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Your data matches 47 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
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording 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]
=> [2,1] => [[1],[2]]
=> 1
[[.,.],.]
=> [1,1,0,0]
=> [1,2] => [[1,2]]
=> 0
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [2,3,1] => [[1,2],[3]]
=> 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [3,1,2] => [[1,3],[2]]
=> 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,2,3] => [[1,2,3]]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [[1,2],[3,4]]
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [[1,2,4],[3]]
=> 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [[1,3],[2,4]]
=> 2
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [[1,3,4],[2]]
=> 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[1,3,4],[2]]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [[1,2,3],[4]]
=> 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [[1,2,4],[3]]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 2
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [[1,2,5],[3,4]]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [[1,3,4],[2,5]]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [[1,2,4],[3,5]]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [[1,2,3],[4,5]]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [[1,3,5],[2,4]]
=> 2
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [[1,2,5],[3,4]]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [[1,2,4],[3,5]]
=> 2
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[1,2,3],[4,5]]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [[1,2,4,5],[3]]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [[1,3,5],[2,4]]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [[1,2,5],[3,4]]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [[1,3,4],[2,5]]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [[1,2,4],[3,5]]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[1,2,3],[4,5]]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [[1,3,5],[2,4]]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [[1,2,5],[3,4]]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [[1,3,5],[2,4]]
=> 2
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [[1,2,5],[3,4]]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [[1,3,4],[2,5]]
=> 2
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [[1,3,4,5],[2]]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [[1,3,4],[2,5]]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [[1,3,4,5],[2]]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [[1,3,4,5],[2]]
=> 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: St000288
(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
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1] => => ? = 0
[.,[.,.]]
=> [1,0,1,0]
=> [2,1] => 1 => 1
[[.,.],.]
=> [1,1,0,0]
=> [1,2] => 0 => 0
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [2,1,3] => 10 => 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [2,3,1] => 01 => 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [3,1,2] => 01 => 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,3,2] => 01 => 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,2,3] => 00 => 0
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 101 => 2
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 001 => 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 100 => 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 010 => 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 001 => 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 011 => 2
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 001 => 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => 010 => 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 001 => 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 010 => 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 001 => 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 001 => 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 001 => 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 000 => 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 1010 => 2
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => 0010 => 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => 1001 => 2
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => 0011 => 2
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => 0001 => 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => 1001 => 2
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => 0001 => 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => 1001 => 2
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => 1000 => 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => 0101 => 2
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 0001 => 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => 0100 => 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 0010 => 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0001 => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => 0110 => 2
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => 0010 => 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => 0101 => 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => 0011 => 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0001 => 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => 0101 => 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => 0001 => 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => 0011 => 2
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 0001 => 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => 0101 => 2
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => 0100 => 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => 0011 => 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => 0010 => 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 0001 => 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => 0101 => 2
[[[.,[.,.]],.],[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,0,1,1,0,1,1,0,0,0]
=> [4,6,7,1,2,8,3,5] => ? => ? = 2
[[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,4,2,6,5,8,7,10,9] => 011010101 => ? = 5
[[[.,.],.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [4,1,5,2,6,3,8,7,10,9] => 001110101 => ? = 5
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000389
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1] => => ? = 0
[.,[.,.]]
=> [1,0,1,0]
=> [2,1] => 1 => 1
[[.,.],.]
=> [1,1,0,0]
=> [1,2] => 0 => 0
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [2,1,3] => 10 => 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [2,3,1] => 01 => 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [3,1,2] => 10 => 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,3,2] => 01 => 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,2,3] => 00 => 0
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 101 => 2
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 010 => 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 100 => 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 010 => 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 001 => 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 101 => 2
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 010 => 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => 100 => 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 100 => 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 010 => 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 001 => 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 010 => 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 001 => 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 000 => 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 1010 => 2
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => 0100 => 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => 1001 => 2
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => 0101 => 2
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => 0010 => 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => 1010 => 2
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => 0100 => 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => 1001 => 2
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => 1000 => 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => 0101 => 2
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 0010 => 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => 0100 => 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 0010 => 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0001 => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => 1010 => 2
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => 0100 => 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => 1001 => 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => 0101 => 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0010 => 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => 1010 => 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => 0100 => 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => 1010 => 2
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 0100 => 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => 1001 => 2
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => 1000 => 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => 1001 => 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => 1000 => 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1000 => 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => 0101 => 2
[.,[[[.,.],.],[[.,[[.,.],.]],.]]]
