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Your data matches 44 different statistics following compositions of up to 3 maps.
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Matching statistic: St000546
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(load all 71 compositions to match this statistic)
St000546: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 2
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 1
[2,4,1,3] => 0
[2,4,3,1] => 1
[3,1,2,4] => 0
[3,1,4,2] => 0
[3,2,1,4] => 0
[3,2,4,1] => 1
[3,4,1,2] => 1
[3,4,2,1] => 2
[4,1,2,3] => 1
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 2
[4,3,1,2] => 2
[4,3,2,1] => 3
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 0
[1,2,5,4,3] => 0
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 0
[1,3,5,4,2] => 0
[1,4,2,3,5] => 0
[1,4,2,5,3] => 0
[1,4,3,2,5] => 0
[1,4,3,5,2] => 0
[1,4,5,2,3] => 0
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: St000288
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 91%
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 91%
Values
[1] => [1] => => => ? = 0
[1,2] => [2,1] => 0 => 0 => 0
[2,1] => [1,2] => 1 => 1 => 1
[1,2,3] => [3,2,1] => 00 => 00 => 0
[1,3,2] => [2,3,1] => 00 => 00 => 0
[2,1,3] => [3,1,2] => 00 => 00 => 0
[2,3,1] => [1,3,2] => 10 => 10 => 1
[3,1,2] => [2,1,3] => 01 => 01 => 1
[3,2,1] => [1,2,3] => 11 => 11 => 2
[1,2,3,4] => [4,3,2,1] => 000 => 000 => 0
[1,2,4,3] => [3,4,2,1] => 000 => 000 => 0
[1,3,2,4] => [4,2,3,1] => 000 => 000 => 0
[1,3,4,2] => [2,4,3,1] => 000 => 000 => 0
[1,4,2,3] => [3,2,4,1] => 000 => 000 => 0
[1,4,3,2] => [2,3,4,1] => 000 => 000 => 0
[2,1,3,4] => [4,3,1,2] => 000 => 000 => 0
[2,1,4,3] => [3,4,1,2] => 000 => 000 => 0
[2,3,1,4] => [4,1,3,2] => 000 => 000 => 0
[2,3,4,1] => [1,4,3,2] => 100 => 010 => 1
[2,4,1,3] => [3,1,4,2] => 000 => 000 => 0
[2,4,3,1] => [1,3,4,2] => 100 => 010 => 1
[3,1,2,4] => [4,2,1,3] => 000 => 000 => 0
[3,1,4,2] => [2,4,1,3] => 000 => 000 => 0
[3,2,1,4] => [4,1,2,3] => 000 => 000 => 0
[3,2,4,1] => [1,4,2,3] => 100 => 010 => 1
[3,4,1,2] => [2,1,4,3] => 010 => 100 => 1
[3,4,2,1] => [1,2,4,3] => 110 => 110 => 2
[4,1,2,3] => [3,2,1,4] => 001 => 001 => 1
[4,1,3,2] => [2,3,1,4] => 001 => 001 => 1
[4,2,1,3] => [3,1,2,4] => 001 => 001 => 1
[4,2,3,1] => [1,3,2,4] => 101 => 101 => 2
[4,3,1,2] => [2,1,3,4] => 011 => 011 => 2
[4,3,2,1] => [1,2,3,4] => 111 => 111 => 3
[1,2,3,4,5] => [5,4,3,2,1] => 0000 => 0000 => 0
[1,2,3,5,4] => [4,5,3,2,1] => 0000 => 0000 => 0
[1,2,4,3,5] => [5,3,4,2,1] => 0000 => 0000 => 0
[1,2,4,5,3] => [3,5,4,2,1] => 0000 => 0000 => 0
[1,2,5,3,4] => [4,3,5,2,1] => 0000 => 0000 => 0
[1,2,5,4,3] => [3,4,5,2,1] => 0000 => 0000 => 0
[1,3,2,4,5] => [5,4,2,3,1] => 0000 => 0000 => 0
[1,3,2,5,4] => [4,5,2,3,1] => 0000 => 0000 => 0
[1,3,4,2,5] => [5,2,4,3,1] => 0000 => 0000 => 0
[1,3,4,5,2] => [2,5,4,3,1] => 0000 => 0000 => 0
[1,3,5,2,4] => [4,2,5,3,1] => 0000 => 0000 => 0
[1,3,5,4,2] => [2,4,5,3,1] => 0000 => 0000 => 0
[1,4,2,3,5] => [5,3,2,4,1] => 0000 => 0000 => 0
[1,4,2,5,3] => [3,5,2,4,1] => 0000 => 0000 => 0
[1,4,3,2,5] => [5,2,3,4,1] => 0000 => 0000 => 0
[1,4,3,5,2] => [2,5,3,4,1] => 0000 => 0000 => 0
[1,4,5,2,3] => [3,2,5,4,1] => 0000 => 0000 => 0
[1,4,5,3,2] => [2,3,5,4,1] => 0000 => 0000 => 0
[8,2,3,1,4,5,6,7] => [7,6,5,4,1,3,2,8] => ? => ? => ? = 1
[6,5,7,4,3,2,1,8] => [8,1,2,3,4,7,5,6] => ? => ? => ? = 0
[6,5,4,7,3,2,1,8] => [8,1,2,3,7,4,5,6] => ? => ? => ? = 0
[5,7,6,4,3,2,1,8] => [8,1,2,3,4,6,7,5] => ? => ? => ? = 0
[1,2,3,4,5,7,6,8] => [8,6,7,5,4,3,2,1] => ? => ? => ? = 0
[1,3,2,6,4,5,7,8] => [8,7,5,4,6,2,3,1] => ? => ? => ? = 0
[1,3,2,7,4,8,5,6] => [6,5,8,4,7,2,3,1] => ? => ? => ? = 0
[1,3,2,7,4,5,6,8] => [8,6,5,4,7,2,3,1] => ? => ? => ? = 0
[1,3,2,8,4,5,6,7] => [7,6,5,4,8,2,3,1] => ? => ? => ? = 0
[1,4,2,6,3,5,7,8] => [8,7,5,3,6,2,4,1] => ? => ? => ? = 0
[1,4,2,7,3,8,5,6] => [6,5,8,3,7,2,4,1] => ? => ? => ? = 0
[1,4,2,7,3,5,6,8] => [8,6,5,3,7,2,4,1] => ? => ? => ? = 0
[1,4,2,3,5,7,6,8] => [8,6,7,5,3,2,4,1] => ? => ? => ? = 0
[1,4,2,3,5,6,7,8] => [8,7,6,5,3,2,4,1] => ? => ? => ? = 0
[1,5,2,6,3,4,7,8] => [8,7,4,3,6,2,5,1] => ? => ? => ? = 0
[1,5,2,3,4,7,6,8] => [8,6,7,4,3,2,5,1] => ? => ? => ? = 0
[1,2,3,5,4,8,6,7] => [7,6,8,4,5,3,2,1] => ? => ? => ? = 0
[1,5,2,3,4,6,7,8] => [8,7,6,4,3,2,5,1] => ? => ? => ? = 0
[1,6,2,3,4,8,5,7] => [7,5,8,4,3,2,6,1] => ? => ? => ? = 0
[1,6,2,3,4,5,7,8] => [8,7,5,4,3,2,6,1] => ? => ? => ? = 0
[1,7,2,3,4,8,5,6] => [6,5,8,4,3,2,7,1] => ? => ? => ? = 0
[1,2,3,7,4,8,5,6] => [6,5,8,4,7,3,2,1] => ? => ? => ? = 0
[1,2,3,7,4,5,6,8] => [8,6,5,4,7,3,2,1] => ? => ? => ? = 0
[10,9,4,3,6,5,8,7,2,1] => [1,2,7,8,5,6,3,4,9,10] => 110000011 => 000011011 => ? = 4
[10,9,6,5,4,3,8,7,2,1] => [1,2,7,8,3,4,5,6,9,10] => 110000011 => 000011011 => ? = 4
[10,9,8,5,4,7,6,3,2,1] => [1,2,3,6,7,4,5,8,9,10] => 111000111 => 001110111 => ? = 6
[10,9,4,3,8,7,6,5,2,1] => [1,2,5,6,7,8,3,4,9,10] => 110000011 => 000011011 => ? = 4
[] => [] => => => ? = 0
[1,8,2,4,3,5,6,7] => [7,6,5,3,4,2,8,1] => ? => ? => ? = 0
[1,7,2,4,3,5,6,8] => [8,6,5,3,4,2,7,1] => ? => ? => ? = 0
[1,6,2,4,3,5,7,8] => [8,7,5,3,4,2,6,1] => ? => ? => ? = 0
[1,5,2,4,3,6,7,8] => [8,7,6,3,4,2,5,1] => ? => ? => ? = 0
[1,5,2,8,3,6,4,7] => [7,4,6,3,8,2,5,1] => ? => ? => ? = 0
[1,4,2,8,3,6,5,7] => [7,5,6,3,8,2,4,1] => ? => ? => ? = 0
[1,3,2,8,4,6,5,7] => [7,5,6,4,8,2,3,1] => ? => ? => ? = 0
[1,2,3,7,4,6,5,8] => [8,5,6,4,7,3,2,1] => ? => ? => ? = 0
[1,3,2,7,4,6,5,8] => [8,5,6,4,7,2,3,1] => ? => ? => ? = 0
[1,4,2,7,3,6,5,8] => [8,5,6,3,7,2,4,1] => ? => ? => ? = 0
[1,8,2,6,3,5,4,7] => [7,4,5,3,6,2,8,1] => ? => ? => ? = 0
[1,8,2,5,3,6,4,7] => [7,4,6,3,5,2,8,1] => ? => ? => ? = 0
[1,7,2,5,3,6,4,8] => [8,4,6,3,5,2,7,1] => ? => ? => ? = 0
[1,6,2,5,3,7,4,8] => [8,4,7,3,5,2,6,1] => ? => ? => ? = 0
[1,5,2,4,3,7,6,8] => [8,6,7,3,4,2,5,1] => ? => ? => ? = 0
[1,6,2,4,3,7,5,8] => [8,5,7,3,4,2,6,1] => ? => ? => ? = 0
[1,7,2,4,3,6,5,8] => [8,5,6,3,4,2,7,1] => ? => ? => ? = 0
[1,8,2,4,3,6,5,7] => [7,5,6,3,4,2,8,1] => ? => ? => ? = 0
[1,8,2,3,4,6,5,7] => [7,5,6,4,3,2,8,1] => ? => ? => ? = 0
[1,8,2,4,3,7,5,6] => [6,5,7,3,4,2,8,1] => ? => ? => ? = 0
[1,6,2,4,3,8,5,7] => [7,5,8,3,4,2,6,1] => ? => ? => ? = 0
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St001462
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 73% ●values known / values provided: 79%●distinct values known / distinct values provided: 73%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 73% ●values known / values provided: 79%●distinct values known / distinct values provided: 73%
Values
[1] => [1] => [1] => [[1]]
=> 1 = 0 + 1
[1,2] => [2,1] => [2,1] => [[1],[2]]
=> 1 = 0 + 1
