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Your data matches 143 different statistics following compositions of up to 3 maps.
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Matching statistic: St000007
(load all 81 compositions to match this statistic)
(load all 81 compositions to match this statistic)
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%
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => 1
[.,[.,.]]
=> [2,1] => 2
[[.,.],.]
=> [1,2] => 1
[.,[.,[.,.]]]
=> [3,2,1] => 3
[.,[[.,.],.]]
=> [2,3,1] => 2
[[.,.],[.,.]]
=> [3,1,2] => 2
[[.,[.,.]],.]
=> [2,1,3] => 1
[[[.,.],.],.]
=> [1,2,3] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 4
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => 3
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 2
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => 3
[[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => 2
[[[.,.],.],[.,.]]
=> [4,1,2,3] => 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 5
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 4
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 4
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 3
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 4
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 3
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 3
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 3
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 2
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 4
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 3
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 3
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 3
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 3
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 2
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 2
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 2
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 2
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 2
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 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
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%
St000546: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => 0 = 1 - 1
[.,[.,.]]
=> [2,1] => 1 = 2 - 1
[[.,.],.]
=> [1,2] => 0 = 1 - 1
[.,[.,[.,.]]]
=> [3,2,1] => 2 = 3 - 1
[.,[[.,.],.]]
=> [2,3,1] => 1 = 2 - 1
[[.,.],[.,.]]
=> [3,1,2] => 1 = 2 - 1
[[.,[.,.]],.]
=> [2,1,3] => 0 = 1 - 1
[[[.,.],.],.]
=> [1,2,3] => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 3 = 4 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2 = 3 - 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => 2 = 3 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 1 = 2 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => 2 = 3 - 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => 1 = 2 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0 = 1 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 0 = 1 - 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => 0 = 1 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 0 = 1 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 4 = 5 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 3 = 4 - 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 3 = 4 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 2 = 3 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 2 = 3 - 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 3 = 4 - 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 2 = 3 - 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 2 = 3 - 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 2 = 3 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 1 = 2 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 1 = 2 - 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 3 = 4 - 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2 = 3 - 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 1 = 2 - 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1 = 2 - 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 1 = 2 - 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: St000297
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00109: Permutations —descent word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => => => ? = 1 - 1
[.,[.,.]]
=> [2,1] => 1 => 1 => 1 = 2 - 1
[[.,.],.]
=> [1,2] => 0 => 0 => 0 = 1 - 1
[.,[.,[.,.]]]
=> [3,2,1] => 11 => 11 => 2 = 3 - 1
[.,[[.,.],.]]
=> [2,3,1] => 01 => 10 => 1 = 2 - 1
[[.,.],[.,.]]
=> [1,3,2] => 01 => 10 => 1 = 2 - 1
[[.,[.,.]],.]
=> [2,1,3] => 10 => 01 => 0 = 1 - 1
[[[.,.],.],.]
=> [1,2,3] => 00 => 00 => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 111 => 111 => 3 = 4 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 011 => 110 => 2 = 3 - 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => 011 => 110 => 2 = 3 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 101 => 101 => 1 = 2 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 001 => 100 => 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => 011 => 110 => 2 = 3 - 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => 001 => 100 => 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => 101 => 101 => 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => 001 => 100 => 1 = 2 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 110 => 011 => 0 = 1 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 010 => 010 => 0 = 1 - 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => 010 => 010 => 0 = 1 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 100 => 001 => 0 = 1 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 1111 => 1111 => 4 = 5 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 0111 => 1110 => 3 = 4 - 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 0111 => 1110 => 3 = 4 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 1011 => 1101 => 2 = 3 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 0011 => 1100 => 2 = 3 - 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 0111 => 1110 => 3 = 4 - 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 0011 => 1100 => 2 = 3 - 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 1011 => 1101 => 2 = 3 - 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 0011 => 1100 => 2 = 3 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 1101 => 1011 => 1 = 2 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 0101 => 1010 => 1 = 2 - 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 0101 => 1010 => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 1001 => 1001 => 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 0001 => 1000 => 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 0111 => 1110 => 3 = 4 - 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 0011 => 1100 => 2 = 3 - 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 0011 => 1100 => 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 0101 => 1010 => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 0001 => 1000 => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 1011 => 1101 => 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 1001 => 1001 => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 0011 => 1100 => 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 0001 => 1000 => 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 1101 => 1011 => 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 0101 => 1010 => 1 = 2 - 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 0101 => 1010 => 1 = 2 - 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 1001 => 1001 => 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 0001 => 1000 => 1 = 2 - 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1110 => 0111 => 0 = 1 - 1
[[[.,.],.],[[[.,.],[[.,.],.]],.]]
=> [1,2,4,6,7,5,8,3] => ? => ? => ? = 2 - 1
[[[.,[.,.]],.],[[[.,[.,.]],.],.]]