=> [1,0,1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [2,6,7,1,3,8,4,5] => ? => ? = 2
[.,[[.,[[.,[[.,.],.]],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [2,1,3,5,6,8,4,7] => ? => ? = 2
[[.,[[.,[.,.]],.]],[.,[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [3,6,1,2,8,4,5,7] => ? => ? = 2
[[[.,[[.,.],.]],[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0,1,0]
=> [4,1,7,2,3,8,5,6] => ? => ? = 3
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [5,1,7,2,3,4,6,8] => ? => ? = 2
[[.,[[[.,[.,.]],.],[.,.]]],[.,.]]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [3,1,2,7,4,5,6,8] => ? => ? = 2
Description
The number of runs of ones of odd length in a binary word.
Matching statistic: St000390
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ 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%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ 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
[.,.]
=> [1,0]
=> [1] => => ? = 0
[.,[.,.]]
=> [1,0,1,0]
=> [2,1] => 1 => 1
[[.,.],.]
=> [1,1,0,0]
=> [1,2] => 0 => 0
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [2,1,3] => 10 => 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [2,3,1] => 01 => 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [3,1,2] => 10 => 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,3,2] => 01 => 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,2,3] => 00 => 0
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 101 => 2
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 010 => 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 100 => 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 010 => 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 001 => 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 101 => 2
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 010 => 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => 100 => 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 100 => 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 010 => 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 001 => 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 010 => 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 001 => 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 000 => 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 1010 => 2
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => 0100 => 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => 1001 => 2
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => 0101 => 2
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => 0010 => 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => 1010 => 2
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => 0100 => 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => 1001 => 2
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => 1000 => 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => 0101 => 2
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 0010 => 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => 0100 => 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 0010 => 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0001 => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => 1010 => 2
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => 0100 => 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => 1001 => 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => 0101 => 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0010 => 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => 1010 => 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => 0100 => 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => 1010 => 2
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 0100 => 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => 1001 => 2
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => 1000 => 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => 1001 => 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => 1000 => 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1000 => 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => 0101 => 2
[.,[[[.,.],.],[[.,[[.,.],.]],.]]]
=> [1,0,1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [2,6,7,1,3,8,4,5] => ? => ? = 2
[.,[[.,[[.,[[.,.],.]],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [2,1,3,5,6,8,4,7] => ? => ? = 2
[[.,[[.,[.,.]],.]],[.,[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [3,6,1,2,8,4,5,7] => ? => ? = 2
[[[.,[[.,.],.]],[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0,1,0]
=> [4,1,7,2,3,8,5,6] => ? => ? = 3
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [5,1,7,2,3,4,6,8] => ? => ? = 2
[[.,[[[.,[.,.]],.],[.,.]]],[.,.]]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [3,1,2,7,4,5,6,8] => ? => ? = 2
Description
The number of runs of ones in a binary word.
Matching statistic: St000142
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1] => [1]
=> 0
[.,[.,.]]
=> [1,0,1,0]
=> [2,1] => [2]
=> 1
[[.,.],.]
=> [1,1,0,0]
=> [1,2] => [1,1]
=> 0
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,3,2] => [2,1]
=> 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 2
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 2
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [2,1,1]
=> 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [2,1,1]
=> 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 2
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,2,1]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,2,1]
=> 2
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [2,2,1]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [2,2,1]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [2,2,1]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [2,2,1]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [2,2,1]
=> 2
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> 2
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [2,2,1]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 1
[.,[.,[[.,[[.,.],.]],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,5,7,8,6] => ?
=> ? = 2
[.,[[[.,.],.],[[.,[[.,.],.]],.]]]
=> [1,0,1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [2,6,7,1,3,8,4,5] => ?
=> ? = 2
[.,[[.,[.,[.,.]]],[.,[[.,.],.]]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,7,4,6,8] => ?
=> ? = 2
[.,[[.,[[.,[[.,.],.]],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [2,1,3,5,6,8,4,7] => ?