[2,1] => [1,2] => [1,2] => [[1,2]]
=> 2 = 1 + 1
[1,2,3] => [3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 1 = 0 + 1
[1,3,2] => [2,3,1] => [3,2,1] => [[1],[2],[3]]
=> 1 = 0 + 1
[2,1,3] => [3,1,2] => [3,2,1] => [[1],[2],[3]]
=> 1 = 0 + 1
[2,3,1] => [1,3,2] => [1,3,2] => [[1,2],[3]]
=> 2 = 1 + 1
[3,1,2] => [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 2 = 1 + 1
[3,2,1] => [1,2,3] => [1,2,3] => [[1,2,3]]
=> 3 = 2 + 1
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[1,2,4,3] => [3,4,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[1,3,2,4] => [4,2,3,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[1,3,4,2] => [2,4,3,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[1,4,2,3] => [3,2,4,1] => [4,2,3,1] => [[1,3],[2],[4]]
=> 1 = 0 + 1
[1,4,3,2] => [2,3,4,1] => [4,2,3,1] => [[1,3],[2],[4]]
=> 1 = 0 + 1
[2,1,3,4] => [4,3,1,2] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[2,1,4,3] => [3,4,1,2] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[2,3,1,4] => [4,1,3,2] => [4,2,3,1] => [[1,3],[2],[4]]
=> 1 = 0 + 1
[2,3,4,1] => [1,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[2,4,1,3] => [3,1,4,2] => [4,2,3,1] => [[1,3],[2],[4]]
=> 1 = 0 + 1
[2,4,3,1] => [1,3,4,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[3,1,2,4] => [4,2,1,3] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[3,1,4,2] => [2,4,1,3] => [3,4,1,2] => [[1,2],[3,4]]
=> 1 = 0 + 1
[3,2,1,4] => [4,1,2,3] => [4,2,3,1] => [[1,3],[2],[4]]
=> 1 = 0 + 1
[3,2,4,1] => [1,4,2,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 2 = 1 + 1
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 3 = 2 + 1
[4,1,2,3] => [3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2 = 1 + 1
[4,1,3,2] => [2,3,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2 = 1 + 1
[4,2,1,3] => [3,1,2,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2 = 1 + 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 3 = 2 + 1
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 3 = 2 + 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,2,3,5,4] => [4,5,3,2,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,2,4,3,5] => [5,3,4,2,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,2,4,5,3] => [3,5,4,2,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,2,5,3,4] => [4,3,5,2,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,2,5,4,3] => [3,4,5,2,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,3,2,4,5] => [5,4,2,3,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,3,2,5,4] => [4,5,2,3,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,3,4,2,5] => [5,2,4,3,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,3,4,5,2] => [2,5,4,3,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,3,5,2,4] => [4,2,5,3,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,3,5,4,2] => [2,4,5,3,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,4,2,3,5] => [5,3,2,4,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,4,2,5,3] => [3,5,2,4,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,4,3,5,2] => [2,5,3,4,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,4,5,2,3] => [3,2,5,4,1] => [5,2,4,3,1] => [[1,3],[2],[4],[5]]
=> 1 = 0 + 1
[6,5,7,4,3,2,8,1] => [1,8,2,3,4,7,5,6] => [1,8,3,4,5,7,6,2] => ?
=> ? = 1 + 1
[8,7,6,4,5,3,1,2] => [2,1,3,5,4,6,7,8] => [2,1,3,5,4,6,7,8] => ?
=> ? = 5 + 1
[8,7,6,4,5,1,2,3] => [3,2,1,5,4,6,7,8] => [3,2,1,5,4,6,7,8] => ?
=> ? = 4 + 1
[8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8] => [3,2,1,6,5,4,7,8] => ?
=> ? = 3 + 1
[7,8,4,5,6,1,2,3] => [3,2,1,6,5,4,8,7] => [3,2,1,6,5,4,8,7] => ?
=> ? = 2 + 1
[8,6,5,4,3,1,2,7] => [7,2,1,3,4,5,6,8] => [7,3,2,4,5,6,1,8] => ?
=> ? = 1 + 1
[8,6,5,4,2,1,3,7] => [7,3,1,2,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ? = 1 + 1
[7,6,4,5,3,2,1,8] => [8,1,2,3,5,4,6,7] => [8,2,3,4,6,5,7,1] => ?
=> ? = 0 + 1
[6,7,5,4,2,3,1,8] => [8,1,3,2,4,5,7,6] => [8,2,4,3,5,6,7,1] => ?
=> ? = 0 + 1
[2,6,8,7,5,4,3,1] => [1,3,4,5,7,8,6,2] => [1,8,3,4,7,6,5,2] => ?
=> ? = 1 + 1
[8,2,7,4,6,5,3,1] => [1,3,5,6,4,7,2,8] => [1,7,5,4,3,6,2,8] => ?
=> ? = 2 + 1
[2,1,4,3,6,5,8,7,10,9] => [9,10,7,8,5,6,3,4,1,2] => [10,9,8,7,6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 0 + 1
[4,3,2,1,6,5,8,7,10,9] => [9,10,7,8,5,6,1,2,3,4] => [10,9,8,7,6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 0 + 1
[6,3,2,5,4,1,8,7,10,9] => [9,10,7,8,1,4,5,2,3,6] => [10,9,8,7,5,6,4,3,2,1] => [[1,6],[2],[3],[4],[5],[7],[8],[9],[10]]
=> ? = 0 + 1
[8,3,2,5,4,7,6,1,10,9] => [9,10,1,6,7,4,5,2,3,8] => [10,9,3,8,7,6,5,4,2,1] => [[1,4],[2],[3],[5],[6],[7],[8],[9],[10]]
=> ? = 0 + 1
[10,3,2,5,4,7,6,9,8,1] => [1,8,9,6,7,4,5,2,3,10] => [1,9,8,7,6,5,4,3,2,10] => [[1,2,10],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 2 + 1
[2,1,6,5,4,3,8,7,10,9] => [9,10,7,8,3,4,5,6,1,2] => [10,9,8,7,6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 0 + 1
[6,5,4,3,2,1,8,7,10,9] => [9,10,7,8,1,2,3,4,5,6] => [10,9,8,7,5,6,4,3,2,1] => [[1,6],[2],[3],[4],[5],[7],[8],[9],[10]]
=> ? = 0 + 1
[8,5,4,3,2,7,6,1,10,9] => [9,10,1,6,7,2,3,4,5,8] => [10,9,3,8,7,6,5,4,2,1] => [[1,4],[2],[3],[5],[6],[7],[8],[9],[10]]
=> ? = 0 + 1
[10,5,4,3,2,7,6,9,8,1] => [1,8,9,6,7,2,3,4,5,10] => [1,9,8,7,6,5,4,3,2,10] => [[1,2,10],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 2 + 1
[2,1,8,5,4,7,6,3,10,9] => [9,10,3,6,7,4,5,8,1,2] => [10,9,8,7,6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 0 + 1
[8,7,4,3,6,5,2,1,10,9] => [9,10,1,2,5,6,3,4,7,8] => [10,9,3,4,8,7,6,5,2,1] => ?
=> ? = 0 + 1
[10,7,4,3,6,5,2,9,8,1] => [1,8,9,2,5,6,3,4,7,10] => [1,9,8,4,7,6,5,3,2,10] => ?
=> ? = 2 + 1
[2,1,10,5,4,7,6,9,8,3] => [3,8,9,6,7,4,5,10,1,2] => [10,9,7,6,5,4,3,8,2,1] => [[1,8],[2],[3],[4],[5],[6],[7],[9],[10]]
=> ? = 0 + 1
[10,9,4,3,6,5,8,7,2,1] => [1,2,7,8,5,6,3,4,9,10] => [1,2,8,7,6,5,4,3,9,10] => ?
=> ? = 4 + 1
[8,9,5,6,3,4,10,1,2,7] => [7,2,1,10,4,3,6,5,9,8] => [8,3,2,10,6,5,7,1,9,4] => ?
=> ? = 0 + 1
[2,1,4,3,8,7,6,5,10,9] => [9,10,5,6,7,8,3,4,1,2] => [10,9,8,7,6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 0 + 1
[4,3,2,1,8,7,6,5,10,9] => [9,10,5,6,7,8,1,2,3,4] => [10,9,8,7,6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 0 + 1
[8,3,2,7,6,5,4,1,10,9] => [9,10,1,4,5,6,7,2,3,8] => [10,9,3,8,7,6,5,4,2,1] => [[1,4],[2],[3],[5],[6],[7],[8],[9],[10]]
=> ? = 0 + 1
[10,3,2,7,6,5,4,9,8,1] => [1,8,9,4,5,6,7,2,3,10] => [1,9,8,7,6,5,4,3,2,10] => [[1,2,10],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 2 + 1
[2,1,8,7,6,5,4,3,10,9] => [9,10,3,4,5,6,7,8,1,2] => [10,9,8,7,5,6,4,3,2,1] => [[1,6],[2],[3],[4],[5],[7],[8],[9],[10]]
=> ? = 0 + 1
[8,7,6,5,4,3,2,1,10,9] => [9,10,1,2,3,4,5,6,7,8] => [10,9,3,4,5,6,7,8,2,1] => ?
