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Mp00017: Binary trees to 312-avoiding permutationPermutations
St000054: 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
[[.,.],[.,.]]
=> [1,3,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => 2
[[[.,.],.],.]
=> [1,2,3] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 4
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 3
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 2
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 5
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 4
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 3
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 4
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 3
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 2
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 2
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 3
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 4
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 3
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 3
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 2
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 3
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 1
Description
The first entry of the permutation. This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1]. This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals $$ \frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1). $$
Mp00017: Binary trees to 312-avoiding permutationPermutations
Mp00131: Permutations descent bottomsBinary words
St000297: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => => ? = 1 - 1
[.,[.,.]]
=> [2,1] => 1 => 1 = 2 - 1
[[.,.],.]
=> [1,2] => 0 => 0 = 1 - 1
[.,[.,[.,.]]]
=> [3,2,1] => 11 => 2 = 3 - 1
[.,[[.,.],.]]
=> [2,3,1] => 10 => 1 = 2 - 1
[[.,.],[.,.]]
=> [1,3,2] => 01 => 0 = 1 - 1
[[.,[.,.]],.]
=> [2,1,3] => 10 => 1 = 2 - 1
[[[.,.],.],.]
=> [1,2,3] => 00 => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 111 => 3 = 4 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 110 => 2 = 3 - 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => 101 => 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 110 => 2 = 3 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 100 => 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => 011 => 0 = 1 - 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => 010 => 0 = 1 - 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => 101 => 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => 001 => 0 = 1 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 110 => 2 = 3 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 100 => 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => 010 => 0 = 1 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 100 => 1 = 2 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => 000 => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 1111 => 4 = 5 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 1110 => 3 = 4 - 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 1101 => 2 = 3 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 1110 => 3 = 4 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 1100 => 2 = 3 - 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 1011 => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 1010 => 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 1101 => 2 = 3 - 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 1001 => 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 1110 => 3 = 4 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 1100 => 2 = 3 - 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 1010 => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 1100 => 2 = 3 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1000 => 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 0111 => 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 0110 => 0 = 1 - 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 0101 => 0 = 1 - 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 0110 => 0 = 1 - 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 0100 => 0 = 1 - 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 1011 => 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 1010 => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 0011 => 0 = 1 - 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 0010 => 0 = 1 - 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 1101 => 2 = 3 - 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 1001 => 1 = 2 - 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 0101 => 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 1001 => 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 0001 => 0 = 1 - 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1110 => 3 = 4 - 1
Description
The number of leading ones in a binary word.
Mp00014: Binary trees to 132-avoiding permutationPermutations
Mp00064: Permutations reversePermutations
Mp00130: Permutations descent topsBinary words
St000326: Binary words ⟶ ℤResult quality: 99% values known / values provided: 99%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
[[.,.],[.,.]]
=> [3,1,2] => [2,1,3] => 10 => 1
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => 01 => 2
[[[.,.],.],.]
=> [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
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => 010 => 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => 001 => 3
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => 011 => 2
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,3,4] => 100 => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => 101 => 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,2,4] => 010 => 2
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,2,1,4] => 110 => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => 001 => 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => 011 => 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [4,2,1,3] => 101 => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => 011 => 2
[[[[.,.],.],.],.]
=> [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
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,4,3,5] => 0010 => 3
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => 0001 => 4
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => 0011 => 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,3,2,4,5] => 0100 => 2
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,3,2,5,4] => 0101 => 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,4,2,3,5] => 0010 => 3
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => 0110 => 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => 0001 => 4
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => 0011 => 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,5,3,2,4] => 0101 => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => 0011 => 3
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => 0111 => 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [2,1,3,4,5] => 1000 => 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,3,5,4] => 1001 => 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [2,1,4,3,5] => 1010 => 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [2,1,5,3,4] => 1001 => 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [2,1,5,4,3] => 1011 => 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [3,1,2,4,5] => 0100 => 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,1,2,5,4] => 0101 => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,2,1,4,5] => 1100 => 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [3,2,1,5,4] => 1101 => 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,1,2,3,5] => 0010 => 3
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,1,3,2,5] => 0110 => 2
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [4,2,1,3,5] => 1010 => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [4,3,1,2,5] => 0110 => 2
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [4,3,2,1,5] => 1110 => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [5,1,2,3,4] => 0001 => 4
[[.,.],[[[.,[.,.]],[.,.]],[.,.]]]
=> [8,6,4,3,5,7,1,2] => [2,1,7,5,3,4,6,8] => ? => ? = 1
[[.,[[.,[.,.]],.]],[[.,.],[.,.]]]
