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Your data matches 16 different statistics following compositions of up to 3 maps.
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Matching statistic: St000027
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(load all 2 compositions to match this statistic)
St000027: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0
[1,0,1,0]
=> 2
[1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> 6
[1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 4
[1,1,0,1,0,0]
=> 3
[1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> 12
[1,0,1,0,1,1,0,0]
=> 6
[1,0,1,1,0,0,1,0]
=> 8
[1,0,1,1,0,1,0,0]
=> 7
[1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> 10
[1,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> 9
[1,1,0,1,0,1,0,0]
=> 8
[1,1,0,1,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> 6
[1,1,1,0,0,1,0,0]
=> 5
[1,1,1,0,1,0,0,0]
=> 4
[1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 20
[1,0,1,0,1,0,1,1,0,0]
=> 12
[1,0,1,0,1,1,0,0,1,0]
=> 14
[1,0,1,0,1,1,0,1,0,0]
=> 13
[1,0,1,0,1,1,1,0,0,0]
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> 16
[1,0,1,1,0,0,1,1,0,0]
=> 8
[1,0,1,1,0,1,0,0,1,0]
=> 15
[1,0,1,1,0,1,0,1,0,0]
=> 14
[1,0,1,1,0,1,1,0,0,0]
=> 7
[1,0,1,1,1,0,0,0,1,0]
=> 10
[1,0,1,1,1,0,0,1,0,0]
=> 9
[1,0,1,1,1,0,1,0,0,0]
=> 8
[1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> 18
[1,1,0,0,1,0,1,1,0,0]
=> 10
[1,1,0,0,1,1,0,0,1,0]
=> 12
[1,1,0,0,1,1,0,1,0,0]
=> 11
[1,1,0,0,1,1,1,0,0,0]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> 17
[1,1,0,1,0,0,1,1,0,0]
=> 9
[1,1,0,1,0,1,0,0,1,0]
=> 16
[1,1,0,1,0,1,0,1,0,0]
=> 15
[1,1,0,1,0,1,1,0,0,0]
=> 8
[1,1,0,1,1,0,0,0,1,0]
=> 11
[1,1,0,1,1,0,0,1,0,0]
=> 10
[1,1,0,1,1,0,1,0,0,0]
=> 9
[1,1,0,1,1,1,0,0,0,0]
=> 3
Description
The major index of a Dyck path.
This is the sum over all $i+j$ for which $(i,j)$ is a valley of $D$.
The generating function of the major index yields '''MacMahon''' 's $q$-Catalan numbers
$$\sum_{D \in \mathfrak{D}_n} q^{\operatorname{maj}(D)} = \frac{1}{[n+1]_q}\begin{bmatrix} 2n \\ n \end{bmatrix}_q,$$
where $[k]_q := 1+q+\ldots+q^{k-1}$ is the usual $q$-extension of the integer $k$, $[k]_q!:= [1]_q[2]_q \cdots [k]_q$ is the $q$-factorial of $k$ and $\left[\begin{smallmatrix} k \\ l \end{smallmatrix}\right]_q:=[k]_q!/[l]_q![k-l]_q!$ is the $q$-binomial coefficient.
The major index was first studied by P.A.MacMahon in [1], where he proved this generating function identity.
There is a bijection $\psi$ between Dyck paths and '''noncrossing permutations''' which simultaneously sends the area of a Dyck path [[St000012]] to the number of inversions [[St000018]], and the major index of the Dyck path to $n(n-1)$ minus the sum of the major index and the major index of the inverse [2].
For the major index on other collections, see [[St000004]] for permutations and [[St000290]] for binary words.
Matching statistic: St000979
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(load all 2 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000979: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000979: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 6
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 4
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 3
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 12
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 6
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 8
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 7
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 10
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 9
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 8
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 6
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 5
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 4
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 20
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 12
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 14
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 13
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 16
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 8
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 15
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 14
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 7
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 10
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 9
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 8
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 18
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 10
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 12
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 11
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 17
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 9
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 16
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 15
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 8
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 11
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 10
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 9
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
Description
Half of MacMahon's equal index of a Dyck path.
This is half the sum of the positions of double (up- or down-)steps of a Dyck path, see [1, p. 135].
Matching statistic: St001696
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001696: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001696: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1],[2]]
=> 0
[1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[1,1,0,0]
=> [[1,2],[3,4]]
=> 0
[1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 6
[1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 2
[1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 4
[1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 3
[1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 0
[1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 12
[1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 6
[1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 8
[1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 7
[1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2
[1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 10
[1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 4
[1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 9
[1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> 8
[1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> 3
[1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 6
[1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 5
[1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> 4
[1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> 20
[1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> 12
[1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> 14
[1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> 13
[1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> 16
[1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> 8
[1,0,1,1,0,1,0,0,1,0]
=> [[1,3,4,6,9],[2,5,7,8,10]]
=> 15
[1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> 14
[1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> 7
[1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> 10
[1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> 9
[1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> 8
[1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> 18
[1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,7,8],[3,4,6,9,10]]
=> 10
[1,1,0,0,1,1,0,0,1,0]
=> [[1,2,5,6,9],[3,4,7,8,10]]
=> 12
[1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> 11
[1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [[1,2,4,7,9],[3,5,6,8,10]]
=> 17
[1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> 9
[1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> 16
[1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> 15
[1,1,0,1,0,1,1,0,0,0]
=> [[1,2,4,6,7],[3,5,8,9,10]]
=> 8
[1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> 11
[1,1,0,1,1,0,0,1,0,0]
=> [[1,2,4,5,8],[3,6,7,9,10]]
=> 10
[1,1,0,1,1,0,1,0,0,0]
=> [[1,2,4,5,7],[3,6,8,9,10]]
=> 9
[1,1,0,1,1,1,0,0,0,0]
=> [[1,2,4,5,6],[3,7,8,9,10]]
=> 3
Description
The natural major index of a standard Young tableau.
A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
The natural major index of a tableau with natural descent set $D$ is then $\sum_{d\in D} d$.
Matching statistic: St001379
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001379: Permutations ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 91%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001379: Permutations ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [2,1] => 2
[1,1,0,0]
=> [2,1] => [2,1] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [3,2,1] => 6
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => [2,1,3] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,3,1] => 4
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [1,3,2] => 3
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 12
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,3,4,1] => [3,2,1,4] => 6
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,3,1,4] => [3,2,4,1] => 8
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => [3,1,4,2] => 7
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,4,2,1] => [2,1,3,4] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 10
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,2,4,1] => [2,3,1,4] => 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [2,4,3,1] => 9
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 8
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,3,1] => [1,3,2,4] => 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 6
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,2,1,3] => [1,3,4,2] => 5
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,3,1,2] => [1,2,4,3] => 4
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 20
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,3,4,5,1] => [4,3,2,1,5] => 12
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,3,4,1,5] => [4,3,2,5,1] => 14
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,3,5,1,4] => [4,3,1,5,2] => 13
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,4,5,2,1] => [3,2,1,4,5] => 6
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,3,1,4,5] => [4,3,5,2,1] => 16
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,4,2,5,1] => [3,2,4,1,5] => 8
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,1,3,5] => [4,2,5,3,1] => 15
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,1,3,4] => [4,1,5,3,2] => 14
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,5,2,4,1] => [3,1,4,2,5] => 7
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,4,2,1,5] => [3,2,4,5,1] => 10
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,5,2,1,4] => [3,1,4,5,2] => 9
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [3,5,4,1,2] => [3,1,2,5,4] => 8
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,5,3,2,1] => [2,1,3,4,5] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [4,5,3,2,1] => 18
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2,4,5,1] => [3,4,2,1,5] => 10
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2,4,1,5] => [3,4,2,5,1] => 12
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2,5,1,4] => [3,4,1,5,2] => 11
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [4,3,5,2,1] => [2,3,1,4,5] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,5,4,2,1] => 17
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,2,3,5,1] => [2,4,3,1,5] => 9
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [2,5,4,3,1] => 16
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,5,4,3,2] => 15
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,4,1] => [1,4,3,2,5] => 8
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,1,5] => [2,4,3,5,1] => 11
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,2,3,1,4] => [1,4,3,5,2] => 10
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [5,2,4,1,3] => [1,4,2,5,3] => 9
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,3,4,2,1] => [1,3,2,4,5] => 3
