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Your data matches 53 different statistics following compositions of up to 3 maps.
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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: 100% ●values known / values provided: 100%●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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1] => [[1]]
=> 0
[.,[.,.]]
=> [1,0,1,0]
=> [1,2] => [[1,2]]
=> 0
[[.,.],.]
=> [1,1,0,0]
=> [2,1] => [[1],[2]]
=> 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,2,3] => [[1,2,3]]
=> 0
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [2,3,1] => [[1,3],[2]]
=> 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3,1,2] => [[1,2],[3]]
=> 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [[1,2,4],[3]]
=> 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [[1,2,3],[4]]
=> 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[1,3,4],[2]]
=> 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[1,2,4],[3]]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[1,3,4],[2]]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [[1,3],[2,4]]
=> 2
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [[1,2],[3,4]]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [[1,2,3],[4]]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [[1,2,3,5],[4]]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [[1,2,3,4],[5]]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [[1,2,4,5],[3]]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [[1,2,3,5],[4]]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [[1,2,4,5],[3]]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [[1,2,4],[3,5]]
=> 2
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [[1,2,3],[4,5]]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [[1,2,3,4],[5]]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[1,3,5],[2,4]]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[1,3,4],[2,5]]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[1,3,4,5],[2]]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[1,3,4],[2,5]]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [[1,2,4,5],[3]]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [[1,2,4],[3,5]]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[1,3,4,5],[2]]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[1,3,5],[2,4]]
=> 2
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [[1,2,5],[3,4]]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [[1,2,5],[3,4]]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [[1,2,3,5],[4]]
=> 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: St000291
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1] => 1 => 0
[.,[.,.]]
=> [1,0,1,0]
=> [1,1] => 11 => 0
[[.,.],.]
=> [1,1,0,0]
=> [2] => 10 => 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1] => 111 => 0
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,2] => 110 => 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1] => 101 => 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [2,1] => 101 => 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3] => 100 => 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => 1101 => 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 1011 => 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => 1011 => 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => 1010 => 2
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => 1001 => 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => 1001 => 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4] => 1000 => 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 11101 => 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 11011 => 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 11011 => 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 11010 => 2
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 11001 => 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 11001 => 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 10101 => 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 10110 => 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 10010 => 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 10111 => 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 10101 => 2
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => 10011 => 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => 10011 => 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 10001 => 1
Description
The number of descents of a binary word.
Matching statistic: St000659
(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
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[[.,.],.]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 2
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 2
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 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)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000919: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000919: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1,1,0,0]
=> [.,[.,.]]
=> 0
[.,[.,.]]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [.,[.,[.,.]]]
=> 0
[[.,.],.]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [.,[[.,.],.]]
=> 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,.]]]]
=> 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [.,[.,[[.,.],.]]]
=> 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [.,[[.,.],[.,[.,.]]]]
=> 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> 2
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> 2
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [.,[.,[[[.,.],[.,.]],.]]]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [.,[.,[[[.,[.,.]],.],.]]]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [.,[.,[[[[.,.],.],.],.]]]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [.,[[.,.],[.,[[.,.],.]]]]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [.,[[.,.],[[.,[.,.]],.]]]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [.,[[.,.],[[[.,.],.],.]]]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [.,[[.,[.,.]],[.,[.,.]]]]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[[.,.],.]]]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [.,[[[.,.],.],[[.,.],.]]]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> 2
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [.,[[[[.,.],.],.],[.,.]]]
=> 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: St000985
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000985: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000985: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [.,.]
=> [1] => ([],1)
=> 0
[.,[.,.]]
=> [[.,.],.]
=> [1,2] => ([],2)
=> 0
[[.,.],.]
=> [.,[.,.]]
=> [2,1] => ([(0,1)],2)
=> 1
[.,[.,[.,.]]]
=> [[[.,.],.],.]
=> [1,2,3] => ([],3)
=> 0
[.,[[.,.],.]]
=> [[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> 1
[[.,.],[.,.]]
