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Your data matches 71 different statistics following compositions of up to 3 maps.
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Matching statistic: St000007
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1] => 1
[[],[]]
=> [[.,.],.]
=> [1,2] => 1
[[[]]]
=> [.,[.,.]]
=> [2,1] => 2
[[],[],[]]
=> [[[.,.],.],.]
=> [1,2,3] => 1
[[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => 2
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => 1
[[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 2
[[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 3
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 1
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 2
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 1
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 3
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 2
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 3
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 4
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 1
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 2
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 1
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 2
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 3
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 1
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 2
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => 1
[[],[[[]]],[]]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 1
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 2
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 3
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 2
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 3
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 4
[[[]],[],[],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 1
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 2
[[[]],[[]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 1
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 2
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 3
[[[],[]],[],[]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => 1
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 1
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 2
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 2
[[[],[],[]],[]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 1
[[[],[[]]],[]]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => 1
[[[[]],[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => 1
[[[[],[]]],[]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 1
[[[[[]]]],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 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].
Matching statistic: St000546
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000546: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000546: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1] => 0 = 1 - 1
[[],[]]
=> [[.,.],.]
=> [1,2] => 0 = 1 - 1
[[[]]]
=> [.,[.,.]]
=> [2,1] => 1 = 2 - 1
[[],[],[]]
=> [[[.,.],.],.]
=> [1,2,3] => 0 = 1 - 1
[[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => 1 = 2 - 1
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => 0 = 1 - 1
[[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 1 = 2 - 1
[[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 2 = 3 - 1
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 0 = 1 - 1
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 1 = 2 - 1
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 0 = 1 - 1
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 1 = 2 - 1
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 2 = 3 - 1
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 0 = 1 - 1
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1 = 2 - 1
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 0 = 1 - 1
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0 = 1 - 1
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1 = 2 - 1
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 2 = 3 - 1
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 1 = 2 - 1
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2 = 3 - 1
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 3 = 4 - 1
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 0 = 1 - 1
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 1 = 2 - 1
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 0 = 1 - 1
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 1 = 2 - 1
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 2 = 3 - 1
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 0 = 1 - 1
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1 = 2 - 1
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => 0 = 1 - 1
[[],[[[]]],[]]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 0 = 1 - 1
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 1 = 2 - 1
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2 = 3 - 1
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 1 = 2 - 1
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2 = 3 - 1
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 3 = 4 - 1
[[[]],[],[],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 0 = 1 - 1
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 1 = 2 - 1
[[[]],[[]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 0 = 1 - 1
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 1 = 2 - 1
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 2 = 3 - 1
[[[],[]],[],[]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => 0 = 1 - 1
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 0 = 1 - 1
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 1 = 2 - 1
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 1 = 2 - 1
[[[],[],[]],[]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 0 = 1 - 1
[[[],[[]]],[]]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => 0 = 1 - 1
[[[[]],[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => 0 = 1 - 1
[[[[],[]]],[]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 0 = 1 - 1
[[[[[]]]],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 0 = 1 - 1
Description
The number of global descents of a permutation.
The global descents are the integers in the set
$$C(\pi)=\{i\in [n-1] : \forall 1 \leq j \leq i < k \leq n :\quad \pi(j) > \pi(k)\}.$$
In particular, if $i\in C(\pi)$ then $i$ is a descent.
For the number of global ascents, see [[St000234]].
Matching statistic: St000382
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00048: Ordered trees —left-right symmetry⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [[]]
=> [1,0]
=> [1] => 1
[[],[]]
=> [[],[]]
=> [1,0,1,0]
=> [1,1] => 1
[[[]]]
=> [[[]]]
=> [1,1,0,0]
=> [2] => 2
[[],[],[]]
=> [[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[[],[[]]]
=> [[[]],[]]
=> [1,1,0,0,1,0]
=> [2,1] => 2
[[[]],[]]
=> [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,2] => 1
[[[],[]]]
=> [[[],[]]]
=> [1,1,0,1,0,0]
=> [2,1] => 2
[[[[]]]]
=> [[[[]]]]
=> [1,1,1,0,0,0]
=> [3] => 3
[[],[],[],[]]
=> [[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[[],[],[[]]]
=> [[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[[],[[]],[]]
=> [[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[[],[[],[]]]
=> [[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 2
[[],[[[]]]]
=> [[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[[[]],[],[]]
=> [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1
[[[]],[[]]]
=> [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[[[],[]],[]]
=> [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => 1
[[[[]]],[]]
=> [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1
[[[],[],[]]]
=> [[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => 2
[[[],[[]]]]
=> [[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => 3
[[[[]],[]]]
=> [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => 2
[[[[],[]]]]
=> [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => 3
[[[[[]]]]]
=> [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[[],[],[],[],[]]
=> [[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1
[[],[],[],[[]]]
=> [[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 2
[[],[],[[]],[]]
=> [[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 1
[[],[],[[],[]]]
=> [[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 2
[[],[],[[[]]]]
=> [[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 3
[[],[[]],[],[]]
=> [[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 1
[[],[[]],[[]]]
=> [[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 2
[[],[[],[]],[]]
=> [[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 1
[[],[[[]]],[]]
=> [[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[[],[[],[],[]]]
=> [[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 2
[[],[[],[[]]]]
=> [[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => 3
[[],[[[]],[]]]
=> [[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 2
[[],[[[],[]]]]
=> [[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => 3
[[],[[[[]]]]]
=> [[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4
[[[]],[],[],[]]
=> [[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 1
[[[]],[],[[]]]
=> [[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2
[[[]],[[]],[]]
=> [[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 1
[[[]],[[],[]]]
=> [[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 2
[[[]],[[[]]]]
=> [[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 3
[[[],[]],[],[]]
=> [[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 1
[[[[]]],[],[]]
=> [[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 1
[[[],[]],[[]]]
=> [[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 2
[[[[]]],[[]]]
=> [[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 2
[[[],[],[]],[]]
=> [[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 1
[[[],[[]]],[]]
=> [[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 1
[[[[]],[]],[]]
=> [[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 1
[[[[],[]]],[]]
=> [[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 1
[[[[[]]]],[]]
=> [[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 1
Description
The first part of an integer composition.
