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Your data matches 114 different statistics following compositions of up to 3 maps.
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Matching statistic: St000228
(load all 36 compositions to match this statistic)
(load all 36 compositions to match this statistic)
Mp00186: Skew partitions —dominating partition⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3],[]]
=> [3]
=> 3
[[3,1],[1]]
=> [3]
=> 3
[[3,2],[2]]
=> [3]
=> 3
[[1,1,1],[]]
=> [1,1,1]
=> 3
[[3,2,1],[2,1]]
=> [3]
=> 3
[[4],[]]
=> [4]
=> 4
[[4,1],[1]]
=> [4]
=> 4
[[2,2],[]]
=> [2,2]
=> 4
[[4,2],[2]]
=> [4]
=> 4
[[4,2,1],[2,1]]
=> [4]
=> 4
[[4,3],[3]]
=> [4]
=> 4
[[2,2,1],[1]]
=> [2,2]
=> 4
[[4,3,1],[3,1]]
=> [4]
=> 4
[[4,3,2],[3,2]]
=> [4]
=> 4
[[1,1,1,1],[]]
=> [1,1,1,1]
=> 4
[[2,2,1,1],[1,1]]
=> [2,2]
=> 4
[[4,3,2,1],[3,2,1]]
=> [4]
=> 4
[[5],[]]
=> [5]
=> 5
[[5,1],[1]]
=> [5]
=> 5
[[5,2],[2]]
=> [5]
=> 5
[[5,2,1],[2,1]]
=> [5]
=> 5
[[5,3],[3]]
=> [5]
=> 5
[[5,3,1],[3,1]]
=> [5]
=> 5
[[5,3,2],[3,2]]
=> [5]
=> 5
[[5,3,2,1],[3,2,1]]
=> [5]
=> 5
[[5,4],[4]]
=> [5]
=> 5
[[5,4,1],[4,1]]
=> [5]
=> 5
[[5,4,2],[4,2]]
=> [5]
=> 5
[[5,4,2,1],[4,2,1]]
=> [5]
=> 5
[[5,4,3],[4,3]]
=> [5]
=> 5
[[5,4,3,1],[4,3,1]]
=> [5]
=> 5
[[5,4,3,2],[4,3,2]]
=> [5]
=> 5
[[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> 5
[[5,4,3,2,1],[4,3,2,1]]
=> [5]
=> 5
[[6],[]]
=> [6]
=> 6
[[6,1],[1]]
=> [6]
=> 6
[[6,2],[2]]
=> [6]
=> 6
[[6,2,1],[2,1]]
=> [6]
=> 6
[[3,3],[]]
=> [3,3]
=> 6
[[6,3],[3]]
=> [6]
=> 6
[[6,3,1],[3,1]]
=> [6]
=> 6
[[6,3,2],[3,2]]
=> [6]
=> 6
[[6,3,2,1],[3,2,1]]
=> [6]
=> 6
[[6,4],[4]]
=> [6]
=> 6
[[3,3,1],[1]]
=> [3,3]
=> 6
[[6,4,1],[4,1]]
=> [6]
=> 6
[[2,2,2],[]]
=> [2,2,2]
=> 6
[[6,4,2],[4,2]]
=> [6]
=> 6
[[3,3,1,1],[1,1]]
=> [3,3]
=> 6
[[6,4,2,1],[4,2,1]]
=> [6]
=> 6
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000293
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00186: Skew partitions —dominating partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3],[]]
=> [3]
=> 1000 => 3
[[3,1],[1]]
=> [3]
=> 1000 => 3
[[3,2],[2]]
=> [3]
=> 1000 => 3
[[1,1,1],[]]
=> [1,1,1]
=> 1110 => 3
[[3,2,1],[2,1]]
=> [3]
=> 1000 => 3
[[4],[]]
=> [4]
=> 10000 => 4
[[4,1],[1]]
=> [4]
=> 10000 => 4
[[2,2],[]]
=> [2,2]
=> 1100 => 4
[[4,2],[2]]
=> [4]
=> 10000 => 4
[[4,2,1],[2,1]]
=> [4]
=> 10000 => 4
[[4,3],[3]]
=> [4]
=> 10000 => 4
[[2,2,1],[1]]
=> [2,2]
=> 1100 => 4
[[4,3,1],[3,1]]
=> [4]
=> 10000 => 4
[[4,3,2],[3,2]]
=> [4]
=> 10000 => 4
[[1,1,1,1],[]]
=> [1,1,1,1]
=> 11110 => 4
[[2,2,1,1],[1,1]]
=> [2,2]
=> 1100 => 4
[[4,3,2,1],[3,2,1]]
=> [4]
=> 10000 => 4
[[5],[]]
=> [5]
=> 100000 => 5
[[5,1],[1]]
=> [5]
=> 100000 => 5
[[5,2],[2]]
=> [5]
=> 100000 => 5
[[5,2,1],[2,1]]
=> [5]
=> 100000 => 5
[[5,3],[3]]
=> [5]
=> 100000 => 5
[[5,3,1],[3,1]]
=> [5]
=> 100000 => 5
[[5,3,2],[3,2]]
=> [5]
=> 100000 => 5
[[5,3,2,1],[3,2,1]]
=> [5]
=> 100000 => 5
[[5,4],[4]]
=> [5]
=> 100000 => 5
[[5,4,1],[4,1]]
=> [5]
=> 100000 => 5
[[5,4,2],[4,2]]
=> [5]
=> 100000 => 5
[[5,4,2,1],[4,2,1]]
=> [5]
=> 100000 => 5
[[5,4,3],[4,3]]
=> [5]
=> 100000 => 5
[[5,4,3,1],[4,3,1]]
=> [5]
=> 100000 => 5
[[5,4,3,2],[4,3,2]]
=> [5]
=> 100000 => 5
[[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> 111110 => 5
[[5,4,3,2,1],[4,3,2,1]]
=> [5]
=> 100000 => 5
[[6],[]]
=> [6]
=> 1000000 => 6
[[6,1],[1]]
=> [6]
=> 1000000 => 6
[[6,2],[2]]
=> [6]
=> 1000000 => 6
[[6,2,1],[2,1]]
=> [6]
=> 1000000 => 6
[[3,3],[]]
=> [3,3]
=> 11000 => 6
[[6,3],[3]]
=> [6]
=> 1000000 => 6
[[6,3,1],[3,1]]
=> [6]
=> 1000000 => 6
[[6,3,2],[3,2]]
=> [6]
=> 1000000 => 6
[[6,3,2,1],[3,2,1]]
=> [6]
=> 1000000 => 6
[[6,4],[4]]
=> [6]
=> 1000000 => 6
[[3,3,1],[1]]
=> [3,3]
=> 11000 => 6
[[6,4,1],[4,1]]
=> [6]
=> 1000000 => 6
[[2,2,2],[]]
=> [2,2,2]
=> 11100 => 6
[[6,4,2],[4,2]]
=> [6]
=> 1000000 => 6
[[3,3,1,1],[1,1]]
=> [3,3]
=> 11000 => 6
[[6,4,2,1],[4,2,1]]
=> [6]
=> 1000000 => 6
Description
The number of inversions of a binary word.
Matching statistic: St000395
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00181: Skew partitions —row lengths⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3],[]]
=> [3] => [1,1,1,0,0,0]
=> 3
[[3,1],[1]]
=> [2,1] => [1,1,0,0,1,0]
=> 3
[[3,2],[2]]
=> [1,2] => [1,0,1,1,0,0]
=> 3
[[1,1,1],[]]
=> [1,1,1] => [1,0,1,0,1,0]
=> 3
[[3,2,1],[2,1]]
=> [1,1,1] => [1,0,1,0,1,0]
=> 3
[[4],[]]
=> [4] => [1,1,1,1,0,0,0,0]
=> 4
[[4,1],[1]]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 4
[[2,2],[]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 4
[[4,2],[2]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 4
[[4,2,1],[2,1]]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[[4,3],[3]]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 4