=> [2,1,3,6,5,7,8,4] => ? => ? => ? = 2 - 1
[[[.,[.,.]],[.,[.,.]]],[[.,.],.]]
=> [2,1,5,4,3,7,8,6] => ? => ? => ? = 2 - 1
[[[.,.],[[.,.],[[.,.],.]]],[.,.]]
=> [1,3,5,6,4,2,8,7] => ? => ? => ? = 2 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000382
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1] => 1
[.,[.,.]]
=> [2,1] => [1,2] => [2] => 2
[[.,.],.]
=> [1,2] => [2,1] => [1,1] => 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [3] => 3
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [2,1] => 2
[[.,.],[.,.]]
=> [1,3,2] => [2,3,1] => [2,1] => 2
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => [1,2] => 1
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => [1,1,1] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [4] => 4
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => [3,1] => 3
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,3,4,2] => [3,1] => 3
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => [2,2] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => [2,1,1] => 2
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,3,4,1] => [3,1] => 3
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,3,1] => [2,1,1] => 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,4,1,2] => [2,2] => 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,4,2,1] => [2,1,1] => 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => [1,3] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => [1,2,1] => 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [4,2,3,1] => [1,2,1] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => [1,1,2] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [5] => 5
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => [4,1] => 4
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,2,4,5,3] => [4,1] => 4
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => [3,2] => 3
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [3,1,1] => 3
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,3,4,5,2] => [4,1] => 4
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,3,5,4,2] => [3,1,1] => 3
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,4,5,2,3] => [3,2] => 3
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,4,5,3,2] => [3,1,1] => 3
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => [2,3] => 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => [2,2,1] => 2
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,5,3,4,2] => [2,2,1] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => [2,1,2] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => [2,1,1,1] => 2
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [2,3,4,5,1] => [4,1] => 4
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [2,3,5,4,1] => [3,1,1] => 3
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [2,4,5,3,1] => [3,1,1] => 3
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [2,5,3,4,1] => [2,2,1] => 2
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [2,5,4,3,1] => [2,1,1,1] => 2
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [3,4,5,1,2] => [3,2] => 3
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [3,5,4,1,2] => [2,1,2] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [3,4,5,2,1] => [3,1,1] => 3
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => 2
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [4,5,1,2,3] => [2,3] => 2
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [4,5,1,3,2] => [2,2,1] => 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [4,5,2,3,1] => [2,2,1] => 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [4,5,3,1,2] => [2,1,2] => 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => 2
[[[.,.],[.,.]],[[.,[[.,.],.]],.]]
=> [1,3,2,6,7,5,8,4] => [4,8,5,7,6,2,3,1] => ? => ? = 2
[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]]
=> [1,3,5,7,9,10,8,6,4,2] => [2,4,6,8,10,9,7,5,3,1] => ? => ? = 5
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
=> [1,2,3,4,5,6,7,8,9,11,10] => [10,11,9,8,7,6,5,4,3,2,1] => [2,1,1,1,1,1,1,1,1,1] => ? = 2
[[[[[[[.,.],[.,.]],.],.],.],.],[.,.]]
=> [1,3,2,4,5,6,7,9,8] => [8,9,7,6,5,4,2,3,1] => ? => ? = 2
[.,[[[[[[[.,.],[.,.]],.],.],.],.],.]]
=> [2,4,3,5,6,7,8,9,1] => [1,9,8,7,6,5,3,4,2] => ? => ? = 2
[.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,9,10,11,1] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,1,1,1,1,1,1,1,1] => ? = 2
[[[[[[[[.,.],.],.],.],.],[.,.]],.],.]
=> [1,2,3,4,5,7,6,8,9] => [9,8,6,7,5,4,3,2,1] => ? => ? = 1
Description
The first part of an integer composition.
Matching statistic: St000288
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => => ? = 1 - 1
[.,[.,.]]
=> [2,1] => [1,2] => 1 => 1 = 2 - 1
[[.,.],.]
=> [1,2] => [2,1] => 0 => 0 = 1 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => 11 => 2 = 3 - 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1,3] => 01 => 1 = 2 - 1
[[.,.],[.,.]]
=> [3,1,2] => [1,3,2] => 10 => 1 = 2 - 1
[[.,[.,.]],.]