=> ? = 0 + 1
[9,7,6,5,4,3,2,10,1,8] => [8,1,10,2,3,4,5,6,7,9] => [9,2,10,4,5,6,7,8,1,3] => [[1,3,5,6,7,8],[2,4],[9,10]]
=> ? = 0 + 1
[10,7,6,5,4,3,2,9,8,1] => [1,8,9,2,3,4,5,6,7,10] => [1,9,8,4,5,6,7,3,2,10] => ?
=> ? = 2 + 1
[2,1,10,7,6,5,4,9,8,3] => [3,8,9,4,5,6,7,10,1,2] => [10,9,7,6,5,4,3,8,2,1] => [[1,8],[2],[3],[4],[5],[6],[7],[9],[10]]
=> ? = 0 + 1
[10,9,6,5,4,3,8,7,2,1] => [1,2,7,8,3,4,5,6,9,10] => [1,2,8,7,5,6,4,3,9,10] => ?
=> ? = 4 + 1
[2,1,4,3,10,7,6,9,8,5] => [5,8,9,6,7,10,3,4,1,2] => [10,9,8,7,5,6,4,3,2,1] => [[1,6],[2],[3],[4],[5],[7],[8],[9],[10]]
=> ? = 0 + 1
[4,3,2,1,10,7,6,9,8,5] => [5,8,9,6,7,10,1,2,3,4] => [10,9,8,7,5,6,4,3,2,1] => [[1,6],[2],[3],[4],[5],[7],[8],[9],[10]]
=> ? = 0 + 1
[10,3,2,9,6,5,8,7,4,1] => [1,4,7,8,5,6,9,2,3,10] => [1,9,8,6,5,4,7,3,2,10] => ?
=> ? = 2 + 1
[2,1,10,9,6,5,8,7,4,3] => [3,4,7,8,5,6,9,10,1,2] => [10,9,6,5,4,3,7,8,2,1] => ?
=> ? = 0 + 1
[10,9,8,5,4,7,6,3,2,1] => [1,2,3,6,7,4,5,8,9,10] => [1,2,3,7,6,5,4,8,9,10] => ?
=> ? = 6 + 1
[6,7,8,9,10,1,2,3,4,5] => [5,4,3,2,1,10,9,8,7,6] => [5,4,3,2,1,10,9,8,7,6] => [[1,6],[2,7],[3,8],[4,9],[5,10]]
=> ? = 1 + 1
[7,3,2,8,9,10,1,4,5,6] => [6,5,4,1,10,9,8,2,3,7] => [9,8,4,3,10,7,6,2,1,5] => ?
=> ? = 0 + 1
[8,5,4,3,2,9,10,1,6,7] => [7,6,1,10,9,2,3,4,5,8] => [9,8,3,10,7,6,5,2,1,4] => ?
=> ? = 0 + 1
[2,1,4,3,6,5,10,9,8,7] => [7,8,9,10,5,6,3,4,1,2] => [10,9,8,7,6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 0 + 1
[4,3,2,1,6,5,10,9,8,7] => [7,8,9,10,5,6,1,2,3,4] => [10,9,8,7,6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 0 + 1
[6,3,2,5,4,1,10,9,8,7] => [7,8,9,10,1,4,5,2,3,6] => [10,9,8,7,5,6,4,3,2,1] => [[1,6],[2],[3],[4],[5],[7],[8],[9],[10]]
=> ? = 0 + 1
[10,3,2,5,4,9,8,7,6,1] => [1,6,7,8,9,4,5,2,3,10] => [1,9,8,7,6,5,4,3,2,10] => [[1,2,10],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 2 + 1
[2,1,6,5,4,3,10,9,8,7] => [7,8,9,10,3,4,5,6,1,2] => [10,9,8,7,6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 0 + 1
[6,5,4,3,2,1,10,9,8,7] => [7,8,9,10,1,2,3,4,5,6] => [10,9,8,7,5,6,4,3,2,1] => [[1,6],[2],[3],[4],[5],[7],[8],[9],[10]]
=> ? = 0 + 1
Description
The number of factors of a standard tableaux under concatenation.
The concatenation of two standard Young tableaux $T_1$ and $T_2$ is obtained by adding the largest entry of $T_1$ to each entry of $T_2$, and then appending the rows of the result to $T_1$, see [1, dfn 2.10].
This statistic returns the maximal number of standard tableaux such that their concatenation is the given tableau.
Matching statistic: St000439
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 91%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 91%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 2 = 0 + 2
[1,2] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2 = 0 + 2
[2,1] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 3 = 1 + 2
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 0 + 2
[1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 0 + 2
[2,1,3] => [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[2,3,1] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 3 = 1 + 2
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 1 + 2
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4 = 2 + 2
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 0 + 2
[2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 0 + 2
[2,4,3,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[3,2,1,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[3,2,4,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 1 + 2
[3,4,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 2 + 2
[4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[4,1,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[4,2,1,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[4,3,1,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 3 + 2
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,2,4,3,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,2,5,3,4] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,3,2,4,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,3,2,5,4] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,3,4,2,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,3,5,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,3,5,4,2] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,4,2,3,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,4,2,5,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,4,3,5,2] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,4,5,2,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[7,8,5,6,4,3,2,1] => [2,1,4,3,5,6,7,8] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 5 + 2
[7,8,6,4,5,3,2,1] => [2,1,3,5,4,6,7,8] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 5 + 2
[8,6,7,4,5,3,2,1] => [1,3,2,5,4,6,7,8] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 5 + 2
[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]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> ? = 5 + 2
[6,7,8,3,4,5,2,1] => [3,2,1,6,5,4,7,8] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 3 + 2
[8,7,5,6,4,2,3,1] => [1,2,4,3,5,7,6,8] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 5 + 2
[8,6,5,7,3,2,4,1] => [1,3,4,2,6,7,5,8] => [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 3 + 2
[6,7,8,5,2,3,4,1] => [3,2,1,4,7,6,5,8] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 3 + 2
[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]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 + 2
[7,8,3,2,4,5,6,1] => [2,1,6,7,5,4,3,8] => [1,1,0,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 2 + 2
[8,7,5,6,4,3,1,2] => [1,2,4,3,5,6,8,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> ? = 5 + 2
[8,7,5,6,3,4,1,2] => [1,2,4,3,6,5,8,7] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> ? = 4 + 2
[5,6,7,8,3,4,1,2] => [4,3,2,1,6,5,8,7] => [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]
=> ? = 2 + 2
[7,8,3,4,5,6,1,2] => [2,1,6,5,4,3,8,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 2 + 2
[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]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 3 + 2
[8,5,6,7,4,1,2,3] => [1,4,3,2,5,8,7,6] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 3 + 2
[8,7,6,4,5,1,2,3] => [1,2,3,5,4,8,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> ? = 4 + 2
[8,6,7,4,5,1,2,3] => [1,3,2,5,4,8,7,6] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 3 + 2
[8,7,4,5,6,1,2,3] => [1,2,5,4,3,8,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 3 + 2
[7,8,4,5,6,1,2,3] => [2,1,5,4,3,8,7,6] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 2 + 2
[6,7,4,5,8,1,2,3] => [3,2,5,4,1,8,7,6] => [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1 + 2
[7,6,5,8,3,2,1,4] => [2,3,4,1,6,7,8,5] => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[7,5,6,8,2,3,1,4] => [2,4,3,1,7,6,8,5] => [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 2
[6,7,5,8,3,1,2,4] => [3,2,4,1,6,8,7,5] => [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 2
[6,5,7,8,2,1,3,4] => [3,4,2,1,7,8,6,5] => [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 1 + 2
[8,7,5,6,1,2,3,4] => [1,2,4,3,8,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 3 + 2
[7,8,5,6,1,2,3,4] => [2,1,4,3,8,7,6,5] => [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]
=> ? = 2 + 2
[8,5,6,7,1,2,3,4] => [1,4,3,2,8,7,6,5] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 + 2
[8,7,6,1,2,3,4,5] => [1,2,3,8,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 + 2
[7,8,5,4,2,1,3,6] => [2,1,4,5,7,8,6,3] => [1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 1 + 2
[7,8,5,4,1,2,3,6] => [2,1,4,5,8,7,6,3] => [1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 1 + 2
[8,5,6,3,4,1,2,7] => [1,4,3,6,5,8,7,2] => [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 1 + 2
[8,6,5,4,2,1,3,7] => [1,3,4,5,7,8,6,2] => [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 1 + 2
[8,6,4,3,2,1,5,7] => [1,3,5,6,7,8,4,2] => [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 2
[8,6,4,2,1,3,5,7] => [1,3,5,7,8,6,4,2] => [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 1 + 2
[6,5,8,7,4,3,2,1] => [3,4,1,2,5,6,7,8] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 4 + 2
[8,3,2,5,4,7,6,1] => [1,6,7,4,5,2,3,8] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 2 + 2
[8,3,2,7,6,5,4,1] => [1,6,7,2,3,4,5,8] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 2 + 2
[8,3,7,4,6,5,2,1] => [1,6,2,5,3,4,7,8] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 + 2
[8,5,4,3,2,7,6,1] => [1,4,5,6,7,2,3,8] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[8,7,4,3,6,5,2,1] => [1,2,5,6,3,4,7,8] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> ? = 4 + 2
[1,3,2,6,4,5,7,8] => [8,6,7,3,5,4,2,1] => ?
=> ?