[[2,2,1],[1]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[[4,3,1],[3,1]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[[4,3,2],[3,2]]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 4
[[1,1,1,1],[]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[[2,2,1,1],[1,1]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[[4,3,2,1],[3,2,1]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[[5],[]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[5,1],[1]]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5
[[5,2],[2]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 5
[[5,2,1],[2,1]]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 5
[[5,3],[3]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 5
[[5,3,1],[3,1]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 5
[[5,3,2],[3,2]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5
[[5,3,2,1],[3,2,1]]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 5
[[5,4],[4]]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 5
[[5,4,1],[4,1]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 5
[[5,4,2],[4,2]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 5
[[5,4,2,1],[4,2,1]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 5
[[5,4,3],[4,3]]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[[5,4,3,1],[4,3,1]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 5
[[5,4,3,2],[4,3,2]]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 5
[[1,1,1,1,1],[]]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[[5,4,3,2,1],[4,3,2,1]]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[[6],[]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[[6,1],[1]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[[6,2],[2]]
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 6
[[6,2,1],[2,1]]
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 6
[[3,3],[]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 6
[[6,3],[3]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 6
[[6,3,1],[3,1]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 6
[[6,3,2],[3,2]]
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 6
[[6,3,2,1],[3,2,1]]
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 6
[[6,4],[4]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 6
[[3,3,1],[1]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 6
[[6,4,1],[4,1]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 6
[[2,2,2],[]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 6
[[6,4,2],[4,2]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 6
[[3,3,1,1],[1,1]]
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 6
[[6,4,2,1],[4,2,1]]
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 6
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St000645
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00186: Skew partitions —dominating partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3],[]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[[3,1],[1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[[3,2],[2]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[[1,1,1],[]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[[3,2,1],[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[[4],[]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[[4,1],[1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[[2,2],[]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 4
[[4,2],[2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[[4,2,1],[2,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[[4,3],[3]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[[2,2,1],[1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 4
[[4,3,1],[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[[4,3,2],[3,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[[2,2,1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 4
[[4,3,2,1],[3,2,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[[5],[]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[[5,1],[1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[[5,2],[2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[[5,2,1],[2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[[5,3],[3]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[[5,3,1],[3,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[[5,3,2],[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[[5,3,2,1],[3,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[[5,4],[4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[[5,4,1],[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[[5,4,2],[4,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[[5,4,2,1],[4,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[[5,4,3],[4,3]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[[5,4,3,1],[4,3,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[[5,4,3,2],[4,3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[[5,4,3,2,1],[4,3,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[[6],[]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 6