=> [2,1,3] => [2,3,1] => 00 => 0 = 1 - 1
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => 00 => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => 111 => 3 = 4 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,1,3,4] => 011 => 2 = 3 - 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => 101 => 2 = 3 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,3,1,4] => 001 => 1 = 2 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,2,1,4] => 001 => 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,2,4,3] => 110 => 2 = 3 - 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => 010 => 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,3,4,2] => 100 => 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,4,3,2] => 100 => 1 = 2 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,3,4,1] => 000 => 0 = 1 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,4,1] => 000 => 0 = 1 - 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,4,3,1] => 000 => 0 = 1 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [3,4,2,1] => 000 => 0 = 1 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => 000 => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 5 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [2,1,3,4,5] => 0111 => 3 = 4 - 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,3,2,4,5] => 1011 => 3 = 4 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,3,1,4,5] => 0011 => 2 = 3 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,2,1,4,5] => 0011 => 2 = 3 - 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,2,4,3,5] => 1101 => 3 = 4 - 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [2,1,4,3,5] => 0101 => 2 = 3 - 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,3,4,2,5] => 1001 => 2 = 3 - 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => 1001 => 2 = 3 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [2,3,4,1,5] => 0001 => 1 = 2 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,2,4,1,5] => 0001 => 1 = 2 - 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [2,4,3,1,5] => 0001 => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,4,2,1,5] => 0001 => 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [4,3,2,1,5] => 0001 => 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,2,3,5,4] => 1110 => 3 = 4 - 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,3,5,4] => 0110 => 2 = 3 - 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,3,2,5,4] => 1010 => 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [2,3,1,5,4] => 0010 => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,2,1,5,4] => 0010 => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,2,4,5,3] => 1100 => 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [2,1,4,5,3] => 0100 => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,5,4,3] => 1100 => 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [2,1,5,4,3] => 0100 => 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,3,4,5,2] => 1000 => 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,4,3,5,2] => 1000 => 1 = 2 - 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,3,5,4,2] => 1000 => 1 = 2 - 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,4,5,3,2] => 1000 => 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,5,4,3,2] => 1000 => 1 = 2 - 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [2,3,4,5,1] => 0000 => 0 = 1 - 1
[[[[[[[.,[.,.]],.],.],.],.],.],[.,.]]
=> [9,2,1,3,4,5,6,7,8] => [1,8,9,7,6,5,4,3,2] => ? => ? = 2 - 1
[[[[[[[[.,[.,.]],.],.],.],.],.],.],[.,.]]
=> [10,2,1,3,4,5,6,7,8,9] => [1,9,10,8,7,6,5,4,3,2] => ? => ? = 2 - 1
[[[[[[[.,.],[.,.]],.],.],.],.],[.,.]]
=> [9,3,1,2,4,5,6,7,8] => [1,7,9,8,6,5,4,3,2] => ? => ? = 2 - 1
[[[[[[.,[[.,.],.]],.],.],.],.],[.,.]]
=> [9,2,3,1,4,5,6,7,8] => [1,8,7,9,6,5,4,3,2] => ? => ? = 2 - 1
[.,[[[[[[[.,.],[.,.]],.],.],.],.],.]]
=> [4,2,3,5,6,7,8,9,1] => [6,8,7,5,4,3,2,1,9] => ? => ? = 2 - 1
[[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]
=> [10,9,8,7,6,5,4,3,2,1,11] => [2,3,4,5,6,7,8,9,10,11,1] => 0000000000 => ? = 1 - 1
[[[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.],.]
=> [2,1,3,4,5,6,7,8,9,10,11] => [10,11,9,8,7,6,5,4,3,2,1] => 0000000000 => ? = 1 - 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000326
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ 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%
Mp00064: Permutations —reverse⟶ 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] => [1] => => ? = 1
[.,[.,.]]
=> [2,1] => [1,2] => 0 => 2
[[.,.],.]
=> [1,2] => [2,1] => 1 => 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => 00 => 3
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => 01 => 2
[[.,.],[.,.]]
=> [1,3,2] => [2,3,1] => 01 => 2
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => 10 => 1
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => 11 => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => 000 => 4
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => 001 => 3
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,3,4,2] => 001 => 3
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => 010 => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => 011 => 2
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,3,4,1] => 001 => 3
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,3,1] => 011 => 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,4,1,2] => 010 => 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,4,2,1] => 011 => 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => 100 => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => 101 => 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [4,2,3,1] => 101 => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => 110 => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => 111 => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0000 => 5
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => 0001 => 4
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,2,4,5,3] => 0001 => 4
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => 0010 => 3
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => 0011 => 3
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,3,4,5,2] => 0001 => 4
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,3,5,4,2] => 0011 => 3
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,4,5,2,3] => 0010 => 3
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,4,5,3,2] => 0011 => 3
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => 0100 => 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => 0101 => 2
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,5,3,4,2] => 0101 => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => 0110 => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => 0111 => 2
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [2,3,4,5,1] => 0001 => 4
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [2,3,5,4,1] => 0011 => 3
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [2,4,5,3,1] => 0011 => 3
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [2,5,3,4,1] => 0101 => 2
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [2,5,4,3,1] => 0111 => 2
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [3,4,5,1,2] => 0010 => 3
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [3,5,4,1,2] => 0110 => 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [3,4,5,2,1] => 0011 => 3
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [3,5,4,2,1] => 0111 => 2
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [4,5,1,2,3] => 0100 => 2
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [4,5,1,3,2] => 0101 => 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [4,5,2,3,1] => 0101 => 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [4,5,3,1,2] => 0110 => 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [4,5,3,2,1] => 0111 => 2
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [5,1,2,3,4] => 1000 => 1
[.,[[[[.,[.,.]],.],.],[[.,.],.]]]
