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Your data matches 53 different statistics following compositions of up to 3 maps.
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Matching statistic: St000507
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [[1]]
=> 1
[.,[.,.]]
=> [2,1] => [2,1] => [[1],[2]]
=> 1
[[.,.],.]
=> [1,2] => [1,2] => [[1,2]]
=> 2
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 1
[.,[[.,.],.]]
=> [2,3,1] => [2,3,1] => [[1,2],[3]]
=> 2
[[.,.],[.,.]]
=> [3,1,2] => [3,1,2] => [[1,3],[2]]
=> 2
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 2
[[[.,.],.],.]
=> [1,2,3] => [1,3,2] => [[1,2],[3]]
=> 2
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,4,2,1] => [[1,2],[3],[4]]
=> 2
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [4,2,3,1] => [[1,3],[2],[4]]
=> 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,4,1] => [[1,3],[2],[4]]
=> 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [4,3,1,2] => [[1,4],[2],[3]]
=> 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,4,1,2] => [[1,2],[3,4]]
=> 3
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [4,2,1,3] => [[1,4],[2],[3]]
=> 2
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [4,1,3,2] => [[1,3],[2],[4]]
=> 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,4,1,3] => [[1,2],[3,4]]
=> 3
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,4,2] => [[1,3],[2,4]]
=> 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,5,3,2,1] => [[1,2],[3],[4],[5]]
=> 2
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [5,3,4,2,1] => [[1,3],[2],[4],[5]]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,5,2,1] => [[1,3],[2],[4],[5]]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,5,4,2,1] => [[1,2],[3],[4],[5]]
=> 2
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [5,4,2,3,1] => [[1,4],[2],[3],[5]]
=> 2
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,5,2,3,1] => [[1,2],[3,4],[5]]
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [5,2,4,3,1] => [[1,3],[2],[4],[5]]
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,5,1] => [[1,4],[2],[3],[5]]
=> 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,5,2,4,1] => [[1,2],[3,4],[5]]
=> 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,5,3,1] => [[1,3],[2,4],[5]]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,5,4,1] => [[1,3],[2,4],[5]]
=> 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [[1,5],[2],[3],[4]]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,5,3,1,2] => [[1,2],[3,5],[4]]
=> 3
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [5,3,4,1,2] => [[1,3],[2,5],[4]]
=> 3
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,5,1,2] => [[1,3],[2,5],[4]]
=> 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,5,4,1,2] => [[1,2],[3,5],[4]]
=> 3
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [5,4,2,1,3] => [[1,5],[2],[3],[4]]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,5,2,1,3] => [[1,2],[3,5],[4]]
=> 3
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [5,4,1,3,2] => [[1,4],[2],[3],[5]]
=> 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,5,1,3,2] => [[1,2],[3,4],[5]]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [5,3,2,1,4] => [[1,5],[2],[3],[4]]
=> 2
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [5,2,4,1,3] => [[1,3],[2,5],[4]]
=> 3
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [5,3,1,4,2] => [[1,4],[2,5],[3]]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [5,2,1,4,3] => [[1,4],[2,5],[3]]
=> 2
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [5,1,4,3,2] => [[1,3],[2],[4],[5]]
=> 2
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St001280
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1]
=> 0 = 1 - 1
[.,[.,.]]
=> [2,1] => [1,1] => [1,1]
=> 0 = 1 - 1
[[.,.],.]
=> [1,2] => [2] => [2]
=> 1 = 2 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1] => [1,1,1]
=> 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [2,1]
=> 1 = 2 - 1
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => [2,1]
=> 1 = 2 - 1
[[.,[.,.]],.]
=> [2,1,3] => [1,2] => [2,1]
=> 1 = 2 - 1
[[[.,.],.],.]
=> [1,2,3] => [3] => [3]
=> 1 = 2 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1] => [1,1,1,1]
=> 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,1,1] => [2,1,1]
=> 1 = 2 - 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,2,1] => [2,1,1]
=> 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,1] => [2,1,1]
=> 1 = 2 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,1] => [3,1]
=> 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,2] => [2,1,1]
=> 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,2] => [2,2]
=> 2 = 3 - 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,2] => [2,1,1]
=> 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,3] => [3,1]
=> 1 = 2 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,2] => [2,1,1]
=> 1 = 2 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,2] => [2,2]
=> 2 = 3 - 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,3] => [3,1]
=> 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3] => [3,1]
=> 1 = 2 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4] => [4]
=> 1 = 2 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1] => [1,1,1,1,1]
=> 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [2,1,1,1] => [2,1,1,1]
=> 1 = 2 - 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,1,1] => [2,1,1,1]
=> 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,1,1] => [2,1,1,1]
=> 1 = 2 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1] => [3,1,1]
=> 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,2,1] => [2,1,1,1]
=> 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [2,2,1] => [2,2,1]
=> 2 = 3 - 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,2,1] => [2,1,1,1]
=> 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,3,1] => [3,1,1]
=> 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,2,1] => [2,1,1,1]
=> 1 = 2 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [2,2,1] => [2,2,1]
=> 2 = 3 - 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,3,1] => [3,1,1]
=> 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,3,1] => [3,1,1]
=> 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [4,1] => [4,1]
=> 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,2] => [2,1,1,1]
=> 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,2] => [2,2,1]
=> 2 = 3 - 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,2,2] => [2,2,1]
=> 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,2,2] => [2,2,1]
=> 2 = 3 - 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,2] => [3,2]
=> 2 = 3 - 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,2] => [2,1,1,1]
=> 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [2,1,2] => [2,2,1]
=> 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,3] => [3,1,1]
=> 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [2,3] => [3,2]
=> 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,2] => [2,1,1,1]
=> 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,2,2] => [2,2,1]
=> 2 = 3 - 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,3] => [3,1,1]
=> 1 = 2 - 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,3] => [3,1,1]
=> 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,4] => [4,1]
=> 1 = 2 - 1
[.,[[[[[.,.],.],[[.,.],.]],.],.]]
=> [5,6,2,3,4,7,8,1] => ? => ?
=> ? = 3 - 1
[[.,.],[[.,[[.,[.,.]],.]],[.,.]]]
=> [8,5,4,6,3,7,1,2] => ? => ?
=> ? = 4 - 1
[[.,.],[[[.,[[.,.],.]],[.,.]],.]]
=> [7,4,5,3,6,8,1,2] => ? => ?
=> ? = 4 - 1
[[[.,.],[[.,.],[[.,[.,.]],.]]],.]
=> [6,5,7,3,4,1,2,8] => ? => ?
=> ? = 4 - 1
[[[.,.],[[.,[[.,.],[.,.]]],.]],.]
=> [6,4,5,3,7,1,2,8] => ? => ?
=> ? = 4 - 1
[[[.,[.,[[.,.],.]]],[[.,.],.]],.]
=> [6,7,3,4,2,1,5,8] => ? => ?
=> ? = 4 - 1
[[[.,[[.,.],[.,.]]],[.,[.,.]]],.]
=> [7,6,4,2,3,1,5,8] => ? => ?
=> ? = 3 - 1
[[[.,[[.,.],[.,.]]],[[.,.],.]],.]
=> [6,7,4,2,3,1,5,8] => ? => ?
=> ? = 4 - 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000390
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00109: Permutations —descent word⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => => => ? = 1 - 1
[.,[.,.]]
=> [2,1] => 1 => 0 => 0 = 1 - 1
[[.,.],.]
=> [1,2] => 0 => 1 => 1 = 2 - 1
[.,[.,[.,.]]]
=> [3,2,1] => 11 => 00 => 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => 01 => 10 => 1 = 2 - 1
[[.,.],[.,.]]
=> [3,1,2] => 10 => 11 => 1 = 2 - 1
[[.,[.,.]],.]
=> [2,1,3] => 10 => 11 => 1 = 2 - 1
[[[.,.],.],.]
=> [1,2,3] => 00 => 01 => 1 = 2 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 111 => 000 => 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 011 => 100 => 1 = 2 - 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => 101 => 110 => 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 101 => 110 => 1 = 2 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 001 => 010 => 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => 110 => 111 => 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => 010 => 101 => 2 = 3 - 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => 110 => 111 => 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => 100 => 011 => 1 = 2 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 110 => 111 => 1 = 2 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 010 => 101 => 2 = 3 - 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => 100 => 011 => 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 100 => 011 => 1 = 2 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => 000 => 001 => 1 = 2 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 1111 => 0000 => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 0111 => 1000 => 1 = 2 - 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 1011 => 1100 => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 1011 => 1100 => 1 = 2 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 0011 => 0100 => 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 1101 => 1110 => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 0101 => 1010 => 2 = 3 - 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 1101 => 1110 => 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 1001 => 0110 => 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 1101 => 1110 => 1 = 2 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 0101 => 1010 => 2 = 3 - 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 1001 => 0110 => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 1001 => 0110 => 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 0001 => 0010 => 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1110 => 1111 => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 0110 => 1011 => 2 = 3 - 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 1010 => 1101 => 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 1010 => 1101 => 2 = 3 - 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 0010 => 0101 => 2 = 3 - 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 1110 => 1111 => 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 0110 => 1011 => 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 1100 => 0111 => 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 0100 => 1001 => 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 1110 => 1111 => 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 1010 => 1101 => 2 = 3 - 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1100 => 0111 => 1 = 2 - 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 1100 => 0111 => 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 1000 => 0011 => 1 = 2 - 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1110 => 1111 => 1 = 2 - 1
[.,[.,[.,[[.,.],[[[.,.],.],.]]]]]