=> ? = 0 + 2
[1,3,2,7,4,5,6,8] => [8,6,7,2,5,4,3,1] => ?
=> ?
=> ? = 0 + 2
[1,4,2,3,5,8,6,7] => [8,5,7,6,4,1,3,2] => ?
=> ?
=> ? = 0 + 2
[1,2,3,5,4,8,6,7] => [8,7,6,4,5,1,3,2] => ?
=> ?
=> ? = 0 + 2
[1,2,3,6,4,5,7,8] => [8,7,6,3,5,4,2,1] => ?
=> ?
=> ? = 0 + 2
[1,2,3,7,4,8,5,6] => [8,7,6,2,5,1,4,3] => ?
=> ?
=> ? = 0 + 2
[8,4,5,2,3,7,6,1] => [1,5,4,7,6,2,3,8] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 2 + 2
[8,6,4,3,7,2,5,1] => [1,3,5,6,2,7,4,8] => [1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 2 + 2
[7,8,4,3,6,5,1,2] => [2,1,5,6,3,4,8,7] => [1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0,1,0]
=> ? = 2 + 2
Description
The position of the first down step of a Dyck path.
Matching statistic: St000237
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 91%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 91%
Values
[1] => [1] => [1,0]
=> [1] => 0
[1,2] => [2,1] => [1,1,0,0]
=> [1,2] => 0
[2,1] => [1,2] => [1,0,1,0]
=> [2,1] => 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[2,1,3] => [2,3,1] => [1,1,0,1,0,0]
=> [3,1,2] => 0
[2,3,1] => [2,1,3] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [2,3,1] => 2
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 0
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 0
[2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 0
[2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 0
[2,4,3,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 0
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 0
[3,2,1,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 0
[3,2,4,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[3,4,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 2
[4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[4,1,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[4,2,1,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[4,3,1,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 3
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,2,4,3,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,2,5,3,4] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,3,2,4,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,3,2,5,4] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,3,4,2,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,3,5,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,3,5,4,2] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,4,2,3,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,4,2,5,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,4,3,5,2] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,4,5,2,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[2,3,4,1,5,6,7] => [6,5,4,7,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 0
[2,3,4,1,5,7,6] => [6,5,4,7,3,1,2] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 0
[2,3,4,5,1,6,7] => [6,5,4,3,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,3,7,4,5,6] => ? = 0
[2,3,4,5,1,7,6] => [6,5,4,3,7,1,2] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,3,7,4,5,6] => ? = 0
[2,3,4,5,6,1,7] => [6,5,4,3,2,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,2,3,4,7,5,6] => ? = 0
[2,3,4,5,7,1,6] => [6,5,4,3,1,7,2] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,2,3,4,7,5,6] => ? = 0
[2,3,4,6,1,7,5] => [6,5,4,2,7,1,3] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,3,7,4,5,6] => ? = 0
[2,3,4,6,7,1,5] => [6,5,4,2,1,7,3] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,2,3,4,7,5,6] => ? = 0
[2,3,4,7,1,5,6] => [6,5,4,1,7,3,2] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,3,7,4,5,6] => ? = 0
[2,3,5,6,7,1,4] => [6,5,3,2,1,7,4] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,2,3,4,7,5,6] => ? = 0
[2,5,7,1,3,4,6] => [6,3,1,7,5,4,2] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 0
[2,6,5,4,3,1,7] => [6,2,3,4,5,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,2,3,4,7,5,6] => ? = 0
[2,6,7,1,3,4,5] => [6,2,1,7,5,4,3] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 0
[3,4,1,2,5,6,7] => [5,4,7,6,3,2,1] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5,7] => ? = 0
[3,5,6,1,2,4,7] => [5,3,2,7,6,4,1] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,2,6,3,4,5,7] => ? = 0
[3,5,6,4,2,1,7] => [5,3,2,4,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,2,3,6,7,4,5] => ? = 0
[3,6,1,2,4,5,7] => [5,2,7,6,4,3,1] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5,7] => ? = 0
[3,6,5,4,2,1,7] => [5,2,3,4,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,2,3,6,7,4,5] => ? = 0
[4,2,5,1,3,6,7] => [4,6,3,7,5,2,1] => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [5,1,7,2,3,4,6] => ? = 0
[4,3,6,5,2,1,7] => [4,5,2,3,6,7,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [5,1,2,6,7,3,4] => ? = 0
[4,5,1,6,2,3,7] => [4,3,7,2,6,5,1] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,5,2,3,4,6,7] => ? = 0
[4,5,3,6,2,1,7] => [4,3,5,2,6,7,1] => [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,5,2,6,7,3,4] => ? = 0
[4,5,6,1,2,3,7] => [4,3,2,7,6,5,1] => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,2,5,3,4,6,7] => ? = 0
[4,5,6,3,2,1,7] => [4,3,2,5,6,7,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,2,5,6,7,3,4] => ? = 0
[4,6,1,2,3,5,7] => [4,2,7,6,5,3,1] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,5,2,3,4,6,7] => ? = 0
[4,6,1,2,3,7,5] => [4,2,7,6,5,1,3] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,5,2,3,4,6,7] => ? = 0
[4,6,5,3,2,1,7] => [4,2,3,5,6,7,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,2,5,6,7,3,4] => ? = 0
[5,2,6,7,1,3,4] => [3,6,2,1,7,5,4] => [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [4,1,2,7,3,5,6] => ? = 0
[5,3,2,6,1,4,7] => [3,5,6,2,7,4,1] => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,6,1,7,2,3,5] => ? = 0
[5,3,6,7,1,2,4] => [3,5,2,1,7,6,4] => [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [4,1,2,6,3,5,7] => ? = 0
[5,4,3,2,1,6,7] => [3,4,5,6,7,2,1] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,7,1,2,3] => ? = 0
[5,4,3,2,1,7,6] => [3,4,5,6,7,1,2] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,7,1,2,3] => ? = 0
[5,4,3,2,6,1,7] => [3,4,5,6,2,7,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [4,5,6,1,7,2,3] => ? = 0
[5,4,3,6,2,1,7] => [3,4,5,2,6,7,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [4,5,1,6,7,2,3] => ? = 0
[5,4,6,3,2,1,7] => [3,4,2,5,6,7,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,1,5,6,7,2,3] => ? = 0
[5,4,6,7,1,2,3] => [3,4,2,1,7,6,5] => [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [4,1,2,5,3,6,7] => ? = 1
[5,6,1,2,3,4,7] => [3,2,7,6,5,4,1] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,4,2,3,5,6,7] => ? = 0
[5,6,1,2,3,7,4] => [3,2,7,6,5,1,4] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,4,2,3,5,6,7] => ? = 0
[5,6,3,4,1,2,7] => [3,2,5,4,7,6,1] => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,4,2,6,3,5,7] => ? = 0
[5,6,4,3,2,1,7] => [3,2,4,5,6,7,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,5,6,7,2,3] => ? = 0
[5,6,7,1,2,3,4] => [3,2,1,7,6,5,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,5,6,7] => ? = 1
[6,2,4,5,3,1,7] => [2,6,4,3,5,7,1] => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,2,4,7,5,6] => ? = 0
[6,2,5,4,3,1,7] => [2,6,3,4,5,7,1] => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,2,4,7,5,6] => ? = 0
[6,3,2,5,4,1,7] => [2,5,6,3,4,7,1] => [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [3,6,1,2,7,4,5] => ? = 0
[6,3,4,2,5,1,7] => [2,5,4,6,3,7,1] => [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,6,2,7,4,5] => ? = 0
[6,3,4,5,2,1,7] => [2,5,4,3,6,7,1] => [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [3,1,2,6,7,4,5] => ? = 0
[6,3,4,7,1,2,5] => [2,5,4,1,7,6,3] => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [3,1,2,6,4,5,7] => ? = 0
[6,3,5,4,2,1,7] => [2,5,3,4,6,7,1] => [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [3,1,2,6,7,4,5] => ? = 0
[6,4,3,2,5,1,7] => [2,4,5,6,3,7,1] => [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [3,5,6,1,7,2,4] => ? = 0
[6,4,3,2,7,1,5] => [2,4,5,6,1,7,3] => [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [3,5,6,1,7,2,4] => ? = 0
Description
The number of small exceedances.