[[6,1],[1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 6
[[6,2],[2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 6
[[6,2,1],[2,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 6
[[3,3],[]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 6
[[6,3],[3]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 6
[[6,3,1],[3,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 6
[[6,3,2],[3,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 6
[[6,3,2,1],[3,2,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 6
[[6,4],[4]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 6
[[3,3,1],[1]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 6
[[6,4,1],[4,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 6
[[2,2,2],[]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 6
[[6,4,2],[4,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 6
[[3,3,1,1],[1,1]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 6
[[6,4,2,1],[4,2,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 6
Description
The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between.
For a Dyck path $D = D_1 \cdots D_{2n}$ with peaks in positions $i_1 < \ldots < i_k$ and valleys in positions $j_1 < \ldots < j_{k-1}$, this statistic is given by
$$
\sum_{a=1}^{k-1} (j_a-i_a)(i_{a+1}-j_a)
$$
Matching statistic: St001020
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00181: Skew partitions —row lengths⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001020: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001020: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3],[]]
=> [3] => [1,1,1,0,0,0]
=> 3
[[3,1],[1]]
=> [2,1] => [1,1,0,0,1,0]
=> 3
[[3,2],[2]]
=> [1,2] => [1,0,1,1,0,0]
=> 3
[[1,1,1],[]]
=> [1,1,1] => [1,0,1,0,1,0]
=> 3
[[3,2,1],[2,1]]
=> [1,1,1] => [1,0,1,0,1,0]
=> 3
[[4],[]]
=> [4] => [1,1,1,1,0,0,0,0]
=> 4
[[4,1],[1]]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 4
[[2,2],[]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 4
[[4,2],[2]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 4
[[4,2,1],[2,1]]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[[4,3],[3]]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 4
[[2,2,1],[1]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[[4,3,1],[3,1]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[[4,3,2],[3,2]]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 4
[[1,1,1,1],[]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[[2,2,1,1],[1,1]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[[4,3,2,1],[3,2,1]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[[5],[]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[5,1],[1]]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5
[[5,2],[2]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 5
[[5,2,1],[2,1]]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 5
[[5,3],[3]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 5
[[5,3,1],[3,1]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 5
[[5,3,2],[3,2]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5
[[5,3,2,1],[3,2,1]]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 5
[[5,4],[4]]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 5
[[5,4,1],[4,1]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 5
[[5,4,2],[4,2]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 5
[[5,4,2,1],[4,2,1]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 5
[[5,4,3],[4,3]]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[[5,4,3,1],[4,3,1]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 5
[[5,4,3,2],[4,3,2]]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 5
[[1,1,1,1,1],[]]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[[5,4,3,2,1],[4,3,2,1]]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[[6],[]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[[6,1],[1]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[[6,2],[2]]
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 6
[[6,2,1],[2,1]]
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 6
[[3,3],[]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 6
[[6,3],[3]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 6
[[6,3,1],[3,1]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 6
[[6,3,2],[3,2]]
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 6
[[6,3,2,1],[3,2,1]]
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 6
[[6,4],[4]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 6
[[3,3,1],[1]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 6