=> ? = 2
[.,[[.,[[[.,.],[.,[.,.]]],.]],.]]
=> [1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,3,5,1,6,4,7,8] => ?
=> ? = 2
[.,[[[.,[[.,[.,[.,.]]],.]],.],.]]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ? = 2
[[.,.],[[[.,.],[.,[[.,.],.]]],.]]
=> [1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,4,5,1,2,8,6,7] => ?
=> ? = 2
[[.,[.,[.,.]]],[.,[[.,[.,.]],.]]]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [3,5,1,7,2,4,8,6] => ?
=> ? = 3
[[.,[[.,[.,.]],.]],[.,[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [3,6,1,2,8,4,5,7] => ?
=> ? = 2
[[[[.,[.,.]],.],.],[[.,[.,.]],.]]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> [5,7,1,8,2,3,4,6] => ?
=> ? = 2
[[.,[[.,.],[.,[.,.]]]],[.,[.,.]]]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [3,1,6,2,4,7,5,8] => ?
=> ? = 3
[[[[.,[.,.]],.],[.,.]],[.,[.,.]]]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> [5,1,7,2,3,8,4,6] => ?
=> ? = 3
[[[[.,[.,.]],.],[.,.]],[[.,.],.]]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,1,0,0]
=> [5,7,1,2,3,8,4,6] => ?
=> ? = 2
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [5,1,7,2,3,4,6,8] => ?
=> ? = 2
[[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> [5,1,2,6,3,4,8,7] => ?
=> ? = 3
[[[[.,[.,[.,.]]],.],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> [5,1,2,7,3,4,6,8] => ?
=> ? = 2
[[[[[.,.],.],[.,.]],[[.,.],.]],.]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> [1,6,7,2,3,4,8,5] => ?
=> ? = 2
[[[.,[[.,.],[.,[.,[.,.]]]]],.],.]
=> [1,1,1,0,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,7,5,8,6] => ?
=> ? = 3
[[[[.,.],[.,[.,[[.,.],.]]]],.],.]
=> [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> [1,2,5,6,3,4,8,7] => ?
=> ? = 2
[[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.],.]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,6,5,8,7,10,9] => ?
=> ? = 4
[[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.],.]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,2,3,4,6,5,8,7,10,9] => ?
=> ? = 3
[[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,8,7,10,9] => ?
=> ? = 2
Description
The number of even parts of a partition.
Matching statistic: St001251
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1] => [1]
=> 0
[.,[.,.]]
=> [1,0,1,0]
=> [2,1] => [2]
=> 1
[[.,.],.]
=> [1,1,0,0]
=> [1,2] => [1,1]
=> 0
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,3,2] => [2,1]
=> 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 2
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 2
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [2,1,1]
=> 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [2,1,1]
=> 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 2
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,2,1]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,2,1]
=> 2
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [2,2,1]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [2,2,1]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [2,2,1]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [2,2,1]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [2,2,1]
=> 2
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> 2
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [2,2,1]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 1
[.,[.,[[.,[[.,.],.]],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,5,7,8,6] => ?
=> ? = 2
[.,[[[.,.],.],[[.,[[.,.],.]],.]]]
=> [1,0,1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [2,6,7,1,3,8,4,5] => ?
=> ? = 2
[.,[[.,[.,[.,.]]],[.,[[.,.],.]]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,7,4,6,8] => ?
=> ? = 2
[.,[[.,[[.,[[.,.],.]],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [2,1,3,5,6,8,4,7] => ?
=> ? = 2
[.,[[.,[[[.,.],[.,[.,.]]],.]],.]]
=> [1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,3,5,1,6,4,7,8] => ?
=> ? = 2
[.,[[[.,[[.,[.,[.,.]]],.]],.],.]]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ? = 2
[[.,.],[[[.,.],[.,[[.,.],.]]],.]]
=> [1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,4,5,1,2,8,6,7] => ?
=> ? = 2
[[.,[.,[.,.]]],[.,[[.,[.,.]],.]]]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [3,5,1,7,2,4,8,6] => ?
=> ? = 3
[[.,[[.,[.,.]],.]],[.,[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [3,6,1,2,8,4,5,7] => ?
=> ? = 2
[[[[.,[.,.]],.],.],[[.,[.,.]],.]]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> [5,7,1,8,2,3,4,6] => ?