=> [8,6,7,3,2,4,1,5] => [5,1,4,2,3,7,6,8] => ? => ? = 3
[[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]],[.,.]]
=> [12,10,8,6,4,2,1,3,5,7,9,11] => [11,9,7,5,3,1,2,4,6,8,10,12] => ? => ? = 2
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.
Mp00017: Binary trees to 312-avoiding permutationPermutations
Mp00059: Permutations Robinson-Schensted insertion tableauStandard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 99% values known / values provided: 99%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
[[.,.],[.,.]]
=> [1,3,2] => [[1,2],[3]]
=> 1
[[.,[.,.]],.]
=> [2,1,3] => [[1,3],[2]]
=> 2
[[[.,.],.],.]
=> [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
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [[1,3],[2],[4]]
=> 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[1,4],[2],[3]]
=> 3
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [[1,3,4],[2]]
=> 2
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [[1,2],[3],[4]]
=> 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [[1,2,4],[3]]
=> 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [[1,3,4],[2]]
=> 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 2
[[[[.,.],.],.],.]
=> [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
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [[1,4],[2],[3],[5]]
=> 3
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[1,5],[2],[3],[4]]
=> 4
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[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
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [[1,4],[2,5],[3]]
=> 3
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [[1,3,4],[2],[5]]
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[1,5],[2],[3],[4]]
=> 4
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [[1,4,5],[2],[3]]
=> 3
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [[1,3,5],[2],[4]]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [[1,4,5],[2],[3]]
=> 3
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [[1,3,4,5],[2]]
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [[1,2,5],[3],[4]]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [[1,2,4],[3],[5]]
=> 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [[1,2,5],[3],[4]]
=> 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [[1,2,4,5],[3]]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [[1,3,5],[2,4]]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [[1,2,3,5],[4]]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 3
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [[1,3,4],[2,5]]
=> 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
=> [1,2,3,4,5,6,7,8,9,11,10] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 1
[.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,9,10,11,1] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 2
[[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]
=> [10,9,8,7,6,5,4,3,2,1,11] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 10
[[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [1,11,10,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 1
[[[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.],.]
=> [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 2
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Mp00020: Binary trees to Tamari-corresponding Dyck pathDyck paths
Mp00102: Dyck paths rise compositionInteger compositions
St000382: Integer compositions ⟶ ℤResult quality: 96% values known / values provided: 96%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1] => 1
[.,[.,.]]
=> [1,1,0,0]
=> [2] => 2
[[.,.],.]
=> [1,0,1,0]
=> [1,1] => 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [3] => 3
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [2,1] => 2
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,2] => 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2,1] => 2
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => 3
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => 2
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => 3
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => 2
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 2
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => 4
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => 3
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => 4
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => 3
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 2
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 2
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => 3
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 2
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => 4
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => 3
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 2
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => 3
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => 2
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 3
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 2
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 1
[.,[[[[[[[.,[.,.]],.],.],.],.],.],.]]
=> ?
=> ? => ? = 3
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
=> ?
=> ? => ? = 1
[.,[[[[[[.,[[.,.],.]],.],.],.],.],.]]
=> ?
=> ? => ? = 3
[.,[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]]
=> ?
=> ? => ? = 3
[.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]]
=> ?
=> ? => ? = 2
[[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ?
=> ? => ? = 1
[[.,.],[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> ?
=> ? => ? = 1
[[.,.],[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> ?
=> ? => ? = 1
[[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ?
=> ? => ? = 1
[[.,[.,.]],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ?
=> ? => ? = 2
[[.,[.,[.,[.,.]]]],[.,[.,[.,[.,[.,.]]]]]]
=> ?
=> ? => ? = 4
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,[.,.]]]]
=> ?
=> ? => ? = 6
[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],[.,.]]
=> ?
=> ? => ? = 8
[[[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.],.]
=> ?
=> ? => ? = 2
[[[.,[[[[[[.,.],.],.],.],.],.]],.],.]
=> ?
=> ? => ? = 2
[[.,[[[[[[[[.,.],.],.],.],.],.],.],.]],.]
=> ?
=> ? => ? = 2
Description
The first part of an integer composition.
Mp00017: Binary trees to 312-avoiding permutationPermutations
Mp00068: Permutations Simion-Schmidt mapPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St000439: Dyck paths ⟶ ℤResult quality: 89% values known / values provided: 89%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1,0]
=> 2 = 1 + 1
[.,[.,.]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 3 = 2 + 1
[[.,.],.]
=> [1,2] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[.,[[.,.],.]]
=> [2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 3 = 2 + 1
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 4 = 3 + 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 5 = 4 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 4 = 3 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]
=> [4,3,6,5,8,7,2,1] => [4,3,8,7,6,5,2,1] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 + 1
[.,[[[.,[.,.]],[.,[.,.]]],[.,.]]]