=> [[.,.],[.,.]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[.,[.,.]],.]
=> [.,[[.,.],.]]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[[[.,.],.],.]
=> [.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[.,[.,[.,[.,.]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> 0
[.,[.,[[.,.],.]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> 1
[.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 1
[.,[[.,[.,.]],.]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1
[.,[[[.,.],.],.]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1
[[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
[[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[.,[.,[.,.]]],.]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[.,[[.,.],.]],.]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[[.,[.,.]],.],.]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => ([],5)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 1
[.,[.,[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 1
[.,[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 1
[.,[[[.,.],.],[.,.]]]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 1
[.,[[.,[[.,.],.]],.]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[.,[[[.,[.,.]],.],.]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[.,[[[[.,.],.],.],.]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[.,.],[.,[[.,.],.]]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[.,.],[[[.,.],.],.]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
[[.,[.,.]],[[.,.],.]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[[[.,.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[[.,.],.],[[.,.],.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
[[.,[[.,.],.]],[.,.]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[[[.,.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[[[.,[.,.]],.],[.,.]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[[[[.,.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
Description
The number of positive eigenvalues of the adjacency matrix of the graph.
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: 100% ●values known / values provided: 100%●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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1]
=> 0
[.,[.,.]]
=> [2,1] => [1,1] => [1,1]
=> 0
[[.,.],.]
=> [1,2] => [2] => [2]
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1] => [1,1,1]
=> 0
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [2,1]
=> 1
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => [2,1]
=> 1
[[.,[.,.]],.]
=> [2,1,3] => [1,2] => [2,1]
=> 1
[[[.,.],.],.]
=> [1,2,3] => [3] => [3]
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1] => [1,1,1,1]
=> 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,1,1] => [2,1,1]
=> 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,2,1] => [2,1,1]
=> 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,1] => [2,1,1]
=> 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,1] => [3,1]
=> 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,2] => [2,1,1]
=> 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,2] => [2,2]
=> 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,2] => [2,1,1]
=> 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,3] => [3,1]
=> 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,2] => [2,1,1]
=> 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,2] => [2,2]
=> 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,3] => [3,1]
=> 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3] => [3,1]
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4] => [4]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1] => [1,1,1,1,1]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [2,1,1,1] => [2,1,1,1]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,1,1] => [2,1,1,1]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,1,1] => [2,1,1,1]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1] => [3,1,1]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,2,1] => [2,1,1,1]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [2,2,1] => [2,2,1]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,2,1] => [2,1,1,1]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,3,1] => [3,1,1]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,2,1] => [2,1,1,1]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [2,2,1] => [2,2,1]
=> 2
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,3,1] => [3,1,1]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,3,1] => [3,1,1]
=> 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [4,1] => [4,1]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,2] => [2,1,1,1]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,2] => [2,2,1]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,2,2] => [2,2,1]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,2,2] => [2,2,1]
=> 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,2] => [3,2]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,2] => [2,1,1,1]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [2,1,2] => [2,2,1]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,3] => [3,1,1]
=> 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [2,3] => [3,2]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,2] => [2,1,1,1]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,2,2] => [2,2,1]
=> 2
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,3] => [3,1,1]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,3] => [3,1,1]
=> 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,4] => [4,1]
=> 1
Description
The number of parts of an integer partition that are at least two.
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1]
=> 1 = 0 + 1
[.,[.,.]]
=> [2,1] => [2,1] => [2]
=> 1 = 0 + 1
[[.,.],.]
=> [1,2] => [1,2] => [1,1]
=> 2 = 1 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [3]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [2,3,1] => [2,3,1] => [2,1]
=> 2 = 1 + 1
[[.,.],[.,.]]