=> [6,7,8,4,5,3,2,1] => ? => ? => ? = 3 - 1
[.,[.,[[.,[[.,[.,.]],.]],[.,.]]]]
=> [8,5,4,6,3,7,2,1] => ? => ? => ? = 3 - 1
[.,[.,[[.,[[.,.],[.,[.,.]]]],.]]]
=> [7,6,4,5,3,8,2,1] => ? => ? => ? = 3 - 1
[.,[[.,.],[[[.,.],[.,[.,.]]],.]]]
=> [7,6,4,5,8,2,3,1] => ? => ? => ? = 3 - 1
[.,[[[[[.,.],.],[[.,.],.]],.],.]]
=> [5,6,2,3,4,7,8,1] => ? => ? => ? = 3 - 1
[[.,.],[.,[[[.,.],.],[[.,.],.]]]]
=> [7,8,4,5,6,3,1,2] => ? => ? => ? = 4 - 1
[[.,.],[[.,[[.,[.,.]],.]],[.,.]]]
=> [8,5,4,6,3,7,1,2] => ? => ? => ? = 4 - 1
[[.,.],[[.,[[.,[.,[.,.]]],.]],.]]
=> [6,5,4,7,3,8,1,2] => ? => ? => ? = 4 - 1
[[.,.],[[.,[[[.,.],[.,.]],.]],.]]
=> [6,4,5,7,3,8,1,2] => ? => ? => ? = 4 - 1
[[.,.],[[[.,[[.,.],.]],[.,.]],.]]
=> [7,4,5,3,6,8,1,2] => ? => ? => ? = 4 - 1
[[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [7,5,4,6,8,2,1,3] => ? => ? => ? = 3 - 1
[[[.,.],.],[[.,[.,[.,.]]],[.,.]]]
=> [8,6,5,4,7,1,2,3] => ? => ? => ? = 3 - 1
[[[[.,[.,[.,.]]],.],[.,.]],[.,.]]
=> [8,6,3,2,1,4,5,7] => ? => ? => ? = 2 - 1
[[[.,.],[[.,.],[[.,[.,.]],.]]],.]
=> [6,5,7,3,4,1,2,8] => ? => ? => ? = 4 - 1
[[[.,.],[[.,[[.,.],[.,.]]],.]],.]
=> [6,4,5,3,7,1,2,8] => ? => ? => ? = 4 - 1
[[[[.,.],.],[[.,[[.,.],.]],.]],.]
=> [5,6,4,7,1,2,3,8] => ? => ? => ? = 4 - 1
[[[[.,.],[.,.]],[[[.,.],.],.]],.]
=> [5,6,7,3,1,2,4,8] => ? => ? => ? = 3 - 1
[[[.,[.,[[.,.],.]]],[[.,.],.]],.]
=> [6,7,3,4,2,1,5,8] => ? => ? => ? = 4 - 1
[[[.,[[.,.],[.,.]]],[.,[.,.]]],.]
=> [7,6,4,2,3,1,5,8] => ? => ? => ? = 3 - 1
[[[.,[[.,.],[.,.]]],[[.,.],.]],.]
=> [6,7,4,2,3,1,5,8] => ? => ? => ? = 4 - 1
[[[.,[.,[[[.,.],.],[.,.]]]],.],.]
=> [6,3,4,5,2,1,7,8] => ? => ? => ? = 3 - 1
[[[[.,.],[.,[.,[[.,.],.]]]],.],.]
=> [5,6,4,3,1,2,7,8] => ? => ? => ? = 3 - 1
[[[[.,.],[[.,[.,.]],[.,.]]],.],.]
=> [6,4,3,5,1,2,7,8] => ? => ? => ? = 3 - 1
[[[[.,[.,.]],[[.,[.,.]],.]],.],.]
=> [5,4,6,2,1,3,7,8] => ? => ? => ? = 3 - 1
[[[[[[.,.],.],.],[.,[.,.]]],.],.]
=> [6,5,1,2,3,4,7,8] => ? => ? => ? = 2 - 1
[[[[[[.,.],.],.],[[.,.],.]],.],.]
=> [5,6,1,2,3,4,7,8] => ? => ? => ? = 3 - 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000291
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => 1 => 0 = 1 - 1
[.,[.,.]]
=> [2,1] => [1,1] => 11 => 0 = 1 - 1
[[.,.],.]
=> [1,2] => [2] => 10 => 1 = 2 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1] => 111 => 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => 101 => 1 = 2 - 1
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => 110 => 1 = 2 - 1
[[.,[.,.]],.]
=> [2,1,3] => [1,2] => 110 => 1 = 2 - 1
[[[.,.],.],.]
=> [1,2,3] => [3] => 100 => 1 = 2 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1] => 1111 => 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,1,1] => 1011 => 1 = 2 - 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,2,1] => 1101 => 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,1] => 1101 => 1 = 2 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,1] => 1001 => 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,2] => 1110 => 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,2] => 1010 => 2 = 3 - 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,2] => 1110 => 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,3] => 1100 => 1 = 2 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,2] => 1110 => 1 = 2 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,2] => 1010 => 2 = 3 - 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,3] => 1100 => 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3] => 1100 => 1 = 2 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4] => 1000 => 1 = 2 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1] => 11111 => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [2,1,1,1] => 10111 => 1 = 2 - 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,1,1] => 11011 => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,1,1] => 11011 => 1 = 2 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1] => 10011 => 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,2,1] => 11101 => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [2,2,1] => 10101 => 2 = 3 - 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,2,1] => 11101 => 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,3,1] => 11001 => 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,2,1] => 11101 => 1 = 2 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [2,2,1] => 10101 => 2 = 3 - 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,3,1] => 11001 => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,3,1] => 11001 => 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [4,1] => 10001 => 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,2] => 11110 => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,2] => 10110 => 2 = 3 - 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,2,2] => 11010 => 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,2,2] => 11010 => 2 = 3 - 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,2] => 10010 => 2 = 3 - 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,2] => 11110 => 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [2,1,2] => 10110 => 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,3] => 11100 => 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [2,3] => 10100 => 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,2] => 11110 => 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,2,2] => 11010 => 2 = 3 - 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,3] => 11100 => 1 = 2 - 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,3] => 11100 => 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,4] => 11000 => 1 = 2 - 1
[.,[[[[[.,.],.],[[.,.],.]],.],.]]
=> [5,6,2,3,4,7,8,1] => ? => ? => ? = 3 - 1
[[.,.],[[.,[[.,[.,.]],.]],[.,.]]]
=> [8,5,4,6,3,7,1,2] => ? => ? => ? = 4 - 1
[[.,.],[[[.,[[.,.],.]],[.,.]],.]]
=> [7,4,5,3,6,8,1,2] => ? => ? => ? = 4 - 1
[[[.,.],[[.,.],[[.,[.,.]],.]]],.]
=> [6,5,7,3,4,1,2,8] => ? => ? => ? = 4 - 1
[[[.,.],[[.,[[.,.],[.,.]]],.]],.]
=> [6,4,5,3,7,1,2,8] => ? => ? => ? = 4 - 1
[[[.,[.,[[.,.],.]]],[[.,.],.]],.]
=> [6,7,3,4,2,1,5,8] => ? => ? => ? = 4 - 1
[[[.,[[.,.],[.,.]]],[.,[.,.]]],.]
=> [7,6,4,2,3,1,5,8] => ? => ? => ? = 3 - 1
[[[.,[[.,.],[.,.]]],[[.,.],.]],.]