=> [3,2,6,5,4,8,7,1] => [3,2,8,7,6,5,4,1] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [6,5,4,3,7,2,8,1] => [6,5,4,3,8,2,7,1] => [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> ? = 6 + 1
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [4,3,6,5,7,2,8,1] => [4,3,8,7,6,2,5,1] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 + 1
[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [3,2,6,5,7,4,8,1] => [3,2,8,7,6,5,4,1] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [4,3,5,2,7,6,8,1] => [4,3,8,2,7,6,5,1] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 + 1
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [3,2,5,4,7,6,8,1] => [3,2,8,7,6,5,4,1] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [4,3,2,5,6,7,8,1] => [4,3,2,8,7,6,5,1] => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,1] => [3,2,8,7,6,5,4,1] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [3,2,1,8,7,6,5,4] => [3,2,1,8,7,6,5,4] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
=> [3,2,1,6,5,8,7,4] => [3,2,1,8,7,6,5,4] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[.,[[.,[.,.]],.]],[[.,.],[.,.]]]
=> [3,2,4,1,6,8,7,5] => [3,2,8,1,7,6,5,4] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [3,2,4,1,7,6,8,5] => [3,2,8,1,7,6,5,4] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [5,4,3,2,1,8,7,6] => [5,4,3,2,1,8,7,6] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 + 1
[[.,[[.,[.,.]],[.,.]]],[.,[.,.]]]
=> [3,2,5,4,1,8,7,6] => [3,2,8,7,1,6,5,4] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [3,2,1,4,5,8,7,6] => [3,2,1,8,7,6,5,4] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
=> [4,3,5,2,6,1,8,7] => [4,3,8,2,7,1,6,5] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 + 1
[[.,[[[.,[.,.]],[.,.]],.]],[.,.]]
=> [3,2,5,4,6,1,8,7] => [3,2,8,7,6,1,5,4] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[[[.,[[.,[.,.]],.]],[.,.]],[.,.]]
=> [3,2,4,1,6,5,8,7] => [3,2,8,1,7,6,5,4] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[[.,[[.,[.,.]],[.,[.,[.,.]]]]],.]
=> [3,2,7,6,5,4,1,8] => [3,2,8,7,6,5,1,4] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[[.,[[.,[.,[.,[.,.]]]],[.,.]]],.]
=> [5,4,3,2,7,6,1,8] => [5,4,3,2,8,7,1,6] => [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? = 5 + 1
[[.,[[[.,[.,.]],[.,.]],[.,.]]],.]
=> [3,2,5,4,7,6,1,8] => [3,2,8,7,6,5,1,4] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> [5,4,3,2,1,6,7,8] => [5,4,3,2,1,8,7,6] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 + 1
[[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> [3,2,1,4,5,6,7,8] => [3,2,1,8,7,6,5,4] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]],[.,.]]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 2 + 1
[.,[[[[[[[.,[.,.]],.],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,9,1] => [3,2,9,8,7,6,5,4,1] => [1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 3 + 1
[.,[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,9,10,1] => [3,2,10,9,8,7,6,5,4,1] => [1,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 3 + 1
[.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,9,10,11,1] => [2,11,10,9,8,7,6,5,4,3,1] => [1,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 2 + 1
[[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]],.]
=> [8,9,7,6,5,4,3,2,1,10] => [8,10,7,6,5,4,3,2,1,9] => [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 8 + 1
[[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]],.]
=> [7,6,8,5,4,3,2,1,9] => [7,6,9,5,4,3,2,1,8] => [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> ? = 7 + 1
[[.,[.,[.,[.,.]]]],[.,[.,[.,[.,[.,.]]]]]]
=> [4,3,2,1,10,9,8,7,6,5] => [4,3,2,1,10,9,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 + 1
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,[.,.]]]]
=> [6,5,4,3,2,1,10,9,8,7] => [6,5,4,3,2,1,10,9,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 6 + 1
[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.]
=> [6,5,4,3,2,1,7,8,9] => [6,5,4,3,2,1,9,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 6 + 1
[[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.]
=> [5,4,3,2,1,6,7,8,9] => [5,4,3,2,1,9,8,7,6] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 + 1
[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.]
=> [4,3,2,1,5,6,7,8,9] => [4,3,2,1,9,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[[[[[[[.,[.,[.,.]]],.],.],.],.],.],.]
=> [3,2,1,4,5,6,7,8,9] => [3,2,1,9,8,7,6,5,4] => [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.],.]
=> [7,6,5,4,3,2,1,8,9,10] => [7,6,5,4,3,2,1,10,9,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 7 + 1
[[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.],.]