Matching statistic: St000326
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1] => => ? = 1
[[],[]]
=> [1,0,1,0]
=> [2,1] => 1 => 1
[[[]]]
=> [1,1,0,0]
=> [1,2] => 0 => 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> [2,1,3] => 10 => 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [2,3,1] => 01 => 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [3,1,2] => 10 => 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,3,2] => 01 => 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,2,3] => 00 => 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 101 => 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 010 => 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 100 => 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 010 => 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 001 => 3
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 101 => 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 010 => 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => 100 => 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 100 => 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 010 => 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 001 => 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 010 => 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 001 => 3
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 000 => 4
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 1010 => 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => 0100 => 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => 1001 => 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => 0101 => 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => 0010 => 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => 1010 => 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => 0100 => 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => 1001 => 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => 1000 => 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => 0101 => 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 0010 => 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => 0100 => 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 0010 => 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0001 => 4
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => 1010 => 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => 0100 => 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => 1001 => 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => 0101 => 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0010 => 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => 1010 => 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => 1010 => 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => 0100 => 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 0100 => 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => 1001 => 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => 1000 => 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => 1001 => 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => 1000 => 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1000 => 1
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => 0101 => 2
[[],[],[],[[],[[],[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,6,1,7,3,5,8] => ? => ? = 3
[[],[],[[],[],[]],[],[]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,1,4,3,5,7,6,8] => ? => ? = 1
[[],[],[[],[],[[[]]]]]
=> [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,4,5,7,1,3,6,8] => ? => ? = 4
[[],[[],[[],[]]],[[]]]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [2,5,1,3,4,6,8,7] => ? => ? = 2
[[],[[],[],[],[[],[]]]]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,1,7,4,6,8] => ? => ? = 3
[[[],[],[[],[],[]]],[]]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [3,1,2,5,6,4,8,7] => ? => ? = 1
[[[[],[],[[],[],[]]]]]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,2,4,6,3,7,5,8] => ? => ? = 4
[[],[[],[[],[[],[[],[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,3,5,6,8,1,9,4,7,10] => ? => ? = 5
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: St000439
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> [1,0]
=> 2 = 1 + 1
[[],[]]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[[[]]]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[],[],[],[],[[[[]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[[],[],[],[[],[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[[],[],[],[[],[[],[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[[],[],[],[[],[[[]]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[[],[],[],[[[[[]]]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 5 + 1
[[],[],[[],[],[[],[]]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[[],[],[[],[],[[[]]]]]
=> [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 4 + 1
[[],[],[[],[[],[[]]]]]
=> [1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 4 + 1
[[],[],[[],[[[[]]]]]]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 5 + 1
[[],[],[[[],[],[[]]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 4 + 1
[[],[],[[[[],[],[]]]]]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 4 + 1
[[],[],[[[[[[]]]]]]]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 + 1
[[],[[],[],[],[[],[]]]]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[[],[[],[[],[],[],[]]]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
[[],[[],[[],[[],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 4 + 1
[[],[[],[[],[[[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 5 + 1
[[],[[],[[[[[]]]]]]]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 + 1
[[],[[[],[[[[]]]]]]]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 + 1
[[],[[[[],[[[]]]]]]]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 + 1
[[],[[[[[],[[]]]]]]]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 + 1
[[],[[[[[[],[]]]]]]]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 + 1
[[[[]],[],[],[[],[]]]]
=> [1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[[[[],[],[[],[],[]]]]]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4 + 1
[[],[[],[[],[[],[[],[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 5 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000011
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00048: Ordered trees —left-right symmetry⟶ Ordered trees
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [[]]
=> [.,.]
=> [1,0]
=> 1
[[],[]]
=> [[],[]]
=> [.,[.,.]]
=> [1,1,0,0]
=> 1
[[[]]]
=> [[[]]]
=> [[.,.],.]