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,7,6,5,4,2] => [5,7,4,6,3,2,1] => [3,1,4,2,5,6,7] => ? = 7
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => [4,7,6,2,5,1,3] => [4,1,2,6,3,7,5] => ? = 17
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => [5,7,6,3,4,2,1] => [3,1,2,5,4,6,7] => ? = 8
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,5,4,7,6,3,2] => [5,7,6,3,2,4,1] => [3,1,2,5,6,4,7] => ? = 10
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,5,6,4,7,3,2] => [4,7,6,3,1,2,5] => [4,1,2,5,7,6,3] => ? = 19
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,5,6,7,4,3,2] => [4,7,6,5,1,2,3] => [4,1,2,3,7,6,5] => ? = 18
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => [5,7,6,4,2,3,1] => [3,1,2,4,6,5,7] => ? = 9
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,5,4,3,7,2] => [5,7,4,3,2,1,6] => [3,1,4,5,6,7,2] => ? = 13
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,6,5,4,7,3,2] => [5,7,6,3,2,1,4] => [3,1,2,5,6,7,4] => ? = 12
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => [5,7,6,4,2,1,3] => [3,1,2,4,6,7,5] => ? = 11
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => [5,7,6,4,3,1,2] => [3,1,2,4,5,7,6] => ? = 10
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => [6,7,5,4,3,2,1] => [2,1,3,4,5,6,7] => ? = 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,6,5,4,3] => [6,5,7,4,3,2,1] => [2,3,1,4,5,6,7] => ? = 4
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => ? = 35
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,7,6,1] => [7,2,3,4,5,6,1] => [1,6,5,4,3,2,7] => ? = 24
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,4,6,5,7,1] => [7,2,3,4,5,1,6] => [1,6,5,4,3,7,2] => ? = 26
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,6,7,5,1] => [7,2,3,4,6,1,5] => [1,6,5,4,2,7,3] => ? = 25
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,7,6,5,1] => [7,3,4,5,6,2,1] => [1,5,4,3,2,6,7] => ? = 15
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [2,3,5,4,6,7,1] => [7,2,3,4,1,5,6] => [1,6,5,4,7,3,2] => ? = 28
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [2,3,5,4,7,6,1] => [7,3,4,5,2,6,1] => [1,5,4,3,6,2,7] => ? = 17
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [2,3,5,6,4,7,1] => [7,2,3,5,1,4,6] => [1,6,5,3,7,4,2] => ? = 27
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,5,6,7,4,1] => [7,2,3,6,1,4,5] => [1,6,5,2,7,4,3] => ? = 26
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,5,7,6,4,1] => [7,3,4,6,2,5,1] => [1,5,4,2,6,3,7] => ? = 16
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,4,7,1] => [7,3,4,5,2,1,6] => [1,5,4,3,6,7,2] => ? = 19
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,6,5,7,4,1] => [7,3,4,6,2,1,5] => [1,5,4,2,6,7,3] => ? = 18
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,6,7,5,4,1] => [7,3,4,6,5,1,2] => [1,5,4,2,3,7,6] => ? = 17
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,4,3,5,6,7,1] => [7,2,3,1,4,5,6] => [1,6,5,7,4,3,2] => ? = 30
[1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [2,4,3,5,7,6,1] => [7,3,4,2,5,6,1] => [1,5,4,6,3,2,7] => ? = 19
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,4,3,6,5,7,1] => [7,3,4,2,5,1,6] => [1,5,4,6,3,7,2] => ? = 21
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [2,4,3,6,7,5,1] => [7,3,4,2,6,1,5] => [1,5,4,6,2,7,3] => ? = 20
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,4,5,3,6,7,1] => [7,2,4,1,3,5,6] => [1,6,4,7,5,3,2] => ? = 29
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [2,4,5,3,7,6,1] => [7,3,5,2,4,6,1] => [1,5,3,6,4,2,7] => ? = 18
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [2,4,5,6,3,7,1] => [7,2,5,1,3,4,6] => [1,6,3,7,5,4,2] => ? = 28
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,4,5,6,7,3,1] => [7,2,6,1,3,4,5] => [1,6,2,7,5,4,3] => ? = 27
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [2,4,5,7,6,3,1] => [7,3,6,2,4,5,1] => [1,5,2,6,4,3,7] => ? = 17
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [2,4,6,5,3,7,1] => [7,3,5,2,4,1,6] => [1,5,3,6,4,7,2] => ? = 20
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [2,4,6,5,7,3,1] => [7,3,6,2,4,1,5] => [1,5,2,6,4,7,3] => ? = 19
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,4,6,7,5,3,1] => [7,3,6,2,5,1,4] => [1,5,2,6,3,7,4] => ? = 18
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [2,5,4,3,6,7,1] => [7,3,4,2,1,5,6] => [1,5,4,6,7,3,2] => ? = 23
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,5,4,6,3,7,1] => [7,3,5,2,1,4,6] => [1,5,3,6,7,4,2] => ? = 22
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,5,4,6,7,3,1] => [7,3,6,2,1,4,5] => [1,5,2,6,7,4,3] => ? = 21
[1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [2,5,6,4,3,7,1] => [7,3,5,4,1,2,6] => [1,5,3,4,7,6,2] => ? = 21
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [2,5,6,4,7,3,1] => [7,3,6,4,1,2,5] => [1,5,2,4,7,6,3] => ? = 20
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,5,6,7,4,3,1] => [7,3,6,5,1,2,4] => [1,5,2,3,7,6,4] => ? = 19
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,6,5,4,3,1,7] => [6,4,5,3,2,1,7] => [2,4,3,5,6,7,1] => ? = 15
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,1] => [7,2,1,3,4,5,6] => [1,6,7,5,4,3,2] => ? = 32
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [3,2,4,5,7,6,1] => [7,3,2,4,5,6,1] => [1,5,6,4,3,2,7] => ? = 21
[1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [3,2,4,6,5,7,1] => [7,3,2,4,5,1,6] => [1,5,6,4,3,7,2] => ? = 23
[1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [3,2,4,6,7,5,1] => [7,3,2,4,6,1,5] => [1,5,6,4,2,7,3] => ? = 22
[1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [3,2,5,4,6,7,1] => [7,3,2,4,1,5,6] => [1,5,6,4,7,3,2] => ? = 25
Description
The number of inversions plus the major index of a permutation.
This is, the sum of [[St000004]] and [[St000018]].
Matching statistic: St000825
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000825: Permutations ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 91%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000825: Permutations ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> [1,0]
=> [1] => 0
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 6
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 4
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 3
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 12
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 6
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 8
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 7
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 10
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 4
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 9
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 8
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 3
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 6
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 5
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 20
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 12
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 14
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 13
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 6
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 16
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 8
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 15
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => 14
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 7
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 10
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 9
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 8
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 18
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 10
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 12
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 11
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 17
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 9
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => 16
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 15
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 8
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 11
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 10
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => 9
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 3
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [3,2,4,1,5,6,7] => ? = 7
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [4,3,5,6,2,7,1] => ? = 17
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [3,2,4,5,1,6,7] => ? = 8
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,6,3,7] => ? = 10
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [3,2,5,6,7,4,1] => ? = 19
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [4,3,5,6,7,2,1] => ? = 18
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [3,2,4,5,6,1,7] => ? = 9
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,3,4,6,7,5] => ? = 13
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [2,1,3,5,6,7,4] => ? = 12
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [2,1,4,5,6,7,3] => ? = 11
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => ? = 35
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [4,6,5,3,2,7,1] => ? = 26
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [5,6,4,3,2,7,1] => ? = 25
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [4,5,3,2,1,6,7] => ? = 15
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [3,6,5,4,7,2,1] => ? = 28
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [3,5,4,2,6,1,7] => ? = 17
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [4,6,5,3,7,2,1] => ? = 27
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,6,4,3,7,2,1] => ? = 26
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [4,5,3,2,6,1,7] => ? = 16
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [3,4,2,1,6,7,5] => ? = 19
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [3,5,4,2,6,7,1] => ? = 18
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [4,5,3,2,6,7,1] => ? = 17
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [3,4,2,1,5,6,7] => ? = 8
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,6,5,7,4,3,1] => ? = 30
[1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [2,5,4,6,3,1,7] => ? = 19
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,4,3,6,5,7,1] => ? = 21
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,5,4,6,3,7,1] => ? = 20
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [2,4,3,5,1,6,7] => ? = 10
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [3,6,5,7,4,2,1] => ? = 29
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [3,5,4,6,2,1,7] => ? = 18
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [4,6,5,7,3,2,1] => ? = 28
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,6,4,7,3,2,1] => ? = 27
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [3,4,2,6,5,7,1] => ? = 20
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [3,5,4,6,2,7,1] => ? = 19
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [4,5,3,6,2,7,1] => ? = 18
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [3,4,2,5,1,6,7] => ? = 9
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [2,3,1,6,7,5,4] => ? = 23
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1,5,6,4,7] => ? = 12
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [2,4,3,6,7,5,1] => ? = 22
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,5,4,6,7,3,1] => ? = 21
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [2,4,3,5,6,1,7] => ? = 11
[1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [3,4,2,6,7,5,1] => ? = 21
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [3,5,4,6,7,2,1] => ? = 20
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [4,5,3,6,7,2,1] => ? = 19
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [3,4,2,5,6,1,7] => ? = 10
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [2,3,1,4,5,7,6] => ? = 15
[1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [2,3,1,4,6,7,5] => ? = 14
[1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,6,7,4] => ? = 13
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,4,3,5,6,7,1] => ? = 12
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,5,6,4,3,2,7] => ? = 21
Description
The sum of the major and the inverse major index of a permutation.
This statistic is the sum of [[St000004]] and [[St000305]].