=> [6,5,4,3,2,1,7,8,9,10] => [6,5,4,3,2,1,10,9,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 6 + 1
[[[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.],.]
=> [5,4,3,2,1,6,7,8,9,10] => [5,4,3,2,1,10,9,8,7,6] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5 + 1
[[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.],.]
=> [4,3,2,1,5,6,7,8,9,10] => [4,3,2,1,10,9,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 + 1
[[[[[[[[.,[.,[.,.]]],.],.],.],.],.],.],.]
=> [3,2,1,4,5,6,7,8,9,10] => [3,2,1,10,9,8,7,6,5,4] => [1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 3 + 1
Description
The position of the first down step of a Dyck path.
Mp00018: Binary trees left border symmetryBinary trees
Mp00012: Binary trees to Dyck path: up step, left tree, down step, right treeDyck paths
St000011: Dyck paths ⟶ ℤResult quality: 89% values known / values provided: 89%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [.,.]
=> [1,0]
=> 1
[.,[.,.]]
=> [.,[.,.]]
=> [1,0,1,0]
=> 2
[[.,.],.]
=> [[.,.],.]
=> [1,1,0,0]
=> 1
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 3
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[[[.,.],.],.]
=> [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 4
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 3
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 3
[.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 2
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 1
[[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 1
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 1
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3
[[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 1
[[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 2
[[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 6
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [.,[.,[[.,[[.,[.,[.,.]]],.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3
[.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]
=> [.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> ? = 4
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2
[.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [.,[[[.,[[[.,[.,.]],.],.]],.],.]]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 2
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 6
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 6
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [.,[[.,.],[[[.,.],[.,.]],[.,.]]]]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 4
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 3
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 3
[[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [[.,[[.,[[.,[[.,.],.]],.]],.]],.]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 1
[[.,.],[[[.,[.,.]],[.,.]],[.,.]]]
=> [[.,[[[.,[.,.]],[.,.]],[.,.]]],.]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 1
[[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [[.,[[[.,.],[.,.]],[.,.]]],[.,.]]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
=> [[.,[[.,[.,.]],[.,.]]],[.,[.,.]]]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 3
[[.,[.,[[.,.],.]]],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],[.,[[.,.],.]]]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 3
[[.,[[.,.],[.,.]]],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],[[.,[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> ? = 2
[[.,[[.,[.,.]],.]],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],[[.,.],[.,.]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> ? = 3
[[.,[[[.,.],.],.]],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],[[[.,.],.],.]]
=> [1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 2
[[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> [[[.,[[.,.],[.,.]]],[.,.]],[.,.]]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 3
[[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
=> [[[.,[.,.]],[[.,.],[.,.]]],[.,.]]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 2
[[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 4
[[[.,[[.,[.,.]],.]],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],[[.,.],[.,.]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 3
[[.,[[.,[.,.]],[.,[.,[.,.]]]]],.]
=> [[.,.],[[.,[.,[.,[.,.]]]],[.,.]]]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
[[.,[[[.,[.,.]],[.,.]],[.,.]]],.]
=> [[.,.],[[[.,[.,.]],[.,.]],[.,.]]]
=> [1,1,0,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> ? = 3
[[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 1
[[[[.,.],[[.,.],.]],[[.,.],.]],.]
=> [[[[.,.],[[.,.],.]],[[.,.],.]],.]
=> [1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 1
[[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 1
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
=> [[[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.],.]
=> ?
=> ? = 1
[.,[[[[[[.,[[.,.],.]],.],.],.],.],.]]
=> [.,[[[[[[.,.],.],.],.],.],[[.,.],.]]]
=> ?
=> ? = 3
[.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]]
=> [.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]]
=> ?
=> ? = 2
[[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]],.]
=> [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ?
=> ? = 7
[[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]
=> [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ?
=> ? = 10
[[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]],.]
=> [[.,.],[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> ?
=> ? = 8
[[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]],.]
=> [[.,.],[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]],.]
=> ?
=> ? = 1
[[.,[.,.]],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],[.,.]]
=> ?
=> ? = 2
[[.,[.,[.,[.,.]]]],[.,[.,[.,[.,[.,.]]]]]]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,[.,.]]]]
=> ?
=> ? = 4
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],[.,[.,[.,[.,[.,.]]]]]]
=> ?
=> ? = 6
[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],[.,.]]
=> [[.,[.,.]],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ?
=> ? = 8
[[[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.],.]
=> [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
=> ?
=> ? = 2
[[[.,[[[[[[.,.],.],.],.],.],.]],.],.]
=> [[[.,.],.],[[[[[[.,.],.],.],.],.],.]]
=> ?
=> ? = 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.