=> [6,7,4,2,3,1,5,8] => ? => ? => ? = 4 - 1
[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [10,9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 1 - 1
[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]
=> [9,10,8,7,6,5,4,3,2,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 2 - 1
[.,[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]]
=> [8,9,10,7,6,5,4,3,2,1] => [3,1,1,1,1,1,1,1] => 1001111111 => ? = 2 - 1
[.,[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]]
=> [7,8,9,10,6,5,4,3,2,1] => [4,1,1,1,1,1,1] => 1000111111 => ? = 2 - 1
[.,[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]]
=> [6,7,8,9,10,5,4,3,2,1] => [5,1,1,1,1,1] => 1000011111 => ? = 2 - 1
[.,[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]]]
=> [5,6,7,8,9,10,4,3,2,1] => [6,1,1,1,1] => 1000001111 => ? = 2 - 1
[.,[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]]
=> [4,5,6,7,8,9,10,3,2,1] => [7,1,1,1] => 1000000111 => ? = 2 - 1
[.,[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]]
=> [3,4,5,6,7,8,9,10,2,1] => [8,1,1] => 1000000011 => ? = 2 - 1
[.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,9,10,1] => [9,1] => 1000000001 => ? = 2 - 1
[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]]
=> [10,8,6,4,2,1,3,5,7,9] => [1,1,1,1,1,5] => 1111110000 => ? = 2 - 1
[.,[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,9,10,1] => [1,8,1] => 1100000001 => ? = 2 - 1
[[[[[[[[.,.],.],.],.],.],.],.],[[.,.],.]]
=> [9,10,1,2,3,4,5,6,7,8] => [2,8] => 1010000000 => ? = 3 - 1
[[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.],.]
=> [8,7,6,5,4,3,2,1,9,10] => [1,1,1,1,1,1,1,3] => 1111111100 => ? = 2 - 1
[[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.],.]
=> [7,6,5,4,3,2,1,8,9,10] => [1,1,1,1,1,1,4] => 1111111000 => ? = 2 - 1
[[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.],.]
=> [6,5,4,3,2,1,7,8,9,10] => [1,1,1,1,1,5] => 1111110000 => ? = 2 - 1
[[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.],.]
=> [4,3,2,1,5,6,7,8,9,10] => [1,1,1,7] => 1111000000 => ? = 2 - 1
[[[[[[[[.,[.,[.,.]]],.],.],.],.],.],.],.]
=> [3,2,1,4,5,6,7,8,9,10] => [1,1,8] => 1110000000 => ? = 2 - 1
[[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.]
=> [2,1,3,4,5,6,7,8,9,10] => [1,9] => 1100000000 => ? = 2 - 1
[[[[[[[[[.,.],[.,.]],.],.],.],.],.],.],.]
=> [3,1,2,4,5,6,7,8,9,10] => [1,9] => 1100000000 => ? = 2 - 1
[[[[[[[[[.,.],.],[.,.]],.],.],.],.],.],.]
=> [4,1,2,3,5,6,7,8,9,10] => [1,9] => 1100000000 => ? = 2 - 1
[[[[[[[[[.,.],.],.],[.,.]],.],.],.],.],.]
=> [5,1,2,3,4,6,7,8,9,10] => [1,9] => 1100000000 => ? = 2 - 1
[[[[[[[[[.,.],.],.],.],[.,.]],.],.],.],.]
=> [6,1,2,3,4,5,7,8,9,10] => [1,9] => 1100000000 => ? = 2 - 1
[[[[[[[[[.,.],.],.],.],.],[.,.]],.],.],.]
=> [7,1,2,3,4,5,6,8,9,10] => [1,9] => 1100000000 => ? = 2 - 1
[[[[[[[[[.,.],.],.],.],.],.],[.,.]],.],.]
=> [8,1,2,3,4,5,6,7,9,10] => [1,9] => 1100000000 => ? = 2 - 1
[[[[[[[[[.,.],.],.],.],.],.],.],[.,.]],.]
=> [9,1,2,3,4,5,6,7,8,10] => [1,9] => 1100000000 => ? = 2 - 1
[[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> [9,8,7,6,5,4,3,1,2,10] => [1,1,1,1,1,1,1,3] => 1111111100 => ? = 2 - 1
[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]],.]
=> [9,7,6,5,4,3,2,1,8,10] => [1,1,1,1,1,1,1,3] => 1111111100 => ? = 2 - 1
[.,[.,[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]]]
=> [8,7,9,10,6,5,4,3,2,1] => [1,3,1,1,1,1,1,1] => 1100111111 => ? = 2 - 1
Description
The number of descents of a binary word.
Matching statistic: St000010
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1]
=> 1
[.,[.,.]]
=> [2,1] => [2,1] => [2]
=> 1
[[.,.],.]
=> [1,2] => [1,2] => [1,1]
=> 2
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [3]
=> 1
[.,[[.,.],.]]
=> [2,3,1] => [2,3,1] => [2,1]
=> 2
[[.,.],[.,.]]
=> [3,1,2] => [3,1,2] => [2,1]
=> 2
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,1]
=> 2
[[[.,.],.],.]
=> [1,2,3] => [1,3,2] => [2,1]
=> 2
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [4]
=> 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,4,2,1] => [3,1]
=> 2
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [4,2,3,1] => [3,1]
=> 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,4,1] => [3,1]
=> 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,4,3,1] => [3,1]
=> 2
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [4,3,1,2] => [3,1]
=> 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,4,1,2] => [2,1,1]
=> 3
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [4,2,1,3] => [3,1]
=> 2
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [4,1,3,2] => [3,1]
=> 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [3,1]
=> 2
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,4,1,3] => [2,1,1]
=> 3
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,4,2] => [2,2]
=> 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,4,3] => [2,2]
=> 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,4,3,2] => [3,1]
=> 2
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [5]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,5,3,2,1] => [4,1]
=> 2
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [5,3,4,2,1] => [4,1]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,5,2,1] => [4,1]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,5,4,2,1] => [4,1]
=> 2
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [5,4,2,3,1] => [4,1]
=> 2
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,5,2,3,1] => [3,1,1]
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [5,3,2,4,1] => [4,1]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [5,2,4,3,1] => [4,1]
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,5,1] => [4,1]
=> 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,5,2,4,1] => [3,1,1]
=> 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,5,3,1] => [3,2]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,5,4,1] => [3,2]
=> 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,5,4,3,1] => [4,1]
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [4,1]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,5,3,1,2] => [3,1,1]
=> 3
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [5,3,4,1,2] => [3,1,1]
=> 3
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,5,1,2] => [3,1,1]
=> 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,5,4,1,2] => [3,1,1]
=> 3
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [5,4,2,1,3] => [4,1]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,5,2,1,3] => [3,1,1]
=> 3
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [5,4,1,3,2] => [4,1]
=> 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,5,1,3,2] => [3,1,1]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [5,3,2,1,4] => [4,1]
=> 2
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [5,2,4,1,3] => [3,1,1]
=> 3
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [5,3,1,4,2] => [3,2]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [5,2,1,4,3] => [3,2]
=> 2
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [5,1,4,3,2] => [4,1]
=> 2
[.,[.,[[[.,.],.],[.,[.,[.,.]]]]]]
=> [8,7,6,3,4,5,2,1] => [8,7,6,3,5,4,2,1] => ?
=> ? = 2
[.,[.,[[[.,.],.],[[.,.],[.,.]]]]]
=> [8,6,7,3,4,5,2,1] => [8,6,7,3,5,4,2,1] => ?
=> ? = 3
[.,[.,[[[.,.],.],[[.,[.,.]],.]]]]
=> [7,6,8,3,4,5,2,1] => [7,6,8,3,5,4,2,1] => ?
=> ? = 3
[.,[.,[[.,[[.,.],[.,[.,.]]]],.]]]
=> [7,6,4,5,3,8,2,1] => [7,6,4,8,3,5,2,1] => ?
=> ? = 3
[.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> [8,7,5,6,4,2,3,1] => [8,7,5,6,4,2,3,1] => ?
=> ? = 3
[.,[[.,.],[[[.,.],.],[.,[.,.]]]]]
=> [8,7,4,5,6,2,3,1] => [8,7,4,6,5,2,3,1] => ?
=> ? = 3
[.,[[[[.,.],.],.],[.,[.,[.,.]]]]]
=> [8,7,6,2,3,4,5,1] => [8,7,6,2,5,4,3,1] => ?
=> ? = 2
[.,[[[[.,.],.],.],[[.,.],[.,.]]]]
=> [8,6,7,2,3,4,5,1] => [8,6,7,2,5,4,3,1] => ?
=> ? = 3
[.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [7,8,2,3,4,5,6,1] => [7,8,2,6,5,4,3,1] => ?
=> ? = 3
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [6,4,3,5,7,2,8,1] => [6,4,3,8,7,2,5,1] => ?
=> ? = 3
[[.,.],[.,[[.,[[.,.],[.,.]]],.]]]
=> [7,5,6,4,8,3,1,2] => [7,5,8,4,6,3,1,2] => ?
=> ? = 4
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
=> [8,5,3,4,6,7,1,2] => [8,5,3,7,6,4,1,2] => ?
=> ? = 3
[[.,[.,.]],[[[.,.],.],[.,[.,.]]]]