=> ? = 2
[[.,[[.,.],[.,[.,.]]]],[.,[.,.]]]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [3,1,6,2,4,7,5,8] => ?
=> ? = 3
[[[[.,[.,.]],.],[.,.]],[.,[.,.]]]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> [5,1,7,2,3,8,4,6] => ?
=> ? = 3
[[[[.,[.,.]],.],[.,.]],[[.,.],.]]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,1,0,0]
=> [5,7,1,2,3,8,4,6] => ?
=> ? = 2
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [5,1,7,2,3,4,6,8] => ?
=> ? = 2
[[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> [5,1,2,6,3,4,8,7] => ?
=> ? = 3
[[[[.,[.,[.,.]]],.],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> [5,1,2,7,3,4,6,8] => ?
=> ? = 2
[[[[[.,.],.],[.,.]],[[.,.],.]],.]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> [1,6,7,2,3,4,8,5] => ?
=> ? = 2
[[[.,[[.,.],[.,[.,[.,.]]]]],.],.]
=> [1,1,1,0,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,7,5,8,6] => ?
=> ? = 3
[[[[.,.],[.,[.,[[.,.],.]]]],.],.]
=> [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> [1,2,5,6,3,4,8,7] => ?
=> ? = 2
[[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.],.]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,6,5,8,7,10,9] => ?
=> ? = 4
[[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.],.]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,2,3,4,6,5,8,7,10,9] => ?
=> ? = 3
[[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,8,7,10,9] => ?
=> ? = 2
Description
The number of parts of a partition that are not congruent 1 modulo 3.
Matching statistic: St001252
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001252: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001252: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1] => [1]
=> 0
[.,[.,.]]
=> [1,0,1,0]
=> [2,1] => [2]
=> 1
[[.,.],.]
=> [1,1,0,0]
=> [1,2] => [1,1]
=> 0
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,3,2] => [2,1]
=> 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 2
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 2
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [2,1,1]
=> 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [2,1,1]
=> 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 2
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,2,1]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,2,1]
=> 2
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [2,2,1]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [2,2,1]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [2,2,1]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [2,2,1]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [2,2,1]
=> 2
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> 2
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [2,2,1]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 1
[.,[.,[[.,[[.,.],.]],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,5,7,8,6] => ?
=> ? = 2
[.,[[[.,.],.],[[.,[[.,.],.]],.]]]
=> [1,0,1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [2,6,7,1,3,8,4,5] => ?
=> ? = 2
[.,[[.,[.,[.,.]]],[.,[[.,.],.]]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,7,4,6,8] => ?
=> ? = 2
[.,[[.,[[.,[[.,.],.]],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [2,1,3,5,6,8,4,7] => ?
=> ? = 2
[.,[[.,[[[.,.],[.,[.,.]]],.]],.]]
=> [1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,3,5,1,6,4,7,8] => ?
=> ? = 2
[.,[[[.,[[.,[.,[.,.]]],.]],.],.]]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ? = 2
[[.,.],[[[.,.],[.,[[.,.],.]]],.]]
=> [1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,4,5,1,2,8,6,7] => ?
=> ? = 2
[[.,[.,[.,.]]],[.,[[.,[.,.]],.]]]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [3,5,1,7,2,4,8,6] => ?
=> ? = 3
[[.,[[.,[.,.]],.]],[.,[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [3,6,1,2,8,4,5,7] => ?
=> ? = 2
[[[[.,[.,.]],.],.],[[.,[.,.]],.]]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> [5,7,1,8,2,3,4,6] => ?
=> ? = 2
[[.,[[.,.],[.,[.,.]]]],[.,[.,.]]]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [3,1,6,2,4,7,5,8] => ?
=> ? = 3
[[[[.,[.,.]],.],[.,.]],[.,[.,.]]]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> [5,1,7,2,3,8,4,6] => ?
=> ? = 3
[[[[.,[.,.]],.],[.,.]],[[.,.],.]]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,1,0,0]
=> [5,7,1,2,3,8,4,6] => ?
=> ? = 2
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [5,1,7,2,3,4,6,8] => ?
=> ? = 2
[[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> [5,1,2,6,3,4,8,7] => ?
=> ? = 3
[[[[.,[.,[.,.]]],.],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> [5,1,2,7,3,4,6,8] => ?