=> [3,2,4,5,7,8,6,1] => [1,6,8,7,5,4,2,3] => ? => ? = 3
[[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> [1,2,5,6,8,7,4,3] => [3,4,7,8,6,5,2,1] => ? => ? = 4
[[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> [1,2,4,5,8,7,6,3] => [3,6,7,8,5,4,2,1] => ? => ? = 4
[[[.,.],[.,.]],[[.,[[.,.],.]],.]]
=> [1,3,2,6,7,5,8,4] => [4,8,5,7,6,2,3,1] => ? => ? = 2
[[[.,.],[[.,.],[[.,.],.]]],[.,.]]
=> [1,3,5,6,4,2,8,7] => [7,8,2,4,6,5,3,1] => ? => ? = 2
[[.,[.,[[.,[.,[.,.]]],[.,.]]]],.]
=> [5,4,3,7,6,2,1,8] => [8,1,2,6,7,3,4,5] => ? => ? = 1
[[.,[[.,[.,.]],[[.,[.,.]],.]]],.]
=> [3,2,6,5,7,4,1,8] => [8,1,4,7,5,6,2,3] => ? => ? = 1
[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]]
=> [1,3,5,7,9,10,8,6,4,2] => [2,4,6,8,10,9,7,5,3,1] => ? => ? = 5
[.,[[[[[[[.,.],[.,.]],.],.],.],.],.]]
=> [2,4,3,5,6,7,8,9,1] => [1,9,8,7,6,5,3,4,2] => ? => ? = 2
[[[[[[[[.,.],.],.],.],.],[.,.]],.],.]
=> [1,2,3,4,5,7,6,8,9] => [9,8,6,7,5,4,3,2,1] => ? => ? = 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: St000010
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1] => [1]
=> 1
[.,[.,.]]
=> [1,0,1,0]
=> [1,1] => [1,1]
=> 2
[[.,.],.]
=> [1,1,0,0]
=> [2] => [2]
=> 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> 3
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,2] => [2,1]
=> 2
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1] => [2,1]
=> 2
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [3] => [3]
=> 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3] => [3]
=> 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> 4
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> 3
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> 3
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => [3,1]
=> 2
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 2
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> 3
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => [3,1]
=> 2
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> 2
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [4] => [4]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [4] => [4]
=> 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [4] => [4]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [4] => [4]
=> 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 5
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> 4
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> 4
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [3,1,1]
=> 3
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> 4
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 3
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 3
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1]
=> 2
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [4,1]
=> 2
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1]
=> 2
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 4
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> 3
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> 3
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [3,2]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> 3
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> 3
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> 2
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> 2
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> 2
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [4,1]
=> 2
[[[[[[[.,[.,.]],.],.],.],.],.],[.,.]]
=> ?
=> ? => ?
=> ? = 2
[.,[[[[[[[.,[.,.]],.],.],.],.],.],.]]
=> ?
=> ? => ?
=> ? = 2
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
=> ?
=> ? => ?
=> ? = 2
[[[[[[[[.,[.,.]],.],.],.],.],.],.],[.,.]]
=> ?
=> ? => ?
=> ? = 2
[.,[[[[[[.,[[.,.],.]],.],.],.],.],.]]
=> ?
=> ? => ?
=> ? = 2
[.,[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]]
=> ?
=> ? => ?
=> ? = 2
[.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]]
=> ?
=> ? => ?
=> ? = 2
[[[[[[[.,.],.],.],.],.],.],[[.,.],.]]
=> ?
=> ? => ?
=> ? = 2
[[[[[[.,.],.],.],.],.],[[[.,.],.],.]]
=> ?
=> ? => ?
=> ? = 2
[[[[[.,.],.],.],.],[[[[.,.],.],.],.]]
=> ?
=> ? => ?
=> ? = 2
[[[[[[[[.,.],.],.],.],.],.],.],[[.,.],.]]
=> ?
=> ? => ?
=> ? = 2
[[[[[[[.,.],.],.],.],.],.],[[[.,.],.],.]]
=> ?
=> ? => ?
=> ? = 2
[[[[[[.,.],.],.],.],.],[[[[.,.],.],.],.]]
=> ?
=> ? => ?
=> ? = 2
[[[[[.,.],.],.],.],[[[[[.,.],.],.],.],.]]
=> ?
=> ? => ?
=> ? = 2
[[[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.],.]
=> ?
=> ? => ?
=> ? = 1
[[[.,[[[[[[.,.],.],.],.],.],.]],.],.]
=> ?
=> ? => ?