[[6,4,1],[4,1]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 6
[[2,2,2],[]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 6
[[6,4,2],[4,2]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 6
[[3,3,1,1],[1,1]]
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 6
[[6,4,2,1],[4,2,1]]
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 6
Description
Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001034
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00186: Skew partitions —dominating partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3],[]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[3,1],[1]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[3,2],[2]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[1,1,1],[]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[[3,2,1],[2,1]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[4],[]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[4,1],[1]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[2,2],[]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 4
[[4,2],[2]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[4,2,1],[2,1]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[4,3],[3]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[2,2,1],[1]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 4
[[4,3,1],[3,1]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[4,3,2],[3,2]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 4
[[2,2,1,1],[1,1]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 4
[[4,3,2,1],[3,2,1]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[5],[]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[5,1],[1]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[5,2],[2]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[5,2,1],[2,1]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[5,3],[3]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[5,3,1],[3,1]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[5,3,2],[3,2]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[5,3,2,1],[3,2,1]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[5,4],[4]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[5,4,1],[4,1]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[5,4,2],[4,2]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[5,4,2,1],[4,2,1]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[5,4,3],[4,3]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[5,4,3,1],[4,3,1]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[5,4,3,2],[4,3,2]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 5
[[5,4,3,2,1],[4,3,2,1]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[6],[]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[[6,1],[1]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[[6,2],[2]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[[6,2,1],[2,1]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[[3,3],[]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 6
[[6,3],[3]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[[6,3,1],[3,1]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[[6,3,2],[3,2]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[[6,3,2,1],[3,2,1]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[[6,4],[4]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[[3,3,1],[1]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 6
[[6,4,1],[4,1]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[[2,2,2],[]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 6
[[6,4,2],[4,2]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[[3,3,1,1],[1,1]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 6
[[6,4,2,1],[4,2,1]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
Description
The area of the parallelogram polyomino associated with the Dyck path.
The (bivariate) generating function is given in [1].
Matching statistic: St000998
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00181: Skew partitions —row lengths⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3],[]]
=> [3] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[[3,1],[1]]
=> [2,1] => [1,1,0,0,1,0]
=> 4 = 3 + 1
[[3,2],[2]]
=> [1,2] => [1,0,1,1,0,0]
=> 4 = 3 + 1
[[1,1,1],[]]
=> [1,1,1] => [1,0,1,0,1,0]
=> 4 = 3 + 1
[[3,2,1],[2,1]]
=> [1,1,1] => [1,0,1,0,1,0]
=> 4 = 3 + 1
[[4],[]]