=> ? = 2
[[[[[.,.],.],[.,.]],[[.,.],.]],.]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> [1,6,7,2,3,4,8,5] => ?
=> ? = 2
[[[.,[[.,.],[.,[.,[.,.]]]]],.],.]
=> [1,1,1,0,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,7,5,8,6] => ?
=> ? = 3
[[[[.,.],[.,[.,[[.,.],.]]]],.],.]
=> [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> [1,2,5,6,3,4,8,7] => ?
=> ? = 2
[[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.],.]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,6,5,8,7,10,9] => ?
=> ? = 4
[[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.],.]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,2,3,4,6,5,8,7,10,9] => ?
=> ? = 3
[[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,8,7,10,9] => ?
=> ? = 2
Description
Half the sum of the even parts of a partition.
Matching statistic: St001280
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1] => [1]
=> 0
[.,[.,.]]
=> [1,0,1,0]
=> [2,1] => [2]
=> 1
[[.,.],.]
=> [1,1,0,0]
=> [1,2] => [1,1]
=> 0
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,3,2] => [2,1]
=> 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 2
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 2
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [2,1,1]
=> 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [2,1,1]
=> 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 2
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,2,1]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,2,1]
=> 2
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [2,2,1]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [2,2,1]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [2,2,1]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [2,2,1]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [2,2,1]
=> 2
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> 2
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [2,2,1]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 1
[.,[.,[[.,[[.,.],.]],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,5,7,8,6] => ?
=> ? = 2
[.,[[[.,.],.],[[.,[[.,.],.]],.]]]
=> [1,0,1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [2,6,7,1,3,8,4,5] => ?
=> ? = 2
[.,[[.,[.,[.,.]]],[.,[[.,.],.]]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,7,4,6,8] => ?
=> ? = 2
[.,[[.,[[.,[[.,.],.]],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [2,1,3,5,6,8,4,7] => ?
=> ? = 2
[.,[[.,[[[.,.],[.,[.,.]]],.]],.]]
=> [1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,3,5,1,6,4,7,8] => ?
=> ? = 2
[.,[[[.,[[.,[.,[.,.]]],.]],.],.]]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ? = 2
[[.,.],[[[.,.],[.,[[.,.],.]]],.]]
=> [1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,4,5,1,2,8,6,7] => ?
=> ? = 2
[[.,[.,[.,.]]],[.,[[.,[.,.]],.]]]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [3,5,1,7,2,4,8,6] => ?
=> ? = 3
[[.,[[.,[.,.]],.]],[.,[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [3,6,1,2,8,4,5,7] => ?
=> ? = 2
[[[[.,[.,.]],.],.],[[.,[.,.]],.]]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> [5,7,1,8,2,3,4,6] => ?
=> ? = 2
[[.,[[.,.],[.,[.,.]]]],[.,[.,.]]]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [3,1,6,2,4,7,5,8] => ?
=> ? = 3
[[[[.,[.,.]],.],[.,.]],[.,[.,.]]]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> [5,1,7,2,3,8,4,6] => ?
=> ? = 3
[[[[.,[.,.]],.],[.,.]],[[.,.],.]]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,1,0,0]
=> [5,7,1,2,3,8,4,6] => ?
=> ? = 2
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [5,1,7,2,3,4,6,8] => ?
=> ? = 2
[[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> [5,1,2,6,3,4,8,7] => ?
=> ? = 3
[[[[.,[.,[.,.]]],.],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> [5,1,2,7,3,4,6,8] => ?
=> ? = 2
[[[[[.,.],.],[.,.]],[[.,.],.]],.]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> [1,6,7,2,3,4,8,5] => ?
=> ? = 2
[[[.,[[.,.],[.,[.,[.,.]]]]],.],.]
=> [1,1,1,0,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,7,5,8,6] => ?
=> ? = 3
[[[[.,.],[.,[.,[[.,.],.]]]],.],.]
=> [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> [1,2,5,6,3,4,8,7] => ?
=> ? = 2
[[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.],.]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,6,5,8,7,10,9] => ?
=> ? = 4
[[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.],.]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,2,3,4,6,5,8,7,10,9] => ?
=> ? = 3
[[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,8,7,10,9] => ?
=> ? = 2
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001657
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1] => [1]
=> 0
[.,[.,.]]
=> [1,0,1,0]
=> [2,1] => [2]
=> 1
[[.,.],.]
