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Your data matches 53 different statistics following compositions of up to 3 maps.
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Matching statistic: St000654
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000654: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000654: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => 1
[[.,.],.]
=> [1,2] => 2
[.,[.,[.,.]]]
=> [3,2,1] => 1
[.,[[.,.],.]]
=> [2,3,1] => 2
[[.,.],[.,.]]
=> [1,3,2] => 2
[[.,[.,.]],.]
=> [2,1,3] => 1
[[[.,.],.],.]
=> [1,2,3] => 3
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 3
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => 2
[[.,.],[[.,.],.]]
=> [1,3,4,2] => 3
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => 3
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => 4
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 2
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 3
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 2
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 3
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 3
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 2
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 4
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 2
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 3
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 3
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 2
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 4
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 3
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 4
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 4
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1
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 $\pi(n+1)=0$.
Matching statistic: St000025
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000326
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00109: Permutations —descent word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => 1 => 1
[[.,.],.]
=> [1,2] => 0 => 2
[.,[.,[.,.]]]
=> [3,2,1] => 11 => 1
[.,[[.,.],.]]
=> [2,3,1] => 01 => 2
[[.,.],[.,.]]
=> [1,3,2] => 01 => 2
[[.,[.,.]],.]
=> [2,1,3] => 10 => 1
[[[.,.],.],.]
=> [1,2,3] => 00 => 3
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 111 => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 011 => 2
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => 011 => 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 101 => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 001 => 3
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => 011 => 2
[[.,.],[[.,.],.]]
=> [1,3,4,2] => 001 => 3
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => 101 => 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => 001 => 3
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 110 => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 010 => 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => 010 => 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 100 => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => 000 => 4
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 1111 => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 0111 => 2
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 0111 => 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 1011 => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 0011 => 3
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 0111 => 2
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 0011 => 3
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 1011 => 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 0011 => 3
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 1101 => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 0101 => 2
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 0101 => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 1001 => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 0001 => 4
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 0111 => 2
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 0011 => 3
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 0011 => 3
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 0101 => 2
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 0001 => 4
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 1011 => 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 1001 => 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 0011 => 3
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 0001 => 4
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 1101 => 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 0101 => 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 0101 => 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 1001 => 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 0001 => 4
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1110 => 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000382
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1,1] => 1
[[.,.],.]
=> [1,2] => [2] => 2
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1] => 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => 2
[[.,.],[.,.]]
=> [1,3,2] => [2,1] => 2
[[.,[.,.]],.]
=> [2,1,3] => [1,2] => 1
[[[.,.],.],.]
=> [1,2,3] => [3] => 3
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,1,1] => 2
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,1,1] => 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,1] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,1] => 3
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,1] => 2
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [3,1] => 3
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [1,2,1] => 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,1] => 3
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,2] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,2] => 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [2,2] => 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4] => 4
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1] => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [2,1,1,1] => 2
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [2,1,1,1] => 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,1,1] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1] => 3
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,1,1,1] => 2
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,1,1] => 3
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,2,1,1] => 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [3,1,1] => 3
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,2,1] => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [2,2,1] => 2
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,2,1] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,3,1] => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [4,1] => 4
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [2,1,1,1] => 2
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [3,1,1] => 3
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [3,1,1] => 3
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [2,2,1] => 2
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [4,1] => 4
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [1,2,1,1] => 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [1,3,1] => 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [3,1,1] => 3
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [4,1] => 4
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [1,1,2,1] => 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,2,1] => 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [2,2,1] => 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [1,3,1] => 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [4,1] => 4
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [1,1,1,2] => 1
Description
The first part of an integer composition.
Matching statistic: St000678
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [[.,.],.]
=> [1,1,0,0]
=> 1
[[.,.],.]
=> [.,[.,.]]
=> [1,0,1,0]
=> 2
[.,[.,[.,.]]]
=> [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 1
[.,[[.,.],.]]
=> [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 2
[[.,.],[.,.]]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[[.,[.,.]],.]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[[[.,.],.],.]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 3
[.,[.,[.,[.,.]]]]
=> [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 1
[.,[.,[[.,.],.]]]
=> [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 2
[.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 2
[.,[[.,[.,.]],.]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 1
[.,[[[.,.],.],.]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 3
[[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 2
[[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3
[[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3
[[.,[.,[.,.]]],.]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
[[.,[[.,.],.]],.]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[[[.,[.,.]],.],.]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[.,[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[.,[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[.,[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[.,[[[[.,.],.],.],.]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[[.,.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[[[.,.],.],[[.,.],.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 4
[[.,[.,[.,.]]],[.,.]]
=> [[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[[.,.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[[[[.,.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[[.,[.,[.,[.,.]]]],.]
=> [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000439
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4 = 3 + 1
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4 = 3 + 1
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4 = 3 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 3 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5 = 4 + 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000011
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 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]
=> 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]
=> 2
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 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]
=> 3
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 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]
=> 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]
=> 3
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 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]
=> 2
[[[.,.],[.,.]],.]
=> [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
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 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]
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[.,.],[.,[.,[.,.]]]]