=> [8,7,4,5,6,2,1,3] => [8,7,4,6,5,2,1,3] => ?
=> ? = 3
[[.,[.,.]],[[[[.,.],.],.],[.,.]]]
=> [8,4,5,6,7,2,1,3] => [8,4,7,6,5,2,1,3] => ?
=> ? = 3
[[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> [8,5,6,7,4,1,2,3] => [8,5,7,6,4,1,3,2] => ?
=> ? = 3
[[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> [8,7,6,4,5,1,2,3] => [8,7,6,4,5,1,3,2] => ?
=> ? = 3
[[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> [8,7,4,5,6,1,2,3] => [8,7,4,6,5,1,3,2] => ?
=> ? = 3
[[[.,.],.],[[.,[.,[.,[.,.]]]],.]]
=> [7,6,5,4,8,1,2,3] => [7,6,5,4,8,1,3,2] => ?
=> ? = 3
[[.,[.,[[.,.],.]]],[.,[.,[.,.]]]]
=> [8,7,6,3,4,2,1,5] => [8,7,6,3,5,2,1,4] => ?
=> ? = 3
[[.,[[[.,.],.],.]],[.,[.,[.,.]]]]
=> [8,7,6,2,3,4,1,5] => [8,7,6,2,5,4,1,3] => ?
=> ? = 3
[[[.,[[.,.],.]],.],[.,[.,[.,.]]]]
=> [8,7,6,2,3,1,4,5] => [8,7,6,2,5,1,4,3] => ?
=> ? = 3
[[[.,[[.,.],.]],.],[[.,[.,.]],.]]
=> [7,6,8,2,3,1,4,5] => [7,6,8,2,5,1,4,3] => ?
=> ? = 4
[[.,[.,[.,[[.,.],.]]]],[.,[.,.]]]
=> [8,7,4,5,3,2,1,6] => [8,7,4,6,3,2,1,5] => ?
=> ? = 3
[[.,[[[[.,.],.],.],.]],[.,[.,.]]]
=> [8,7,2,3,4,5,1,6] => [8,7,2,6,5,4,1,3] => ?
=> ? = 3
[[[.,[.,[.,[.,.]]]],.],[.,[.,.]]]
=> [8,7,4,3,2,1,5,6] => [8,7,4,3,2,1,6,5] => ?
=> ? = 2
[[.,[[.,[.,.]],[.,[.,.]]]],[.,.]]
=> [8,6,5,3,2,4,1,7] => [8,6,5,3,2,7,1,4] => ?
=> ? = 3
[[[.,[[.,.],.]],[.,[.,[.,.]]]],.]
=> [7,6,5,2,3,1,4,8] => [7,6,5,2,8,1,4,3] => ?
=> ? = 3
[[[[[.,.],[.,[.,[.,.]]]],.],.],.]
=> [5,4,3,1,2,6,7,8] => [5,4,3,1,8,7,6,2] => ?
=> ? = 2
[.,[[[[[[[.,.],[.,.]],.],.],.],.],.]]
=> [4,2,3,5,6,7,8,9,1] => [4,2,9,8,7,6,5,3,1] => ?
=> ? = 2
[.,[[[[[[.,[[.,.],.]],.],.],.],.],.]]
=> [3,4,2,5,6,7,8,9,1] => [3,9,2,8,7,6,5,4,1] => ?
=> ? = 3
[.,[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,9,10,1] => [3,2,10,9,8,7,6,5,4,1] => ?
=> ? = 2
[[.,[.,[.,.]]],[.,[.,[.,[.,[.,.]]]]]]
=> [9,8,7,6,5,3,2,1,4] => [9,8,7,6,5,3,2,1,4] => ?
=> ? = 2
[[.,[.,[.,[.,[.,.]]]]],[.,[.,[.,.]]]]
=> [9,8,7,5,4,3,2,1,6] => [9,8,7,5,4,3,2,1,6] => ?
=> ? = 2
[[.,[.,[.,.]]],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [10,9,8,7,6,5,3,2,1,4] => [10,9,8,7,6,5,3,2,1,4] => ?
=> ? = 2
[[.,[.,[.,[.,[.,.]]]]],[.,[.,[.,[.,.]]]]]
=> [10,9,8,7,5,4,3,2,1,6] => [10,9,8,7,5,4,3,2,1,6] => ?
=> ? = 2
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,[.,.]]]
=> [10,9,7,6,5,4,3,2,1,8] => [10,9,7,6,5,4,3,2,1,8] => ?
=> ? = 2
[.,[[[[[[[.,.],.],.],.],.],[.,.]],.]]
=> [8,2,3,4,5,6,7,9,1] => [8,2,9,7,6,5,4,3,1] => ?
=> ? = 2
[[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]],.]
=> [8,7,6,5,4,2,1,3,9] => [8,7,6,5,4,2,1,9,3] => ?
=> ? = 2
[[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]],.]
=> [8,7,5,4,3,2,1,6,9] => [8,7,5,4,3,2,1,9,6] => ?
=> ? = 2
[.,[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]]
=> [7,6,8,9,5,4,3,2,1] => [7,6,9,8,5,4,3,2,1] => ?
=> ? = 2
[.,[.,[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]]]
=> [8,7,9,10,6,5,4,3,2,1] => [8,7,10,9,6,5,4,3,2,1] => ?
=> ? = 2
Description
The length of the partition.
Matching statistic: St000157
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1] => [[1]]
=> 0 = 1 - 1
[.,[.,.]]
=> [1,0,1,0]
=> [1,2] => [[1,2]]
=> 0 = 1 - 1
[[.,.],.]
=> [1,1,0,0]
=> [2,1] => [[1],[2]]
=> 1 = 2 - 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,2,3] => [[1,2,3]]
=> 0 = 1 - 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> 1 = 2 - 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> 1 = 2 - 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [2,3,1] => [[1,3],[2]]
=> 1 = 2 - 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3,1,2] => [[1,2],[3]]
=> 1 = 2 - 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 1 = 2 - 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [[1,2,4],[3]]
=> 1 = 2 - 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [[1,2,3],[4]]
=> 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2 = 3 - 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[1,3,4],[2]]
=> 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[1,3,4],[2]]
=> 1 = 2 - 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [[1,3],[2,4]]
=> 2 = 3 - 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [[1,2],[3,4]]
=> 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 1 = 2 - 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [[1,2,3],[4]]
=> 1 = 2 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1 = 2 - 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [[1,2,3,5],[4]]
=> 1 = 2 - 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [[1,2,3,4],[5]]
=> 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 2 = 3 - 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [[1,2,4,5],[3]]
=> 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [[1,2,3,5],[4]]
=> 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [[1,2,4,5],[3]]
=> 1 = 2 - 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [[1,2,4],[3,5]]
=> 2 = 3 - 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [[1,2,3],[4,5]]
=> 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [[1,2,3,4],[5]]
=> 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 2 = 3 - 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[1,3,5],[2,4]]
=> 2 = 3 - 1
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[1,3,4],[2,5]]
=> 2 = 3 - 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[1,3,4,5],[2]]
=> 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[1,3,4],[2,5]]
=> 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [[1,2,4,5],[3]]
=> 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [[1,2,4],[3,5]]
=> 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[1,3,4,5],[2]]
=> 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[1,3,5],[2,4]]
=> 2 = 3 - 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [[1,2,5],[3,4]]
=> 1 = 2 - 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [[1,2,5],[3,4]]
=> 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [[1,2,3,5],[4]]
=> 1 = 2 - 1
[.,[.,[[.,[.,.]],[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,2,4,5,3,6,7,8] => ?
=> ? = 2 - 1
[.,[.,[[.,[.,.]],[.,[[.,.],.]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6,8,7] => ?
=> ? = 3 - 1
[.,[.,[[.,[[.,[.,.]],.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,2,4,6,7,3,5,8] => ?
=> ? = 3 - 1
[.,[.,[[[.,[.,[.,.]]],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,2,5,6,7,3,4,8] => ?
=> ? = 2 - 1
[.,[.,[[.,[[.,.],[.,[.,.]]]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,2,4,6,3,7,8,5] => ?
=> ? = 3 - 1
[.,[.,[[.,[[.,.],[[.,.],.]]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,2,4,6,3,8,5,7] => ?
=> ? = 4 - 1
[.,[.,[[.,[[.,[.,.]],[.,.]]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,2,4,6,7,3,8,5] => ?
=> ? = 3 - 1
[.,[.,[[.,[[.,[.,[.,.]]],.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,2,4,6,7,8,3,5] => ?
=> ? = 3 - 1
[.,[[.,.],[.,[[.,[.,.]],[.,.]]]]]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,3,2,4,6,7,5,8] => ?
=> ? = 3 - 1
[.,[[.,.],[.,[[[.,.],.],[.,.]]]]]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,3,2,4,7,5,6,8] => ?