Matching statistic: St000290
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 88%
Mp00105: Binary words —complement⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 88%
Values
[1,0]
=> 10 => 01 => 0
[1,0,1,0]
=> 1010 => 0101 => 2
[1,1,0,0]
=> 1100 => 0011 => 0
[1,0,1,0,1,0]
=> 101010 => 010101 => 6
[1,0,1,1,0,0]
=> 101100 => 010011 => 2
[1,1,0,0,1,0]
=> 110010 => 001101 => 4
[1,1,0,1,0,0]
=> 110100 => 001011 => 3
[1,1,1,0,0,0]
=> 111000 => 000111 => 0
[1,0,1,0,1,0,1,0]
=> 10101010 => 01010101 => 12
[1,0,1,0,1,1,0,0]
=> 10101100 => 01010011 => 6
[1,0,1,1,0,0,1,0]
=> 10110010 => 01001101 => 8
[1,0,1,1,0,1,0,0]
=> 10110100 => 01001011 => 7
[1,0,1,1,1,0,0,0]
=> 10111000 => 01000111 => 2
[1,1,0,0,1,0,1,0]
=> 11001010 => 00110101 => 10
[1,1,0,0,1,1,0,0]
=> 11001100 => 00110011 => 4
[1,1,0,1,0,0,1,0]
=> 11010010 => 00101101 => 9
[1,1,0,1,0,1,0,0]
=> 11010100 => 00101011 => 8
[1,1,0,1,1,0,0,0]
=> 11011000 => 00100111 => 3
[1,1,1,0,0,0,1,0]
=> 11100010 => 00011101 => 6
[1,1,1,0,0,1,0,0]
=> 11100100 => 00011011 => 5
[1,1,1,0,1,0,0,0]
=> 11101000 => 00010111 => 4
[1,1,1,1,0,0,0,0]
=> 11110000 => 00001111 => 0
[1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 0101010101 => 20
[1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 0101010011 => 12
[1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 0101001101 => 14
[1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 0101001011 => 13
[1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 0101000111 => 6
[1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 0100110101 => 16
[1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 0100110011 => 8
[1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 0100101101 => 15
[1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 0100101011 => 14
[1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 0100100111 => 7
[1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 0100011101 => 10
[1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 0100011011 => 9
[1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 0100010111 => 8
[1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 0100001111 => 2
[1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 0011010101 => 18
[1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 0011010011 => 10
[1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 0011001101 => 12
[1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 0011001011 => 11
[1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 0011000111 => 4
[1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 0010110101 => 17
[1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 0010110011 => 9
[1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 0010101101 => 16
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 0010101011 => 15
[1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 0010100111 => 8
[1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 0010011101 => 11
[1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 0010011011 => 10
[1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 0010010111 => 9
[1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 0010001111 => 3
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> 10110111100000 => 01001000011111 => ? = 7
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> 10111011010000 => 01000100101111 => ? = 17
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> 10111011100000 => 01000100011111 => ? = 8
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> 10111100110000 => 01000011001111 => ? = 10
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> 10111101001000 => 01000010110111 => ? = 19
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> 10111101010000 => 01000010101111 => ? = 18
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> 10111101100000 => 01000010011111 => ? = 9
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> 10111110000100 => 01000001111011 => ? = 13
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> 10111110001000 => 01000001110111 => ? = 12
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> 10111110010000 => 01000001101111 => ? = 11
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 10111110100000 => 01000001011111 => ? = 10
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => 01000000111111 => ? = 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 11001111100000 => 00110000011111 => ? = 4
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 11010101010100 => 00101010101011 => ? = 35
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> 11010101011000 => 00101010100111 => ? = 24
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> 11010101100100 => 00101010011011 => ? = 26
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> 11010101101000 => 00101010010111 => ? = 25
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> 11010101110000 => 00101010001111 => ? = 15
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> 11010110010100 => 00101001101011 => ? = 28
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> 11010110011000 => 00101001100111 => ? = 17
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> 11010110100100 => 00101001011011 => ? = 27
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> 11010110101000 => 00101001010111 => ? = 26
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> 11010110110000 => 00101001001111 => ? = 16
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> 11010111000100 => 00101000111011 => ? = 19
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> 11010111001000 => 00101000110111 => ? = 18
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> 11010111010000 => 00101000101111 => ? = 17
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> 11010111100000 => 00101000011111 => ? = 8
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> 11011001010100 => 00100110101011 => ? = 30
[1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> 11011001011000 => 00100110100111 => ? = 19
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> 11011001100100 => 00100110011011 => ? = 21
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> 11011001101000 => 00100110010111 => ? = 20
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> 11011001110000 => 00100110001111 => ? = 10
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> 11011010010100 => 00100101101011 => ? = 29
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> 11011010011000 => 00100101100111 => ? = 18
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> 11011010100100 => 00100101011011 => ? = 28
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> 11011010101000 => 00100101010111 => ? = 27
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> 11011010110000 => 00100101001111 => ? = 17
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> 11011011000100 => 00100100111011 => ? = 20
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> 11011011001000 => 00100100110111 => ? = 19
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> 11011011010000 => 00100100101111 => ? = 18
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> 11011011100000 => 00100100011111 => ? = 9
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> 11011100010100 => 00100011101011 => ? = 23
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> 11011100011000 => 00100011100111 => ? = 12
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> 11011100100100 => 00100011011011 => ? = 22
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> 11011100101000 => 00100011010111 => ? = 21
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> 11011100110000 => 00100011001111 => ? = 11
[1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> 11011101000100 => 00100010111011 => ? = 21
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> 11011101001000 => 00100010110111 => ? = 20
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> 11011101010000 => 00100010101111 => ? = 19
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> 11011101100000 => 00100010011111 => ? = 10
Description
The major index of a binary word.
This is the sum of the positions of descents, i.e., a one followed by a zero.
For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
Matching statistic: St000330
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 88%
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 88%
Values
[1,0]
=> [[1],[2]]
=> [[1,2]]
=> 0
[1,0,1,0]
=> [[1,3],[2,4]]
=> [[1,2,4],[3]]
=> 2
[1,1,0,0]
=> [[1,2],[3,4]]
=> [[1,2,3,4]]
=> 0
[1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [[1,2,4,6],[3,5]]
=> 6
[1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [[1,2,4,5,6],[3]]
=> 2
[1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [[1,2,3,4,6],[5]]
=> 4
[1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [[1,2,3,5,6],[4]]
=> 3
[1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [[1,2,3,4,5,6]]
=> 0
[1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [[1,2,4,6,8],[3,5,7]]
=> 12
[1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [[1,2,4,6,7,8],[3,5]]
=> 6
[1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [[1,2,4,5,6,8],[3,7]]
=> 8
[1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [[1,2,4,5,7,8],[3,6]]
=> 7
[1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [[1,2,4,5,6,7,8],[3]]
=> 2
[1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [[1,2,3,4,6,8],[5,7]]
=> 10
[1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [[1,2,3,4,7,8],[5,6]]
=> 4
[1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [[1,2,3,5,6,8],[4,7]]
=> 9
[1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [[1,2,3,5,7,8],[4,6]]
=> 8
[1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [[1,2,3,5,6,7,8],[4]]
=> 3
[1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [[1,2,3,4,5,6,8],[7]]
=> 6
[1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [[1,2,3,4,5,7,8],[6]]
=> 5
[1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [[1,2,3,4,6,7,8],[5]]