=> [1,0,1,0]
=> 2
[[],[],[]]
=> [[],[],[]]
=> [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1
[[],[[]]]
=> [[[]],[]]
=> [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2
[[[]],[]]
=> [[],[[]]]
=> [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[[[],[]]]
=> [[[],[]]]
=> [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2
[[[[]]]]
=> [[[[]]]]
=> [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 3
[[],[],[],[]]
=> [[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1
[[],[],[[]]]
=> [[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[[],[[]],[]]
=> [[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[[],[[],[]]]
=> [[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[[],[[[]]]]
=> [[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 3
[[[]],[],[]]
=> [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[[[]],[[]]]
=> [[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[[[],[]],[]]
=> [[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[[[[]]],[]]
=> [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 1
[[[],[],[]]]
=> [[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 2
[[[],[[]]]]
=> [[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 3
[[[[]],[]]]
=> [[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 2
[[[[],[]]]]
=> [[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 3
[[[[[]]]]]
=> [[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 4
[[],[],[],[],[]]
=> [[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[[],[],[],[[]]]
=> [[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[[],[],[[]],[]]
=> [[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[[],[],[[],[]]]
=> [[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[],[],[[[]]]]
=> [[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[[],[[]],[],[]]
=> [[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[[],[[]],[[]]]
=> [[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[[],[[],[]],[]]
=> [[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[[],[[[]]],[]]
=> [[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[[],[[],[],[]]]
=> [[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[],[[],[[]]]]
=> [[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[[],[[[]],[]]]
=> [[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[[],[[[],[]]]]
=> [[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[[],[[[[]]]]]
=> [[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[[]],[],[],[]]
=> [[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[[[]],[],[[]]]
=> [[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[[]],[[]],[]]
=> [[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[[[]],[[],[]]]
=> [[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[[]],[[[]]]]
=> [[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[[[],[]],[],[]]
=> [[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[[[[]]],[],[]]
=> [[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[[[],[]],[[]]]
=> [[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[[[[]]],[[]]]
=> [[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[[[],[],[]],[]]
=> [[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[[[],[[]]],[]]
=> [[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[[[[]],[]],[]]
=> [[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[[[],[]]],[]]
=> [[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[[[[[]]]],[]]
=> [[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[],[],[],[[],[[[]]]]]
=> [[[[[]]],[]],[],[],[]]
=> [[[[.,.],.],[.,.]],[.,[.,[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[],[],[[],[],[[],[]]]]
=> [[[[],[]],[],[]],[],[]]
=> [[[.,[.,.]],[.,[.,.]]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 3
[[],[],[[],[],[[[]]]]]
=> [[[[[]]],[],[]],[],[]]
=> [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 4
[[],[],[[],[[],[]],[]]]
=> [[[],[[],[]],[]],[],[]]
=> [[.,[[.,[.,.]],[.,.]]],[.,[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 2
[[],[],[[],[[],[[]]]]]
=> [[[[[]],[]],[]],[],[]]
=> [[[[.,.],[.,.]],[.,.]],[.,[.,.]]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4
[[],[],[[],[[[[]]]]]]
=> [[[[[[]]]],[]],[],[]]
=> [[[[[.,.],.],.],[.,.]],[.,[.,.]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[[],[],[[[],[],[[]]]]]
=> [[[[[]],[],[]]],[],[]]
=> [[[[.,.],[.,[.,.]]],.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4
[[],[],[[[[],[],[]]]]]
=> [[[[[],[],[]]]],[],[]]
=> [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4
[[],[[]],[[[],[]],[]]]
=> [[[],[[],[]]],[[]],[]]
=> [[.,[[.,[.,.]],.]],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 2
[[],[[],[[],[]]],[[]]]
=> [[[]],[[[],[]],[]],[]]
=> [[.,.],[[[.,[.,.]],[.,.]],[.,.]]]
=> [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 2
[[[]],[[]],[[[[]]]]]
=> [[[[[]]]],[[]],[[]]]
=> [[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 4
[[[]],[[[]]],[[],[]]]
=> [[[],[]],[[[]]],[[]]]
=> [[.,[.,.]],[[[.,.],.],[[.,.],.]]]
=> [1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2
[[[]],[[[[]]]],[[]]]
=> [[[]],[[[[]]]],[[]]]
=> [[.,.],[[[[.,.],.],.],[[.,.],.]]]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2
[[[],[]],[[],[[]],[]]]
=> [[[],[[]],[]],[[],[]]]
=> [[.,[[.,.],[.,.]]],[[.,[.,.]],.]]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 2
[[[[[]]]],[[]],[[]]]
=> [[[]],[[]],[[[[]]]]]
=> [[.,.],[[.,.],[[[[.,.],.],.],.]]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 2
[[[[]],[[[]],[]]],[]]
=> [[],[[[],[[]]],[[]]]]
=> [.,[[[.,[[.,.],.]],[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 1
[[[[[[]]],[[]]]],[]]
=> [[],[[[[]],[[[]]]]]]
=> [.,[[[[.,.],[[[.,.],.],.]],.],.]]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 1
[[[],[],[[],[],[[]]]]]
=> [[[[[]],[],[]],[],[]]]
=> [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4
[[[],[],[[[],[],[]]]]]
=> [[[[[],[],[]]],[],[]]]
=> [[[[.,[.,[.,.]]],.],[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 4
[[[],[[],[[[[]]]]]]]
=> [[[[[[[]]]],[]],[]]]
=> [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 6
[[[],[[[],[[[]]]]]]]
=> [[[[[[[]]],[]]],[]]]
=> [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 6
[[[],[[[[],[[]]]]]]]
=> [[[[[[[]],[]]]],[]]]
=> [[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 6
[[[[]],[],[],[[],[]]]]
=> [[[[],[]],[],[],[[]]]]