=> [3,1,2] => [3,1,2] => [2,1]
=> 2 = 1 + 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,1]
=> 2 = 1 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [4]
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,4,2,1] => [3,1]
=> 2 = 1 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [4,2,3,1] => [3,1]
=> 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,4,1] => [3,1]
=> 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,4,3,1] => [3,1]
=> 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [4,3,1,2] => [3,1]
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,4,1,2] => [2,1,1]
=> 3 = 2 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [4,2,1,3] => [3,1]
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [4,1,3,2] => [3,1]
=> 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [3,1]
=> 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,4,1,3] => [2,1,1]
=> 3 = 2 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,4,2] => [2,2]
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,4,3,2] => [3,1]
=> 2 = 1 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [5]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,5,3,2,1] => [4,1]
=> 2 = 1 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [5,3,4,2,1] => [4,1]
=> 2 = 1 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,5,2,1] => [4,1]
=> 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,5,4,2,1] => [4,1]
=> 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [5,4,2,3,1] => [4,1]
=> 2 = 1 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,5,2,3,1] => [3,1,1]
=> 3 = 2 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [5,3,2,4,1] => [4,1]
=> 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [5,2,4,3,1] => [4,1]
=> 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,5,1] => [4,1]
=> 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,5,2,4,1] => [3,1,1]
=> 3 = 2 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,5,3,1] => [3,2]
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,5,4,1] => [3,2]
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,5,4,3,1] => [4,1]
=> 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [4,1]
=> 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,5,3,1,2] => [3,1,1]
=> 3 = 2 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [5,3,4,1,2] => [3,1,1]
=> 3 = 2 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,5,1,2] => [3,1,1]
=> 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,5,4,1,2] => [3,1,1]
=> 3 = 2 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [5,4,2,1,3] => [4,1]
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,5,2,1,3] => [3,1,1]
=> 3 = 2 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [5,4,1,3,2] => [4,1]
=> 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,5,1,3,2] => [3,1,1]
=> 3 = 2 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [5,3,2,1,4] => [4,1]
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [5,2,4,1,3] => [3,1,1]
=> 3 = 2 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [5,3,1,4,2] => [3,2]
=> 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [5,2,1,4,3] => [3,2]
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [5,1,4,3,2] => [4,1]
=> 2 = 1 + 1
Description
The length of the partition.
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 = 0 + 1
[.,[.,.]]
=> [2,1] => [2,1] => [[1],[2]]
=> 1 = 0 + 1
[[.,.],.]
=> [1,2] => [1,2] => [[1,2]]
=> 2 = 1 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [2,3,1] => [2,3,1] => [[1,2],[3]]
=> 2 = 1 + 1
[[.,.],[.,.]]
=> [3,1,2] => [3,1,2] => [[1,3],[2]]
=> 2 = 1 + 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 2 = 1 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,3,2] => [[1,2],[3]]
=> 2 = 1 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,4,2,1] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [4,2,3,1] => [[1,3],[2],[4]]
=> 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,4,1] => [[1,3],[2],[4]]
=> 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [4,3,1,2] => [[1,4],[2],[3]]
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,4,1,2] => [[1,2],[3,4]]
=> 3 = 2 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [4,2,1,3] => [[1,4],[2],[3]]
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [4,1,3,2] => [[1,3],[2],[4]]
=> 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,4,1,3] => [[1,2],[3,4]]
=> 3 = 2 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,4,2] => [[1,3],[2,4]]
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,4,3] => [[1,3],[2,4]]
=> 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,5,3,2,1] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [5,3,4,2,1] => [[1,3],[2],[4],[5]]
=> 2 = 1 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,5,2,1] => [[1,3],[2],[4],[5]]
=> 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,5,4,2,1] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [5,4,2,3,1] => [[1,4],[2],[3],[5]]
=> 2 = 1 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,5,2,3,1] => [[1,2],[3,4],[5]]
=> 3 = 2 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [5,2,4,3,1] => [[1,3],[2],[4],[5]]
=> 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,5,1] => [[1,4],[2],[3],[5]]
=> 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,5,2,4,1] => [[1,2],[3,4],[5]]
=> 3 = 2 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,5,3,1] => [[1,3],[2,4],[5]]
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,5,4,1] => [[1,3],[2,4],[5]]
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [[1,5],[2],[3],[4]]
=> 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,5,3,1,2] => [[1,2],[3,5],[4]]
=> 3 = 2 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [5,3,4,1,2] => [[1,3],[2,5],[4]]
=> 3 = 2 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,5,1,2] => [[1,3],[2,5],[4]]
=> 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,5,4,1,2] => [[1,2],[3,5],[4]]
=> 3 = 2 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [5,4,2,1,3] => [[1,5],[2],[3],[4]]
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,5,2,1,3] => [[1,2],[3,5],[4]]
=> 3 = 2 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [5,4,1,3,2] => [[1,4],[2],[3],[5]]
=> 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,5,1,3,2] => [[1,2],[3,4],[5]]
=> 3 = 2 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [5,3,2,1,4] => [[1,5],[2],[3],[4]]
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [5,2,4,1,3] => [[1,3],[2,5],[4]]
=> 3 = 2 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [5,3,1,4,2] => [[1,4],[2,5],[3]]
=> 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [5,2,1,4,3] => [[1,4],[2,5],[3]]
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [5,1,4,3,2] => [[1,3],[2],[4],[5]]
=> 2 = 1 + 1
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: St000251
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1,0]
=> {{1}}
=> ? = 0
[.,[.,.]]