=> [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[4,1],[1]]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[[2,2],[]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[[4,2],[2]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[[4,2,1],[2,1]]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[[4,3],[3]]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[[2,2,1],[1]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[[4,3,1],[3,1]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[[4,3,2],[3,2]]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[[1,1,1,1],[]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[[2,2,1,1],[1,1]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[[4,3,2,1],[3,2,1]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[[5],[]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[5,1],[1]]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 6 = 5 + 1
[[5,2],[2]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 6 = 5 + 1
[[5,2,1],[2,1]]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 6 = 5 + 1
[[5,3],[3]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 6 = 5 + 1
[[5,3,1],[3,1]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6 = 5 + 1
[[5,3,2],[3,2]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[[5,3,2,1],[3,2,1]]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[[5,4],[4]]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 6 = 5 + 1
[[5,4,1],[4,1]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 6 = 5 + 1
[[5,4,2],[4,2]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[[5,4,2,1],[4,2,1]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 6 = 5 + 1
[[5,4,3],[4,3]]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 6 = 5 + 1
[[5,4,3,1],[4,3,1]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 6 = 5 + 1
[[5,4,3,2],[4,3,2]]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[[1,1,1,1,1],[]]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[[5,4,3,2,1],[4,3,2,1]]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[[6],[]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7 = 6 + 1
[[6,1],[1]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 7 = 6 + 1
[[6,2],[2]]
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 7 = 6 + 1
[[6,2,1],[2,1]]
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 7 = 6 + 1
[[3,3],[]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 7 = 6 + 1
[[6,3],[3]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 7 = 6 + 1
[[6,3,1],[3,1]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 7 = 6 + 1
[[6,3,2],[3,2]]
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 7 = 6 + 1
[[6,3,2,1],[3,2,1]]
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 7 = 6 + 1
[[6,4],[4]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 7 = 6 + 1
[[3,3,1],[1]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 7 = 6 + 1
[[6,4,1],[4,1]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 7 = 6 + 1
[[2,2,2],[]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 7 = 6 + 1
[[6,4,2],[4,2]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 7 = 6 + 1
[[3,3,1,1],[1,1]]
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 7 = 6 + 1
[[6,4,2,1],[4,2,1]]
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 7 = 6 + 1
Description
Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001643
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00186: Skew partitions —dominating partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001643: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001643: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3],[]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[[3,1],[1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[[3,2],[2]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[[1,1,1],[]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 5 = 3 + 2
[[3,2,1],[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[[4],[]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 4 + 2
[[4,1],[1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 4 + 2
[[2,2],[]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 6 = 4 + 2
[[4,2],[2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 4 + 2
[[4,2,1],[2,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 4 + 2
[[4,3],[3]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 4 + 2
[[2,2,1],[1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 6 = 4 + 2
[[4,3,1],[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 4 + 2
[[4,3,2],[3,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 4 + 2
[[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6 = 4 + 2
[[2,2,1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 6 = 4 + 2
[[4,3,2,1],[3,2,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 4 + 2
[[5],[]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 7 = 5 + 2
[[5,1],[1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 7 = 5 + 2
[[5,2],[2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 7 = 5 + 2
[[5,2,1],[2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 7 = 5 + 2
[[5,3],[3]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 7 = 5 + 2
[[5,3,1],[3,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 7 = 5 + 2