=> [1,1,0,0]
=> [1,2] => [1,1]
=> 0
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,3,2] => [2,1]
=> 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 2
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 2
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [2,1,1]
=> 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [2,1,1]
=> 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 2
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,2,1]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,2,1]
=> 2
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [2,2,1]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [2,2,1]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [2,2,1]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [2,2,1]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [2,2,1]
=> 2
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> 2
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [2,2,1]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 1
[.,[.,[[.,[[.,.],.]],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,5,7,8,6] => ?
=> ? = 2
[.,[[[.,.],.],[[.,[[.,.],.]],.]]]
=> [1,0,1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [2,6,7,1,3,8,4,5] => ?
=> ? = 2
[.,[[.,[.,[.,.]]],[.,[[.,.],.]]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,7,4,6,8] => ?
=> ? = 2
[.,[[.,[[.,[[.,.],.]],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [2,1,3,5,6,8,4,7] => ?
=> ? = 2
[.,[[.,[[[.,.],[.,[.,.]]],.]],.]]
=> [1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,3,5,1,6,4,7,8] => ?
=> ? = 2
[.,[[[.,[[.,[.,[.,.]]],.]],.],.]]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ? = 2
[[.,.],[[[.,.],[.,[[.,.],.]]],.]]
=> [1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,4,5,1,2,8,6,7] => ?
=> ? = 2
[[.,[.,[.,.]]],[.,[[.,[.,.]],.]]]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [3,5,1,7,2,4,8,6] => ?
=> ? = 3
[[.,[[.,[.,.]],.]],[.,[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [3,6,1,2,8,4,5,7] => ?
=> ? = 2
[[[[.,[.,.]],.],.],[[.,[.,.]],.]]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> [5,7,1,8,2,3,4,6] => ?
=> ? = 2
[[.,[[.,.],[.,[.,.]]]],[.,[.,.]]]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [3,1,6,2,4,7,5,8] => ?
=> ? = 3
[[[[.,[.,.]],.],[.,.]],[.,[.,.]]]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> [5,1,7,2,3,8,4,6] => ?
=> ? = 3
[[[[.,[.,.]],.],[.,.]],[[.,.],.]]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,1,0,0]
=> [5,7,1,2,3,8,4,6] => ?
=> ? = 2
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [5,1,7,2,3,4,6,8] => ?
=> ? = 2
[[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> [5,1,2,6,3,4,8,7] => ?
=> ? = 3
[[[[.,[.,[.,.]]],.],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> [5,1,2,7,3,4,6,8] => ?
=> ? = 2
[[[[[.,.],.],[.,.]],[[.,.],.]],.]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> [1,6,7,2,3,4,8,5] => ?
=> ? = 2
[[[.,[[.,.],[.,[.,[.,.]]]]],.],.]
=> [1,1,1,0,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,7,5,8,6] => ?
=> ? = 3
[[[[.,.],[.,[.,[[.,.],.]]]],.],.]
=> [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> [1,2,5,6,3,4,8,7] => ?
=> ? = 2
[[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.],.]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,6,5,8,7,10,9] => ?
=> ? = 4
[[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.],.]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,2,3,4,6,5,8,7,10,9] => ?
=> ? = 3
[[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,8,7,10,9] => ?
=> ? = 2
Description
The number of twos in an integer partition.
The total number of twos in all partitions of $n$ is equal to the total number of singletons [[St001484]] in all partitions of $n-1$, see [1].
Matching statistic: St000321
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000321: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000321: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1] => [1]
=> 1 = 0 + 1
[.,[.,.]]
=> [1,0,1,0]
=> [2,1] => [2]
=> 2 = 1 + 1
[[.,.],.]
=> [1,1,0,0]
=> [1,2] => [1,1]
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> 2 = 1 + 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 2 = 1 + 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 2 = 1 + 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,3,2] => [2,1]
=> 2 = 1 + 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 3 = 2 + 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 2 = 1 + 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 3 = 2 + 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [2,1,1]
=> 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,1]
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [2,1,1]
=> 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 3 = 2 + 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,2,1]
=> 3 = 2 + 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 3 = 2 + 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,2,1]
=> 3 = 2 + 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> 3 = 2 + 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 3 = 2 + 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [2,2,1]
=> 3 = 2 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [2,2,1]
=> 3 = 2 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [2,2,1]
=> 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [2,2,1]
=> 3 = 2 + 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [2,2,1]
=> 3 = 2 + 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> 3 = 2 + 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [2,2,1]
=> 3 = 2 + 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 2 = 1 + 1
[.,[.,[[.,[[.,.],.]],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,5,7,8,6] => ?