=> ? = 1
[[.,[[[[[[[[.,.],.],.],.],.],.],.],.]],.]
=> ?
=> ? => ?
=> ? = 1
Description
The length of the partition.
Matching statistic: St000745
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [.,.]
=> [1] => [[1]]
=> 1
[.,[.,.]]
=> [.,[.,.]]
=> [2,1] => [[1],[2]]
=> 2
[[.,.],.]
=> [[.,.],.]
=> [1,2] => [[1,2]]
=> 1
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [[1],[2],[3]]
=> 3
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> [2,3,1] => [[1,3],[2]]
=> 2
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> [2,1,3] => [[1,3],[2]]
=> 2
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> [1,3,2] => [[1,2],[3]]
=> 1
[[[.,.],.],.]
=> [[[.,.],.],.]
=> [1,2,3] => [[1,2,3]]
=> 1
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 4
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[1,4],[2],[3]]
=> 3
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[1,4],[2],[3]]
=> 3
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [[1,3],[2],[4]]
=> 2
[.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [[1,3,4],[2]]
=> 2
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> 3
[[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [[1,3,4],[2]]
=> 2
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2
[[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 2
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [[1,2],[3],[4]]
=> 1
[[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => [[1,2,4],[3]]
=> 1
[[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 1
[[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 1
[[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 5
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[1,5],[2],[3],[4]]
=> 4
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[1,5],[2],[3],[4]]
=> 4
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [[1,4],[2],[3],[5]]
=> 3
[.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[1,4,5],[2],[3]]
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[1,5],[2],[3],[4]]
=> 4
[.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [[1,4,5],[2],[3]]
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [[1,4],[2,5],[3]]
=> 3
[.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [[1,4,5],[2],[3]]
=> 3
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [[1,3],[2],[4],[5]]
=> 2
[.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [[1,3,5],[2],[4]]
=> 2
[.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [[1,3,5],[2],[4]]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [[1,3,4],[2],[5]]
=> 2
[.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [[1,3,4,5],[2]]
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 4
[[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [[1,4,5],[2],[3]]
=> 3
[[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [[1,4,5],[2],[3]]
=> 3
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [[1,3,5],[2],[4]]
=> 2
[[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [[1,3,4,5],[2]]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 3
[[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [[1,3,4],[2,5]]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> 3
[[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [[1,3,4,5],[2]]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 2
[[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [[1,3,5],[2,4]]
=> 2
[[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 2
[[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 2
[.,[.,[[[.,.],.],[[[.,.],.],.]]]]
=> [.,[.,[[[.,[[[.,.],.],.]],.],.]]]
=> [4,5,6,3,7,8,2,1] => ?
=> ? = 4
[[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> [[[.,[.,[[[.,[.,.]],.],.]]],.],.]
=> [4,3,5,6,2,1,7,8] => ?
=> ? = 4
[[[.,.],.],[[[.,.],[[.,.],.]],.]]
=> [[[.,[[[.,.],[[.,.],.]],.]],.],.]
=> [2,4,5,3,6,1,7,8] => ?
=> ? = 2
[[[.,.],[[.,.],[[.,.],.]]],[.,.]]
=> [[[.,[.,.]],[[.,[[.,.],.]],.]],.]
=> [2,1,5,6,4,7,3,8] => ?
=> ? = 2
[[.,[.,[[.,.],[[.,.],[.,.]]]]],.]
=> [[.,.],[.,[[.,[[.,[.,.]],.]],.]]]
=> [1,6,5,7,4,8,3,2] => ?
=> ? = 1
[[.,[.,[[.,[.,.]],[.,[.,.]]]]],.]
=> [[.,.],[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,6,5,4,8,7,3,2] => ?
=> ? = 1
[[.,[.,[[.,[.,[.,.]]],[.,.]]]],.]
=> [[.,.],[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,5,4,8,7,6,3,2] => ?
=> ? = 1
[[.,[[[.,.],.],[[[.,.],.],.]]],.]
=> [[.,.],[[[.,[[[.,.],.],.]],.],.]]
=> [1,4,5,6,3,7,8,2] => ?
=> ? = 1
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
=> [[[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.],.]
=> [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 2
[[[[[[[[.,[.,.]],.],.],.],.],.],.],[.,.]]
=> [[[[[[[[.,[.,.]],.],.],.],.],.],.],[.,.]]
=> [2,1,3,4,5,6,7,8,10,9] => [[1,3,4,5,6,7,8,9],[2,10]]
=> ? = 2
[.,[[[[[[[.,.],[.,.]],.],.],.],.],.]]
=> [.,[[[[[[[.,.],.],.],.],.],[.,.]],.]]
=> ? => ?
=> ? = 2
[.,[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]]
=> [.,[[[[[[[[.,.],.],.],.],.],.],.],[.,.]]]