Mp00018: Binary trees left border symmetryBinary trees
Mp00017: Binary trees to 312-avoiding permutationPermutations
St000007: Permutations ⟶ ℤResult quality: 84% values known / values provided: 84%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [.,.]
=> [1] => 1
[.,[.,.]]
=> [.,[.,.]]
=> [2,1] => 2
[[.,.],.]
=> [[.,.],.]
=> [1,2] => 1
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 3
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> [2,3,1] => 2
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> [2,1,3] => 1
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> [1,3,2] => 2
[[[.,.],.],.]
=> [[[.,.],.],.]
=> [1,2,3] => 1
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 4
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 3
[.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1
[[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2
[[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 3
[[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => 2
[[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => 1
[[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => 2
[[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 5
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 4
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 3
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 4
[.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 3
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 2
[.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 2
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 3
[.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 2
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 4
[.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 3
[.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 2
[.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 3
[.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 2
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1
[[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 1
[[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => 1
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => 1
[[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 1
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 2
[[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 2
[[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 1
[[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => 1
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 3
[[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 2
[[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 1
[[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 2
[[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 1
[.,[[.,[[[[.,.],.],.],.]],.]]
=> [.,[[.,.],[[[[.,.],.],.],.]]]
=> [2,4,5,6,7,3,1] => ? = 3
[.,[[[.,[[.,[.,.]],.]],.],.]]
=> [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => ? = 4
[.,[[[.,[[[.,.],.],.]],.],.]]
=> [.,[[[.,.],.],[[[.,.],.],.]]]
=> [2,3,5,6,7,4,1] => ? = 3
[.,[[[[.,[.,[.,.]]],.],.],.]]
=> [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [2,3,4,7,6,5,1] => ? = 4
[.,[[[[.,[[.,.],.]],.],.],.]]
=> [.,[[[[.,.],.],.],[[.,.],.]]]
=> [2,3,4,6,7,5,1] => ? = 3
[.,[[[[[.,[.,.]],.],.],.],.]]
=> [.,[[[[[.,.],.],.],.],[.,.]]]
=> [2,3,4,5,7,6,1] => ? = 3
[[.,[[[[.,.],.],.],.]],[.,.]]
=> [[.,[.,.]],[[[[.,.],.],.],.]]
=> [2,1,4,5,6,7,3] => ? = 2
[[[.,[[[.,.],.],.]],.],[.,.]]
=> [[[.,[.,.]],.],[[[.,.],.],.]]
=> [2,1,3,5,6,7,4] => ? = 2
[[.,[[[[[.,.],.],.],.],.]],.]
=> [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[[[.,[.,[.,[.,[.,.]]]]],.],.]
=> [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [1,2,7,6,5,4,3] => ? = 5
[[[.,[[[[.,.],.],.],.]],.],.]
=> [[[.,.],.],[[[[.,.],.],.],.]]
=> [1,2,4,5,6,7,3] => ? = 2
[[[[.,[.,[.,[.,.]]]],.],.],.]
=> [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,7,6,5,4] => ? = 4
[[[[.,[[[.,.],.],.]],.],.],.]
=> [[[[.,.],.],.],[[[.,.],.],.]]
=> [1,2,3,5,6,7,4] => ? = 2
[[[[[.,[.,[.,.]]],.],.],.],.]
=> [[[[[.,.],.],.],.],[.,[.,.]]]
=> [1,2,3,4,7,6,5] => ? = 3
[[[[[.,[[.,.],.]],.],.],.],.]
=> [[[[[.,.],.],.],.],[[.,.],.]]
=> [1,2,3,4,6,7,5] => ? = 2
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [6,8,7,5,4,3,2,1] => ? = 7
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [4,8,7,6,5,3,2,1] => ? = 7
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [4,6,8,7,5,3,2,1] => ? = 6
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [.,[.,[[.,[[.,[.,[.,.]]],.]],.]]]
=> [6,5,4,7,3,8,2,1] => ? = 3
[.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [.,[[[.,[[[.,[.,.]],.],.]],.],.]]
=> [4,3,5,6,2,7,8,1] => ? = 2
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [2,8,7,6,5,4,3,1] => ? = 7
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> [2,4,8,7,6,5,3,1] => ? = 6
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> [2,4,6,8,7,5,3,1] => ? = 5
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [.,[[.,.],[[[.,.],[.,.]],[.,.]]]]
=> [2,4,6,5,8,7,3,1] => ? = 4
[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [.,[[[.,.],[[.,.],[.,.]]],[.,.]]]
=> [2,4,6,5,3,8,7,1] => ? = 3
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> [2,4,3,6,8,7,5,1] => ? = 4
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> [2,4,3,6,5,8,7,1] => ? = 3
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> [2,3,4,6,7,8,5,1] => ? = 3
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [.,[[[[[.,.],.],.],.],[.,[.,.]]]]