=> ? = 3 - 1
[.,[[.,.],[[.,[[.,.],[.,.]]],.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,7,4,8,6] => ?
=> ? = 4 - 1
[.,[[.,.],[[[.,.],[.,[.,.]]],.]]]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,7,8,5] => ?
=> ? = 3 - 1
[.,[[.,[.,.]],[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,4,2,6,5,7,8] => ?
=> ? = 3 - 1
[.,[[.,[.,.]],[[[.,.],.],[.,.]]]]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,4,2,7,5,6,8] => ?
=> ? = 3 - 1
[.,[[.,[.,[.,.]]],[.,[[.,.],.]]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6,8,7] => ?
=> ? = 3 - 1
[.,[[.,[[.,.],.]],[[.,[.,.]],.]]]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4,7,8,6] => ?
=> ? = 4 - 1
[.,[[[.,.],[.,.]],[[.,[.,.]],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,4,2,5,3,7,8,6] => ?
=> ? = 3 - 1
[.,[[[.,[.,.]],.],[.,[[.,.],.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [1,4,5,2,3,6,8,7] => ?
=> ? = 3 - 1
[.,[[.,[[.,.],[.,.]]],[.,[.,.]]]]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,3,5,2,6,4,7,8] => ?
=> ? = 3 - 1
[.,[[.,[[.,.],[.,.]]],[[.,.],.]]]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,3,5,2,6,4,8,7] => ?
=> ? = 4 - 1
[.,[[[.,[.,[.,.]]],.],[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [1,4,5,6,2,3,8,7] => ?
=> ? = 3 - 1
[.,[[.,[[[.,.],.],[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,3,6,2,4,7,5,8] => ?
=> ? = 3 - 1
[.,[[[.,.],[.,[[.,.],.]]],[.,.]]]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,4,2,5,7,3,6,8] => ?
=> ? = 3 - 1
[.,[[[[.,[[.,.],.]],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,5,7,2,3,4,6,8] => ?
=> ? = 3 - 1
[.,[[.,[.,[.,[[.,.],[.,.]]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,5,7,2,8,6] => ?
=> ? = 3 - 1
[.,[[.,[.,[[.,.],[.,[.,.]]]]],.]]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,4,6,2,7,8,5] => ?
=> ? = 3 - 1
[.,[[.,[.,[[.,.],[[.,.],.]]]],.]]
=> [1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,4,6,2,8,5,7] => ?
=> ? = 4 - 1
[.,[[.,[.,[[[.,.],.],[.,.]]]],.]]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,4,7,2,5,8,6] => ?
=> ? = 3 - 1
[.,[[.,[.,[[.,[.,[.,.]]],.]]],.]]
=> [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,4,6,7,8,2,5] => ?
=> ? = 3 - 1
[.,[[.,[.,[[[.,.],[.,.]],.]]],.]]
=> [1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,4,7,2,8,5,6] => ?
=> ? = 3 - 1
[.,[[.,[[.,.],[.,[.,[.,.]]]]],.]]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,3,5,2,6,7,8,4] => ?
=> ? = 3 - 1
[.,[[.,[[.,[.,.]],[.,[.,.]]]],.]]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,3,5,6,2,7,8,4] => ?
=> ? = 3 - 1
[.,[[.,[[[.,.],[[.,.],.]],.]],.]]
=> [1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,6,2,8,4,5,7] => ?
=> ? = 4 - 1
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,6,7,2,8,4,5] => ?
=> ? = 3 - 1
[.,[[.,[[[.,[[.,.],.]],.],.]],.]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,3,6,8,2,4,5,7] => ?
=> ? = 4 - 1
[.,[[[.,.],[[.,[.,.]],[.,.]]],.]]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,4,2,6,7,3,8,5] => ?
=> ? = 3 - 1
[.,[[[.,[.,[.,.]]],[[.,.],.]],.]]
=> [1,0,1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,4,5,6,2,8,3,7] => ?
=> ? = 3 - 1
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,6,7,2,3,8,5] => ?
=> ? = 3 - 1
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,5,6,2,7,3,8,4] => ?
=> ? = 2 - 1
[.,[[[.,[.,[[[.,.],.],.]]],.],.]]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,4,5,8,2,3,6,7] => ?
=> ? = 3 - 1
[[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,4,6,7,5,8] => ?
=> ? = 3 - 1
[[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,5,4,6,7,8] => ?
=> ? = 3 - 1
[[.,.],[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4,6,8,7] => ?
=> ? = 4 - 1
[[.,.],[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6,8] => ?
=> ? = 4 - 1
[[.,.],[.,[[.,.],[[[.,.],.],.]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,8,6,7] => ?
=> ? = 4 - 1
[[.,.],[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,3,5,6,4,7,8] => ?
=> ? = 3 - 1
[[.,.],[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,3,6,4,7,5,8] => ?
=> ? = 3 - 1
[[.,.],[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,3,6,7,4,5,8] => ?
=> ? = 3 - 1
[[.,.],[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,1,3,5,7,4,8,6] => ?
=> ? = 4 - 1
[[.,.],[.,[[[.,.],[.,[.,.]]],.]]]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,1,3,6,4,7,8,5] => ?
=> ? = 3 - 1
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000659
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 84%●distinct values known / distinct values provided: 71%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 84%●distinct values known / distinct values provided: 71%
Values
[.,.]
=> [1] => [.,.]
=> [1,0]
=> ? = 1 - 1
[.,[.,.]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0 = 1 - 1
[[.,.],.]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1 = 2 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[.,.],[.,.]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[[.,[.,.]],.]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[[[.,.],.],.]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [6,7,8,5,4,3,2,1] => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [5,6,7,8,4,3,2,1] => [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[.,[.,[.,[[.,.],[[.,[.,.]],.]]]]]
=> [7,6,8,4,5,3,2,1] => ?
=> ?
=> ? = 3 - 1
[.,[.,[.,[[.,.],[[[.,.],.],.]]]]]
=> [6,7,8,4,5,3,2,1] => ?
=> ?
=> ? = 3 - 1
[.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [8,6,5,4,7,3,2,1] => ?
=> ?
=> ? = 2 - 1
[.,[.,[.,[[[.,[.,.]],.],[.,.]]]]]
=> [8,5,4,6,7,3,2,1] => ?
=> ?
=> ? = 2 - 1
[.,[.,[.,[[.,[[[.,.],.],.]],.]]]]
=> [5,6,7,4,8,3,2,1] => [[[[[.,[.,[.,.]]],[.,.]],.],.],.]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [4,5,6,7,8,3,2,1] => [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> [5,6,7,8,3,4,2,1] => [[[[.,[.,[.,[.,.]]]],[.,.]],.],.]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[.,[.,[[.,[.,.]],[[.,.],[.,.]]]]]
=> [8,6,7,4,3,5,2,1] => ?
=> ?
=> ? = 3 - 1
[.,[.,[[.,[.,.]],[[[.,.],.],.]]]]
=> [6,7,8,4,3,5,2,1] => [[[[[.,[.,[.,.]]],.],[.,.]],.],.]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[.,[.,[[[.,.],.],[[[.,.],.],.]]]]
=> [6,7,8,3,4,5,2,1] => [[[[.,[.,[.,.]]],[.,[.,.]]],.],.]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 - 1
[.,[.,[[.,[[.,.],.]],[.,[.,.]]]]]
=> [8,7,4,5,3,6,2,1] => ?
=> ?
=> ? = 3 - 1
[.,[.,[[[.,[.,[.,.]]],.],[.,.]]]]
=> [8,5,4,3,6,7,2,1] => ?
=> ?
=> ? = 2 - 1
[.,[.,[[.,[.,[[.,[.,.]],.]]],.]]]
=> [6,5,7,4,3,8,2,1] => ?
=> ?
=> ? = 3 - 1
[.,[.,[[.,[.,[[[.,.],.],.]]],.]]]
=> [5,6,7,4,3,8,2,1] => [[[[[.,[.,[.,.]]],.],[.,.]],.],.]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[.,[.,[[.,[[[[.,.],.],.],.]],.]]]
=> [4,5,6,7,3,8,2,1] => [[[[.,[.,[.,[.,.]]]],[.,.]],.],.]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[.,[.,[[[.,[[[.,.],.],.]],.],.]]]
=> [4,5,6,3,7,8,2,1] => ?
=> ?
=> ? = 3 - 1
[.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> [3,4,5,6,7,8,2,1] => [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[.,[[.,.],[.,[[[[.,.],.],.],.]]]]
=> [5,6,7,8,4,2,3,1] => [[[[.,[.,[.,[.,.]]]],.],[.,.]],.]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 3 - 1
[.,[[.,.],[[.,[.,.]],[.,[.,.]]]]]
=> [8,7,5,4,6,2,3,1] => ?
=> ?