=> 4
[1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [[1,2,3,4,5,6,7,8]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> [[1,2,4,6,8,10],[3,5,7,9]]
=> 20
[1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> [[1,2,4,6,8,9,10],[3,5,7]]
=> 12
[1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> [[1,2,4,6,7,8,10],[3,5,9]]
=> 14
[1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> [[1,2,4,6,7,9,10],[3,5,8]]
=> 13
[1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> [[1,2,4,6,7,8,9,10],[3,5]]
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> [[1,2,4,5,6,8,10],[3,7,9]]
=> 16
[1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> [[1,2,4,5,6,9,10],[3,7,8]]
=> 8
[1,0,1,1,0,1,0,0,1,0]
=> [[1,3,4,6,9],[2,5,7,8,10]]
=> [[1,2,4,5,7,8,10],[3,6,9]]
=> 15
[1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> [[1,2,4,5,7,9,10],[3,6,8]]
=> 14
[1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> [[1,2,4,5,7,8,9,10],[3,6]]
=> 7
[1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> [[1,2,4,5,6,7,8,10],[3,9]]
=> 10
[1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> [[1,2,4,5,6,7,9,10],[3,8]]
=> 9
[1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> [[1,2,4,5,6,8,9,10],[3,7]]
=> 8
[1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> [[1,2,4,5,6,7,8,9,10],[3]]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> [[1,2,3,4,6,8,10],[5,7,9]]
=> 18
[1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,7,8],[3,4,6,9,10]]
=> [[1,2,3,4,6,9,10],[5,7,8]]
=> 10
[1,1,0,0,1,1,0,0,1,0]
=> [[1,2,5,6,9],[3,4,7,8,10]]
=> [[1,2,3,4,7,8,10],[5,6,9]]
=> 12
[1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> [[1,2,3,4,7,9,10],[5,6,8]]
=> 11
[1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> [[1,2,3,4,7,8,9,10],[5,6]]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [[1,2,4,7,9],[3,5,6,8,10]]
=> [[1,2,3,5,6,8,10],[4,7,9]]
=> 17
[1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> [[1,2,3,5,6,9,10],[4,7,8]]
=> 9
[1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> [[1,2,3,5,7,8,10],[4,6,9]]
=> 16
[1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> [[1,2,3,5,7,9,10],[4,6,8]]
=> 15
[1,1,0,1,0,1,1,0,0,0]
=> [[1,2,4,6,7],[3,5,8,9,10]]
=> [[1,2,3,5,7,8,9,10],[4,6]]
=> 8
[1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> [[1,2,3,5,6,7,8,10],[4,9]]
=> 11
[1,1,0,1,1,0,0,1,0,0]
=> [[1,2,4,5,8],[3,6,7,9,10]]
=> [[1,2,3,5,6,7,9,10],[4,8]]
=> 10
[1,1,0,1,1,0,1,0,0,0]
=> [[1,2,4,5,7],[3,6,8,9,10]]
=> [[1,2,3,5,6,8,9,10],[4,7]]
=> 9
[1,1,0,1,1,1,0,0,0,0]
=> [[1,2,4,5,6],[3,7,8,9,10]]
=> [[1,2,3,5,6,7,8,9,10],[4]]
=> 3
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,6,7,8,9],[2,5,10,11,12,13,14]]
=> [[1,2,4,5,7,8,9,10,11,12,13,14],[3,6]]
=> ? = 7
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,7,8,10],[2,6,9,11,12,13,14]]
=> [[1,2,4,5,6,8,9,11,12,13,14],[3,7,10]]
=> ? = 17
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,7,8,9],[2,6,10,11,12,13,14]]
=> [[1,2,4,5,6,8,9,10,11,12,13,14],[3,7]]
=> ? = 8
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [[1,3,4,5,6,9,10],[2,7,8,11,12,13,14]]
=> [[1,2,4,5,6,7,8,11,12,13,14],[3,9,10]]
=> ? = 10
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [[1,3,4,5,6,8,11],[2,7,9,10,12,13,14]]
=> [[1,2,4,5,6,7,9,10,12,13,14],[3,8,11]]
=> ? = 19
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8,10],[2,7,9,11,12,13,14]]
=> [[1,2,4,5,6,7,9,11,12,13,14],[3,8,10]]
=> ? = 18
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,8,9],[2,7,10,11,12,13,14]]
=> [[1,2,4,5,6,7,9,10,11,12,13,14],[3,8]]
=> ? = 9
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,3,4,5,6,7,12],[2,8,9,10,11,13,14]]
=> [[1,2,4,5,6,7,8,9,10,11,13,14],[3,12]]
=> ? = 13
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [[1,3,4,5,6,7,11],[2,8,9,10,12,13,14]]
=> [[1,2,4,5,6,7,8,9,10,12,13,14],[3,11]]
=> ? = 12
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [[1,3,4,5,6,7,10],[2,8,9,11,12,13,14]]
=> [[1,2,4,5,6,7,8,9,11,12,13,14],[3,10]]
=> ? = 11
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[1,3,4,5,6,7,9],[2,8,10,11,12,13,14]]
=> [[1,2,4,5,6,7,8,10,11,12,13,14],[3,9]]
=> ? = 10
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> [[1,2,4,5,6,7,8,9,10,11,12,13,14],[3]]
=> ? = 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,2,5,6,7,8,9],[3,4,10,11,12,13,14]]
=> [[1,2,3,4,7,8,9,10,11,12,13,14],[5,6]]
=> ? = 4
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8,10,12],[3,5,7,9,11,13,14]]
=> [[1,2,3,5,7,9,11,13,14],[4,6,8,10,12]]
=> ? = 35
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [[1,2,4,6,8,10,11],[3,5,7,9,12,13,14]]
=> [[1,2,3,5,7,9,11,12,13,14],[4,6,8,10]]
=> ? = 24
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [[1,2,4,6,8,9,12],[3,5,7,10,11,13,14]]
=> [[1,2,3,5,7,9,10,11,13,14],[4,6,8,12]]
=> ? = 26
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [[1,2,4,6,8,9,11],[3,5,7,10,12,13,14]]
=> [[1,2,3,5,7,9,10,12,13,14],[4,6,8,11]]
=> ? = 25
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [[1,2,4,6,8,9,10],[3,5,7,11,12,13,14]]
=> [[1,2,3,5,7,9,10,11,12,13,14],[4,6,8]]
=> ? = 15
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [[1,2,4,6,7,10,12],[3,5,8,9,11,13,14]]
=> [[1,2,3,5,7,8,9,11,13,14],[4,6,10,12]]
=> ? = 28
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [[1,2,4,6,7,10,11],[3,5,8,9,12,13,14]]
=> [[1,2,3,5,7,8,9,12,13,14],[4,6,10,11]]
=> ? = 17
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [[1,2,4,6,7,9,12],[3,5,8,10,11,13,14]]
=> [[1,2,3,5,7,8,10,11,13,14],[4,6,9,12]]
=> ? = 27
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [[1,2,4,6,7,9,11],[3,5,8,10,12,13,14]]
=> [[1,2,3,5,7,8,10,12,13,14],[4,6,9,11]]
=> ? = 26
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [[1,2,4,6,7,9,10],[3,5,8,11,12,13,14]]
=> [[1,2,3,5,7,8,10,11,12,13,14],[4,6,9]]
=> ? = 16
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [[1,2,4,6,7,8,12],[3,5,9,10,11,13,14]]
=> [[1,2,3,5,7,8,9,10,11,13,14],[4,6,12]]
=> ? = 19
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [[1,2,4,6,7,8,11],[3,5,9,10,12,13,14]]
=> [[1,2,3,5,7,8,9,10,12,13,14],[4,6,11]]
=> ? = 18
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [[1,2,4,6,7,8,10],[3,5,9,11,12,13,14]]
=> [[1,2,3,5,7,8,9,11,12,13,14],[4,6,10]]
=> ? = 17
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[1,2,4,6,7,8,9],[3,5,10,11,12,13,14]]
=> [[1,2,3,5,7,8,9,10,11,12,13,14],[4,6]]
=> ? = 8
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [[1,2,4,5,8,10,12],[3,6,7,9,11,13,14]]
=> [[1,2,3,5,6,7,9,11,13,14],[4,8,10,12]]
=> ? = 30
[1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [[1,2,4,5,8,10,11],[3,6,7,9,12,13,14]]
=> [[1,2,3,5,6,7,9,12,13,14],[4,8,10,11]]
=> ? = 19
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [[1,2,4,5,8,9,12],[3,6,7,10,11,13,14]]
=> [[1,2,3,5,6,7,10,11,13,14],[4,8,9,12]]
=> ? = 21
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [[1,2,4,5,8,9,11],[3,6,7,10,12,13,14]]
=> [[1,2,3,5,6,7,10,12,13,14],[4,8,9,11]]
=> ? = 20
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [[1,2,4,5,8,9,10],[3,6,7,11,12,13,14]]
=> [[1,2,3,5,6,7,10,11,12,13,14],[4,8,9]]
=> ? = 10
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [[1,2,4,5,7,10,12],[3,6,8,9,11,13,14]]
=> [[1,2,3,5,6,8,9,11,13,14],[4,7,10,12]]
=> ? = 29
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [[1,2,4,5,7,10,11],[3,6,8,9,12,13,14]]
=> [[1,2,3,5,6,8,9,12,13,14],[4,7,10,11]]
=> ? = 18
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [[1,2,4,5,7,9,12],[3,6,8,10,11,13,14]]
=> [[1,2,3,5,6,8,10,11,13,14],[4,7,9,12]]
=> ? = 28
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [[1,2,4,5,7,9,11],[3,6,8,10,12,13,14]]
=> [[1,2,3,5,6,8,10,12,13,14],[4,7,9,11]]
=> ? = 27
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [[1,2,4,5,7,9,10],[3,6,8,11,12,13,14]]
=> [[1,2,3,5,6,8,10,11,12,13,14],[4,7,9]]
=> ? = 17
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [[1,2,4,5,7,8,12],[3,6,9,10,11,13,14]]
=> [[1,2,3,5,6,8,9,10,11,13,14],[4,7,12]]
=> ? = 20
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [[1,2,4,5,7,8,11],[3,6,9,10,12,13,14]]
=> [[1,2,3,5,6,8,9,10,12,13,14],[4,7,11]]
=> ? = 19
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [[1,2,4,5,7,8,10],[3,6,9,11,12,13,14]]
=> [[1,2,3,5,6,8,9,11,12,13,14],[4,7,10]]
=> ? = 18
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [[1,2,4,5,7,8,9],[3,6,10,11,12,13,14]]
=> [[1,2,3,5,6,8,9,10,11,12,13,14],[4,7]]
=> ? = 9
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [[1,2,4,5,6,10,12],[3,7,8,9,11,13,14]]
=> [[1,2,3,5,6,7,8,9,11,13,14],[4,10,12]]
=> ? = 23
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [[1,2,4,5,6,10,11],[3,7,8,9,12,13,14]]
=> [[1,2,3,5,6,7,8,9,12,13,14],[4,10,11]]
=> ? = 12
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [[1,2,4,5,6,9,12],[3,7,8,10,11,13,14]]
=> [[1,2,3,5,6,7,8,10,11,13,14],[4,9,12]]
=> ? = 22
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [[1,2,4,5,6,9,11],[3,7,8,10,12,13,14]]
=> [[1,2,3,5,6,7,8,10,12,13,14],[4,9,11]]
=> ? = 21
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [[1,2,4,5,6,9,10],[3,7,8,11,12,13,14]]
=> [[1,2,3,5,6,7,8,11,12,13,14],[4,9,10]]
=> ? = 11
[1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [[1,2,4,5,6,8,12],[3,7,9,10,11,13,14]]
=> [[1,2,3,5,6,7,9,10,11,13,14],[4,8,12]]
=> ? = 21
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [[1,2,4,5,6,8,11],[3,7,9,10,12,13,14]]
=> [[1,2,3,5,6,7,9,10,12,13,14],[4,8,11]]
=> ? = 20
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [[1,2,4,5,6,8,10],[3,7,9,11,12,13,14]]
=> [[1,2,3,5,6,7,9,11,12,13,14],[4,8,10]]
=> ? = 19
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [[1,2,4,5,6,8,9],[3,7,10,11,12,13,14]]
=> [[1,2,3,5,6,7,9,10,11,12,13,14],[4,8]]
=> ? = 10
Description
The (standard) major index of a standard tableau.
A descent of a standard tableau $T$ is an index $i$ such that $i+1$ appears in a row strictly below the row of $i$. The (standard) major index is the the sum of the descents.