=> [1,0,1,0]
=> [1,0,1,0]
=> {{1},{2}}
=> 0
[[.,.],.]
=> [1,1,0,0]
=> [1,1,0,0]
=> {{1,2}}
=> 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> 2
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 2
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> 1
Description
The number of nonsingleton blocks of a set partition.
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
Mp00064: Permutations —reverse⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => => ? = 0
[.,[.,.]]
=> [2,1] => [1,2] => 0 => 0
[[.,.],.]
=> [1,2] => [2,1] => 1 => 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => 00 => 0
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => 01 => 1
[[.,.],[.,.]]
=> [3,1,2] => [2,1,3] => 10 => 1
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => 10 => 1
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => 11 => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => 000 => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => 001 => 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => 010 => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => 010 => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => 011 => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,3,4] => 100 => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => 101 => 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,2,4] => 100 => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,2,1,4] => 110 => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => 100 => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => 101 => 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [4,2,1,3] => 110 => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => 110 => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => 111 => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0000 => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => 0001 => 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,4,3,5] => 0010 => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => 0010 => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => 0011 => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,3,2,4,5] => 0100 => 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,3,2,5,4] => 0101 => 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,4,2,3,5] => 0100 => 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => 0110 => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => 0100 => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => 0101 => 2
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,5,3,2,4] => 0110 => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => 0110 => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => 0111 => 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [2,1,3,4,5] => 1000 => 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,3,5,4] => 1001 => 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [2,1,4,3,5] => 1010 => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [2,1,5,3,4] => 1010 => 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [2,1,5,4,3] => 1011 => 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [3,1,2,4,5] => 1000 => 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,1,2,5,4] => 1001 => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,2,1,4,5] => 1100 => 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [3,2,1,5,4] => 1101 => 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,1,2,3,5] => 1000 => 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,1,3,2,5] => 1010 => 2
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [4,2,1,3,5] => 1100 => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [4,3,1,2,5] => 1100 => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [4,3,2,1,5] => 1110 => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [5,1,2,3,4] => 1000 => 1
Description
The number of runs of ones in a binary word.
The following 43 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001354The number of series nodes in the modular decomposition of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000670The reversal length of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001737The number of descents of type 2 in a permutation. St000386The number of factors DDU in a Dyck path. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000834The number of right outer peaks of a permutation. St000703The number of deficiencies of a permutation. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000354The number of recoils 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. St000742The number of big ascents of a permutation after prepending zero. St000035The number of left outer peaks of a permutation. St000245The number of ascents 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. 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. 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. 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. St000455The second largest eigenvalue of a graph if it is integral. St000647The number of big descents of a permutation. 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. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000920The logarithmic height of a Dyck path.
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