=> ? = 2 + 1
[.,[[[.,.],.],[[.,[[.,.],.]],.]]]
=> [1,0,1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [2,6,7,1,3,8,4,5] => ?
=> ? = 2 + 1
[.,[[.,[.,[.,.]]],[.,[[.,.],.]]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,7,4,6,8] => ?
=> ? = 2 + 1
[.,[[.,[[.,[[.,.],.]],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [2,1,3,5,6,8,4,7] => ?
=> ? = 2 + 1
[.,[[.,[[[.,.],[.,[.,.]]],.]],.]]
=> [1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,3,5,1,6,4,7,8] => ?
=> ? = 2 + 1
[.,[[[.,[[.,[.,[.,.]]],.]],.],.]]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ? = 2 + 1
[[.,.],[[[.,.],[.,[[.,.],.]]],.]]
=> [1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,4,5,1,2,8,6,7] => ?
=> ? = 2 + 1
[[.,[.,[.,.]]],[.,[[.,[.,.]],.]]]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [3,5,1,7,2,4,8,6] => ?
=> ? = 3 + 1
[[.,[[.,[.,.]],.]],[.,[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [3,6,1,2,8,4,5,7] => ?
=> ? = 2 + 1
[[[[.,[.,.]],.],.],[[.,[.,.]],.]]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> [5,7,1,8,2,3,4,6] => ?
=> ? = 2 + 1
[[.,[[.,.],[.,[.,.]]]],[.,[.,.]]]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [3,1,6,2,4,7,5,8] => ?
=> ? = 3 + 1
[[[[.,[.,.]],.],[.,.]],[.,[.,.]]]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> [5,1,7,2,3,8,4,6] => ?
=> ? = 3 + 1
[[[[.,[.,.]],.],[.,.]],[[.,.],.]]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,1,0,0]
=> [5,7,1,2,3,8,4,6] => ?
=> ? = 2 + 1
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [5,1,7,2,3,4,6,8] => ?
=> ? = 2 + 1
[[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> [5,1,2,6,3,4,8,7] => ?
=> ? = 3 + 1
[[[[.,[.,[.,.]]],.],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> [5,1,2,7,3,4,6,8] => ?
=> ? = 2 + 1
[[[[[.,.],.],[.,.]],[[.,.],.]],.]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> [1,6,7,2,3,4,8,5] => ?
=> ? = 2 + 1
[[[.,[[.,.],[.,[.,[.,.]]]]],.],.]
=> [1,1,1,0,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,7,5,8,6] => ?
=> ? = 3 + 1
[[[[.,.],[.,[.,[[.,.],.]]]],.],.]
=> [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> [1,2,5,6,3,4,8,7] => ?
=> ? = 2 + 1
[[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.],.]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,6,5,8,7,10,9] => ?
=> ? = 4 + 1
[[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.],.]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,2,3,4,6,5,8,7,10,9] => ?
=> ? = 3 + 1
[[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,8,7,10,9] => ?
=> ? = 2 + 1
Description
The number of integer partitions of n that are dominated by an integer partition.
A partition $\lambda = (\lambda_1,\ldots,\lambda_n) \vdash n$ dominates a partition $\mu = (\mu_1,\ldots,\mu_n) \vdash n$ if $\sum_{i=1}^k (\lambda_i - \mu_i) \geq 0$ for all $k$.
The following 37 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000345The number of refinements of a partition. St000935The number of ordered refinements of an integer partition. St001389The number of partitions of the same length below the given integer partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000919The number of maximal left branches of a binary tree. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000884The number of isolated descents of a permutation. St000035The number of left outer peaks of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000662The staircase size of the code of a permutation. St000245The number of ascents of a permutation. St000834The number of right outer peaks of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in 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. St001427The number of descents of a signed permutation. St000021The number of descents of a permutation. St000325The width of the tree associated to a permutation. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000619The number of cyclic descents of a permutation. St000023The number of inner peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000920The logarithmic height of a Dyck path. St001624The breadth of a lattice.
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