[[5,3,2],[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 7 = 5 + 2
[[5,3,2,1],[3,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 7 = 5 + 2
[[5,4],[4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 7 = 5 + 2
[[5,4,1],[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 7 = 5 + 2
[[5,4,2],[4,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 7 = 5 + 2
[[5,4,2,1],[4,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 7 = 5 + 2
[[5,4,3],[4,3]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 7 = 5 + 2
[[5,4,3,1],[4,3,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 7 = 5 + 2
[[5,4,3,2],[4,3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 7 = 5 + 2
[[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 7 = 5 + 2
[[5,4,3,2,1],[4,3,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 7 = 5 + 2
[[6],[]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 8 = 6 + 2
[[6,1],[1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 8 = 6 + 2
[[6,2],[2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 8 = 6 + 2
[[6,2,1],[2,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 8 = 6 + 2
[[3,3],[]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 8 = 6 + 2
[[6,3],[3]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 8 = 6 + 2
[[6,3,1],[3,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 8 = 6 + 2
[[6,3,2],[3,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 8 = 6 + 2
[[6,3,2,1],[3,2,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 8 = 6 + 2
[[6,4],[4]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 8 = 6 + 2
[[3,3,1],[1]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 8 = 6 + 2
[[6,4,1],[4,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 8 = 6 + 2
[[2,2,2],[]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 8 = 6 + 2
[[6,4,2],[4,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 8 = 6 + 2
[[3,3,1,1],[1,1]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 8 = 6 + 2
[[6,4,2,1],[4,2,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 8 = 6 + 2
Description
The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001838
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00186: Skew partitions —dominating partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001838: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
St001838: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3],[]]
=> [3]
=> 1000 => 5 = 3 + 2
[[3,1],[1]]
=> [3]
=> 1000 => 5 = 3 + 2
[[3,2],[2]]
=> [3]
=> 1000 => 5 = 3 + 2
[[1,1,1],[]]
=> [1,1,1]
=> 1110 => 5 = 3 + 2
[[3,2,1],[2,1]]
=> [3]
=> 1000 => 5 = 3 + 2
[[4],[]]
=> [4]
=> 10000 => 6 = 4 + 2
[[4,1],[1]]
=> [4]
=> 10000 => 6 = 4 + 2
[[2,2],[]]
=> [2,2]
=> 1100 => 6 = 4 + 2
[[4,2],[2]]
=> [4]
=> 10000 => 6 = 4 + 2
[[4,2,1],[2,1]]
=> [4]
=> 10000 => 6 = 4 + 2
[[4,3],[3]]
=> [4]
=> 10000 => 6 = 4 + 2
[[2,2,1],[1]]
=> [2,2]
=> 1100 => 6 = 4 + 2
[[4,3,1],[3,1]]
=> [4]
=> 10000 => 6 = 4 + 2
[[4,3,2],[3,2]]
=> [4]
=> 10000 => 6 = 4 + 2
[[1,1,1,1],[]]
=> [1,1,1,1]
=> 11110 => 6 = 4 + 2
[[2,2,1,1],[1,1]]
=> [2,2]
=> 1100 => 6 = 4 + 2
[[4,3,2,1],[3,2,1]]
=> [4]
=> 10000 => 6 = 4 + 2
[[5],[]]
=> [5]
=> 100000 => 7 = 5 + 2
[[5,1],[1]]
=> [5]
=> 100000 => 7 = 5 + 2
[[5,2],[2]]
=> [5]
=> 100000 => 7 = 5 + 2
[[5,2,1],[2,1]]
=> [5]
=> 100000 => 7 = 5 + 2
[[5,3],[3]]
=> [5]
=> 100000 => 7 = 5 + 2
[[5,3,1],[3,1]]
=> [5]
=> 100000 => 7 = 5 + 2
[[5,3,2],[3,2]]
=> [5]
=> 100000 => 7 = 5 + 2
[[5,3,2,1],[3,2,1]]
=> [5]
=> 100000 => 7 = 5 + 2
[[5,4],[4]]
=> [5]
=> 100000 => 7 = 5 + 2
[[5,4,1],[4,1]]
=> [5]
=> 100000 => 7 = 5 + 2
[[5,4,2],[4,2]]
=> [5]
=> 100000 => 7 = 5 + 2
[[5,4,2,1],[4,2,1]]
=> [5]
=> 100000 => 7 = 5 + 2
[[5,4,3],[4,3]]
=> [5]
=> 100000 => 7 = 5 + 2
[[5,4,3,1],[4,3,1]]
=> [5]
=> 100000 => 7 = 5 + 2
[[5,4,3,2],[4,3,2]]
=> [5]
=> 100000 => 7 = 5 + 2
[[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> 111110 => 7 = 5 + 2
[[5,4,3,2,1],[4,3,2,1]]
=> [5]
=> 100000 => 7 = 5 + 2
[[6],[]]
=> [6]
=> 1000000 => 8 = 6 + 2
[[6,1],[1]]
=> [6]
=> 1000000 => 8 = 6 + 2
[[6,2],[2]]
=> [6]
=> 1000000 => 8 = 6 + 2
[[6,2,1],[2,1]]
=> [6]
=> 1000000 => 8 = 6 + 2
[[3,3],[]]
=> [3,3]
=> 11000 => 8 = 6 + 2
[[6,3],[3]]
=> [6]
=> 1000000 => 8 = 6 + 2
[[6,3,1],[3,1]]
=> [6]
=> 1000000 => 8 = 6 + 2
[[6,3,2],[3,2]]
=> [6]
=> 1000000 => 8 = 6 + 2
[[6,3,2,1],[3,2,1]]
=> [6]
=> 1000000 => 8 = 6 + 2
[[6,4],[4]]
=> [6]
=> 1000000 => 8 = 6 + 2
[[3,3,1],[1]]
=> [3,3]
=> 11000 => 8 = 6 + 2
[[6,4,1],[4,1]]
=> [6]
=> 1000000 => 8 = 6 + 2
[[2,2,2],[]]
=> [2,2,2]
=> 11100 => 8 = 6 + 2
[[6,4,2],[4,2]]
=> [6]
=> 1000000 => 8 = 6 + 2
[[3,3,1,1],[1,1]]
=> [3,3]
=> 11000 => 8 = 6 + 2
[[6,4,2,1],[4,2,1]]
=> [6]
=> 1000000 => 8 = 6 + 2
Description
The number of nonempty primitive factors of a binary word.
