Your data matches 63 different statistics following compositions of up to 3 maps.
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Mp00169: Signed permutations odd cycle typeInteger partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[-1,-2,-3] => [1,1,1]
=> 3
[2,3,-1] => [3]
=> 3
[2,-3,1] => [3]
=> 3
[-2,3,1] => [3]
=> 3
[-2,-3,-1] => [3]
=> 3
[3,1,-2] => [3]
=> 3
[3,-1,2] => [3]
=> 3
[-3,1,2] => [3]
=> 3
[-3,-1,-2] => [3]
=> 3
[1,-2,-3,-4] => [1,1,1]
=> 3
[-1,2,-3,-4] => [1,1,1]
=> 3
[-1,-2,3,-4] => [1,1,1]
=> 3
[-1,-2,-3,4] => [1,1,1]
=> 3
[-1,-2,-3,-4] => [1,1,1,1]
=> 4
[1,3,4,-2] => [3]
=> 3
[1,3,-4,2] => [3]
=> 3
[1,-3,4,2] => [3]
=> 3
[1,-3,-4,-2] => [3]
=> 3
[1,4,2,-3] => [3]
=> 3
[1,4,-2,3] => [3]
=> 3
[1,-4,2,3] => [3]
=> 3
[1,-4,-2,-3] => [3]
=> 3
[2,-1,4,-3] => [2,2]
=> 4
[2,-1,-4,3] => [2,2]
=> 4
[-2,1,4,-3] => [2,2]
=> 4
[-2,1,-4,3] => [2,2]
=> 4
[2,3,-1,4] => [3]
=> 3
[2,-3,1,4] => [3]
=> 3
[-2,3,1,4] => [3]
=> 3
[-2,-3,-1,4] => [3]
=> 3
[2,3,4,-1] => [4]
=> 4
[2,3,-4,1] => [4]
=> 4
[2,-3,4,1] => [4]
=> 4
[2,-3,-4,-1] => [4]
=> 4
[-2,3,4,1] => [4]
=> 4
[-2,3,-4,-1] => [4]
=> 4
[-2,-3,4,-1] => [4]
=> 4
[-2,-3,-4,1] => [4]
=> 4
[2,4,1,-3] => [4]
=> 4
[2,4,-1,3] => [4]
=> 4
[2,-4,1,3] => [4]
=> 4
[2,-4,-1,-3] => [4]
=> 4
[-2,4,1,3] => [4]
=> 4
[-2,4,-1,-3] => [4]
=> 4
[-2,-4,1,-3] => [4]
=> 4
[-2,-4,-1,3] => [4]
=> 4
[2,4,3,-1] => [3]
=> 3
[2,-4,3,1] => [3]
=> 3
[-2,4,3,1] => [3]
=> 3
[-2,-4,3,-1] => [3]
=> 3
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00095: Integer partitions to binary wordBinary words
St000293: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[-1,-2,-3] => [1,1,1]
=> 1110 => 3
[2,3,-1] => [3]
=> 1000 => 3
[2,-3,1] => [3]
=> 1000 => 3
[-2,3,1] => [3]
=> 1000 => 3
[-2,-3,-1] => [3]
=> 1000 => 3
[3,1,-2] => [3]
=> 1000 => 3
[3,-1,2] => [3]
=> 1000 => 3
[-3,1,2] => [3]
=> 1000 => 3
[-3,-1,-2] => [3]
=> 1000 => 3
[1,-2,-3,-4] => [1,1,1]
=> 1110 => 3
[-1,2,-3,-4] => [1,1,1]
=> 1110 => 3
[-1,-2,3,-4] => [1,1,1]
=> 1110 => 3
[-1,-2,-3,4] => [1,1,1]
=> 1110 => 3
[-1,-2,-3,-4] => [1,1,1,1]
=> 11110 => 4
[1,3,4,-2] => [3]
=> 1000 => 3
[1,3,-4,2] => [3]
=> 1000 => 3
[1,-3,4,2] => [3]
=> 1000 => 3
[1,-3,-4,-2] => [3]
=> 1000 => 3
[1,4,2,-3] => [3]
=> 1000 => 3
[1,4,-2,3] => [3]
=> 1000 => 3
[1,-4,2,3] => [3]
=> 1000 => 3
[1,-4,-2,-3] => [3]
=> 1000 => 3
[2,-1,4,-3] => [2,2]
=> 1100 => 4
[2,-1,-4,3] => [2,2]
=> 1100 => 4
[-2,1,4,-3] => [2,2]
=> 1100 => 4
[-2,1,-4,3] => [2,2]
=> 1100 => 4
[2,3,-1,4] => [3]
=> 1000 => 3
[2,-3,1,4] => [3]
=> 1000 => 3
[-2,3,1,4] => [3]
=> 1000 => 3
[-2,-3,-1,4] => [3]
=> 1000 => 3
[2,3,4,-1] => [4]
=> 10000 => 4
[2,3,-4,1] => [4]
=> 10000 => 4
[2,-3,4,1] => [4]
=> 10000 => 4
[2,-3,-4,-1] => [4]
=> 10000 => 4
[-2,3,4,1] => [4]
=> 10000 => 4
[-2,3,-4,-1] => [4]
=> 10000 => 4
[-2,-3,4,-1] => [4]
=> 10000 => 4
[-2,-3,-4,1] => [4]
=> 10000 => 4
[2,4,1,-3] => [4]
=> 10000 => 4
[2,4,-1,3] => [4]
=> 10000 => 4
[2,-4,1,3] => [4]
=> 10000 => 4
[2,-4,-1,-3] => [4]
=> 10000 => 4
[-2,4,1,3] => [4]
=> 10000 => 4
[-2,4,-1,-3] => [4]
=> 10000 => 4
[-2,-4,1,-3] => [4]
=> 10000 => 4
[-2,-4,-1,3] => [4]
=> 10000 => 4
[2,4,3,-1] => [3]
=> 1000 => 3
[2,-4,3,1] => [3]
=> 1000 => 3
[-2,4,3,1] => [3]
=> 1000 => 3
[-2,-4,3,-1] => [3]
=> 1000 => 3