=> [[[.,[.,.]],[.,[.,[[.,.],.]]]],.]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 3
[[[[]],[[[]],[[]]]]]
=> [[[[[]],[[]]],[[]]]]
=> [[[[.,.],[[.,.],.]],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 4
[[[[[[[]]]],[]],[]]]
=> [[[],[[],[[[[]]]]]]]
=> [[.,[[.,[[[[.,.],.],.],.]],.]],.]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[[[],[],[[],[],[]]]]]
=> [[[[[],[],[]],[],[]]]]
=> [[[[.,[.,[.,.]]],[.,[.,.]]],.],.]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 4
[[[[],[[],[[[]]]]]]]
=> [[[[[[[]]],[]],[]]]]
=> [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 6
[[[[],[[[],[[]]]]]]]
=> [[[[[[[]],[]]],[]]]]
=> [[[[[[.,.],[.,.]],.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 6
[[[[[],[[],[[]]]]]]]
=> [[[[[[[]],[]],[]]]]]
=> [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 6
[[[[[],[[[],[]]]]]]]
=> [[[[[[[],[]]],[]]]]]
=> [[[[[[.,[.,.]],.],[.,.]],.],.],.]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 6
[[[[[[[]],[]],[]]]]]
=> [[[[[],[[],[[]]]]]]]
=> [[[[.,[[.,[[.,.],.]],.]],.],.],.]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 4
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: St000678
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> [1,0]
=> ? = 1
[[],[]]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[[[]]]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[],[],[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[[],[],[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
[[],[],[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
[[],[],[],[],[],[[],[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
[[],[],[],[],[],[[[]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 3
[[],[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
[[],[],[],[],[[],[]],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
[[],[],[],[],[[],[],[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 2
[[],[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 3
[[],[],[],[],[[[[]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[[],[],[],[[]],[],[],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[[],[],[],[[],[],[]],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[[],[],[],[[],[],[],[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 2
[[],[],[],[[],[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 3
[[],[],[],[[],[[],[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 3
[[],[],[],[[],[[[]]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[[],[],[],[[[[[]]]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[[],[],[[]],[],[],[[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[[],[],[[[]]],[[],[]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[[],[],[[],[],[]],[],[]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[[],[],[[],[],[[],[]]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 3
[[],[],[[],[],[[[]]]]]
=> [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 4
[[],[],[[],[[],[]],[]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[[],[],[[],[[],[[]]]]]
=> [1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 4
[[],[],[[],[[[[]]]]]]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[[],[],[[[],[],[[]]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 4
[[],[],[[[[],[],[]]]]]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 4
[[],[],[[[[[[]]]]]]]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[[],[[],[]],[],[],[],[]]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[[],[[],[],[],[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[],[[],[[],[],[],[]]]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3
[[],[[],[[],[[],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 4
[[],[[],[[[[[]]]]]]]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[[],[[[],[[[[]]]]]]]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[[],[[[[],[[[]]]]]]]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[[],[[[[[],[[]]]]]]]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[[],[[[[[[],[]]]]]]]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[[],[[],[[],[[],[[],[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000383
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 88%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 88%
Values
[[]]
=> [1,0]
=> [1] => [1] => 1
[[],[]]
=> [1,0,1,0]
=> [2,1] => [1,1] => 1
[[[]]]
=> [1,1,0,0]
=> [1,2] => [2] => 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> [2,3,1] => [2,1] => 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [2,1,3] => [1,2] => 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,3,2] => [2,1] => 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [3,1,2] => [1,2] => 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,2,3] => [3] => 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [3,1] => 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,2] => 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [1,2,1] => 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,2] => 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [1,3] => 3
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [3,1] => 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [2,2] => 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [1,2,1] => 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [3,1] => 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,2] => 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [1,3] => 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,2] => 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,3] => 3
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4] => 4
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [4,1] => 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [3,2] => 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,2,1] => 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [3,2] => 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,3] => 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [1,3,1] => 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [1,2,2] => 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,2,1] => 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [1,3,1] => 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,2] => 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,3] => 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [1,2,2] => 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,3] => 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [1,4] => 4