This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St000925
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 64%
Mp00069: Permutations —complement⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 64%
Values
[1] => [1] => [1] => {{1}}
=> ? = 0 + 1
[1,2] => [1,2] => [2,1] => {{1,2}}
=> 1 = 0 + 1
[2,1] => [2,1] => [1,2] => {{1},{2}}
=> 2 = 1 + 1
[1,2,3] => [1,3,2] => [3,1,2] => {{1,2,3}}
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => [3,1,2] => {{1,2,3}}
=> 1 = 0 + 1
[2,1,3] => [2,1,3] => [2,3,1] => {{1,2,3}}
=> 1 = 0 + 1
[2,3,1] => [2,3,1] => [2,1,3] => {{1,2},{3}}
=> 2 = 1 + 1
[3,1,2] => [3,1,2] => [1,3,2] => {{1},{2,3}}
=> 2 = 1 + 1
[3,2,1] => [3,2,1] => [1,2,3] => {{1},{2},{3}}
=> 3 = 2 + 1
[1,2,3,4] => [1,4,3,2] => [4,1,2,3] => {{1,2,3,4}}
=> 1 = 0 + 1
[1,2,4,3] => [1,4,3,2] => [4,1,2,3] => {{1,2,3,4}}
=> 1 = 0 + 1
[1,3,2,4] => [1,4,3,2] => [4,1,2,3] => {{1,2,3,4}}
=> 1 = 0 + 1
[1,3,4,2] => [1,4,3,2] => [4,1,2,3] => {{1,2,3,4}}
=> 1 = 0 + 1
[1,4,2,3] => [1,4,3,2] => [4,1,2,3] => {{1,2,3,4}}
=> 1 = 0 + 1
[1,4,3,2] => [1,4,3,2] => [4,1,2,3] => {{1,2,3,4}}
=> 1 = 0 + 1
[2,1,3,4] => [2,1,4,3] => [3,4,1,2] => {{1,3},{2,4}}
=> 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => {{1,3},{2,4}}
=> 1 = 0 + 1
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => {{1,2,3,4}}
=> 1 = 0 + 1
[2,3,4,1] => [2,4,3,1] => [3,1,2,4] => {{1,2,3},{4}}
=> 2 = 1 + 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => {{1,2,3,4}}
=> 1 = 0 + 1
[2,4,3,1] => [2,4,3,1] => [3,1,2,4] => {{1,2,3},{4}}
=> 2 = 1 + 1
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => {{1,2,3,4}}
=> 1 = 0 + 1
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => {{1,2,3,4}}
=> 1 = 0 + 1
[3,2,1,4] => [3,2,1,4] => [2,3,4,1] => {{1,2,3,4}}
=> 1 = 0 + 1
[3,2,4,1] => [3,2,4,1] => [2,3,1,4] => {{1,2,3},{4}}
=> 2 = 1 + 1
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => {{1,2},{3,4}}
=> 2 = 1 + 1
[3,4,2,1] => [3,4,2,1] => [2,1,3,4] => {{1,2},{3},{4}}
=> 3 = 2 + 1
[4,1,2,3] => [4,1,3,2] => [1,4,2,3] => {{1},{2,3,4}}
=> 2 = 1 + 1
[4,1,3,2] => [4,1,3,2] => [1,4,2,3] => {{1},{2,3,4}}
=> 2 = 1 + 1
[4,2,1,3] => [4,2,1,3] => [1,3,4,2] => {{1},{2,3,4}}
=> 2 = 1 + 1
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> 3 = 2 + 1
[4,3,1,2] => [4,3,1,2] => [1,2,4,3] => {{1},{2},{3,4}}
=> 3 = 2 + 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4 = 3 + 1
[1,2,3,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1 = 0 + 1
[1,2,3,5,4] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1 = 0 + 1
[1,2,4,3,5] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1 = 0 + 1
[1,2,4,5,3] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1 = 0 + 1
[1,2,5,3,4] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1 = 0 + 1
[1,2,5,4,3] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1 = 0 + 1
[1,3,2,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1 = 0 + 1
[1,3,2,5,4] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1 = 0 + 1
[1,3,4,2,5] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1 = 0 + 1
[1,3,4,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1 = 0 + 1
[1,3,5,2,4] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1 = 0 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1 = 0 + 1
[1,4,2,3,5] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1 = 0 + 1
[1,4,2,5,3] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1 = 0 + 1
[1,4,3,2,5] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1 = 0 + 1
[1,4,3,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1 = 0 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1 = 0 + 1
[1,4,5,3,2] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1 = 0 + 1
[8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 7 + 1
[7,6,8,5,4,3,2,1] => [7,6,8,5,4,3,2,1] => [2,3,1,4,5,6,7,8] => {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 5 + 1
[7,8,5,6,4,3,2,1] => [7,8,5,6,4,3,2,1] => [2,1,4,3,5,6,7,8] => {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 5 + 1
[7,8,6,4,5,3,2,1] => [7,8,6,4,5,3,2,1] => [2,1,3,5,4,6,7,8] => {{1,2},{3},{4,5},{6},{7},{8}}
=> ? = 5 + 1
[8,6,7,4,5,3,2,1] => [8,6,7,4,5,3,2,1] => [1,3,2,5,4,6,7,8] => {{1},{2,3},{4,5},{6},{7},{8}}
=> ? = 5 + 1
[7,6,5,4,8,3,2,1] => [7,6,5,4,8,3,2,1] => [2,3,4,5,1,6,7,8] => {{1,2,3,4,5},{6},{7},{8}}
=> ? = 3 + 1
[6,5,7,4,8,3,2,1] => [6,5,8,4,7,3,2,1] => [3,4,1,5,2,6,7,8] => {{1,3},{2,4,5},{6},{7},{8}}
=> ? = 3 + 1
[7,8,6,5,3,4,2,1] => [7,8,6,5,3,4,2,1] => [2,1,3,4,6,5,7,8] => {{1,2},{3},{4},{5,6},{7},{8}}
=> ? = 5 + 1
[8,6,7,5,3,4,2,1] => [8,6,7,5,3,4,2,1] => [1,3,2,4,6,5,7,8] => {{1},{2,3},{4},{5,6},{7},{8}}
=> ? = 5 + 1
[8,7,5,6,3,4,2,1] => [8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => {{1},{2},{3,4},{5,6},{7},{8}}
=> ? = 5 + 1
[6,7,8,3,4,5,2,1] => [6,8,7,3,5,4,2,1] => [3,1,2,6,4,5,7,8] => ?
=> ? = 3 + 1
[7,8,6,5,4,2,3,1] => [7,8,6,5,4,2,3,1] => [2,1,3,4,5,7,6,8] => {{1,2},{3},{4},{5},{6,7},{8}}
=> ? = 5 + 1
[8,6,7,5,4,2,3,1] => [8,6,7,5,4,2,3,1] => [1,3,2,4,5,7,6,8] => {{1},{2,3},{4},{5},{6,7},{8}}
=> ? = 5 + 1
[8,7,5,6,4,2,3,1] => [8,7,5,6,4,2,3,1] => [1,2,4,3,5,7,6,8] => {{1},{2},{3,4},{5},{6,7},{8}}
=> ? = 5 + 1
[8,7,6,4,5,2,3,1] => [8,7,6,4,5,2,3,1] => [1,2,3,5,4,7,6,8] => {{1},{2},{3},{4,5},{6,7},{8}}
=> ? = 5 + 1
[8,6,5,7,3,2,4,1] => [8,6,5,7,3,2,4,1] => [1,3,4,2,6,7,5,8] => {{1},{2,3,4},{5,6,7},{8}}
=> ? = 3 + 1
[6,7,8,5,2,3,4,1] => [6,8,7,5,2,4,3,1] => [3,1,2,4,7,5,6,8] => {{1,2,3},{4},{5,6,7},{8}}
=> ? = 3 + 1
[8,5,6,7,2,3,4,1] => [8,5,7,6,2,4,3,1] => [1,4,2,3,7,5,6,8] => {{1},{2,3,4},{5,6,7},{8}}
=> ? = 3 + 1
[7,8,3,2,4,5,6,1] => [7,8,3,2,6,5,4,1] => [2,1,6,7,3,4,5,8] => ?
=> ? = 2 + 1
[6,5,7,4,3,2,8,1] => [6,5,8,4,3,2,7,1] => [3,4,1,5,6,7,2,8] => {{1,3},{2,4,5,6,7},{8}}
=> ? = 1 + 1
[6,5,4,3,7,2,8,1] => [6,5,4,3,8,2,7,1] => [3,4,5,6,1,7,2,8] => {{1,3,5},{2,4,6,7},{8}}
=> ? = 1 + 1
[5,4,6,3,7,2,8,1] => [5,4,8,3,7,2,6,1] => [4,5,1,6,2,7,3,8] => {{1,3,4,6,7},{2,5},{8}}
=> ? = 1 + 1
[3,4,2,5,6,7,8,1] => [3,8,2,7,6,5,4,1] => [6,1,7,2,3,4,5,8] => {{1,2,4,6},{3,5,7},{8}}
=> ? = 1 + 1
[4,2,3,5,6,7,8,1] => [4,2,8,7,6,5,3,1] => [5,7,1,2,3,4,6,8] => {{1,3,5},{2,4,6,7},{8}}
=> ? = 1 + 1
[8,7,6,5,4,3,1,2] => [8,7,6,5,4,3,1,2] => [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 6 + 1
[7,8,6,5,4,3,1,2] => [7,8,6,5,4,3,1,2] => [2,1,3,4,5,6,8,7] => {{1,2},{3},{4},{5},{6},{7,8}}
=> ? = 5 + 1
[8,6,7,5,4,3,1,2] => [8,6,7,5,4,3,1,2] => [1,3,2,4,5,6,8,7] => {{1},{2,3},{4},{5},{6},{7,8}}
=> ? = 5 + 1
[8,7,5,6,4,3,1,2] => [8,7,5,6,4,3,1,2] => [1,2,4,3,5,6,8,7] => {{1},{2},{3,4},{5},{6},{7,8}}
=> ? = 5 + 1
[8,7,6,4,5,3,1,2] => [8,7,6,4,5,3,1,2] => [1,2,3,5,4,6,8,7] => {{1},{2},{3},{4,5},{6},{7,8}}
=> ? = 5 + 1
[8,7,6,5,3,4,1,2] => [8,7,6,5,3,4,1,2] => [1,2,3,4,6,5,8,7] => {{1},{2},{3},{4},{5,6},{7,8}}
=> ? = 5 + 1
[8,7,5,6,3,4,1,2] => [8,7,5,6,3,4,1,2] => [1,2,4,3,6,5,8,7] => {{1},{2},{3,4},{5,6},{7,8}}
=> ? = 4 + 1
[7,8,5,6,3,4,1,2] => [7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => {{1,2},{3,4},{5,6},{7,8}}
=> ? = 3 + 1
[5,6,7,8,3,4,1,2] => [5,8,7,6,3,4,1,2] => [4,1,2,3,6,5,8,7] => {{1,2,3,4},{5,6},{7,8}}
=> ? = 2 + 1
[7,8,3,4,5,6,1,2] => [7,8,3,6,5,4,1,2] => [2,1,6,3,4,5,8,7] => {{1,2},{3,4,5,6},{7,8}}
=> ? = 2 + 1
[8,7,6,5,4,2,1,3] => [8,7,6,5,4,2,1,3] => [1,2,3,4,5,7,8,6] => {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 5 + 1
[7,6,8,5,4,2,1,3] => [7,6,8,5,4,2,1,3] => [2,3,1,4,5,7,8,6] => {{1,2,3},{4},{5},{6,7,8}}
=> ? = 3 + 1
[8,7,6,5,4,1,2,3] => [8,7,6,5,4,1,3,2] => [1,2,3,4,5,8,6,7] => {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 5 + 1
[6,7,8,5,4,1,2,3] => [6,8,7,5,4,1,3,2] => [3,1,2,4,5,8,6,7] => {{1,2,3},{4},{5},{6,7,8}}
=> ? = 3 + 1
[8,5,6,7,4,1,2,3] => [8,5,7,6,4,1,3,2] => [1,4,2,3,5,8,6,7] => {{1},{2,3,4},{5},{6,7,8}}
=> ? = 3 + 1
[8,7,6,4,5,1,2,3] => [8,7,6,4,5,1,3,2] => [1,2,3,5,4,8,6,7] => {{1},{2},{3},{4,5},{6,7,8}}
=> ? = 4 + 1
[8,6,7,4,5,1,2,3] => [8,6,7,4,5,1,3,2] => [1,3,2,5,4,8,6,7] => {{1},{2,3},{4,5},{6,7,8}}
=> ? = 3 + 1
[8,7,4,5,6,1,2,3] => [8,7,4,6,5,1,3,2] => [1,2,5,3,4,8,6,7] => {{1},{2},{3,4,5},{6,7,8}}
=> ? = 3 + 1
[7,8,4,5,6,1,2,3] => [7,8,4,6,5,1,3,2] => [2,1,5,3,4,8,6,7] => ?
