Dyck path.
Mp00051: Ordered trees to Dyck pathDyck paths
Mp00120: Dyck paths Lalanne-Kreweras involutionDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
St000007: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> [1] => 1 = 2 - 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 1 = 2 - 1
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 2 = 3 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1 = 2 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2 = 3 - 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2 = 3 - 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => 1 = 2 - 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3 = 4 - 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1 = 2 - 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2 = 3 - 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2 = 3 - 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1 = 2 - 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3 = 4 - 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3 = 4 - 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1 = 2 - 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 3 = 4 - 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1 = 2 - 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2 = 3 - 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2 = 3 - 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1 = 2 - 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 4 = 5 - 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1 = 2 - 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 2 = 3 - 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 2 = 3 - 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 1 = 2 - 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 3 = 4 - 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2 = 3 - 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 3 = 4 - 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => 1 = 2 - 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 3 = 4 - 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 1 = 2 - 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 2 = 3 - 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 2 = 3 - 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => 1 = 2 - 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 4 = 5 - 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 2 = 3 - 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 3 = 4 - 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 3 = 4 - 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 2 = 3 - 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 4 = 5 - 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 1 = 2 - 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 3 = 4 - 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 2 = 3 - 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 4 = 5 - 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 1 = 2 - 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 2 = 3 - 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2 = 3 - 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 1 = 2 - 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 4 = 5 - 1
Description
The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Mp00051: Ordered trees to Dyck pathDyck paths
Mp00120: Dyck paths Lalanne-Kreweras involutionDyck paths
Mp00101: Dyck paths decomposition reverseDyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1 = 2 - 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2 = 3 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[[],[],[],[],[]]
=> [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]
=> 1 = 2 - 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 4 - 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3 = 4 - 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3 = 4 - 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 4 - 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 4 - 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 5 - 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 4 - 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 5 - 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
Description
The number of initial rises of a Dyck path. In other words, this is the height of the first peak of $D$.
Mp00051: Ordered trees to Dyck pathDyck paths
Mp00229: Dyck paths Delest-ViennotDyck paths
Mp00026: Dyck paths to ordered treeOrdered trees
St000084: Ordered trees ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> [[]]
=> 1 = 2 - 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> [[[]]]
=> 1 = 2 - 1
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [[],[]]
=> 2 = 3 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [[[],[]]]
=> 1 = 2 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[[]],[]]
=> 2 = 3 - 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [[],[[]]]
=> 2 = 3 - 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> 1 = 2 - 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [[],[],[]]
=> 3 = 4 - 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> 1 = 2 - 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> 2 = 3 - 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 2 = 3 - 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> 1 = 2 - 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 3 = 4 - 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 2 = 3 - 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 3 = 4 - 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 1 = 2 - 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 3 = 4 - 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 1 = 2 - 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 2 = 3 - 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 2 = 3 - 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> 1 = 2 - 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 4 = 5 - 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> 1 = 2 - 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[[],[],[]],[]]
=> 2 = 3 - 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[[],[]],[[]]]
=> 2 = 3 - 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[[],[],[[]]]]
=> 1 = 2 - 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[[],[]],[],[]]
=> 3 = 4 - 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[[]],[[],[]]]
=> 2 = 3 - 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 3 = 4 - 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[[],[[]],[]]]
=> 1 = 2 - 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 3 = 4 - 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> 1 = 2 - 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[[],[[]]],[]]
=> 2 = 3 - 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> 2 = 3 - 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[[],[[],[]]]]
=> 1 = 2 - 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[[]],[],[],[]]
=> 4 = 5 - 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[],[[],[],[]]]
=> 2 = 3 - 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[],[[],[]],[]]
=> 3 = 4 - 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 3 = 4 - 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[],[[],[[]]]]
=> 2 = 3 - 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> 4 = 5 - 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[[]],[],[]]]
=> 1 = 2 - 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> 3 = 4 - 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[[[]],[]],[]]
=> 2 = 3 - 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> 4 = 5 - 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[[]]],[]]]
=> 1 = 2 - 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[[[]]],[[]]]
=> 2 = 3 - 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> 2 = 3 - 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[[[],[]],[]]]
=> 1 = 2 - 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 4 = 5 - 1
Description
The number of subtrees.
Matching statistic: St000287
Mp00051: Ordered trees to Dyck pathDyck paths
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
Mp00160: Permutations graph of inversionsGraphs
St000287: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1] => ([],1)
=> 1 = 2 - 1
[[],[]]
=> [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[[]]]
=> [1,1,0,0]
=> [1,2] => ([],2)
=> 2 = 3 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 2 = 3 - 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> 2 = 3 - 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 3 = 4 - 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 3 - 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 3 = 4 - 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 3 = 4 - 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 3 = 4 - 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1 = 2 - 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> 4 = 5 - 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 3 = 4 - 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 3 = 4 - 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 3 = 4 - 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 4 = 5 - 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 3 = 4 - 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 3 = 4 - 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 4 = 5 - 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 3 = 4 - 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 4 = 5 - 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 4 = 5 - 1
Description
The number of connected components of a graph.
The following 25 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000314The number of left-to-right-maxima of a permutation. St000843The decomposition number of a perfect matching. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000214The number of adjacencies of a permutation. St000237The number of small exceedances. St000546The number of global descents of a permutation. St000061The number of nodes on the left branch of a binary tree. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000717The number of ordinal summands of a poset. St000925The number of topologically connected components of a set partition. St000288The number of ones in a binary word. St000654The first descent of a permutation. St000022The number of fixed points of a permutation. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000989The number of final rises 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. St001810The number of fixed points of a permutation smaller than its largest moved point. St001889The size of the connectivity set of a signed permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. 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. St001904The length of the initial strictly increasing segment of a parking function.