=> ? = 3 - 1
[.,[[.,.],[[.,[.,[.,.]]],[.,.]]]]
=> [8,6,5,4,7,2,3,1] => ?
=> ?
=> ? = 3 - 1
[.,[[.,.],[[.,[.,[.,[.,.]]]],.]]]
=> [7,6,5,4,8,2,3,1] => ?
=> ?
=> ? = 3 - 1
[.,[[.,.],[[[[[.,.],.],.],.],.]]]
=> [4,5,6,7,8,2,3,1] => [[[.,[.,[.,[.,[.,.]]]]],[.,.]],.]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[.,[[.,[.,.]],[.,[[[.,.],.],.]]]]
=> [6,7,8,5,3,2,4,1] => [[[[[.,[.,[.,.]]],.],.],[.,.]],.]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3 - 1
[.,[[.,[.,.]],[[[[.,.],.],.],.]]]
=> [5,6,7,8,3,2,4,1] => [[[[.,[.,[.,[.,.]]]],.],[.,.]],.]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 3 - 1
[.,[[[.,.],.],[[[[.,.],.],.],.]]]
=> [5,6,7,8,2,3,4,1] => [[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 3 - 1
[.,[[.,[.,[.,.]]],[.,[[.,.],.]]]]
=> [7,8,6,4,3,2,5,1] => ?
=> ?
=> ? = 3 - 1
[.,[[.,[.,[.,.]]],[[.,.],[.,.]]]]
=> [8,6,7,4,3,2,5,1] => ?
=> ?
=> ? = 3 - 1
[.,[[.,[.,[.,.]]],[[[.,.],.],.]]]
=> [6,7,8,4,3,2,5,1] => ?
=> ?
=> ? = 3 - 1
[.,[[.,[[.,.],.]],[[[.,.],.],.]]]
=> [6,7,8,3,4,2,5,1] => [[[[.,[.,[.,.]]],[.,.]],[.,.]],.]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
[.,[[[.,.],[.,.]],[[.,[.,.]],.]]]
=> [7,6,8,4,2,3,5,1] => ?
=> ?
=> ? = 3 - 1
[.,[[[.,.],[.,.]],[[[.,.],.],.]]]
=> [6,7,8,4,2,3,5,1] => [[[[.,[.,[.,.]]],.],[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 - 1
[.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> [6,7,8,2,3,4,5,1] => [[[.,[.,[.,.]]],[.,[.,[.,.]]]],.]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 3 - 1
[.,[[.,[.,[.,[.,.]]]],[.,[.,.]]]]
=> [8,7,5,4,3,2,6,1] => ?
=> ?
=> ? = 2 - 1
[.,[[.,[.,[.,[[.,.],.]]]],[.,.]]]
=> [8,5,6,4,3,2,7,1] => ?
=> ?
=> ? = 3 - 1
[.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [8,6,4,5,3,2,7,1] => ?
=> ?
=> ? = 3 - 1
[.,[[.,[[.,.],[.,[.,.]]]],[.,.]]]
=> [8,6,5,3,4,2,7,1] => ?
=> ?
=> ? = 3 - 1
[.,[[.,[[.,[.,[.,.]]],.]],[.,.]]]
=> [8,5,4,3,6,2,7,1] => ?
=> ?
=> ? = 3 - 1
[.,[[.,[[[[.,.],.],.],.]],[.,.]]]
=> [8,3,4,5,6,2,7,1] => ?
=> ?
=> ? = 3 - 1
[.,[[[[.,.],[[.,.],.]],.],[.,.]]]
=> [8,4,5,2,3,6,7,1] => ?
=> ?
=> ? = 3 - 1
[.,[[.,[.,[.,[[.,.],[.,.]]]]],.]]
=> [7,5,6,4,3,2,8,1] => ?
=> ?
=> ? = 3 - 1
[.,[[.,[.,[[.,[.,[.,.]]],.]]],.]]
=> [6,5,4,7,3,2,8,1] => ?
=> ?
=> ? = 3 - 1
[.,[[.,[.,[[[[.,.],.],.],.]]],.]]
=> [4,5,6,7,3,2,8,1] => [[[[.,[.,[.,[.,.]]]],.],[.,.]],.]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 3 - 1
[.,[[.,[[.,.],[[[.,.],.],.]]],.]]
=> [5,6,7,3,4,2,8,1] => ?
=> ?
=> ? = 4 - 1
[.,[[.,[[.,[[[.,.],.],.]],.]],.]]
=> [4,5,6,3,7,2,8,1] => [[[[.,[.,[.,.]]],[.,.]],[.,.]],.]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
[.,[[.,[[[[.,[.,.]],.],.],.]],.]]
=> [4,3,5,6,7,2,8,1] => ?
=> ?
=> ? = 3 - 1
[.,[[.,[[[[[.,.],.],.],.],.]],.]]
=> [3,4,5,6,7,2,8,1] => [[[.,[.,[.,[.,[.,.]]]]],[.,.]],.]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[.,[[[.,.],[.,[[[.,.],.],.]]],.]]
=> [5,6,7,4,2,3,8,1] => [[[[.,[.,[.,.]]],.],[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 - 1
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St000919
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000919: Binary trees ⟶ ℤResult quality: 71% ●values known / values provided: 73%●distinct values known / distinct values provided: 71%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000919: Binary trees ⟶ ℤResult quality: 71% ●values known / values provided: 73%●distinct values known / distinct values provided: 71%
Values
[.,.]
=> [1,0]
=> [1] => [.,.]
=> ? = 1 - 1
[.,[.,.]]
=> [1,0,1,0]
=> [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[[.,.],.]
=> [1,1,0,0]
=> [2,1] => [[.,.],.]
=> 1 = 2 - 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> 1 = 2 - 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> 1 = 2 - 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [2,3,1] => [[.,.],[.,.]]
=> 1 = 2 - 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> 1 = 2 - 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1 = 2 - 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> 1 = 2 - 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> 2 = 3 - 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> 1 = 2 - 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> 2 = 3 - 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 1 = 2 - 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> 1 = 2 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1 = 2 - 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> 1 = 2 - 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 2 = 3 - 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> 1 = 2 - 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> 2 = 3 - 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> 2 = 3 - 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> 2 = 3 - 1
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> 2 = 3 - 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> 2 = 3 - 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> 1 = 2 - 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
=> 1 = 2 - 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> 1 = 2 - 1
[.,[.,[[.,[.,.]],[.,[[.,.],.]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6,8,7] => ?
=> ? = 3 - 1
[.,[.,[[.,[[.,[.,.]],.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,2,4,6,7,3,5,8] => ?
=> ? = 3 - 1
[.,[.,[[[.,[.,[.,.]]],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,2,5,6,7,3,4,8] => ?
=> ? = 2 - 1
[.,[.,[[.,[[.,.],[.,[.,.]]]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,2,4,6,3,7,8,5] => ?
=> ? = 3 - 1
[.,[.,[[.,[[.,[.,.]],[.,.]]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,2,4,6,7,3,8,5] => ?
=> ? = 3 - 1
[.,[[.,.],[.,[[[.,.],.],[.,.]]]]]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,3,2,4,7,5,6,8] => ?
=> ? = 3 - 1
[.,[[.,.],[[.,[[.,.],[.,.]]],.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,7,4,8,6] => ?
=> ? = 4 - 1
[.,[[.,.],[[[.,.],[.,[.,.]]],.]]]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,7,8,5] => ?
=> ? = 3 - 1
[.,[[.,[.,.]],[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,4,2,6,5,7,8] => ?
=> ? = 3 - 1
[.,[[.,[.,.]],[[[.,.],.],[.,.]]]]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,4,2,7,5,6,8] => ?
=> ? = 3 - 1
[.,[[.,[.,[.,.]]],[.,[[.,.],.]]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6,8,7] => ?
=> ? = 3 - 1
[.,[[.,[[.,.],.]],[[.,[.,.]],.]]]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4,7,8,6] => ?
=> ? = 4 - 1
[.,[[[.,.],[.,.]],[[.,[.,.]],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,4,2,5,3,7,8,6] => ?
=> ? = 3 - 1
[.,[[[.,[.,.]],.],[.,[[.,.],.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [1,4,5,2,3,6,8,7] => ?
=> ? = 3 - 1
[.,[[.,[[.,.],[.,.]]],[.,[.,.]]]]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,3,5,2,6,4,7,8] => ?
=> ? = 3 - 1
[.,[[[.,[.,[.,.]]],.],[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [1,4,5,6,2,3,8,7] => ?
=> ? = 3 - 1
[.,[[[[.,[[.,.],.]],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,5,7,2,3,4,6,8] => ?
=> ? = 3 - 1
[.,[[.,[.,[.,[[.,.],[.,.]]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,5,7,2,8,6] => ?