Matching statistic: St000169
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
St000169: Standard tableaux ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 88%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
St000169: Standard tableaux ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 88%
Values
[1,0]
=> [1,0]
=> [[1],[2]]
=> [[1,2]]
=> 0
[1,0,1,0]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> [[1,2,4],[3]]
=> 2
[1,1,0,0]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> [[1,2,3,4]]
=> 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [[1,2,4,6],[3,5]]
=> 6
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [[1,2,3,4,6],[5]]
=> 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [[1,2,4,5,6],[3]]
=> 4
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [[1,2,3,5,6],[4]]
=> 3
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [[1,2,3,4,5,6]]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [[1,2,4,6,8],[3,5,7]]
=> 12
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [[1,2,3,4,6,8],[5,7]]
=> 6
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [[1,2,4,5,6,8],[3,7]]
=> 8
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [[1,2,3,5,6,8],[4,7]]
=> 7
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [[1,2,3,4,5,6,8],[7]]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [[1,2,4,6,7,8],[3,5]]
=> 10
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [[1,2,3,4,7,8],[5,6]]
=> 4
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [[1,2,4,5,7,8],[3,6]]
=> 9
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [[1,2,3,5,7,8],[4,6]]
=> 8
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [[1,2,3,4,5,7,8],[6]]
=> 3
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [[1,2,4,5,6,7,8],[3]]
=> 6
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [[1,2,3,5,6,7,8],[4]]
=> 5
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [[1,2,3,4,6,7,8],[5]]
=> 4
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [[1,2,3,4,5,6,7,8]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> [[1,2,4,6,8,10],[3,5,7,9]]
=> 20
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> [[1,2,3,4,6,8,10],[5,7,9]]
=> 12
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> [[1,2,4,5,6,8,10],[3,7,9]]
=> 14
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[1,2,4,7,9],[3,5,6,8,10]]
=> [[1,2,3,5,6,8,10],[4,7,9]]
=> 13
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[1,2,3,7,9],[4,5,6,8,10]]
=> [[1,2,3,4,5,6,8,10],[7,9]]
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> [[1,2,4,6,7,8,10],[3,5,9]]
=> 16
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[1,2,5,6,9],[3,4,7,8,10]]
=> [[1,2,3,4,7,8,10],[5,6,9]]
=> 8
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[1,3,4,6,9],[2,5,7,8,10]]
=> [[1,2,4,5,7,8,10],[3,6,9]]
=> 15
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> [[1,2,3,5,7,8,10],[4,6,9]]
=> 14
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> [[1,2,3,4,5,7,8,10],[6,9]]
=> 7
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> [[1,2,4,5,6,7,8,10],[3,9]]
=> 10
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> [[1,2,3,5,6,7,8,10],[4,9]]
=> 9
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> [[1,2,3,4,6,7,8,10],[5,9]]
=> 8
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> [[1,2,3,4,5,6,7,8,10],[9]]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> [[1,2,4,6,8,9,10],[3,5,7]]
=> 18
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,7,8],[3,4,6,9,10]]
=> [[1,2,3,4,6,9,10],[5,7,8]]
=> 10
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> [[1,2,4,5,6,9,10],[3,7,8]]
=> 12
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> [[1,2,3,5,6,9,10],[4,7,8]]
=> 11
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> [[1,2,3,4,5,6,9,10],[7,8]]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> [[1,2,4,6,7,9,10],[3,5,8]]
=> 17
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> [[1,2,3,4,7,9,10],[5,6,8]]
=> 9
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> [[1,2,4,5,7,9,10],[3,6,8]]
=> 16
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> [[1,2,3,5,7,9,10],[4,6,8]]
=> 15
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[1,2,3,6,8],[4,5,7,9,10]]
=> [[1,2,3,4,5,7,9,10],[6,8]]
=> 8
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> [[1,2,4,5,6,7,9,10],[3,8]]
=> 11
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[1,2,4,5,8],[3,6,7,9,10]]
=> [[1,2,3,5,6,7,9,10],[4,8]]
=> 10
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[1,2,3,5,8],[4,6,7,9,10]]
=> [[1,2,3,4,6,7,9,10],[5,8]]
=> 9
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[1,2,3,4,8],[5,6,7,9,10]]
=> [[1,2,3,4,5,6,7,9,10],[8]]
=> 3
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [[1,2,3,4,5,10,13],[6,7,8,9,11,12,14]]
=> [[1,2,3,4,5,6,7,8,9,11,12,14],[10,13]]
=> ? = 7
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [[1,2,3,4,6,9,13],[5,7,8,10,11,12,14]]
=> [[1,2,3,4,5,7,8,10,11,12,14],[6,9,13]]
=> ? = 17
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [[1,2,3,4,5,9,13],[6,7,8,10,11,12,14]]
=> [[1,2,3,4,5,6,7,8,10,11,12,14],[9,13]]
=> ? = 8
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,7,8,13],[5,6,9,10,11,12,14]]
=> [[1,2,3,4,5,6,9,10,11,12,14],[7,8,13]]
=> ? = 10
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,5,6,8,13],[4,7,9,10,11,12,14]]
=> [[1,2,3,4,6,7,9,10,11,12,14],[5,8,13]]
=> ? = 19
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,8,13],[5,7,9,10,11,12,14]]
=> [[1,2,3,4,5,7,9,10,11,12,14],[6,8,13]]
=> ? = 18
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,5,8,13],[6,7,9,10,11,12,14]]
=> [[1,2,3,4,5,6,7,9,10,11,12,14],[8,13]]
=> ? = 9
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,4,5,6,7,13],[3,8,9,10,11,12,14]]
=> [[1,2,3,5,6,7,8,9,10,11,12,14],[4,13]]
=> ? = 13
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,5,6,7,13],[4,8,9,10,11,12,14]]
=> [[1,2,3,4,6,7,8,9,10,11,12,14],[5,13]]
=> ? = 12
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,6,7,13],[5,8,9,10,11,12,14]]
=> [[1,2,3,4,5,7,8,9,10,11,12,14],[6,13]]
=> ? = 11
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,7,13],[6,8,9,10,11,12,14]]
=> [[1,2,3,4,5,6,8,9,10,11,12,14],[7,13]]
=> ? = 10
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12,14]]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,14],[13]]
=> ? = 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10,13,14]]
=> [[1,2,3,4,5,6,7,8,9,10,13,14],[11,12]]
=> ? = 4
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8,10,12],[3,5,7,9,11,13,14]]
=> [[1,2,3,5,7,9,11,13,14],[4,6,8,10,12]]
=> ? = 35
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,3,6,8,10,12],[4,5,7,9,11,13,14]]
=> [[1,2,3,4,5,7,9,11,13,14],[6,8,10,12]]
=> ? = 24
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [[1,2,4,5,8,10,12],[3,6,7,9,11,13,14]]
=> [[1,2,3,5,6,7,9,11,13,14],[4,8,10,12]]
=> ? = 26
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [[1,2,3,5,8,10,12],[4,6,7,9,11,13,14]]
=> [[1,2,3,4,6,7,9,11,13,14],[5,8,10,12]]
=> ? = 25
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [[1,2,3,4,8,10,12],[5,6,7,9,11,13,14]]
=> [[1,2,3,4,5,6,7,9,11,13,14],[8,10,12]]
=> ? = 15
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [[1,2,4,6,7,10,12],[3,5,8,9,11,13,14]]
=> [[1,2,3,5,7,8,9,11,13,14],[4,6,10,12]]
=> ? = 28
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [[1,2,3,6,7,10,12],[4,5,8,9,11,13,14]]
=> [[1,2,3,4,5,8,9,11,13,14],[6,7,10,12]]
=> ? = 17
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [[1,2,4,5,7,10,12],[3,6,8,9,11,13,14]]
=> [[1,2,3,5,6,8,9,11,13,14],[4,7,10,12]]
=> ? = 27
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [[1,2,3,5,7,10,12],[4,6,8,9,11,13,14]]
=> [[1,2,3,4,6,8,9,11,13,14],[5,7,10,12]]
=> ? = 26
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [[1,2,3,4,7,10,12],[5,6,8,9,11,13,14]]
=> [[1,2,3,4,5,6,8,9,11,13,14],[7,10,12]]
=> ? = 16
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [[1,2,4,5,6,10,12],[3,7,8,9,11,13,14]]
=> [[1,2,3,5,6,7,8,9,11,13,14],[4,10,12]]
=> ? = 19
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [[1,2,3,5,6,10,12],[4,7,8,9,11,13,14]]
=> [[1,2,3,4,6,7,8,9,11,13,14],[5,10,12]]
=> ? = 18
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [[1,2,3,4,6,10,12],[5,7,8,9,11,13,14]]
=> [[1,2,3,4,5,7,8,9,11,13,14],[6,10,12]]
=> ? = 17
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [[1,2,3,4,5,10,12],[6,7,8,9,11,13,14]]