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [4,1] => 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [3,2] => 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [2,2,1] => 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [3,2] => 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [2,3] => 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [1,3,1] => 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [4,1] => 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [1,2,2] => 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [3,2] => 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [2,2,1] => 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [1,3,1] => 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [2,2,1] => 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [1,3,1] => 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [4,1] => 1
[[],[],[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,7,1,8] => [6,2] => ? = 2
[[],[],[],[],[],[[],[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,6,8,1,7] => [6,2] => ? = 2
[[],[],[],[],[],[[[]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,5,6,1,7,8] => [5,3] => ? = 3
[[],[],[],[],[[],[],[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,5,7,8,1,6] => [6,2] => ? = 2
[[],[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,4,5,7,1,6,8] => [5,3] => ? = 3
[[],[],[],[[],[],[],[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,6,7,8,1,5] => [6,2] => ? = 2
[[],[],[],[[],[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,7,1,5,8] => [5,3] => ? = 3
[[],[],[],[[],[[],[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,4,6,8,1,5,7] => [5,3] => ? = 3
[[],[],[],[[[[[]]]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6,7,8] => [3,5] => ? = 5
[[],[],[[],[],[[],[]]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,6,8,1,4,7] => [5,3] => ? = 3
[[],[],[[],[[],[]],[]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,3,5,7,1,8,4,6] => [4,2,2] => ? = 2
[[],[],[[],[[[[]]]]]]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,5,1,4,6,7,8] => [3,5] => ? = 5
[[],[],[[[[[[]]]]]]]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7,8] => [2,6] => ? = 6
[[],[[],[]],[],[],[],[]]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,4,1,5,6,7,8,3] => [2,5,1] => ? = 1
[[],[[],[],[],[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,7,8,1,3] => [6,2] => ? = 2
[[],[[],[],[],[[],[]]]]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,4,5,6,8,1,3,7] => [5,3] => ? = 3
[[],[[],[[],[],[],[]]]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,4,6,7,8,1,3,5] => [5,3] => ? = 3
[[],[[],[[],[[[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,4,6,1,3,5,7,8] => [3,5] => ? = 5
[[],[[],[[[[[]]]]]]]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,4,1,3,5,6,7,8] => [2,6] => ? = 6
[[],[[[],[[[[]]]]]]]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,5,1,3,4,6,7,8] => [2,6] => ? = 6
[[],[[[[],[[[]]]]]]]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,6,1,3,4,5,7,8] => [2,6] => ? = 6
[[],[[[[[],[[]]]]]]]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,7,1,3,4,5,6,8] => [2,6] => ? = 6
[[],[[[[[[],[]]]]]]]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7] => [2,6] => ? = 6
[[[]],[],[],[],[],[[]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,6,7,2,8] => [6,2] => ? = 2
[[[]],[[]],[[[[]]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,5,4,6,7,8] => [2,2,4] => ? = 4
[[[]],[[[[[]],[]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,3,2,8,4,5,6,7] => [2,2,4] => ? = 4
[[[]],[[[[[[]]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,2,4,5,6,7,8] => [2,6] => ? = 6
[[[[]]],[],[],[],[[]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,4,5,6,7,3,8] => [6,2] => ? = 2
[[[[[]]]],[[]],[[]]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6,8] => [4,2,2] => ? = 2
[[[[[[[]]]]]],[[]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,2,3,4,5,7,6,8] => [6,2] => ? = 2
[[[],[[[[],[]]]]],[]]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [3,7,1,2,4,5,8,6] => [2,5,1] => ? = 1
[[[[],[],[[[]]]]],[]]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0]
=> [4,5,1,2,3,6,8,7] => [2,5,1] => ? = 1
[[[[],[[],[[]]]]],[]]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,1,2,3,5,8,7] => [2,5,1] => ? = 1
[[[[[]],[[[]]]]],[]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0,1,0]
=> [1,5,2,3,4,6,8,7] => [2,5,1] => ? = 1
[[[[[[],[],[]]]]],[]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [6,7,1,2,3,4,8,5] => [2,5,1] => ? = 1
[[[[[[[]],[]]]]],[]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,7,2,3,4,5,8,6] => [2,5,1] => ? = 1
[[[],[],[],[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,8,1,2] => [6,2] => ? = 2
[[[],[[],[[[[]]]]]]]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,5,1,2,4,6,7,8] => [2,6] => ? = 6
[[[],[[[],[[[]]]]]]]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [3,6,1,2,4,5,7,8] => [2,6] => ? = 6
[[[],[[[[],[[]]]]]]]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,7,1,2,4,5,6,8] => [2,6] => ? = 6
[[[],[[[[[],[]]]]]]]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,8,1,2,4,5,6,7] => [2,6] => ? = 6
[[[[]],[],[],[],[],[]]]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,4,5,6,7,8,2,3] => [6,2] => ? = 2
[[[[]],[],[],[[],[]]]]
=> [1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,4,5,6,8,2,3,7] => [5,3] => ? = 3
[[[[]],[[[]],[[]]]]]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,4,2,7,3,5,6,8] => [2,2,4] => ? = 4
[[[[[]]],[],[],[],[]]]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,2,5,6,7,8,3,4] => [6,2] => ? = 2
[[[[[[[]]]],[]],[]]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,2,3,7,4,8,5,6] => [4,2,2] => ? = 2
[[[[[[[[]]]]]],[]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,2,3,4,5,8,6,7] => [6,2] => ? = 2
[[[[],[[],[[[]]]]]]]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,6,1,2,3,5,7,8] => [2,6] => ? = 6
[[[[],[[[],[[]]]]]]]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> [4,7,1,2,3,5,6,8] => [2,6] => ? = 6
[[[[],[[[[],[]]]]]]]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> [4,8,1,2,3,5,6,7] => [2,6] => ? = 6
Description
The last part of an integer composition.