A word $u$ is a factor of a word $w$ if $w = p u s$ for words $p$ and $s$. A word is primitive, if it is not of the form $u^k$ for a word $u$ and an integer $k\geq 2$.
Apparently, the maximal number of nonempty primitive factors a binary word of length $n$ can have is given by [[oeis:A131673]].
Matching statistic: St000018
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00186: Skew partitions —dominating partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3],[]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[[3,1],[1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[[3,2],[2]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[[1,1,1],[]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[[3,2,1],[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[[4],[]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[[4,1],[1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[[2,2],[]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
[[4,2],[2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[[4,2,1],[2,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[[4,3],[3]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[[2,2,1],[1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
[[4,3,1],[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[[4,3,2],[3,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[[2,2,1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
[[4,3,2,1],[3,2,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[[5],[]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5
[[5,1],[1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5
[[5,2],[2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5
[[5,2,1],[2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5
[[5,3],[3]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5
[[5,3,1],[3,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5
[[5,3,2],[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5
[[5,3,2,1],[3,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5
[[5,4],[4]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5
[[5,4,1],[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5
[[5,4,2],[4,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5
[[5,4,2,1],[4,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5
[[5,4,3],[4,3]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5
[[5,4,3,1],[4,3,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5
[[5,4,3,2],[4,3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5
[[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5
[[5,4,3,2,1],[4,3,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5
[[6],[]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 6
[[6,1],[1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 6
[[6,2],[2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 6
[[6,2,1],[2,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 6
[[3,3],[]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 6
[[6,3],[3]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 6
[[6,3,1],[3,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 6
[[6,3,2],[3,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 6
[[6,3,2,1],[3,2,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 6
[[6,4],[4]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 6
[[3,3,1],[1]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 6
[[6,4,1],[4,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 6
[[2,2,2],[]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 6
[[6,4,2],[4,2]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 6
[[3,3,1,1],[1,1]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 6
[[6,4,2,1],[4,2,1]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 6
Description
The number of inversions of a permutation.
This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.
The following 104 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000246The number of non-inversions of a permutation. St000290The major index of a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000564The number of occurrences of the pattern {{1},{2}} in a set partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000883The number of longest increasing subsequences of a permutation. St001437The flex of a binary word. St001523The degree of symmetry of a Dyck path. St001746The coalition number of a graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000296The length of the symmetric border of a binary word. St000806The semiperimeter of the associated bargraph. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St000110The number of permutations less than or equal to a permutation in left weak order. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001645The pebbling number of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000271The chromatic index of a graph. St000288The number of ones in a binary word. St000336The leg major index of a standard tableau. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001342The number of vertices in the center of a graph. St000189The number of elements in the poset. St000144The pyramid weight of the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000064The number of one-box pattern of a permutation. St000171The degree of the graph. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000656The number of cuts of a poset. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001717The largest size of an interval in a poset. St000026The position of the first return of a Dyck path. St000058The order of a permutation. St000060The greater neighbor of the maximum. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001911A descent variant minus the number of inversions. St001883The mutual visibility number of a graph. St000890The number of nonzero entries in an alternating sign matrix. St001925The minimal number of zeros in a row of an alternating sign matrix. St001725The harmonious chromatic number of a graph. St000019The cardinality of the support of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000924The number of topologically connected components of a perfect matching. St001759The Rajchgot index of a permutation. St001959The product of the heights of the peaks of a Dyck path. St000029The depth of a permutation. St000809The reduced reflection length of the permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001622The number of join-irreducible elements of a lattice. St000197The number of entries equal to positive one in the alternating sign matrix. St000209Maximum difference of elements in cycles. St000210Minimum over maximum difference of elements in cycles. St000216The absolute length of a permutation. St001077The prefix exchange distance of a permutation. St001480The number of simple summands of the module J^2/J^3. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001958The degree of the polynomial interpolating the values of a permutation. St000327The number of cover relations in a poset. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000673The number of non-fixed points of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001468The smallest fixpoint of a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000719The number of alignments in a perfect matching. St001684The reduced word complexity of a permutation. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000744The length of the path to the largest entry in a standard Young tableau. St000044The number of vertices of the unicellular map given by a perfect matching.
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