=> [[1,2,3,4,5,6,7,8,9,11,13,14],[10,12]]
=> ? = 8
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [[1,2,4,6,8,9,12],[3,5,7,10,11,13,14]]
=> [[1,2,3,5,7,9,10,11,13,14],[4,6,8,12]]
=> ? = 30
[1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [[1,2,3,6,8,9,12],[4,5,7,10,11,13,14]]
=> [[1,2,3,4,5,7,10,11,13,14],[6,8,9,12]]
=> ? = 19
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [[1,2,4,5,8,9,12],[3,6,7,10,11,13,14]]
=> [[1,2,3,5,6,7,10,11,13,14],[4,8,9,12]]
=> ? = 21
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [[1,2,3,5,8,9,12],[4,6,7,10,11,13,14]]
=> [[1,2,3,4,6,7,10,11,13,14],[5,8,9,12]]
=> ? = 20
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [[1,2,3,4,8,9,12],[5,6,7,10,11,13,14]]
=> [[1,2,3,4,5,6,7,10,11,13,14],[8,9,12]]
=> ? = 10
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [[1,2,4,6,7,9,12],[3,5,8,10,11,13,14]]
=> [[1,2,3,5,7,8,10,11,13,14],[4,6,9,12]]
=> ? = 29
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [[1,2,3,6,7,9,12],[4,5,8,10,11,13,14]]
=> [[1,2,3,4,5,8,10,11,13,14],[6,7,9,12]]
=> ? = 18
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [[1,2,4,5,7,9,12],[3,6,8,10,11,13,14]]
=> [[1,2,3,5,6,8,10,11,13,14],[4,7,9,12]]
=> ? = 28
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [[1,2,3,5,7,9,12],[4,6,8,10,11,13,14]]
=> [[1,2,3,4,6,8,10,11,13,14],[5,7,9,12]]
=> ? = 27
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [[1,2,3,4,7,9,12],[5,6,8,10,11,13,14]]
=> [[1,2,3,4,5,6,8,10,11,13,14],[7,9,12]]
=> ? = 17
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [[1,2,4,5,6,9,12],[3,7,8,10,11,13,14]]
=> [[1,2,3,5,6,7,8,10,11,13,14],[4,9,12]]
=> ? = 20
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [[1,2,3,5,6,9,12],[4,7,8,10,11,13,14]]
=> [[1,2,3,4,6,7,8,10,11,13,14],[5,9,12]]
=> ? = 19
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [[1,2,3,4,6,9,12],[5,7,8,10,11,13,14]]
=> [[1,2,3,4,5,7,8,10,11,13,14],[6,9,12]]
=> ? = 18
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [[1,2,3,4,5,9,12],[6,7,8,10,11,13,14]]
=> [[1,2,3,4,5,6,7,8,10,11,13,14],[9,12]]
=> ? = 9
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [[1,2,4,6,7,8,12],[3,5,9,10,11,13,14]]
=> [[1,2,3,5,7,8,9,10,11,13,14],[4,6,12]]
=> ? = 23
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [[1,2,3,6,7,8,12],[4,5,9,10,11,13,14]]
=> [[1,2,3,4,5,8,9,10,11,13,14],[6,7,12]]
=> ? = 12
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [[1,2,4,5,7,8,12],[3,6,9,10,11,13,14]]
=> [[1,2,3,5,6,8,9,10,11,13,14],[4,7,12]]
=> ? = 22
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [[1,2,3,5,7,8,12],[4,6,9,10,11,13,14]]
=> [[1,2,3,4,6,8,9,10,11,13,14],[5,7,12]]
=> ? = 21
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [[1,2,3,4,7,8,12],[5,6,9,10,11,13,14]]
=> [[1,2,3,4,5,6,9,10,11,13,14],[7,8,12]]
=> ? = 11
[1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [[1,2,4,5,6,8,12],[3,7,9,10,11,13,14]]
=> [[1,2,3,5,6,7,9,10,11,13,14],[4,8,12]]
=> ? = 21
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [[1,2,3,5,6,8,12],[4,7,9,10,11,13,14]]
=> [[1,2,3,4,6,7,9,10,11,13,14],[5,8,12]]
=> ? = 20
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [[1,2,3,4,6,8,12],[5,7,9,10,11,13,14]]
=> [[1,2,3,4,5,7,9,10,11,13,14],[6,8,12]]
=> ? = 19
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [[1,2,3,4,5,8,12],[6,7,9,10,11,13,14]]
=> [[1,2,3,4,5,6,7,9,10,11,13,14],[8,12]]
=> ? = 10
Description
The cocharge of a standard tableau.
The '''cocharge''' of a standard tableau $T$, denoted $\mathrm{cc}(T)$, is defined to be the cocharge of the reading word of the tableau. The cocharge of a permutation $w_1 w_2\cdots w_n$ can be computed by the following algorithm:
1) Starting from $w_n$, scan the entries right-to-left until finding the entry $1$ with a superscript $0$.
2) Continue scanning until the $2$ is found, and label this with a superscript $1$. Then scan until the $3$ is found, labeling with a $2$, and so on, incrementing the label each time, until the beginning of the word is reached. Then go back to the end and scan again from right to left, and *do not* increment the superscript label for the first number found in the next scan. Then continue scanning and labeling, each time incrementing the superscript only if we have not cycled around the word since the last labeling.
3) The cocharge is defined as the sum of the superscript labels on the letters.
Matching statistic: St000293
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 88%
Mp00096: Binary words —Foata bijection⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 88%
Values
[1,0]
=> 10 => 10 => 01 => 0
[1,0,1,0]
=> 1010 => 1100 => 0110 => 2
[1,1,0,0]
=> 1100 => 0110 => 0011 => 0
[1,0,1,0,1,0]
=> 101010 => 111000 => 011100 => 6
[1,0,1,1,0,0]
=> 101100 => 011010 => 001101 => 2
[1,1,0,0,1,0]
=> 110010 => 101100 => 010110 => 4
[1,1,0,1,0,0]
=> 110100 => 011100 => 001110 => 3
[1,1,1,0,0,0]
=> 111000 => 001110 => 000111 => 0
[1,0,1,0,1,0,1,0]
=> 10101010 => 11110000 => 01111000 => 12
[1,0,1,0,1,1,0,0]
=> 10101100 => 01110010 => 00111001 => 6
[1,0,1,1,0,0,1,0]
=> 10110010 => 10110100 => 01011010 => 8
[1,0,1,1,0,1,0,0]
=> 10110100 => 01110100 => 00111010 => 7
[1,0,1,1,1,0,0,0]
=> 10111000 => 00110110 => 00011011 => 2
[1,1,0,0,1,0,1,0]
=> 11001010 => 11011000 => 01101100 => 10
[1,1,0,0,1,1,0,0]
=> 11001100 => 01011010 => 00101101 => 4
[1,1,0,1,0,0,1,0]
=> 11010010 => 10111000 => 01011100 => 9
[1,1,0,1,0,1,0,0]
=> 11010100 => 01111000 => 00111100 => 8
[1,1,0,1,1,0,0,0]
=> 11011000 => 00111010 => 00011101 => 3
[1,1,1,0,0,0,1,0]
=> 11100010 => 10011100 => 01001110 => 6
[1,1,1,0,0,1,0,0]
=> 11100100 => 01011100 => 00101110 => 5
[1,1,1,0,1,0,0,0]
=> 11101000 => 00111100 => 00011110 => 4
[1,1,1,1,0,0,0,0]
=> 11110000 => 00011110 => 00001111 => 0
[1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 1111100000 => 0111110000 => 20
[1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 0111100010 => 0011110001 => ? = 12
[1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 1011100100 => 0101110010 => 14
[1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 0111100100 => 0011110010 => 13
[1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 0011100110 => 0001110011 => 6
[1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 1101101000 => 0110110100 => 16
[1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 0101101010 => 0010110101 => 8
[1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 1011101000 => 0101110100 => 15
[1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 0111101000 => 0011110100 => 14
[1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 0011101010 => 0001110101 => 7
[1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 1001101100 => 0100110110 => ? = 10
[1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 0101101100 => 0010110110 => 9
[1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 0011101100 => 0001110110 => 8
[1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 0001101110 => 0000110111 => 2
[1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 1110110000 => 0111011000 => 18
[1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 0110110010 => 0011011001 => ? = 10
[1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 1010110100 => 0101011010 => 12
[1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 0110110100 => 0011011010 => ? = 11
[1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 0010110110 => 0001011011 => 4
[1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 1101110000 => 0110111000 => 17
[1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 0101110010 => 0010111001 => ? = 9
[1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 1011110000 => 0101111000 => 16
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 0111110000 => 0011111000 => 15
[1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 0011110010 => 0001111001 => 8
[1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 1001110100 => 0100111010 => 11
[1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 0101110100 => 0010111010 => 10
[1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 0011110100 => 0001111010 => 9
[1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 0001110110 => 0000111011 => 3
[1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 1100111000 => 0110011100 => 14
[1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 0100111010 => 0010011101 => 6