=> ? = 2 + 1
[6,7,4,5,8,1,2,3] => [6,8,4,7,5,1,3,2] => [3,1,5,2,4,8,6,7] => {{1,2,3,4,5},{6,7,8}}
=> ? = 1 + 1
[7,6,5,8,3,2,1,4] => [7,6,5,8,3,2,1,4] => [2,3,4,1,6,7,8,5] => {{1,2,3,4},{5,6,7,8}}
=> ? = 1 + 1
[7,5,6,8,2,3,1,4] => [7,5,8,6,2,4,1,3] => [2,4,1,3,7,5,8,6] => {{1,2,3,4},{5,6,7,8}}
=> ? = 1 + 1
[6,7,5,8,3,1,2,4] => [6,8,5,7,3,1,4,2] => [3,1,4,2,6,8,5,7] => {{1,2,3,4},{5,6,7,8}}
=> ? = 1 + 1
[6,5,7,8,2,1,3,4] => [6,5,8,7,2,1,4,3] => [3,4,1,2,7,8,5,6] => {{1,3},{2,4},{5,7},{6,8}}
=> ? = 1 + 1
[8,7,6,5,1,2,3,4] => [8,7,6,5,1,4,3,2] => [1,2,3,4,8,5,6,7] => {{1},{2},{3},{4},{5,6,7,8}}
=> ? = 4 + 1
Description
The number of topologically connected components of a set partition.
For example, the set partition $\{\{1,5\},\{2,3\},\{4,6\}\}$ has the two connected components $\{1,4,5,6\}$ and $\{2,3\}$.
The number of set partitions with only one block is [[oeis:A099947]].
Matching statistic: St000011
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 91%
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 91%
Values
[1] => [1] => [1] => [1,0]
=> 1 = 0 + 1
[1,2] => [2,1] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[2,1] => [1,2] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[1,2,3] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 1 = 0 + 1
[2,1,3] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,3,1] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,2] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3] => [4,3,1,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2,4] => [4,2,3,1] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,3,4,2] => [4,2,1,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,4,2,3] => [4,1,3,2] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,4,3,2] => [4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,1,3,4] => [3,4,2,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,1,4,3] => [3,4,1,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[2,3,1,4] => [3,2,4,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,3,4,1] => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,4,1,3] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,4,3,1] => [3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,2,4] => [2,4,3,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,1,4,2] => [2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[3,2,1,4] => [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,2,4,1] => [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,1,3,2] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[4,2,1,3] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [5,4,3,1,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3,5] => [5,4,2,3,1] => [3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,2,4,5,3] => [5,4,2,1,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,5,3,4] => [5,4,1,3,2] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,2,5,4,3] => [5,4,1,2,3] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2,4,5] => [5,3,4,2,1] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[1,3,2,5,4] => [5,3,4,1,2] => [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [5,3,2,4,1] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,3,4,5,2] => [5,3,2,1,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,5,2,4] => [5,3,1,4,2] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,3,5,4,2] => [5,3,1,2,4] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,4,2,3,5] => [5,2,4,3,1] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[1,4,2,5,3] => [5,2,4,1,3] => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[1,4,3,5,2] => [5,2,3,1,4] => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,4,5,2,3] => [5,2,1,4,3] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[7,6,8,5,4,3,2,1] => [2,3,1,4,5,6,7,8] => ? => ?
=> ? = 5 + 1
[7,8,5,6,4,3,2,1] => [2,1,4,3,5,6,7,8] => ? => ?
=> ? = 5 + 1
[7,8,6,4,5,3,2,1] => [2,1,3,5,4,6,7,8] => ? => ?
=> ? = 5 + 1
[8,6,7,4,5,3,2,1] => [1,3,2,5,4,6,7,8] => [1,3,2,5,4,6,7,8] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 5 + 1
[6,5,7,4,8,3,2,1] => [3,4,2,5,1,6,7,8] => ? => ?
=> ? = 3 + 1
[7,8,6,5,3,4,2,1] => [2,1,3,4,6,5,7,8] => ? => ?
=> ? = 5 + 1
[8,6,7,5,3,4,2,1] => [1,3,2,4,6,5,7,8] => ? => ?
=> ? = 5 + 1
[8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => [1,2,4,3,6,5,7,8] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 5 + 1
[6,7,8,3,4,5,2,1] => [3,2,1,6,5,4,7,8] => ? => ?
=> ? = 3 + 1
[7,8,6,5,4,2,3,1] => [2,1,3,4,5,7,6,8] => ? => ?
=> ? = 5 + 1
[8,6,7,5,4,2,3,1] => [1,3,2,4,5,7,6,8] => [1,3,2,4,5,7,6,8] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 5 + 1
[8,7,5,6,4,2,3,1] => [1,2,4,3,5,7,6,8] => ? => ?
=> ? = 5 + 1
[8,7,6,4,5,2,3,1] => [1,2,3,5,4,7,6,8] => [1,2,3,5,4,7,6,8] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 + 1
[8,6,5,7,3,2,4,1] => [1,3,4,2,6,7,5,8] => [1,4,2,3,7,5,6,8] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
[6,7,8,5,2,3,4,1] => [3,2,1,4,7,6,5,8] => [3,2,1,4,7,6,5,8] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
[8,5,6,7,2,3,4,1] => [1,4,3,2,7,6,5,8] => [1,4,3,2,7,6,5,8] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
[7,8,3,2,4,5,6,1] => [2,1,6,7,5,4,3,8] => [2,1,7,5,4,3,6,8] => ?
=> ? = 2 + 1
[6,5,7,4,3,2,8,1] => [3,4,2,5,6,7,1,8] => ? => ?
=> ? = 1 + 1
[6,5,4,3,7,2,8,1] => [3,4,5,6,2,7,1,8] => ? => ?
=> ? = 1 + 1
[3,4,5,2,6,7,8,1] => [6,5,4,7,3,2,1,8] => [7,3,2,1,6,5,4,8] => ?
=> ? = 1 + 1
[8,6,7,5,4,3,1,2] => [1,3,2,4,5,6,8,7] => ? => ?
=> ? = 5 + 1
[8,7,5,6,4,3,1,2] => [1,2,4,3,5,6,8,7] => ? => ?
=> ? = 5 + 1
[8,7,6,4,5,3,1,2] => [1,2,3,5,4,6,8,7] => ? => ?
=> ? = 5 + 1
[8,7,6,5,3,4,1,2] => [1,2,3,4,6,5,8,7] => ? => ?
=> ? = 5 + 1
[8,7,5,6,3,4,1,2] => [1,2,4,3,6,5,8,7] => ? => ?
=> ? = 4 + 1
[7,6,8,5,4,2,1,3] => [2,3,1,4,5,7,8,6] => [3,1,2,4,5,8,6,7] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 + 1
[6,7,8,5,4,1,2,3] => [3,2,1,4,5,8,7,6] => [3,2,1,4,5,8,7,6] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 + 1
[8,5,6,7,4,1,2,3] => [1,4,3,2,5,8,7,6] => ? => ?
=> ? = 3 + 1
[8,7,6,4,5,1,2,3] => [1,2,3,5,4,8,7,6] => ? => ?
=> ? = 4 + 1
[8,6,7,4,5,1,2,3] => [1,3,2,5,4,8,7,6] => ? => ?
=> ? = 3 + 1
[8,7,4,5,6,1,2,3] => [1,2,5,4,3,8,7,6] => ? => ?
=> ? = 3 + 1
[7,8,4,5,6,1,2,3] => [2,1,5,4,3,8,7,6] => ? => ?
=> ? = 2 + 1
[8,7,5,6,1,2,3,4] => [1,2,4,3,8,7,6,5] => [1,2,4,3,8,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[7,8,5,4,2,1,3,6] => [2,1,4,5,7,8,6,3] => [2,1,8,6,3,4,5,7] => ?
=> ? = 1 + 1
[8,7,1,2,3,4,5,6] => [1,2,8,7,6,5,4,3] => ? => ?
=> ? = 2 + 1
[8,5,6,3,4,1,2,7] => [1,4,3,6,5,8,7,2] => [1,8,7,2,4,3,6,5] => ?
=> ? = 1 + 1
[7,6,4,5,3,2,1,8] => [2,3,5,4,6,7,8,1] => ? => ?
=> ? = 0 + 1
[7,6,4,3,5,2,1,8] => [2,3,5,6,4,7,8,1] => ? => ?
