Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St000654
Mp00199: Dyck paths prime Dyck pathDyck paths
Mp00140: Dyck paths logarithmic height to pruning numberBinary trees
Mp00017: Binary trees to 312-avoiding permutationPermutations
St000654: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [[.,.],.]
=> [1,2] => 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [[[.,.],.],.]
=> [1,2,3] => 3
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> [1,3,2] => 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 4
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => 3
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 5
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 3
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 4
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 3
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 4
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 4
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[[[.,.],.],.],.],.],.]
=> [1,2,3,4,5,6] => 6
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [[[[[.,.],[.,.]],.],.],.]
=> [1,3,2,4,5,6] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [[[[.,.],[[.,.],.]],.],.]
=> [1,3,4,2,5,6] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [[[[.,[.,.]],[.,.]],.],.]
=> [2,1,4,3,5,6] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [[[[[.,.],.],[.,.]],.],.]
=> [1,2,4,3,5,6] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [[[.,.],[[[.,.],.],.]],.]
=> [1,3,4,5,2,6] => 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> [1,3,5,4,2,6] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [[[.,[.,.]],[[.,.],.]],.]
=> [2,1,4,5,3,6] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [[[.,[.,[.,.]]],[.,.]],.]
=> [3,2,1,5,4,6] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [[[.,[[.,.],.]],[.,.]],.]
=> [2,3,1,5,4,6] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [[[[.,.],.],[[.,.],.]],.]
=> [1,2,4,5,3,6] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [[[[.,[.,.]],.],[.,.]],.]
=> [2,1,3,5,4,6] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [[[[[.,.],.],.],[.,.]],.]
=> [1,2,3,5,4,6] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[[[.,.],[.,.]],[.,.]],.]
=> [1,3,2,5,4,6] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[.,.],.],.],.]]
=> [1,3,4,5,6,2] => 5
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [[.,.],[[[.,.],[.,.]],.]]
=> [1,3,5,4,6,2] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],[[.,.],.]]]
=> [1,3,5,6,4,2] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [[.,.],[[.,[.,.]],[.,.]]]
=> [1,4,3,6,5,2] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[.,.],[[[.,.],.],[.,.]]]
=> [1,3,4,6,5,2] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[.,[.,.]],[[[.,.],.],.]]
=> [2,1,4,5,6,3] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [[.,[.,.]],[[.,.],[.,.]]]
=> [2,1,4,6,5,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [[.,[.,[.,.]]],[[.,.],.]]
=> [3,2,1,5,6,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> [4,3,2,1,6,5] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [[.,[.,[[.,.],.]]],[.,.]]
=> [3,4,2,1,6,5] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [[.,[[.,.],.]],[[.,.],.]]
=> [2,3,1,5,6,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [[.,[[.,[.,.]],.]],[.,.]]
=> [3,2,4,1,6,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [[.,[[[.,.],.],.]],[.,.]]
=> [2,3,4,1,6,5] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [[.,[[.,.],[.,.]]],[.,.]]
=> [2,4,3,1,6,5] => 2
Description
The first descent of a permutation. For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the smallest index $0 < i \leq n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(n+1)=0$.