=> [2,3,4,5,8,7,6,1] => ? = 4
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [2,3,4,5,7,8,6,1] => ? = 3
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [.,[[[[[[.,.],.],.],.],.],[.,.]]]
=> [2,3,4,5,6,8,7,1] => ? = 3
[[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [[.,[[.,[[.,[[.,.],.]],.]],.]],.]
=> [4,5,3,6,2,7,1,8] => ? = 1
[[.,[.,.]],[[.,[[.,[.,.]],.]],.]]
=> [[.,[[.,.],[[.,.],[.,.]]]],[.,.]]
=> [2,4,6,5,3,1,8,7] => ? = 2
[[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [[.,[[[.,.],[.,.]],[.,.]]],[.,.]]
=> [2,4,3,6,5,1,8,7] => ? = 2
[[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> [[[.,[.,[.,[[[.,.],.],.]]]],.],.]
=> [4,5,6,3,2,1,7,8] => ? = 1
[[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
=> [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [2,1,4,6,8,7,5,3] => ? = 4
[[.,[[[.,[.,.]],[.,.]],.]],[.,.]]
=> [[.,[.,.]],[[[.,.],[.,.]],[.,.]]]
=> [2,1,4,6,5,8,7,3] => ? = 3
[[.,[[[[[.,.],.],.],.],.]],[.,.]]
=> [[.,[.,.]],[[[[[.,.],.],.],.],.]]
=> [2,1,4,5,6,7,8,3] => ? = 2
[[.,[[[[[[.,.],.],.],.],.],.]],.]
=> [[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [1,3,4,5,6,7,8,2] => ? = 2
[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,2,8,7,6,5,4,3] => ? = 6
[[[.,[[[[[.,.],.],.],.],.]],.],.]
=> [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> [1,2,4,5,6,7,8,3] => ? = 2
[[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,2,3,8,7,6,5,4] => ? = 5
[[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,4,8,7,6,5] => ? = 4
[[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,2,3,4,5,8,7,6] => ? = 3
[.,[[[[[[[.,[.,.]],.],.],.],.],.],.]]
=> [.,[[[[[[[.,.],.],.],.],.],.],[.,.]]]
=> [2,3,4,5,6,7,9,8,1] => ? = 3
[.,[[[[[[.,[[.,.],.]],.],.],.],.],.]]
=> [.,[[[[[[.,.],.],.],.],.],[[.,.],.]]]
=> [2,3,4,5,6,8,9,7,1] => ? = 3
[.,[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]]
=> [.,[[[[[[[[.,.],.],.],.],.],.],.],[.,.]]]
=> [2,3,4,5,6,7,8,10,9,1] => ? = 3
[[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]],.]
=> [[.,.],[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ? => ? = 7
[[.,.],[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]],.]
=> ? => ? = 1
[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.]
=> [[[.,.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,2,9,8,7,6,5,4,3] => ? = 7
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].
Mp00012: Binary trees to Dyck path: up step, left tree, down step, right treeDyck paths
Mp00027: Dyck paths to partitionInteger partitions
St000759: Integer partitions ⟶ ℤResult quality: 79% values known / values provided: 79%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> []
=> 1
[.,[.,.]]
=> [1,0,1,0]
=> [1]
=> 2
[[.,.],.]
=> [1,1,0,0]
=> []
=> 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [2,1]
=> 3
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1]
=> 2
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> []
=> 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 4
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 3
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 3
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 2
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 2
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 2
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> []
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 5
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 4
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 3
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 4
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 2
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 3
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 4
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 3
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 3
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 1
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 3
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 2
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 2
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ? = 1
[[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> ? = 1
[[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2]
=> ? = 1
[[.,.],[.,[[.,[[.,.],.]],.]]]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2]
=> ? = 1
[[.,.],[[.,[.,[.,[.,.]]]],.]]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2]
=> ? = 1
[[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1]
=> ? = 5
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,4,3,2,1]
=> ? = 6
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,3,2,1]
=> ? = 6
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4,2,2,1]
=> ? = 3
[.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,2,2,1]
=> ? = 4
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,3,1,1]
=> ? = 2
[.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4,1,1,1]
=> ? = 2
[.,[[[.,[.,.]],[.,[.,.]]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [7,5,4,2,1,1,1]
=> ? = 3
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,2,1,1]
=> ? = 6
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,2,1,1]
=> ? = 6
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,3,2,2,1,1]
=> ? = 5
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [5,3,2,2,2,1,1]
=> ? = 4
[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [5,4,4,2,1,1,1]
=> ? = 3
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [6,3,2,2,1,1,1]
=> ? = 4
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [6,4,2,1,1,1,1]
=> ? = 3
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2]
=> ? = 1
[[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,3,2]
=> ? = 1
[[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,3,2]
=> ? = 1
[[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,3,2]
=> ? = 1
[[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4,3,2]
=> ? = 1
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,3,2]
=> ? = 1
[[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,2,2]
=> ? = 1
[[.,.],[[.,[.,.]],[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,2]
=> ? = 1
[[.,.],[[.,[.,[.,[.,.]]]],[.,.]]]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,2]
=> ? = 1
[[.,.],[[[.,[.,.]],[.,.]],[.,.]]]