=> [2,3,4,5,6,7,8,10,9,1] => [[1,3,4,5,6,7,8,9],[2],[10]]
=> ? = 2
[.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]]
=> [.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,9,10,11,1] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 2
[[[[[[.,.],.],.],.],.],[[[.,.],.],.]]
=> [[[[[[.,[[[.,.],.],.]],.],.],.],.],.]
=> ? => ?
=> ? = 2
[[[[[.,.],.],.],.],[[[[.,.],.],.],.]]
=> [[[[[.,[[[[.,.],.],.],.]],.],.],.],.]
=> ? => ?
=> ? = 2
[[[[[[[.,.],.],.],.],.],.],[[[.,.],.],.]]
=> [[[[[[[.,[[[.,.],.],.]],.],.],.],.],.],.]
=> ? => ?
=> ? = 2
[[[[[[.,.],.],.],.],.],[[[[.,.],.],.],.]]
=> [[[[[[.,[[[[.,.],.],.],.]],.],.],.],.],.]
=> ? => ?
=> ? = 2
[[[[[.,.],.],.],.],[[[[[.,.],.],.],.],.]]
=> [[[[[.,[[[[[.,.],.],.],.],.]],.],.],.],.]
=> ? => ?
=> ? = 2
[[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]
=> [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [1,11,10,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 1
[[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]],.]
=> [[.,.],[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ? => ?
=> ? = 1
[[[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.],.]
=> [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
=> [1,2,3,4,5,6,7,8,9,11,10] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 1
[[[[[[[.,.],[[.,.],.]],.],.],.],.],.]
=> [[[[[[[.,.],.],.],.],.],[[.,.],.]],.]
=> ? => ?
=> ? = 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000011
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1,0]
=> 1
[.,[.,.]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 2
[[.,.],.]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[[.,.],[.,.]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 4
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 4
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 4
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 3
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [8,6,7,4,5,3,2,1] => [1,2,3,5,4,7,6,8] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 6
[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [8,6,7,5,3,4,2,1] => [1,2,4,3,5,7,6,8] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 6
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 6
[.,[.,[[[.,.],.],[[[.,.],.],.]]]]
=> [6,7,8,3,4,5,2,1] => [1,2,5,4,3,8,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 4
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [8,6,7,5,4,2,3,1] => [1,3,2,4,5,7,6,8] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 6
[.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> [8,7,5,6,4,2,3,1] => [1,3,2,4,6,5,7,8] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 6
[.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> [8,7,6,4,5,2,3,1] => [1,3,2,5,4,6,7,8] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 6
[.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [8,6,5,7,3,2,4,1] => [1,4,2,3,7,5,6,8] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4
[.,[[[.,.],.],[.,[[[.,.],.],.]]]]
=> [6,7,8,5,2,3,4,1] => [1,4,3,2,5,8,7,6] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4
[.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [8,5,6,7,2,3,4,1] => [1,4,3,2,7,6,5,8] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4
[[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
=> [8,7,6,4,5,3,1,2] => [2,1,3,5,4,6,7,8] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 6
[[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> [8,7,5,6,3,4,1,2] => [2,1,4,3,6,5,7,8] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 5
[[.,[.,.]],[.,[.,[[.,[.,.]],.]]]]
=> [7,6,8,5,4,2,1,3] => [3,1,2,4,5,8,6,7] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4
[[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> [6,7,8,5,4,1,2,3] => [3,2,1,4,5,8,7,6] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4
[[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> [8,5,6,7,4,1,2,3] => [3,2,1,4,7,6,5,8] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 4
[[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> [8,7,6,4,5,1,2,3] => [3,2,1,5,4,6,7,8] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 5
[[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> [8,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,8] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> [8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 4
[[[.,.],.],[[[.,.],.],[[.,.],.]]]
=> [7,8,4,5,6,1,2,3] => [3,2,1,6,5,4,8,7] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> [8,7,5,6,1,2,3,4] => [4,3,2,1,6,5,7,8] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 4
[[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [8,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,8] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 3
[[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> [8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[[[[[[.,[[.,.],.]],.],.],.],.],[.,.]]
=> [9,2,3,1,4,5,6,7,8] => [8,7,6,5,4,1,3,2,9] => ?
=> ? = 2
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: St000733
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [[1]]
=> 1
[.,[.,.]]
=> [2,1] => [2,1] => [[1],[2]]
=> 2
[[.,.],.]
=> [1,2] => [1,2] => [[1,2]]
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 3
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [[1,2],[3]]
=> 2
[[.,.],[.,.]]