Matching statistic: St000759
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000759: Integer partitions ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000759: Integer partitions ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> []
=> 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> 1
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 3
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 4
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 4
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1
[[[[]],[]],[],[[]]]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [6,4,4,1]
=> ? = 2
[[[[],[]],[[]]],[]]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2]
=> ? = 1
[[[[[]]],[[]]],[]]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2]
=> ? = 1
[[[[],[],[[]]]],[]]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,2]
=> ? = 1
[[[[],[[[]]]]],[]]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2]
=> ? = 1
[[[[[]],[[]]]],[]]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> ? = 1
[[[[]],[[]],[],[]]]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,1,1,1]
=> ? = 2
[[],[[]],[[[],[]],[]]]
=> [1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
=> [6,3,3,1,1]
=> ? = 2
[[],[[],[[],[]]],[[]]]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,1]
=> ? = 2
[[],[[],[[],[[[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,2,1,1]
=> ? = 5
[[[]],[],[[[]],[],[]]]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0,1,0]
=> [7,4,1,1,1]
=> ? = 2
[[[]],[[]],[[[[]]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,1]
=> ? = 4
[[[]],[[[]]],[[],[]]]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0,1,0]
=> [7,5,4,1,1]
=> ? = 2
[[[]],[[[[[]],[]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,3,2,1]
=> ? = 4
[[[]],[[[[[[]]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,1]
=> ? = 6
[[[[]]],[],[],[],[],[]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [7,6]
=> ? = 1
[[[],[]],[],[],[],[[]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [6,6,1]
=> ? = 2
[[[[]]],[],[],[],[[]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1]
=> ? = 2
[[[],[]],[[],[[]],[]]]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [6,6,3,1,1,1]
=> ? = 2
[[[[]]],[[[[]]],[]]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,1,1]
=> ? = 2
[[[[[]]]],[],[],[],[]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 1
[[[],[],[]],[],[],[[]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> [5,5,5,1]
=> ? = 2
[[[[[]]]],[[]],[[]]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,1]
=> ? = 2
[[[[[]]]],[[[[]]]]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,1]
=> ? = 4
[[[[[[]]]]],[],[],[]]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4]
=> ? = 1
[[[[[[[]]]]]],[],[]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3]
=> ? = 1
[[[[[[[]]]]]],[[]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,1]
=> ? = 2
[[[],[],[[],[],[]]],[]]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,3,3,2,2,2]
=> ? = 1
[[[],[[[[],[]]]]],[]]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,5,4,3,2,2]
=> ? = 1
[[[[]],[[[]],[]]],[]]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,3,2,2]
=> ? = 1
[[[[],[],[[[]]]]],[]]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,3,3,2]
=> ? = 1
[[[[],[[],[[]]]]],[]]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,3,2]
=> ? = 1
[[[[[]],[[[]]]]],[]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,3,2]
=> ? = 1
[[[[[[]]],[[]]]],[]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,3,2]
=> ? = 1
[[[[[],[[[]]]]]],[]]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,3,2]
=> ? = 1
[[[[[[[]]],[]]]],[]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4,3,2]
=> ? = 1
[[[[[[],[],[]]]]],[]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,3,2]
=> ? = 1
[[[[[[[]],[]]]]],[]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,3,2]
=> ? = 1
[[[[[[[],[]]]]]],[]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2]
=> ? = 1
[[[],[],[[[],[],[]]]]]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,3,3,2,1,1,1]
=> ? = 4
[[[],[[],[[[[]]]]]]]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,2,1,1]
=> ? = 6
[[[],[[[],[[[]]]]]]]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,3,2,1,1]
=> ? = 6
[[[],[[[[],[[]]]]]]]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,2,1,1]
=> ? = 6
[[[],[[[[[],[]]]]]]]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [5,5,4,3,2,1,1]
=> ? = 6
[[[[]],[],[],[],[],[]]]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,1,1,1,1,1]
=> ? = 2
[[[[]],[],[],[[],[]]]]
=> [1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,2,1,1,1,1]
=> ? = 3
[[[[]],[[[]],[[]]]]]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,2,1,1]
=> ? = 4
[[[[],[]],[],[],[],[]]]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,6,1,1,1,1,1]
=> ? = 2
[[[[[]]],[],[],[],[]]]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,1,1,1,1]
=> ? = 2
[[[[],[],[]],[],[],[]]]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [5,5,5,1,1,1,1]
=> ? = 2
Description
The smallest missing part in an integer partition.
In [3], this is referred to as the mex, the minimal excluded part of the partition.
For compositions, this is studied in [sec.3.2., 1].
Matching statistic: St000069
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1] => ([],1)
=> 1
[[],[]]
=> [[.,.],.]
=> [1,2] => ([(0,1)],2)
=> 1
[[[]]]
=> [.,[.,.]]
=> [2,1] => ([],2)
=> 2
[[],[],[]]
=> [[[.,.],.],.]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => ([(1,2)],3)
=> 2
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => ([(1,2)],3)
=> 2
[[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => ([],3)
=> 3
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(2,3)],4)
=> 3
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(2,3)],4)
=> 3
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(2,3)],4)
=> 3
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([],4)
=> 4
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 2
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> 2
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 3
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> 1
[[],[[[]]],[]]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> 2
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 3
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> 2
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> 3
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => ([(3,4)],5)
=> 4
[[[]],[],[],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> 2
[[[]],[[]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> 1
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> 2
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> 3
[[[],[]],[],[]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> 1
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> 2
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> 2
[[[],[],[]],[]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[[[],[[]]],[]]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[[[[]],[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> 1
[[[[],[]]],[]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[[[[[]]]],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[],[],[],[],[[],[]]]
=> [[[[[.,.],.],.],.],[[.,.],.]]
=> [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 2
[[],[],[],[[],[],[]]]
=> [[[[.,.],.],.],[[[.,.],.],.]]
=> [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 2
[[],[],[[],[[]],[]]]
=> [[[.,.],.],[[[.,.],[.,.]],.]]
=> [6,4,5,7,1,2,3] => ([(0,6),(1,3),(2,4),(3,5),(4,6)],7)
=> ? = 2
[[],[],[[],[[],[]]]]
=> [[[.,.],.],[[.,.],[[.,.],.]]]