=> ? = 3 - 1
[.,[[.,[.,[[.,.],[.,[.,.]]]]],.]]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,4,6,2,7,8,5] => ?
=> ? = 3 - 1
[.,[[.,[.,[[.,.],[[.,.],.]]]],.]]
=> [1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,4,6,2,8,5,7] => ?
=> ? = 4 - 1
[.,[[.,[.,[[.,[.,[.,.]]],.]]],.]]
=> [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,4,6,7,8,2,5] => ?
=> ? = 3 - 1
[.,[[.,[.,[[[.,.],[.,.]],.]]],.]]
=> [1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,4,7,2,8,5,6] => ?
=> ? = 3 - 1
[.,[[.,[[.,.],[.,[.,[.,.]]]]],.]]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,3,5,2,6,7,8,4] => ?
=> ? = 3 - 1
[.,[[.,[[[.,.],[[.,.],.]],.]],.]]
=> [1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,6,2,8,4,5,7] => ?
=> ? = 4 - 1
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,6,7,2,8,4,5] => ?
=> ? = 3 - 1
[.,[[.,[[[.,[[.,.],.]],.],.]],.]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,3,6,8,2,4,5,7] => ?
=> ? = 4 - 1
[.,[[[.,.],[[.,[.,.]],[.,.]]],.]]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,4,2,6,7,3,8,5] => ?
=> ? = 3 - 1
[.,[[[.,[.,[.,.]]],[[.,.],.]],.]]
=> [1,0,1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,4,5,6,2,8,3,7] => ?
=> ? = 3 - 1
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,6,7,2,3,8,5] => ?
=> ? = 3 - 1
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,5,6,2,7,3,8,4] => ?
=> ? = 2 - 1
[.,[[[.,[.,[[[.,.],.],.]]],.],.]]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,4,5,8,2,3,6,7] => ?
=> ? = 3 - 1
[[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,4,6,7,5,8] => ?
=> ? = 3 - 1
[[.,.],[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6,8] => ?
=> ? = 4 - 1
[[.,.],[.,[[.,.],[[[.,.],.],.]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,8,6,7] => ?
=> ? = 4 - 1
[[.,.],[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,3,5,6,4,7,8] => ?
=> ? = 3 - 1
[[.,.],[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,3,6,4,7,5,8] => ?
=> ? = 3 - 1
[[.,.],[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,3,6,7,4,5,8] => ?
=> ? = 3 - 1
[[.,.],[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,1,3,5,7,4,8,6] => ?
=> ? = 4 - 1
[[.,.],[.,[[[.,.],[.,[.,.]]],.]]]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,1,3,6,4,7,8,5] => ?
=> ? = 3 - 1
[[.,.],[.,[[[.,[[.,.],.]],.],.]]]
=> [1,1,0,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,1,3,6,8,4,5,7] => ?
=> ? = 4 - 1
[[.,.],[[.,.],[[[.,.],.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,7,5,6,8] => ?
=> ? = 4 - 1
[[.,.],[[.,[.,.]],[[[.,.],.],.]]]
=> [1,1,0,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,1,4,5,3,8,6,7] => ?
=> ? = 4 - 1
[[.,.],[[.,[[.,[.,.]],.]],[.,.]]]
=> [1,1,0,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,1,4,6,7,3,5,8] => ?
=> ? = 4 - 1
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,1,6,3,7,4,5,8] => ?
=> ? = 3 - 1
[[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [2,1,6,7,3,4,5,8] => ?
=> ? = 3 - 1
[[.,.],[[.,[[[.,.],[.,.]],.]],.]]
=> [1,1,0,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,1,4,7,3,8,5,6] => ?
=> ? = 4 - 1
[[.,.],[[[[.,[.,.]],[.,.]],.],.]]
=> [1,1,0,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [2,1,6,7,3,8,4,5] => ?
=> ? = 3 - 1
[[.,[.,.]],[.,[[.,.],[.,[.,.]]]]]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,4,6,5,7,8] => ?
=> ? = 3 - 1
[[.,[.,.]],[.,[[.,[[.,.],.]],.]]]
=> [1,1,0,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,1,4,6,8,5,7] => ?
=> ? = 4 - 1
Description
The number of maximal left branches of a binary tree.
A maximal left branch of a binary tree is an inclusion wise maximal path which consists of left edges only. This statistic records the number of distinct maximal left branches in the tree.
Matching statistic: St000374
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [.,.]
=> [1] => 0 = 1 - 1
[.,[.,.]]
=> [2,1] => [[.,.],.]
=> [1,2] => 0 = 1 - 1
[[.,.],.]
=> [1,2] => [.,[.,.]]
=> [2,1] => 1 = 2 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 1 = 2 - 1
[[.,.],[.,.]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => 1 = 2 - 1
[[.,[.,.]],.]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 1 = 2 - 1
[[[.,.],.],.]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 1 = 2 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1 = 2 - 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 1 = 2 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2 = 3 - 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1 = 2 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2 = 3 - 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1 = 2 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 1 = 2 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 1 = 2 - 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => 1 = 2 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 2 = 3 - 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 1 = 2 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 2 = 3 - 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 2 = 3 - 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 2 = 3 - 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 2 = 3 - 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 2 = 3 - 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1 = 2 - 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 1 = 2 - 1
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [6,7,4,5,3,2,1] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [2,1,4,3,5,6,7] => ? = 3 - 1
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [2,1,4,3,5,6,7] => ? = 3 - 1
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [6,7,5,3,4,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 3 - 1
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [5,6,7,3,4,2,1] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 3 - 1
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [6,7,4,3,5,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 3 - 1
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [6,7,3,4,5,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 3 - 1
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [7,3,4,5,6,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 2 - 1
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 3 - 1
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [4,5,6,3,7,2,1] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 3 - 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [5,6,3,4,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 3 - 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [6,3,4,5,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 2 - 1
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [4,5,3,6,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 3 - 1
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [5,3,4,6,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 2 - 1
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [4,3,5,6,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 2 - 1
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,2,3,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 3 - 1
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [5,6,7,4,2,3,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [3,2,1,4,6,5,7] => ? = 3 - 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [6,7,4,5,2,3,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 4 - 1
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [5,6,4,7,2,3,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 4 - 1
[.,[[.,.],[[[[.,.],.],.],.]]]
=> [4,5,6,7,2,3,1] => [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [6,7,5,3,2,4,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 3 - 1
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> [5,6,7,3,2,4,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [3,2,1,4,6,5,7] => ? = 3 - 1
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> [6,7,5,2,3,4,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 3 - 1
[.,[[[.,.],.],[[[.,.],.],.]]]
=> [5,6,7,2,3,4,1] => [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [3,2,1,6,5,4,7] => ? = 3 - 1
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [6,7,4,3,2,5,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 3 - 1
[.,[[.,[[.,.],.]],[[.,.],.]]]
=> [6,7,3,4,2,5,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 4 - 1
[.,[[[.,.],[.,.]],[[.,.],.]]]
=> [6,7,4,2,3,5,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 3 - 1
[.,[[[.,[.,.]],.],[[.,.],.]]]
=> [6,7,3,2,4,5,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 3 - 1
[.,[[[[.,.],.],.],[[.,.],.]]]
=> [6,7,2,3,4,5,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 3 - 1
[.,[[[[[.,.],.],.],.],[.,.]]]
=> [7,2,3,4,5,6,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 2 - 1
[.,[[.,[.,[.,[[.,.],.]]]],.]]
=> [5,6,4,3,2,7,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 3 - 1
[.,[[.,[.,[[[.,.],.],.]]],.]]
=> [4,5,6,3,2,7,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [3,2,1,4,6,5,7] => ? = 3 - 1
[.,[[.,[[.,.],[[.,.],.]]],.]]
=> [5,6,3,4,2,7,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 4 - 1
[.,[[.,[[.,[[.,.],.]],.]],.]]
=> [4,5,3,6,2,7,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 4 - 1
[.,[[.,[[[[.,.],.],.],.]],.]]
=> [3,4,5,6,2,7,1] => [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[.,[[[.,.],[.,[[.,.],.]]],.]]
=> [5,6,4,2,3,7,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 3 - 1
[.,[[[.,.],[[[.,.],.],.]],.]]
=> [4,5,6,2,3,7,1] => [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [3,2,1,6,5,4,7] => ? = 3 - 1
[.,[[[.,[.,.]],[[.,.],.]],.]]
=> [5,6,3,2,4,7,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 3 - 1
[.,[[[[.,.],.],[[.,.],.]],.]]
=> [5,6,2,3,4,7,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 3 - 1
[.,[[[[[.,.],.],.],[.,.]],.]]
=> [6,2,3,4,5,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 2 - 1
[.,[[[.,[.,[[.,.],.]]],.],.]]
=> [4,5,3,2,6,7,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 3 - 1
[.,[[[.,[[[.,.],.],.]],.],.]]