=> [1,3,2] => [3,1,2] => [[1,2],[3]]
=> 2
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [[1,2,3]]
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 4
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,4,3,2] => [[1,2],[3],[4]]
=> 3
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,1,3,2] => [[1,2],[3],[4]]
=> 3
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 2
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [4,3,1,2] => [[1,2],[3],[4]]
=> 3
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,1,3] => [[1,3],[2,4]]
=> 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [1,4,2,3] => [[1,2,3],[4]]
=> 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [4,1,2,3] => [[1,2,3],[4]]
=> 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [3,1,2,4] => [[1,2,4],[3]]
=> 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 5
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 4
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,1,4,3,2] => [[1,2],[3],[4],[5]]
=> 4
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 3
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,4,1,3,2] => [[1,2],[3],[4],[5]]
=> 4
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,5,1,4,3] => [[1,3],[2,4],[5]]
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,5,2,4,3] => [[1,2,3],[4],[5]]
=> 3
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,1,2,4,3] => [[1,2,3],[4],[5]]
=> 3
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 2
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [3,1,2,5,4] => [[1,2,4],[3,5]]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [5,4,3,1,2] => [[1,2],[3],[4],[5]]
=> 4
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [2,5,4,1,3] => [[1,3],[2,4],[5]]
=> 3
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [5,2,4,1,3] => [[1,3],[2,4],[5]]
=> 3
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [3,2,5,1,4] => [[1,4],[2,5],[3]]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [2,3,5,1,4] => [[1,3,4],[2,5]]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [5,1,4,2,3] => [[1,2,3],[4],[5]]
=> 3
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [1,3,5,2,4] => [[1,2,4],[3,5]]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [5,4,1,2,3] => [[1,2,3],[4],[5]]
=> 3
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [3,5,1,2,4] => [[1,2,4],[3,5]]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [2,1,5,3,4] => [[1,3,4],[2,5]]
=> 2
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [1,2,5,3,4] => [[1,2,3,4],[5]]
=> 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [2,5,1,3,4] => [[1,3,4],[2,5]]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [1,5,2,3,4] => [[1,2,3,4],[5]]
=> 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [5,1,2,3,4] => [[1,2,3,4],[5]]
=> 2
[.,[.,[[[.,.],.],[[[.,.],.],.]]]]
=> [3,4,6,7,8,5,2,1] => [3,4,8,1,2,7,6,5] => ?
=> ? = 4
[.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [3,2,6,5,8,7,4,1] => [3,8,1,4,7,2,6,5] => ?
=> ? = 4
[[[.,.],.],[[[.,.],.],[[.,.],.]]]
=> [1,2,4,5,7,8,6,3] => [5,8,3,4,7,1,2,6] => ?
=> ? = 3
[[[.,.],.],[[[.,.],[[.,.],.]],.]]
=> [1,2,4,6,7,5,8,3] => ? => ?
=> ? = 2
[[.,[[.,.],.]],[[[.,.],[.,.]],.]]
=> [2,3,1,5,7,6,8,4] => [6,1,2,4,5,8,3,7] => ?
=> ? = 2
[[[.,.],[.,.]],[[.,[[.,.],.]],.]]
=> [1,3,2,6,7,5,8,4] => [4,6,2,5,8,1,3,7] => ?
=> ? = 2
[[[.,[.,.]],.],[[[.,[.,.]],.],.]]
=> [2,1,3,6,5,7,8,4] => ? => ?
=> ? = 2
[[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> [1,2,3,5,7,8,6,4] => [5,8,4,7,1,2,3,6] => ?
=> ? = 3
[[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,2,3,5,6,8,7,4] => [8,4,5,7,1,2,3,6] => ?
=> ? = 3
[[[.,[.,.]],[.,[.,.]]],[[.,.],.]]
=> [2,1,5,4,3,7,8,6] => ? => ?
=> ? = 2
[[[[.,.],.],[.,[.,.]]],[[.,.],.]]
=> [1,2,5,4,3,7,8,6] => [4,3,6,8,1,2,5,7] => ?
=> ? = 2
[[[.,.],[[.,.],[[.,.],.]]],[.,.]]
=> [1,3,5,6,4,2,8,7] => ? => ?
=> ? = 2
[[.,[.,[.,[[.,.],[.,[.,.]]]]]],.]
=> [4,7,6,5,3,2,1,8] => [7,6,1,5,4,3,2,8] => ?
=> ? = 1
[[.,[.,[[.,[.,[.,.]]],[.,.]]]],.]
=> [5,4,3,7,6,2,1,8] => [2,1,7,3,6,5,4,8] => ?
=> ? = 1
[[.,[[.,[.,.]],[[.,[.,.]],.]]],.]
=> [3,2,6,5,7,4,1,8] => [4,1,3,7,2,6,5,8] => ?
=> ? = 1
[[.,[[[.,.],.],[[[.,.],.],.]]],.]
=> [2,3,5,6,7,4,1,8] => [3,4,7,1,2,6,5,8] => ?
=> ? = 1
[[[[[.,.],.],.],[[[.,.],.],.]],.]
=> [1,2,3,5,6,7,4,8] => [4,5,7,1,2,3,6,8] => ?
=> ? = 1
[[[[.,.],[[.,.],[[.,.],.]]],.],.]