=> [1,1,0,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,2,2]
=> ? = 1
[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,1]
=> ? = 2
[[.,[.,.]],[[.,[[.,[.,.]],.]],.]]
=> [1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,3,1]
=> ? = 2
[[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,3,3,1]
=> ? = 2
[[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,3]
=> ? = 1
[[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,1]
=> ? = 3
[[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [7,5,4,4,2,1]
=> ? = 3
[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,1]
=> ? = 4
[[.,[.,[[.,.],.]]],[.,[[.,.],.]]]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [6,6,5,2,2,1]
=> ? = 3
[[.,[[.,.],[.,.]]],[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,1,1]
=> ? = 2
[[.,[[.,.],[.,.]]],[[.,[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [6,5,5,3,1,1]
=> ? = 2
[[.,[[.,[.,.]],.]],[[.,.],[.,.]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [7,5,5,2,1,1]
=> ? = 3
[[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [6,5,5,2,1,1]
=> ? = 3
[[.,[[[.,.],.],.]],[[[.,.],.],.]]
=> [1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [5,5,5,1,1,1]
=> ? = 2
[[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,1]
=> ? = 2
[[[.,[.,.]],[.,.]],[[.,.],[.,.]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,1]
=> ? = 2
[[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [6,5,5,3,1]
=> ? = 2
[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,1]
=> ? = 5
[[.,[[.,[.,.]],[.,.]]],[.,[.,.]]]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [7,6,4,2,1,1]
=> ? = 3
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [7,6,2,1]
=> ? = 3
[[[[.,[[.,.],.]],.],.],[[.,.],.]]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> [6,6,1,1]
=> ? = 2
Description
The smallest missing part in an integer partition. In [3], this is referred to as the mex, the minimal excluded part of the partition. For compositions, this is studied in [sec.3.2., 1].
Mp00020: Binary trees to Tamari-corresponding Dyck pathDyck paths
Mp00102: Dyck paths rise compositionInteger compositions
Mp00038: Integer compositions reverseInteger compositions
St000383: Integer compositions ⟶ ℤResult quality: 79% values known / values provided: 79%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1] => [1] => 1
[.,[.,.]]
=> [1,1,0,0]
=> [2] => [2] => 2
[[.,.],.]
=> [1,0,1,0]
=> [1,1] => [1,1] => 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [3] => [3] => 3
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [2,1] => [1,2] => 2
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,2] => [2,1] => 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2,1] => [1,2] => 2
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,1,1] => 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [4] => [4] => 4
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => [1,3] => 3
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => [2,2] => 2
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => [1,3] => 3
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,2] => 2
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1] => 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,2,1] => 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => 2
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1] => 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [1,3] => 3
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,2] => 2
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,2,1] => 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,2] => 2
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => [5] => 5
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => [1,4] => 4
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => [2,3] => 3
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => [1,4] => 4
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => [1,1,3] => 3
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2] => 2
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,2,2] => 2
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => [2,3] => 3
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,1,2] => 2
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => [1,4] => 4
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => [1,1,3] => 3
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,2,2] => 2
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => [1,1,3] => 3
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,1,2] => 2
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,3,1] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1] => 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,3,1] => 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,1,2,1] => 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2] => 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,2,2] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1] => 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,2,1,1] => 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [2,3] => 3
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,1,2] => 2
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1] => 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,1,2] => 2
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1] => 1
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => [8] => ? = 8
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [6,2] => [2,6] => ? = 6
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [6,1,1] => [1,1,6] => ? = 6
[.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [4,2,2] => [2,2,4] => ? = 4
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [6,1,1] => [1,1,6] => ? = 6
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [6,1,1] => [1,1,6] => ? = 6
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [5,1,1,1] => [1,1,1,5] => ? = 5
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [4,2,1,1] => [1,1,2,4] => ? = 4
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1,1] => [1,1,1,1,1,3] => ? = 3
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [4,1,1,1,1] => [1,1,1,1,4] => ? = 4
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1,1] => [1,1,1,1,1,3] => ? = 3
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1,1] => [1,1,1,1,1,3] => ? = 3