=> ? = 0 + 1
[7,6,3,4,5,2,1,8] => [2,3,6,5,4,7,8,1] => ? => ?
=> ? = 0 + 1
[7,5,4,3,6,2,1,8] => [2,4,5,6,3,7,8,1] => ? => ?
=> ? = 0 + 1
[6,7,1,2,3,4,5,8] => [3,2,8,7,6,5,4,1] => ? => ?
=> ? = 0 + 1
[7,1,2,3,4,5,6,8] => [2,8,7,6,5,4,3,1] => ? => ?
=> ? = 0 + 1
[6,5,4,3,2,1,7,8] => [3,4,5,6,7,8,2,1] => ? => ?
=> ? = 0 + 1
[4,5,6,1,2,3,7,8] => [5,4,3,8,7,6,2,1] => ? => ?
=> ? = 0 + 1
[6,1,2,3,4,5,7,8] => [3,8,7,6,5,4,2,1] => ? => ?
=> ? = 0 + 1
[4,3,6,5,8,7,2,1] => [5,6,3,4,1,2,7,8] => [2,4,1,6,3,5,7,8] => [1,1,0,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> ? = 2 + 1
[2,3,5,6,8,7,4,1] => [7,6,4,3,1,2,5,8] => [2,5,7,6,4,3,1,8] => [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 1 + 1
[3,2,6,5,4,8,7,1] => [6,7,3,4,5,1,2,8] => ? => ?
=> ? = 1 + 1
[3,2,6,5,7,4,8,1] => [6,7,3,4,2,5,1,8] => [5,1,4,2,7,3,6,8] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> ? = 1 + 1
[4,3,5,2,7,6,8,1] => [5,6,4,7,2,3,1,8] => [3,1,7,2,6,4,5,8] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1 + 1
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000383
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 64%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 64%
Values
[1] => [1] => ([],1)
=> [1] => 1 = 0 + 1
[1,2] => [2,1] => ([(0,1)],2)
=> [1,1] => 1 = 0 + 1
[2,1] => [1,2] => ([],2)
=> [2] => 2 = 1 + 1
[1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 1 = 0 + 1
[1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> [1,1,1] => 1 = 0 + 1
[2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
=> [1,1,1] => 1 = 0 + 1
[2,3,1] => [1,3,2] => ([(1,2)],3)
=> [1,2] => 2 = 1 + 1
[3,1,2] => [2,1,3] => ([(1,2)],3)
=> [1,2] => 2 = 1 + 1
[3,2,1] => [1,2,3] => ([],3)
=> [3] => 3 = 2 + 1
[1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 1 = 0 + 1
[1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 1 = 0 + 1
[1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 1 = 0 + 1
[1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1 = 0 + 1
[1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1 = 0 + 1
[1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => 1 = 0 + 1
[2,1,3,4] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 1 = 0 + 1
[2,1,4,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => 1 = 0 + 1
[2,3,1,4] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1 = 0 + 1
[2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [2,2] => 2 = 1 + 1
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1 = 0 + 1
[2,4,3,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> [1,1,2] => 2 = 1 + 1
[3,1,2,4] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1 = 0 + 1
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1 = 0 + 1
[3,2,1,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => 1 = 0 + 1
[3,2,4,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> [1,1,2] => 2 = 1 + 1
[3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2] => 2 = 1 + 1
[3,4,2,1] => [1,2,4,3] => ([(2,3)],4)
=> [1,3] => 3 = 2 + 1
[4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [2,2] => 2 = 1 + 1
[4,1,3,2] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> [1,1,2] => 2 = 1 + 1
[4,2,1,3] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> [1,1,2] => 2 = 1 + 1
[4,2,3,1] => [1,3,2,4] => ([(2,3)],4)
=> [1,3] => 3 = 2 + 1
[4,3,1,2] => [2,1,3,4] => ([(2,3)],4)
=> [1,3] => 3 = 2 + 1
[4,3,2,1] => [1,2,3,4] => ([],4)
=> [4] => 4 = 3 + 1
[1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => 1 = 0 + 1
[1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 1 = 0 + 1
[1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 1 = 0 + 1
[1,2,4,5,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 1 = 0 + 1
[1,2,5,3,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 1 = 0 + 1
[1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 1 = 0 + 1
[1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 1 = 0 + 1
[1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 1 = 0 + 1
[1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 1 = 0 + 1
[1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => 1 = 0 + 1
[1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1 = 0 + 1
[1,3,5,4,2] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1 = 0 + 1
[1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 1 = 0 + 1
[1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1 = 0 + 1
[1,4,3,2,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 1 = 0 + 1
[1,4,3,5,2] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1 = 0 + 1
[1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => 1 = 0 + 1
[8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => ([],8)
=> [8] => ? = 7 + 1
[7,6,8,5,4,3,2,1] => [1,2,3,4,5,8,6,7] => ([(5,7),(6,7)],8)
=> ? => ? = 5 + 1
[7,8,5,6,4,3,2,1] => [1,2,3,4,6,5,8,7] => ([(4,7),(5,6)],8)
=> [2,6] => ? = 5 + 1
[7,8,6,4,5,3,2,1] => [1,2,3,5,4,6,8,7] => ([(4,7),(5,6)],8)
=> [2,6] => ? = 5 + 1
[8,6,7,4,5,3,2,1] => [1,2,3,5,4,7,6,8] => ([(4,7),(5,6)],8)
=> [2,6] => ? = 5 + 1
[7,6,5,4,8,3,2,1] => [1,2,3,8,4,5,6,7] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> [1,3,4] => ? = 3 + 1
[6,5,7,4,8,3,2,1] => [1,2,3,8,4,7,5,6] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3 + 1
[7,8,6,5,3,4,2,1] => [1,2,4,3,5,6,8,7] => ([(4,7),(5,6)],8)
=> [2,6] => ? = 5 + 1
[8,6,7,5,3,4,2,1] => [1,2,4,3,5,7,6,8] => ([(4,7),(5,6)],8)
=> [2,6] => ? = 5 + 1
[8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => ([(4,7),(5,6)],8)
=> [2,6] => ? = 5 + 1
[7,8,6,5,4,2,3,1] => [1,3,2,4,5,6,8,7] => ([(4,7),(5,6)],8)
=> [2,6] => ? = 5 + 1
[8,6,7,5,4,2,3,1] => [1,3,2,4,5,7,6,8] => ([(4,7),(5,6)],8)
=> [2,6] => ? = 5 + 1
[8,7,5,6,4,2,3,1] => [1,3,2,4,6,5,7,8] => ([(4,7),(5,6)],8)
=> [2,6] => ? = 5 + 1
[8,7,6,4,5,2,3,1] => [1,3,2,5,4,6,7,8] => ([(4,7),(5,6)],8)
=> [2,6] => ? = 5 + 1
[8,6,5,7,3,2,4,1] => [1,4,2,3,7,5,6,8] => ([(2,7),(3,7),(4,6),(5,6)],8)
=> [2,2,4] => ? = 3 + 1
[7,8,3,2,4,5,6,1] => [1,6,5,4,2,3,8,7] => ([(1,2),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2 + 1
[7,6,5,4,3,2,8,1] => [1,8,2,3,4,5,6,7] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? => ? = 1 + 1
[5,4,6,3,7,2,8,1] => [1,8,2,7,3,6,4,5] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,1,1,1,1,1,2] => ? = 1 + 1
[4,3,2,5,6,7,8,1] => [1,8,7,6,5,2,3,4] => ([(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,2,2] => ? = 1 + 1
[3,4,2,5,6,7,8,1] => [1,8,7,6,5,2,4,3] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,1,1,2] => ? = 1 + 1
[4,2,3,5,6,7,8,1] => [1,8,7,6,5,3,2,4] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,1,1,2] => ? = 1 + 1
[3,2,4,5,6,7,8,1] => [1,8,7,6,5,4,2,3] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [5,1,2] => ? = 1 + 1
[2,3,4,5,6,7,8,1] => [1,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [6,2] => ? = 1 + 1
[8,7,6,5,4,3,1,2] => [2,1,3,4,5,6,7,8] => ([(6,7)],8)
=> ? => ? = 6 + 1
[7,8,6,5,4,3,1,2] => [2,1,3,4,5,6,8,7] => ([(4,7),(5,6)],8)
=> [2,6] => ? = 5 + 1
[8,6,7,5,4,3,1,2] => [2,1,3,4,5,7,6,8] => ([(4,7),(5,6)],8)
=> [2,6] => ? = 5 + 1
[8,7,5,6,4,3,1,2] => [2,1,3,4,6,5,7,8] => ([(4,7),(5,6)],8)
=> [2,6] => ? = 5 + 1
[8,7,6,4,5,3,1,2] => [2,1,3,5,4,6,7,8] => ([(4,7),(5,6)],8)
=> [2,6] => ? = 5 + 1
[8,7,6,5,3,4,1,2] => [2,1,4,3,5,6,7,8] => ([(4,7),(5,6)],8)
=> [2,6] => ? = 5 + 1
[8,7,5,6,3,4,1,2] => [2,1,4,3,6,5,7,8] => ([(2,7),(3,6),(4,5)],8)
=> ? => ? = 4 + 1
[3,4,5,6,7,8,1,2] => [2,1,8,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [5,1,2] => ? = 1 + 1
[8,7,6,5,4,2,1,3] => [3,1,2,4,5,6,7,8] => ([(5,7),(6,7)],8)
=> ? => ? = 5 + 1
[7,6,8,5,4,2,1,3] => [3,1,2,4,5,8,6,7] => ([(2,7),(3,7),(4,6),(5,6)],8)
=> [2,2,4] => ? = 3 + 1
[8,7,6,5,4,1,2,3] => [3,2,1,4,5,6,7,8] => ([(5,6),(5,7),(6,7)],8)
=> ? => ? = 5 + 1
[8,7,6,4,5,1,2,3] => [3,2,1,5,4,6,7,8] => ([(3,4),(5,6),(5,7),(6,7)],8)
=> ? => ? = 4 + 1
[8,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,8] => ([(1,4),(2,3),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3 + 1
[7,8,4,5,6,1,2,3] => [3,2,1,6,5,4,8,7] => ([(0,1),(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
=> ? => ? = 2 + 1
[6,5,7,8,2,1,3,4] => [4,3,1,2,8,7,5,6] => ([(0,6),(0,7),(1,4),(1,5),(2,4),(2,5),(3,6),(3,7),(4,5),(6,7)],8)
=> [4,2,2] => ? = 1 + 1
[8,7,6,5,1,2,3,4] => [4,3,2,1,5,6,7,8] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [3,5] => ? = 4 + 1
[8,7,5,6,1,2,3,4] => [4,3,2,1,6,5,7,8] => ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [3,1,4] => ? = 3 + 1
[5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4),(5,6),(5,7),(6,7)],8)
=> [6,2] => ? = 1 + 1
[8,7,6,4,3,2,1,5] => [5,1,2,3,4,6,7,8] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> [1,3,4] => ? = 3 + 1
[8,7,6,4,2,1,3,5] => [5,3,1,2,4,6,7,8] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3 + 1
[8,7,6,1,2,3,4,5] => [5,4,3,2,1,6,7,8] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3 + 1
[8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8] => ([(1,2),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2 + 1
[6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,2,2] => ? = 1 + 1
[7,8,5,4,1,2,3,6] => [6,3,2,1,4,5,8,7] => ([(0,1),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 1 + 1
[8,7,1,2,3,4,5,6] => [6,5,4,3,2,1,7,8] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2 + 1
[7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [5,1,2] => ? = 1 + 1
[8,6,5,4,3,2,1,7] => [7,1,2,3,4,5,6,8] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? => ? = 1 + 1
Description
The last part of an integer composition.