=> [1,3,5,6,4,2,7,8] => [3,6,2,5,1,4,7,8] => ?
=> ? = 1
[[[[[.,.],.],[[[.,.],.],.]],.],.]
=> [1,2,4,5,6,3,7,8] => [3,4,6,1,2,5,7,8] => ?
=> ? = 1
[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]]
=> [1,3,5,7,9,10,8,6,4,2] => [5,10,4,9,3,8,2,7,1,6] => ?
=> ? = 5
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
=> [1,2,3,4,5,6,7,8,9,11,10] => [11,1,2,3,4,5,6,7,8,9,10] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 2
[.,[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,9,10,1] => [2,1,3,4,5,6,7,8,10,9] => [[1,3,4,5,6,7,8,9],[2,10]]
=> ? = 2
[.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,9,10,11,1] => [1,2,3,4,5,6,7,8,9,11,10] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 2
[[[[[[.,.],.],.],.],.],[[[.,.],.],.]]
=> [1,2,3,4,5,7,8,9,6] => [6,7,9,1,2,3,4,5,8] => [[1,2,3,4,5,8],[6,7,9]]
=> ? = 2
[[[[[.,.],.],.],.],[[[[.,.],.],.],.]]
=> [1,2,3,4,6,7,8,9,5] => [5,6,7,9,1,2,3,4,8] => [[1,2,3,4,8],[5,6,7,9]]
=> ? = 2
[[[[[[[.,.],.],.],.],.],.],[[[.,.],.],.]]
=> [1,2,3,4,5,6,8,9,10,7] => [7,8,10,1,2,3,4,5,6,9] => [[1,2,3,4,5,6,9],[7,8,10]]
=> ? = 2
[[[[[[.,.],.],.],.],.],[[[[.,.],.],.],.]]
=> [1,2,3,4,5,7,8,9,10,6] => [6,7,8,10,1,2,3,4,5,9] => ?
=> ? = 2
[[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]
=> [10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1,11] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 1
[[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]],.]
=> [7,6,8,5,4,3,2,1,9] => [2,1,8,7,6,5,4,3,9] => [[1,3,9],[2,4],[5],[6],[7],[8]]
=> ? = 1
[[[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.],.]
=> [2,1,3,4,5,6,7,8,9,10,11] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 1
Description
The row containing the largest entry of a standard tableau.
The following 133 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000306The bounce count of a Dyck path. St000097The order of the largest clique of the graph. St000439The position of the first down step of a Dyck path. St001581The achromatic number of a graph. St000383The last part of an integer composition. St000759The smallest missing part in an integer partition. St000678The number of up steps after the last double rise of a Dyck path. St000617The number of global maxima of a Dyck path. St000054The first entry of the permutation. St000675The number of centered multitunnels of a Dyck path. St000069The number of maximal elements of a poset. St000098The chromatic number of a graph. St000925The number of topologically connected components of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St000237The number of small exceedances. St001733The number of weak left to right maxima of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St001050The number of terminal closers of a set partition. St000286The number of connected components of the complement of a graph. St000971The smallest closer of a set partition. St000068The number of minimal elements in a poset. St000504The cardinality of the first block of a set partition. St000031The number of cycles in the cycle decomposition of a permutation. St000502The number of successions of a set partitions. St000025The number of initial rises of a Dyck path. St000234The number of global ascents of a permutation. St000026The position of the first return of a Dyck path. St000172The Grundy number of a graph. St000544The cop number of a graph. St001029The size of the core of a graph. 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. St001203We associate to 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 Dyck path as follows:
St001363The Euler characteristic of a graph according to Knill. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St000053The number of valleys of the Dyck path. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001316The domatic number of a graph. St000273The domination number of a graph. St000916The packing number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in 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. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000989The number of final rises of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000738The first entry in the last row of a standard tableau. St000542The number of left-to-right-minima 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. St000654The first descent of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000740The last entry of a permutation. St000990The first ascent of a permutation. St000203The number of external nodes of a binary tree. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000084The number of subtrees. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000056The decomposition (or block) number of a permutation. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001185The number of indecomposable injective modules of grade at least 2 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. St000015The number of peaks of a Dyck path. St000287The number of connected components of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000822The Hadwiger number of the graph. St000996The number of exclusive left-to-right maxima 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. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St001828The Euler characteristic of a graph. St000051The size of the left subtree of a binary tree. St000133The "bounce" of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000331The number of upper interactions of a Dyck path. St000883The number of longest increasing subsequences of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. 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. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001812The biclique partition number of a graph. St000993The multiplicity of the largest part of an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001330The hat guessing number of a graph. St000648The number of 2-excedences of a permutation. St000456The monochromatic index of a connected graph. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000924The number of topologically connected components of a perfect matching. 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. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000898The number of maximal entries in the last diagonal of the monotone triangle. St001889The size of the connectivity set of a signed permutation. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. 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. St000942The number of critical left to right maxima of the parking functions. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function.
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