[1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => 1010111000 => 0101011100 => 13
[1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => 0110111000 => 0011011100 => ? = 12
[1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => 0010111010 => 0001011101 => 5
[1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => 1001111000 => 0100111100 => 12
[1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => 1000111100 => 0100011110 => ? = 8
[1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => 011111000010 => 001111100001 => ? = 20
[1,0,1,0,1,0,1,1,1,0,0,0]
=> 101010111000 => 001111000110 => 000111100011 => ? = 12
[1,0,1,0,1,1,0,0,1,1,0,0]
=> 101011001100 => 010111001010 => 001011100101 => ? = 14
[1,0,1,0,1,1,0,1,1,0,0,0]
=> 101011011000 => 001111001010 => 000111100101 => ? = 13
[1,0,1,0,1,1,1,0,0,0,1,0]
=> 101011100010 => 100111001100 => 010011100110 => ? = 16
[1,0,1,1,0,0,1,0,1,1,0,0]
=> 101100101100 => 011011010010 => 001101101001 => ? = 16
[1,0,1,1,0,1,0,0,1,1,0,0]
=> 101101001100 => 010111010010 => 001011101001 => ? = 15
[1,0,1,1,0,1,0,1,1,0,0,0]
=> 101101011000 => 001111010010 => 000111101001 => ? = 14
[1,0,1,1,1,1,0,0,0,0,1,0]
=> 101111000010 => 100011011100 => 010001101110 => ? = 12
[1,0,1,1,1,1,0,0,0,1,0,0]
=> 101111000100 => 010011011100 => 001001101110 => ? = 11
[1,1,0,0,1,0,1,0,1,1,0,0]
=> 110010101100 => 011101100010 => 001110110001 => ? = 18
[1,1,0,0,1,0,1,1,0,1,0,0]
=> 110010110100 => 011101100100 => 001110110010 => ? = 19
[1,1,0,0,1,1,0,1,0,1,0,0]
=> 110011010100 => 011101101000 => 001110110100 => ? = 20
[1,1,0,0,1,1,1,0,0,0,1,0]
=> 110011100010 => 100101101100 => 010010110110 => ? = 14
[1,1,0,0,1,1,1,0,0,1,0,0]
=> 110011100100 => 010101101100 => 001010110110 => ? = 13
[1,1,0,0,1,1,1,0,1,0,0,0]
=> 110011101000 => 001101101100 => 000110110110 => ? = 12
[1,1,0,1,0,0,1,0,1,1,0,0]
=> 110100101100 => 011011100010 => 001101110001 => ? = 17
[1,1,0,1,0,1,0,0,1,1,0,0]
=> 110101001100 => 010111100010 => 001011110001 => ? = 16
[1,1,0,1,0,1,0,1,1,0,0,0]
=> 110101011000 => 001111100010 => 000111110001 => ? = 15
[1,1,0,1,1,1,0,0,0,0,1,0]
=> 110111000010 => 100011101100 => 010001110110 => ? = 13
[1,1,1,0,0,0,1,0,1,1,0,0]
=> 111000101100 => 011001110010 => 001100111001 => ? = 14
[1,1,1,0,0,0,1,1,0,0,1,0]
=> 111000110010 => 101001110100 => 010100111010 => ? = 16
[1,1,1,0,0,0,1,1,0,1,0,0]
=> 111000110100 => 011001110100 => 001100111010 => ? = 15
[1,1,1,0,0,1,0,0,1,1,0,0]
=> 111001001100 => 010101110010 => 001010111001 => ? = 13
[1,1,1,0,0,1,0,1,0,1,0,0]
=> 111001010100 => 011101110000 => 001110111000 => ? = 21
[1,1,1,0,0,1,0,1,1,0,0,0]
=> 111001011000 => 001101110010 => 000110111001 => ? = 12
[1,1,1,0,0,1,1,0,0,0,1,0]
=> 111001100010 => 100101110100 => 010010111010 => ? = 15
[1,1,1,0,0,1,1,0,0,1,0,0]
=> 111001100100 => 010101110100 => 001010111010 => ? = 14
[1,1,1,0,0,1,1,0,1,0,0,0]
=> 111001101000 => 001101110100 => 000110111010 => ? = 13
[1,1,1,0,1,0,0,0,1,1,0,0]
=> 111010001100 => 010011110010 => 001001111001 => ? = 12
[1,1,1,0,1,0,0,1,1,0,0,0]
=> 111010011000 => 001011110010 => 000101111001 => ? = 11
[1,1,1,0,1,0,1,1,0,0,0,0]
=> 111010110000 => 000111110010 => 000011111001 => ? = 10
[1,1,1,1,0,0,0,0,1,0,1,0]
=> 111100001010 => 110001111000 => 011000111100 => ? = 18
[1,1,1,1,0,0,0,1,0,0,1,0]
=> 111100010010 => 101001111000 => 010100111100 => ? = 17
[1,1,1,1,0,0,0,1,0,1,0,0]
=> 111100010100 => 011001111000 => 001100111100 => ? = 16
[1,1,1,1,0,0,1,0,0,0,1,0]
=> 111100100010 => 100101111000 => 010010111100 => ? = 16
[1,1,1,1,0,0,1,0,0,1,0,0]
=> 111100100100 => 010101111000 => 001010111100 => ? = 15
[1,1,1,1,0,0,1,0,1,0,0,0]
=> 111100101000 => 001101111000 => 000110111100 => ? = 14
[1,1,1,1,1,0,0,0,0,0,1,0]
=> 111110000010 => 100001111100 => 010000111110 => ? = 10
[1,1,1,1,1,0,0,0,0,1,0,0]
=> 111110000100 => 010001111100 => 001000111110 => ? = 9
[1,1,1,1,1,0,0,0,1,0,0,0]
=> 111110001000 => 001001111100 => 000100111110 => ? = 8
[1,1,1,1,1,0,0,1,0,0,0,0]
=> 111110010000 => 000101111100 => 000010111110 => ? = 7
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> 10110111100000 => 00001110101110 => ? => ? = 7
Description
The number of inversions of a binary word.
Matching statistic: St000391
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 58%
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 58%
Values
[1,0]
=> [[1],[2]]
=> [[1,2]]
=> 0 => 0
[1,0,1,0]
=> [[1,3],[2,4]]
=> [[1,2,4],[3]]
=> 010 => 2
[1,1,0,0]
=> [[1,2],[3,4]]
=> [[1,2,3,4]]
=> 000 => 0
[1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [[1,2,4,6],[3,5]]
=> 01010 => 6
[1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [[1,2,4,5,6],[3]]
=> 01000 => 2
[1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [[1,2,3,4,6],[5]]
=> 00010 => 4
[1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [[1,2,3,5,6],[4]]
=> 00100 => 3
[1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [[1,2,3,4,5,6]]
=> 00000 => 0
[1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [[1,2,4,6,8],[3,5,7]]
=> 0101010 => 12
[1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [[1,2,4,6,7,8],[3,5]]
=> 0101000 => 6
[1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [[1,2,4,5,6,8],[3,7]]
=> 0100010 => 8
[1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [[1,2,4,5,7,8],[3,6]]
=> 0100100 => 7
[1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [[1,2,4,5,6,7,8],[3]]
=> 0100000 => 2
[1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [[1,2,3,4,6,8],[5,7]]
=> 0001010 => 10
[1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [[1,2,3,4,7,8],[5,6]]
=> 0001000 => 4
[1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [[1,2,3,5,6,8],[4,7]]
=> 0010010 => 9
[1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [[1,2,3,5,7,8],[4,6]]
=> 0010100 => 8
[1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [[1,2,3,5,6,7,8],[4]]
=> 0010000 => 3
[1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [[1,2,3,4,5,6,8],[7]]
=> 0000010 => 6
[1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [[1,2,3,4,5,7,8],[6]]
=> 0000100 => 5
[1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [[1,2,3,4,6,7,8],[5]]
=> 0001000 => 4
[1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [[1,2,3,4,5,6,7,8]]
=> 0000000 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> [[1,2,4,6,8,10],[3,5,7,9]]
=> 010101010 => 20
[1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> [[1,2,4,6,8,9,10],[3,5,7]]
=> 010101000 => 12
[1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> [[1,2,4,6,7,8,10],[3,5,9]]
=> 010100010 => 14
[1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> [[1,2,4,6,7,9,10],[3,5,8]]
=> 010100100 => 13
[1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> [[1,2,4,6,7,8,9,10],[3,5]]
=> 010100000 => 6
[1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> [[1,2,4,5,6,8,10],[3,7,9]]
=> 010001010 => 16
[1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> [[1,2,4,5,6,9,10],[3,7,8]]
=> 010001000 => 8
[1,0,1,1,0,1,0,0,1,0]
=> [[1,3,4,6,9],[2,5,7,8,10]]
=> [[1,2,4,5,7,8,10],[3,6,9]]
=> 010010010 => 15
[1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> [[1,2,4,5,7,9,10],[3,6,8]]
=> 010010100 => 14
[1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> [[1,2,4,5,7,8,9,10],[3,6]]
=> 010010000 => 7
[1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> [[1,2,4,5,6,7,8,10],[3,9]]
=> 010000010 => 10
[1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> [[1,2,4,5,6,7,9,10],[3,8]]
=> 010000100 => 9
[1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> [[1,2,4,5,6,8,9,10],[3,7]]
=> 010001000 => 8
[1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> [[1,2,4,5,6,7,8,9,10],[3]]
=> 010000000 => 2
[1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> [[1,2,3,4,6,8,10],[5,7,9]]
=> 000101010 => 18
[1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,7,8],[3,4,6,9,10]]
=> [[1,2,3,4,6,9,10],[5,7,8]]
=> 000101000 => 10
[1,1,0,0,1,1,0,0,1,0]
=> [[1,2,5,6,9],[3,4,7,8,10]]
=> [[1,2,3,4,7,8,10],[5,6,9]]
=> 000100010 => 12
[1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> [[1,2,3,4,7,9,10],[5,6,8]]
=> 000100100 => 11
[1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> [[1,2,3,4,7,8,9,10],[5,6]]
=> 000100000 => 4
[1,1,0,1,0,0,1,0,1,0]
=> [[1,2,4,7,9],[3,5,6,8,10]]
=> [[1,2,3,5,6,8,10],[4,7,9]]
=> 001001010 => 17
[1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> [[1,2,3,5,6,9,10],[4,7,8]]
=> 001001000 => 9