=> [3,4,5,2,6,7,1] => [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [3,2,1,6,5,4,7] => ? = 3 - 1
[.,[[[[.,.],[[.,.],.]],.],.]]
=> [4,5,2,3,6,7,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 3 - 1
[.,[[[[[.,.],.],[.,.]],.],.]]
=> [5,2,3,4,6,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 2 - 1
[.,[[[[.,[[.,.],.]],.],.],.]]
=> [3,4,2,5,6,7,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 3 - 1
[.,[[[[[.,.],[.,.]],.],.],.]]
=> [4,2,3,5,6,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 2 - 1
[.,[[[[[.,[.,.]],.],.],.],.]]
=> [3,2,4,5,6,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 2 - 1
[[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,3,1,2] => [[[[[.,[.,.]],.],.],.],[.,.]]
=> [2,1,3,4,5,7,6] => ? = 3 - 1
[[.,.],[.,[.,[[[.,.],.],.]]]]
=> [5,6,7,4,3,1,2] => [[[[.,[.,[.,.]]],.],.],[.,.]]
=> [3,2,1,4,5,7,6] => ? = 3 - 1
[[.,.],[.,[[.,.],[[.,.],.]]]]
=> [6,7,4,5,3,1,2] => [[[[.,[.,.]],[.,.]],.],[.,.]]
=> [2,1,4,3,5,7,6] => ? = 4 - 1
[[.,.],[.,[[.,[[.,.],.]],.]]]
=> [5,6,4,7,3,1,2] => [[[[.,[.,.]],[.,.]],.],[.,.]]
=> [2,1,4,3,5,7,6] => ? = 4 - 1
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000996
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [.,.]
=> [1] => 0 = 1 - 1
[.,[.,.]]
=> [2,1] => [[.,.],.]
=> [1,2] => 0 = 1 - 1
[[.,.],.]
=> [1,2] => [.,[.,.]]
=> [2,1] => 1 = 2 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 1 = 2 - 1
[[.,.],[.,.]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => 1 = 2 - 1
[[.,[.,.]],.]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 1 = 2 - 1
[[[.,.],.],.]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 1 = 2 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1 = 2 - 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 1 = 2 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2 = 3 - 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1 = 2 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2 = 3 - 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1 = 2 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 1 = 2 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 1 = 2 - 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => 1 = 2 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 2 = 3 - 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 1 = 2 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 2 = 3 - 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 2 = 3 - 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 2 = 3 - 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 2 = 3 - 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 2 = 3 - 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1 = 2 - 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 1 = 2 - 1
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [6,7,4,5,3,2,1] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [2,1,4,3,5,6,7] => ? = 3 - 1
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [2,1,4,3,5,6,7] => ? = 3 - 1
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [6,7,5,3,4,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 3 - 1
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [5,6,7,3,4,2,1] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 3 - 1
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [6,7,4,3,5,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 3 - 1
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [6,7,3,4,5,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 3 - 1
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [7,3,4,5,6,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 2 - 1
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 3 - 1
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [4,5,6,3,7,2,1] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 3 - 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [5,6,3,4,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 3 - 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [6,3,4,5,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 2 - 1
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [4,5,3,6,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 3 - 1
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [5,3,4,6,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 2 - 1
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [4,3,5,6,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 2 - 1
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,2,3,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 3 - 1
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [5,6,7,4,2,3,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [3,2,1,4,6,5,7] => ? = 3 - 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [6,7,4,5,2,3,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 4 - 1
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [5,6,4,7,2,3,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 4 - 1
[.,[[.,.],[[[[.,.],.],.],.]]]
=> [4,5,6,7,2,3,1] => [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [6,7,5,3,2,4,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 3 - 1
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> [5,6,7,3,2,4,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [3,2,1,4,6,5,7] => ? = 3 - 1
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> [6,7,5,2,3,4,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 3 - 1
[.,[[[.,.],.],[[[.,.],.],.]]]
=> [5,6,7,2,3,4,1] => [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [3,2,1,6,5,4,7] => ? = 3 - 1
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [6,7,4,3,2,5,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 3 - 1
[.,[[.,[[.,.],.]],[[.,.],.]]]
=> [6,7,3,4,2,5,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 4 - 1
[.,[[[.,.],[.,.]],[[.,.],.]]]
=> [6,7,4,2,3,5,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 3 - 1
[.,[[[.,[.,.]],.],[[.,.],.]]]
=> [6,7,3,2,4,5,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 3 - 1
[.,[[[[.,.],.],.],[[.,.],.]]]
=> [6,7,2,3,4,5,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 3 - 1
[.,[[[[[.,.],.],.],.],[.,.]]]
=> [7,2,3,4,5,6,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 2 - 1
[.,[[.,[.,[.,[[.,.],.]]]],.]]
=> [5,6,4,3,2,7,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 3 - 1
[.,[[.,[.,[[[.,.],.],.]]],.]]
=> [4,5,6,3,2,7,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [3,2,1,4,6,5,7] => ? = 3 - 1
[.,[[.,[[.,.],[[.,.],.]]],.]]
=> [5,6,3,4,2,7,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 4 - 1
[.,[[.,[[.,[[.,.],.]],.]],.]]
=> [4,5,3,6,2,7,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 4 - 1
[.,[[.,[[[[.,.],.],.],.]],.]]
=> [3,4,5,6,2,7,1] => [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[.,[[[.,.],[.,[[.,.],.]]],.]]
=> [5,6,4,2,3,7,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 3 - 1
[.,[[[.,.],[[[.,.],.],.]],.]]
=> [4,5,6,2,3,7,1] => [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [3,2,1,6,5,4,7] => ? = 3 - 1
[.,[[[.,[.,.]],[[.,.],.]],.]]
=> [5,6,3,2,4,7,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 3 - 1
[.,[[[[.,.],.],[[.,.],.]],.]]
=> [5,6,2,3,4,7,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 3 - 1
[.,[[[[[.,.],.],.],[.,.]],.]]
=> [6,2,3,4,5,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 2 - 1
[.,[[[.,[.,[[.,.],.]]],.],.]]
=> [4,5,3,2,6,7,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 3 - 1
[.,[[[.,[[[.,.],.],.]],.],.]]
=> [3,4,5,2,6,7,1] => [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [3,2,1,6,5,4,7] => ? = 3 - 1
[.,[[[[.,.],[[.,.],.]],.],.]]
=> [4,5,2,3,6,7,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 3 - 1
[.,[[[[[.,.],.],[.,.]],.],.]]
=> [5,2,3,4,6,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 2 - 1
[.,[[[[.,[[.,.],.]],.],.],.]]
=> [3,4,2,5,6,7,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 3 - 1
[.,[[[[[.,.],[.,.]],.],.],.]]
=> [4,2,3,5,6,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 2 - 1
[.,[[[[[.,[.,.]],.],.],.],.]]
=> [3,2,4,5,6,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 2 - 1
[[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,3,1,2] => [[[[[.,[.,.]],.],.],.],[.,.]]
=> [2,1,3,4,5,7,6] => ? = 3 - 1
[[.,.],[.,[.,[[[.,.],.],.]]]]
=> [5,6,7,4,3,1,2] => [[[[.,[.,[.,.]]],.],.],[.,.]]
=> [3,2,1,4,5,7,6] => ? = 3 - 1
[[.,.],[.,[[.,.],[[.,.],.]]]]
=> [6,7,4,5,3,1,2] => [[[[.,[.,.]],[.,.]],.],[.,.]]
=> [2,1,4,3,5,7,6] => ? = 4 - 1
[[.,.],[.,[[.,[[.,.],.]],.]]]
=> [5,6,4,7,3,1,2] => [[[[.,[.,.]],[.,.]],.],[.,.]]
=> [2,1,4,3,5,7,6] => ? = 4 - 1
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
The following 43 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000251The number of nonsingleton blocks of a set partition. St000670The reversal length of a permutation. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001354The number of series nodes in the modular decomposition of a graph. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001737The number of descents of type 2 in a permutation. St000834The number of right outer peaks of a permutation. St000035The number of left outer peaks of a permutation. St000386The number of factors DDU in a Dyck path. St000703The number of deficiencies of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000470The number of runs in a permutation. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000354The number of recoils of a permutation. St000245The number of ascents of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001665The number of pure excedances of a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St000213The number of weak exceedances (also weak excedences) of a permutation. St000325The width of the tree associated to a permutation. St000702The number of weak deficiencies of a permutation. St000021The number of descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001840The number of descents of a set partition. St000083The number of left oriented leafs of a binary tree except the first one. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000647The number of big descents of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000646The number of big ascents of a permutation. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001896The number of right descents of a signed permutations. St001597The Frobenius rank of a skew partition. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St000920The logarithmic height of a Dyck path. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
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