=> [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]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [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]
=> 3
[[.,.],[[.,.],[.,.]]]
=> [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]
=> 2
[[.,.],[[[.,.],.],.]]
=> [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]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> [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]
=> 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]
=> 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
[[[.,.],.],[[.,.],.]]
=> [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]
=> 4
[[.,[.,[.,.]]],[.,.]]
=> [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]
=> 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
[[[.,.],[.,.]],[.,.]]
=> [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]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [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]
=> 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]
=> 4
[[.,[.,[.,[.,.]]]],.]
=> [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]
=> 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.
Matching statistic: St000026
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1,1] => [1,0,1,0]
=> 1
[[.,.],.]
=> [1,2] => [2] => [1,1,0,0]
=> 2
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 2
[[.,.],[.,.]]
=> [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[[.,[.,.]],.]
=> [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[[[.,.],.],.]
=> [1,2,3] => [3] => [1,1,1,0,0,0]
=> 3
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
Description
The position of the first return of a Dyck path.
Matching statistic: St000068
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [.,[.,.]]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 1
[[.,.],.]
=> [[.,.],.]
=> [1,1,0,0]
=> ([],2)
=> 2
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> 2
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> ([(1,2)],3)
=> 2
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 1
[[[.,.],.],.]
=> [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> ([],3)
=> 3
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 2
[[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> 3
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> 3
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> 1
[[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> 2
[[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 1
[[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 1
[.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 2
[.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 2
[[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(2,4)],5)
=> 3
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 2
[[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> 4
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 1
[[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(1,4),(2,3),(2,4)],5)
=> 3
[[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(2,4),(3,4)],5)
=> 4
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 1
[[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2
[[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(1,4)],5)
=> 2
[[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> 1
[[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> 4
[[.,[.,[.,[.,.]]]],.]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 1
Description
The number of minimal elements in a poset.
Matching statistic: St000383
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1,1] => [1,1] => 1
[[.,.],.]
=> [1,2] => [2] => [2] => 2
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1] => [1,1,1] => 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [1,2] => 2
[[.,.],[.,.]]
=> [1,3,2] => [2,1] => [1,2] => 2
[[.,[.,.]],.]
=> [2,1,3] => [1,2] => [2,1] => 1
[[[.,.],.],.]
=> [1,2,3] => [3] => [3] => 3
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1] => [1,1,1,1] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,1,1] => [1,1,2] => 2
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,1,1] => [1,1,2] => 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,1] => [1,2,1] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,1] => [1,3] => 3
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,1] => [1,1,2] => 2
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [3,1] => [1,3] => 3
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [1,2,1] => [1,2,1] => 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,1] => [1,3] => 3
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,2] => [2,1,1] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,2] => [2,2] => 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [2,2] => [2,2] => 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3] => [3,1] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4] => [4] => 4
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1] => [1,1,1,1,1] => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [2,1,1,1] => [1,1,1,2] => 2
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [2,1,1,1] => [1,1,1,2] => 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,1,1] => [1,1,2,1] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1] => [1,1,3] => 3
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,1,1,1] => [1,1,1,2] => 2
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,1,1] => [1,1,3] => 3
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,2,1,1] => [1,1,2,1] => 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [3,1,1] => [1,1,3] => 3
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,2,1] => [1,2,1,1] => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [2,2,1] => [1,2,2] => 2
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,2,1] => [1,2,2] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,3,1] => [1,3,1] => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [4,1] => [1,4] => 4
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [2,1,1,1] => [1,1,1,2] => 2
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [3,1,1] => [1,1,3] => 3
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [3,1,1] => [1,1,3] => 3
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [2,2,1] => [1,2,2] => 2
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [4,1] => [1,4] => 4
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [1,2,1,1] => [1,1,2,1] => 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [1,3,1] => [1,3,1] => 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [3,1,1] => [1,1,3] => 3
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [4,1] => [1,4] => 4
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [1,1,2,1] => [1,2,1,1] => 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,2,1] => [1,2,2] => 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [2,2,1] => [1,2,2] => 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [1,3,1] => [1,3,1] => 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [4,1] => [1,4] => 4
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [1,1,1,2] => [2,1,1,1] => 1
Description
The last part of an integer composition.
The following 43 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000971The smallest closer of a set partition. St001050The number of terminal closers of a set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St000297The number of leading ones in a binary word. St000069The number of maximal elements of a poset. St000759The smallest missing part in an integer partition. St000740The last entry of a permutation. St000989The number of final rises of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000054The first entry of the permutation. St000007The number of saliances of the permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000738The first entry in the last row of a standard tableau. St001461The number of topologically connected components of the chord diagram of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000314The number of left-to-right-maxima of a permutation. St000990The first ascent of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000061The number of nodes on the left branch of a binary tree. St000542The number of left-to-right-minima of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000991The number of right-to-left minima of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000051The size of the left subtree of a binary tree. St000133The "bounce" of a permutation. St000331The number of upper interactions of a Dyck path. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000338The number of pixed points of a permutation. St001889The size of the connectivity set of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St000942The number of critical left to right maxima of the parking functions. 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. St001937The size of the center of a parking function.
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