[1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> [[1,2,3,5,7,8,10],[4,6,9]]
=> 001010010 => 16
[1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> [[1,2,3,5,7,9,10],[4,6,8]]
=> 001010100 => 15
[1,1,0,1,0,1,1,0,0,0]
=> [[1,2,4,6,7],[3,5,8,9,10]]
=> [[1,2,3,5,7,8,9,10],[4,6]]
=> 001010000 => 8
[1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> [[1,2,3,5,6,7,8,10],[4,9]]
=> 001000010 => 11
[1,1,0,1,1,0,0,1,0,0]
=> [[1,2,4,5,8],[3,6,7,9,10]]
=> [[1,2,3,5,6,7,9,10],[4,8]]
=> 001000100 => 10
[1,1,0,1,1,0,1,0,0,0]
=> [[1,2,4,5,7],[3,6,8,9,10]]
=> [[1,2,3,5,6,8,9,10],[4,7]]
=> 001001000 => 9
[1,1,0,1,1,1,0,0,0,0]
=> [[1,2,4,5,6],[3,7,8,9,10]]
=> [[1,2,3,5,6,7,8,9,10],[4]]
=> 001000000 => 3
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11],[2,4,6,8,10,12]]
=> [[1,2,4,6,8,10,12],[3,5,7,9,11]]
=> 01010101010 => ? = 30
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,10],[2,4,6,8,11,12]]
=> [[1,2,4,6,8,10,11,12],[3,5,7,9]]
=> 01010101000 => ? = 20
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,7,8,11],[2,4,6,9,10,12]]
=> [[1,2,4,6,8,9,10,12],[3,5,7,11]]
=> 01010100010 => ? = 22
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,7,8,10],[2,4,6,9,11,12]]
=> [[1,2,4,6,8,9,11,12],[3,5,7,10]]
=> 01010100100 => ? = 21
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,7,8,9],[2,4,6,10,11,12]]
=> [[1,2,4,6,8,9,10,11,12],[3,5,7]]
=> 01010100000 => ? = 12
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[1,3,5,6,9,11],[2,4,7,8,10,12]]
=> [[1,2,4,6,7,8,10,12],[3,5,9,11]]
=> 01010001010 => ? = 24
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,3,5,6,9,10],[2,4,7,8,11,12]]
=> [[1,2,4,6,7,8,11,12],[3,5,9,10]]
=> 01010001000 => ? = 14
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[1,3,5,6,8,11],[2,4,7,9,10,12]]
=> [[1,2,4,6,7,9,10,12],[3,5,8,11]]
=> 01010010010 => ? = 23
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,6,8,10],[2,4,7,9,11,12]]
=> [[1,2,4,6,7,9,11,12],[3,5,8,10]]
=> 01010010100 => ? = 22
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[1,3,5,6,8,9],[2,4,7,10,11,12]]
=> [[1,2,4,6,7,9,10,11,12],[3,5,8]]
=> 01010010000 => ? = 13
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[1,3,5,6,7,11],[2,4,8,9,10,12]]
=> [[1,2,4,6,7,8,9,10,12],[3,5,11]]
=> 01010000010 => ? = 16
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,3,5,6,7,10],[2,4,8,9,11,12]]
=> [[1,2,4,6,7,8,9,11,12],[3,5,10]]
=> 01010000100 => ? = 15
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,3,5,6,7,9],[2,4,8,10,11,12]]
=> [[1,2,4,6,7,8,10,11,12],[3,5,9]]
=> 01010001000 => ? = 14
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,3,5,6,7,8],[2,4,9,10,11,12]]
=> [[1,2,4,6,7,8,9,10,11,12],[3,5]]
=> 01010000000 => ? = 6
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[1,3,4,7,9,11],[2,5,6,8,10,12]]
=> [[1,2,4,5,6,8,10,12],[3,7,9,11]]
=> 01000101010 => ? = 26
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,3,4,7,9,10],[2,5,6,8,11,12]]
=> [[1,2,4,5,6,8,11,12],[3,7,9,10]]
=> 01000101000 => ? = 16
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[1,3,4,7,8,11],[2,5,6,9,10,12]]
=> [[1,2,4,5,6,9,10,12],[3,7,8,11]]
=> 01000100010 => ? = 18
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[1,3,4,7,8,10],[2,5,6,9,11,12]]
=> [[1,2,4,5,6,9,11,12],[3,7,8,10]]
=> 01000100100 => ? = 17
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[1,3,4,7,8,9],[2,5,6,10,11,12]]
=> [[1,2,4,5,6,9,10,11,12],[3,7,8]]
=> 01000100000 => ? = 8
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[1,3,4,6,9,11],[2,5,7,8,10,12]]
=> [[1,2,4,5,7,8,10,12],[3,6,9,11]]
=> 01001001010 => ? = 25
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[1,3,4,6,9,10],[2,5,7,8,11,12]]
=> [[1,2,4,5,7,8,11,12],[3,6,9,10]]
=> 01001001000 => ? = 15
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[1,3,4,6,8,11],[2,5,7,9,10,12]]
=> [[1,2,4,5,7,9,10,12],[3,6,8,11]]
=> 01001010010 => ? = 24
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,3,4,6,8,10],[2,5,7,9,11,12]]
=> [[1,2,4,5,7,9,11,12],[3,6,8,10]]
=> 01001010100 => ? = 23
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[1,3,4,6,8,9],[2,5,7,10,11,12]]
=> [[1,2,4,5,7,9,10,11,12],[3,6,8]]
=> 01001010000 => ? = 14
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[1,3,4,6,7,11],[2,5,8,9,10,12]]
=> [[1,2,4,5,7,8,9,10,12],[3,6,11]]
=> 01001000010 => ? = 17
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[1,3,4,6,7,10],[2,5,8,9,11,12]]
=> [[1,2,4,5,7,8,9,11,12],[3,6,10]]
=> 01001000100 => ? = 16
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [[1,3,4,6,7,9],[2,5,8,10,11,12]]
=> [[1,2,4,5,7,8,10,11,12],[3,6,9]]
=> 01001001000 => ? = 15
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [[1,3,4,6,7,8],[2,5,9,10,11,12]]
=> [[1,2,4,5,7,8,9,10,11,12],[3,6]]
=> 01001000000 => ? = 7
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [[1,3,4,5,9,11],[2,6,7,8,10,12]]
=> [[1,2,4,5,6,7,8,10,12],[3,9,11]]
=> 01000001010 => ? = 20
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [[1,3,4,5,9,10],[2,6,7,8,11,12]]
=> [[1,2,4,5,6,7,8,11,12],[3,9,10]]
=> 01000001000 => ? = 10
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [[1,3,4,5,8,11],[2,6,7,9,10,12]]
=> [[1,2,4,5,6,7,9,10,12],[3,8,11]]
=> 01000010010 => ? = 19
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[1,3,4,5,8,10],[2,6,7,9,11,12]]
=> [[1,2,4,5,6,7,9,11,12],[3,8,10]]
=> 01000010100 => ? = 18
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [[1,3,4,5,8,9],[2,6,7,10,11,12]]
=> [[1,2,4,5,6,7,10,11,12],[3,8,9]]
=> 01000010000 => ? = 9
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [[1,3,4,5,7,11],[2,6,8,9,10,12]]
=> [[1,2,4,5,6,8,9,10,12],[3,7,11]]
=> 01000100010 => ? = 18
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,3,4,5,7,10],[2,6,8,9,11,12]]
=> [[1,2,4,5,6,8,9,11,12],[3,7,10]]
=> 01000100100 => ? = 17
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[1,3,4,5,7,9],[2,6,8,10,11,12]]
=> [[1,2,4,5,6,8,10,11,12],[3,7,9]]
=> 01000101000 => ? = 16
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,3,4,5,7,8],[2,6,9,10,11,12]]
=> [[1,2,4,5,6,8,9,10,11,12],[3,7]]
=> 01000100000 => ? = 8
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [[1,3,4,5,6,11],[2,7,8,9,10,12]]
=> [[1,2,4,5,6,7,8,9,10,12],[3,11]]
=> 01000000010 => ? = 12
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[1,3,4,5,6,10],[2,7,8,9,11,12]]
=> [[1,2,4,5,6,7,8,9,11,12],[3,10]]
=> 01000000100 => ? = 11
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [[1,3,4,5,6,9],[2,7,8,10,11,12]]
=> [[1,2,4,5,6,7,8,10,11,12],[3,9]]
=> 01000001000 => ? = 10
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> [[1,2,4,5,6,7,9,10,11,12],[3,8]]
=> 01000010000 => ? = 9
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> [[1,2,4,5,6,7,8,9,10,11,12],[3]]
=> 01000000000 => ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [[1,2,5,7,9,11],[3,4,6,8,10,12]]
=> [[1,2,3,4,6,8,10,12],[5,7,9,11]]
=> 00010101010 => ? = 28
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [[1,2,5,7,9,10],[3,4,6,8,11,12]]
=> [[1,2,3,4,6,8,11,12],[5,7,9,10]]
=> 00010101000 => ? = 18
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [[1,2,5,7,8,11],[3,4,6,9,10,12]]
=> [[1,2,3,4,6,9,10,12],[5,7,8,11]]
=> 00010100010 => ? = 20
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [[1,2,5,7,8,10],[3,4,6,9,11,12]]
=> [[1,2,3,4,6,9,11,12],[5,7,8,10]]
=> 00010100100 => ? = 19
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[1,2,5,7,8,9],[3,4,6,10,11,12]]
=> [[1,2,3,4,6,9,10,11,12],[5,7,8]]
=> 00010100000 => ? = 10
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [[1,2,5,6,9,11],[3,4,7,8,10,12]]
=> [[1,2,3,4,7,8,10,12],[5,6,9,11]]
=> 00010001010 => ? = 22
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [[1,2,5,6,9,10],[3,4,7,8,11,12]]
=> [[1,2,3,4,7,8,11,12],[5,6,9,10]]
=> 00010001000 => ? = 12
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [[1,2,5,6,8,11],[3,4,7,9,10,12]]
=> [[1,2,3,4,7,9,10,12],[5,6,8,11]]
=> 00010010010 => ? = 21
Description
The sum of the positions of the ones in a binary word.
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000008The major index of the composition. St000246The number of non-inversions of a permutation. St001699The major index of a standard tableau minus the weighted size of its shape. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001697The shifted natural comajor index of a standard Young tableau.
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