Matching statistic: St000678
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 73%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 73%
Values
[1] => [1] => [1,0]
=> [1,0]
=> ? = 0 + 1
[1,2] => [2,1] => [1,1,0,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[2,1] => [1,2] => [1,0,1,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,5,3,2] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[7,6,8,5,4,3,2,1] => [1,2,3,4,5,8,6,7] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 5 + 1
[7,8,5,6,4,3,2,1] => [1,2,3,4,6,5,8,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 5 + 1
[7,8,6,4,5,3,2,1] => [1,2,3,5,4,6,8,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 5 + 1
[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]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 5 + 1
[7,6,5,4,8,3,2,1] => [1,2,3,8,4,5,6,7] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 3 + 1
[6,5,7,4,8,3,2,1] => [1,2,3,8,4,7,5,6] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 3 + 1
[7,8,6,5,3,4,2,1] => [1,2,4,3,5,6,8,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 5 + 1
[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]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 5 + 1
[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]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 5 + 1
[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]
=> [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
=> ? = 3 + 1
[7,8,6,5,4,2,3,1] => [1,3,2,4,5,6,8,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 + 1
[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]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 5 + 1
[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]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 5 + 1
[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]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 5 + 1
[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]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> ? = 3 + 1
[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]
=> [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
=> ? = 3 + 1
[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]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> ? = 3 + 1
[7,8,3,2,4,5,6,1] => [1,6,5,4,2,3,8,7] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> ? = 2 + 1
[7,6,5,4,3,2,8,1] => [1,8,2,3,4,5,6,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[6,5,7,4,3,2,8,1] => [1,8,2,3,4,7,5,6] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[6,5,4,3,7,2,8,1] => [1,8,2,7,3,4,5,6] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[5,4,6,3,7,2,8,1] => [1,8,2,7,3,6,4,5] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[3,4,5,2,6,7,8,1] => [1,8,7,6,2,5,4,3] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[5,2,3,4,6,7,8,1] => [1,8,7,6,4,3,2,5] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[4,3,2,5,6,7,8,1] => [1,8,7,6,5,2,3,4] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[3,4,2,5,6,7,8,1] => [1,8,7,6,5,2,4,3] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[4,2,3,5,6,7,8,1] => [1,8,7,6,5,3,2,4] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[3,2,4,5,6,7,8,1] => [1,8,7,6,5,4,2,3] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[2,3,4,5,6,7,8,1] => [1,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[7,8,6,5,4,3,1,2] => [2,1,3,4,5,6,8,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 + 1
[8,6,7,5,4,3,1,2] => [2,1,3,4,5,7,6,8] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 5 + 1
[8,7,5,6,4,3,1,2] => [2,1,3,4,6,5,7,8] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 5 + 1
[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]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 5 + 1
[8,7,6,5,3,4,1,2] => [2,1,4,3,5,6,7,8] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5 + 1
[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]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 4 + 1
[7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3 + 1
[5,6,7,8,3,4,1,2] => [2,1,4,3,8,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 2 + 1
[7,8,3,4,5,6,1,2] => [2,1,6,5,4,3,8,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> ? = 2 + 1
[3,4,5,6,7,8,1,2] => [2,1,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 1 + 1
[8,7,6,5,4,2,1,3] => [3,1,2,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5 + 1
[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]
=> [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
=> ? = 3 + 1
[8,7,6,5,4,1,2,3] => [3,2,1,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5 + 1
[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]
=> [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
=> ? = 3 + 1
[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]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> ? = 3 + 1
[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]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 4 + 1
[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]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 3 + 1
[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]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
[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]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 2 + 1
[6,7,4,5,8,1,2,3] => [3,2,1,8,5,4,7,6] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 1 + 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000675
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000675: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 64%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000675: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 64%
Values
[1] => [1] => [1,0]
=> [1,0]
=> ? = 0 + 1
[1,2] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[2,1] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 2 = 1 + 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,4,5,3,2] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [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]
=> ? = 7 + 1
[7,6,8,5,4,3,2,1] => [1,2,3,4,5,8,6,7] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 5 + 1
[7,8,5,6,4,3,2,1] => [1,2,3,4,6,5,8,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 5 + 1
[7,8,6,4,5,3,2,1] => [1,2,3,5,4,6,8,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 5 + 1
[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]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 5 + 1
[7,6,5,4,8,3,2,1] => [1,2,3,8,4,5,6,7] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 3 + 1
[6,5,7,4,8,3,2,1] => [1,2,3,8,4,7,5,6] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 3 + 1
[7,8,6,5,3,4,2,1] => [1,2,4,3,5,6,8,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 5 + 1
[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]
=> [1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 5 + 1
[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]
=> [1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 5 + 1
[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]
=> [1,1,1,1,0,1,1,0,0,1,0,0,1,0,0,0]
=> ? = 3 + 1
[7,8,6,5,4,2,3,1] => [1,3,2,4,5,6,8,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 5 + 1
[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]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 5 + 1
[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]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 5 + 1
[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]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 5 + 1
[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]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 3 + 1
[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]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 3 + 1
[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]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 3 + 1
[7,8,3,2,4,5,6,1] => [1,6,5,4,2,3,8,7] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> ? = 2 + 1
[7,6,5,4,3,2,8,1] => [1,8,2,3,4,5,6,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[6,5,7,4,3,2,8,1] => [1,8,2,3,4,7,5,6] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[6,5,4,3,7,2,8,1] => [1,8,2,7,3,4,5,6] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[5,4,6,3,7,2,8,1] => [1,8,2,7,3,6,4,5] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[3,4,5,2,6,7,8,1] => [1,8,7,6,2,5,4,3] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[5,2,3,4,6,7,8,1] => [1,8,7,6,4,3,2,5] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[4,3,2,5,6,7,8,1] => [1,8,7,6,5,2,3,4] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[3,4,2,5,6,7,8,1] => [1,8,7,6,5,2,4,3] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[4,2,3,5,6,7,8,1] => [1,8,7,6,5,3,2,4] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[3,2,4,5,6,7,8,1] => [1,8,7,6,5,4,2,3] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[2,3,4,5,6,7,8,1] => [1,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[8,7,6,5,4,2,1,3] => [3,1,2,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 5 + 1
[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]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 3 + 1
[8,7,6,5,4,1,2,3] => [3,2,1,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 5 + 1
[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]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 3 + 1
[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]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 3 + 1
[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]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 4 + 1
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 + 1
[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]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> ? = 3 + 1
[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]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
[6,7,4,5,8,1,2,3] => [3,2,1,8,5,4,7,6] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 1 + 1
[7,6,5,8,3,2,1,4] => [4,1,2,3,8,5,6,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 1 + 1
[7,5,6,8,2,3,1,4] => [4,1,3,2,8,6,5,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 1 + 1
[6,7,5,8,3,1,2,4] => [4,2,1,3,8,5,7,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 1 + 1
[6,5,7,8,2,1,3,4] => [4,3,1,2,8,7,5,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 1 + 1
[8,7,6,5,1,2,3,4] => [4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 4 + 1
[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]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0,1,0]
=> ? = 3 + 1
[7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 2 + 1
[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]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 2 + 1
[5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 1 + 1
Description
The number of centered multitunnels of a Dyck path.
This is the number of factorisations $D = A B C$ of a Dyck path, such that $B$ is a Dyck path and $A$ and $B$ have the same length.
The following 34 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000234The number of global ascents of a permutation. St000544The cop number of a graph. St001363The Euler characteristic of a graph according to Knill. St000717The number of ordinal summands of a poset. St000553The number of blocks of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001828The Euler characteristic of a graph. St000382The first part of an integer composition. St000843The decomposition number of a perfect matching. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001461The number of topologically connected components of the chord diagram of a permutation. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. 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. St000061The number of nodes on the left branch of a binary tree. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. 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 $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St000264The girth of a graph, which is not a tree. St000456The monochromatic index of a connected graph. St000181The number of connected components of the Hasse diagram for the poset. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001889The size of the connectivity set of a signed permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's.
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