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1,1] => [1,1,1,1,1,1,2] => ? = 2
[[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,5,2] => [2,5,1] => ? = 1
[[.,.],[[.,[.,[.,[.,.]]]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,5,2] => [2,5,1] => ? = 1
[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,6] => [6,2] => ? = 2
[[.,[.,.]],[[.,[[.,[.,.]],.]],.]]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,4,1,1] => [1,1,4,2] => ? = 2
[[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,4,1,1] => [1,1,4,1,1] => ? = 1
[[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,5] => [5,3] => ? = 3
[[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,4] => [4,2,2] => ? = 2
[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,3] => [3,5] => ? = 5
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,2] => [2,6] => ? = 6
[[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [4,1,1,2] => [2,1,1,4] => ? = 4
[[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [4,2,2] => [2,2,4] => ? = 4
[[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1] => ? = 1
[[.,[.,[.,[.,[.,[[.,.],.]]]]]],.]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,1] => [1,1,6] => ? = 6
[[.,[.,[.,[.,[[.,.],[.,.]]]]]],.]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,2,1] => [1,2,5] => ? = 5
[[.,[.,[.,[.,[[.,[.,.]],.]]]]],.]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [6,1,1] => [1,1,6] => ? = 6
[[.,[.,[.,[.,[[[.,.],.],.]]]]],.]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [5,1,1,1] => [1,1,1,5] => ? = 5
[[.,[.,[.,[[.,.],[.,[.,.]]]]]],.]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [4,3,1] => [1,3,4] => ? = 4
[[.,[.,[.,[[.,[.,[.,.]]],.]]]],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [6,1,1] => [1,1,6] => ? = 6
[[.,[[.,[.,[.,[.,.]]]],[.,.]]],.]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> [5,2,1] => [1,2,5] => ? = 5
[[.,[[[[[[.,.],.],.],.],.],.]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1,1,1,1] => [1,1,1,1,1,1,2] => ? = 2
[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [6,1,1] => [1,1,6] => ? = 6
[[[.,[[[[[.,.],.],.],.],.]],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1,1,1,1] => [1,1,1,1,1,1,2] => ? = 2
[[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [5,1,1,1] => [1,1,1,5] => ? = 5
[[[[.,[[[[.,.],.],.],.]],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1] => [1,1,1,1,1,1,2] => ? = 2
[[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1] => [1,1,1,1,4] => ? = 4
[[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,1,1,3] => ? = 3
[[[[[[[.,[.,.]],.],.],.],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1] => [1,1,1,1,1,1,2] => ? = 2
[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [9] => [9] => ? = 9
[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,2] => ? = 2
[[[[[[[[[.,.],.],.],.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => ? = 1
[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [10] => [10] => ? = 10
[.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,2] => ? = 2
[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,2,2,2,2] => [2,2,2,2,2] => ? = 2
[[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1] => ? = 1
[[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1] => ? = 1
[.,[[[[[[[.,[.,.]],.],.],.],.],.],.]]
=> ?
=> ? => ? => ? = 3
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
=> ?
=> ? => ? => ? = 1
Description
The last part of an integer composition.
The following 74 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000678The number of up steps after the last double rise of a Dyck path. St000069The number of maximal elements of a poset. St000675The number of centered multitunnels of a Dyck path. St000971The smallest closer of a set partition. St000546The number of global descents of a permutation. St000068The number of minimal elements in a poset. St001050The number of terminal closers of a set partition. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000237The number of small exceedances. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000053The number of valleys of the Dyck path. St000234The number of global ascents of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000031The number of cycles in the cycle decomposition of a permutation. St000738The first entry in the last row of a standard tableau. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000717The number of ordinal summands of a poset. St000740The last entry of a permutation. St000654The first descent of a permutation. St000203The number of external nodes of a binary tree. St000989The number of final rises of a permutation. St000843The decomposition number of a perfect matching. 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. St000702The number of weak deficiencies of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000990The first ascent of a permutation. St000734The last entry in the first row of a standard tableau. St001640The number of ascent tops in the permutation such that all smaller elements appear before. 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. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000051The size of the left subtree of a binary tree. St000015The number of peaks of a Dyck path. St000056The decomposition (or block) number of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000286The number of connected components of the complement of a graph. St000335The difference of lower and upper interactions. St000352The Elizalde-Pak rank of a permutation. St000133The "bounce" of a permutation. St000331The number of upper interactions of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000061The number of nodes on the left branch of a binary tree. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000181The number of connected components of the Hasse diagram for the poset. 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. St000338The number of pixed points of a permutation. St001889The size of the connectivity set of a signed permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000942The number of critical left to right maxima of the parking functions. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. 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. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001621The number of atoms of a lattice.