=> [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? = 3
[[],[[],[]],[[],[]]]
=> [[[.,.],[[.,.],.]],[[.,.],.]]
=> [6,7,3,4,1,2,5] => ([(0,5),(1,4),(2,3),(4,6),(5,6)],7)
=> ? = 2
[[],[[],[],[],[],[]]]
=> [[.,.],[[[[[.,.],.],.],.],.]]
=> [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 2
[[],[],[],[],[],[],[[]]]
=> [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [8,1,2,3,4,5,6,7] => ([(1,7),(3,4),(4,6),(5,3),(6,2),(7,5)],8)
=> ? = 2
[[],[],[],[],[],[[]],[]]
=> [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [7,1,2,3,4,5,6,8] => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 1
[[],[],[],[],[],[[],[]]]
=> [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> [7,8,1,2,3,4,5,6] => ([(0,7),(1,3),(4,6),(5,4),(6,2),(7,5)],8)
=> ? = 2
[[],[],[],[],[],[[[]]]]
=> [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [8,7,1,2,3,4,5,6] => ([(2,7),(4,6),(5,4),(6,3),(7,5)],8)
=> ? = 3
[[],[],[],[],[[]],[],[]]
=> [[[[[[[.,.],.],.],.],[.,.]],.],.]
=> [6,1,2,3,4,5,7,8] => ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ? = 1
[[],[],[],[],[[],[]],[]]
=> [[[[[[.,.],.],.],.],[[.,.],.]],.]
=> [6,7,1,2,3,4,5,8] => ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
=> ? = 1
[[],[],[],[],[[],[],[]]]
=> [[[[[.,.],.],.],.],[[[.,.],.],.]]
=> [6,7,8,1,2,3,4,5] => ([(0,7),(1,6),(4,5),(5,3),(6,4),(7,2)],8)
=> ? = 2
[[],[],[],[],[[],[[]]]]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> [8,6,7,1,2,3,4,5] => ([(1,7),(2,4),(5,6),(6,3),(7,5)],8)
=> ? = 3
[[],[],[],[],[[[[]]]]]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [8,7,6,1,2,3,4,5] => ([(3,4),(4,7),(6,5),(7,6)],8)
=> ? = 4
[[],[],[],[[]],[],[],[]]
=> [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [5,1,2,3,4,6,7,8] => ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ? = 1
[[],[],[],[[],[],[]],[]]
=> [[[[[.,.],.],.],[[[.,.],.],.]],.]
=> [5,6,7,1,2,3,4,8] => ([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
=> ? = 1
[[],[],[[]],[],[],[[]]]
=> [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [8,4,1,2,3,5,6,7] => ([(1,7),(2,5),(4,7),(5,4),(6,3),(7,6)],8)
=> ? = 2
[[],[],[[[]]],[[],[]]]
=> [[[[.,.],.],[.,[.,.]]],[[.,.],.]]
=> [7,8,5,4,1,2,3,6] => ([(0,7),(1,7),(2,4),(3,5),(5,6),(6,7)],8)
=> ? = 2
[[],[],[[],[],[]],[],[]]
=> [[[[[.,.],.],[[[.,.],.],.]],.],.]
=> [4,5,6,1,2,3,7,8] => ([(0,6),(1,5),(2,7),(3,7),(5,2),(6,3),(7,4)],8)
=> ? = 1
[[],[],[[],[],[[],[]]]]
=> [[[.,.],.],[[[.,.],.],[[.,.],.]]]
=> [7,8,4,5,6,1,2,3] => ([(0,5),(1,7),(2,6),(6,3),(7,4)],8)
=> ? = 3
[[],[],[[],[[],[]],[]]]
=> [[[.,.],.],[[[.,.],[[.,.],.]],.]]
=> [6,7,4,5,8,1,2,3] => ([(0,5),(1,4),(2,6),(4,7),(5,7),(6,3)],8)
=> ? = 2
[[],[],[[],[[],[[]]]]]
=> [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> [8,6,7,4,5,1,2,3] => ([(1,5),(2,4),(3,6),(6,7)],8)
=> ? = 4
[[],[],[[],[[[[]]]]]]
=> [[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> [8,7,6,4,5,1,2,3] => ([(3,5),(4,6),(6,7)],8)
=> ? = 5
[[],[],[[[[[[]]]]]]]
=> [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [8,7,6,5,4,1,2,3] => ([(5,6),(6,7)],8)
=> ? = 6
[[],[[]],[],[],[],[[]]]
=> [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [8,3,1,2,4,5,6,7] => ([(1,7),(2,4),(4,7),(5,3),(6,5),(7,6)],8)
=> ? = 2
[[],[[],[]],[],[],[],[]]
=> [[[[[[.,.],[[.,.],.]],.],.],.],.]
=> [3,4,1,2,5,6,7,8] => ([(0,4),(1,3),(3,7),(4,7),(5,2),(6,5),(7,6)],8)
=> ? = 1
[[],[[],[[],[]]],[],[]]
=> [[[[.,.],[[.,.],[[.,.],.]]],.],.]
=> [5,6,3,4,1,2,7,8] => ([(0,6),(1,5),(2,4),(4,7),(5,7),(6,7),(7,3)],8)
=> ? = 1
[[],[[],[[],[]]],[[]]]
=> [[[.,.],[[.,.],[[.,.],.]]],[.,.]]
=> [8,5,6,3,4,1,2,7] => ([(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[],[[[[[]]]]],[[]]]
=> [[[.,.],[.,[.,[.,[.,.]]]]],[.,.]]
=> [8,6,5,4,3,1,2,7] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 2
[[],[[],[],[],[],[],[]]]
=> [[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [3,4,5,6,7,8,1,2] => ([(0,7),(1,3),(4,6),(5,4),(6,2),(7,5)],8)
=> ? = 2
[[],[[],[[],[[[]]]]]]
=> [[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> [8,7,5,6,3,4,1,2] => ([(2,7),(3,6),(4,5)],8)
=> ? = 5
[[],[[[[[[[]]]]]]]]
=> [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [8,7,6,5,4,3,1,2] => ([(6,7)],8)
=> ? = 7
[[[]],[],[],[],[],[[]]]
=> [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [8,2,1,3,4,5,6,7] => ([(1,7),(2,7),(4,5),(5,3),(6,4),(7,6)],8)
=> ? = 2
[[[]],[[]],[[]],[[]]]
=> [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [8,6,4,2,1,3,5,7] => ([(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 2
[[[]],[[]],[[[[]]]]]
=> [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [8,7,6,4,2,1,3,5] => ([(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 4
[[[]],[[[]]],[[],[]]]
=> [[[.,[.,.]],[.,[.,.]]],[[.,.],.]]
=> [7,8,5,4,2,1,3,6] => ([(0,7),(1,7),(2,6),(3,6),(4,5),(6,7)],8)
=> ? = 2
[[[]],[[[[]]]],[[]]]
=> [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [8,6,5,4,2,1,3,7] => ([(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 2
[[[]],[[[[[[]]]]]]]
=> [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [8,7,6,5,4,2,1,3] => ([(5,7),(6,7)],8)
=> ? = 6
[[[],[]],[],[],[],[[]]]
=> [[[[[.,[[.,.],.]],.],.],.],[.,.]]
=> [8,2,3,1,4,5,6,7] => ([(1,7),(2,4),(4,7),(5,3),(6,5),(7,6)],8)
=> ? = 2
[[[[]]],[],[],[],[[]]]
=> [[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> [8,3,2,1,4,5,6,7] => ([(1,7),(2,7),(3,7),(4,6),(6,5),(7,4)],8)
=> ? = 2
[[[],[],[]],[],[],[[]]]
=> [[[[.,[[[.,.],.],.]],.],.],[.,.]]
=> [8,2,3,4,1,5,6,7] => ([(1,7),(2,5),(4,7),(5,4),(6,3),(7,6)],8)
=> ? = 2
[[[[[]]]],[[]],[[]]]
=> [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [8,6,4,3,2,1,5,7] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ? = 2
[[[[[]]]],[[[[]]]]]
=> [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [8,7,6,4,3,2,1,5] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
[[[],[],[],[]],[],[],[]]
=> [[[[.,[[[[.,.],.],.],.]],.],.],.]
=> [2,3,4,5,1,6,7,8] => ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ? = 1
[[[[[[]]]]],[],[],[]]
=> [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> [5,4,3,2,1,6,7,8] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(7,5)],8)
=> ? = 1
[[[],[],[],[],[]],[],[]]
=> [[[.,[[[[[.,.],.],.],.],.]],.],.]
=> [2,3,4,5,6,1,7,8] => ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ? = 1
[[[[[[[]]]]]],[],[]]
=> [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> [6,5,4,3,2,1,7,8] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(7,6)],8)
=> ? = 1
[[[[[[[]]]]]],[[]]]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [8,6,5,4,3,2,1,7] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 2
[[[],[],[],[],[],[]],[]]
=> [[.,[[[[[[.,.],.],.],.],.],.]],.]
=> [2,3,4,5,6,7,1,8] => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 1
Description
The number of maximal elements of a poset.
The following 61 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000617The number of global maxima of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000971The smallest closer of a set partition. St001050The number of terminal closers of a set partition. St000504The cardinality of the first block of a set partition. St001733The number of weak left to right maxima of a Dyck path. St000068The number of minimal elements in a poset. St000237The number of small exceedances. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St000053The number of valleys of the Dyck path. St000234The number of global ascents of a permutation. St000054The first entry of the permutation. St000717The number of ordinal summands of a poset. St000843The decomposition number of a perfect matching. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000031The number of cycles in the cycle decomposition of a permutation. St000989The number of final rises of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000297The number of leading ones in a binary word. St000738The first entry in the last row of a standard tableau. St000542The number of left-to-right-minima of a permutation. St000654The first descent of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001461The number of topologically connected components of the chord diagram of a permutation. St000740The last entry of a permutation. St000203The number of external nodes of a binary tree. St000990The first ascent of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000084The number of subtrees. St000314The number of left-to-right-maxima of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000056The decomposition (or block) number of a permutation. St000286The number of connected components of the complement of a graph. 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. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001481The minimal height of a peak of a Dyck path. St000133The "bounce" of a permutation. St000331The number of upper interactions of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001226The 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. 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. St000061The number of nodes on the left branch of a binary tree. St000993The multiplicity of the largest part of an integer partition. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. 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. St001904The length of the initial strictly increasing segment of a parking function.
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