Description
The number of inversions of a binary word.
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
St000645: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[-1,2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,-3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,-3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,-4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,-4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[2,-1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 4
[2,-1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 4
[-2,1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 4
[-2,1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 4
[2,3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[2,-3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[-2,3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[-2,-3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[2,3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[2,3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[2,-3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[2,-3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[-2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[-2,3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[-2,-3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[-2,-3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[2,4,1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[2,4,-1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[2,-4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[2,-4,-1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[-2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[-2,4,-1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[-2,-4,1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[-2,-4,-1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[2,4,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[2,-4,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[-2,4,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[-2,-4,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 3
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 $$ \sum_{a=1}^{k-1} (j_a-i_a)(i_{a+1}-j_a) $$
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> 3
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> 3
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> 3
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> 3
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> 3
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> 3
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> 3
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> 3
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[-1,-2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[-1,-2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 4
[1,3,4,-2] => [3]
=> [1,0,1,0,1,0]
=> 3
[1,3,-4,2] => [3]
=> [1,0,1,0,1,0]
=> 3
[1,-3,4,2] => [3]
=> [1,0,1,0,1,0]
=> 3
[1,-3,-4,-2] => [3]
=> [1,0,1,0,1,0]
=> 3
[1,4,2,-3] => [3]
=> [1,0,1,0,1,0]
=> 3
[1,4,-2,3] => [3]
=> [1,0,1,0,1,0]
=> 3
[1,-4,2,3] => [3]
=> [1,0,1,0,1,0]
=> 3
[1,-4,-2,-3] => [3]
=> [1,0,1,0,1,0]
=> 3
[2,-1,4,-3] => [2,2]
=> [1,1,1,0,0,0]
=> 4
[2,-1,-4,3] => [2,2]
=> [1,1,1,0,0,0]
=> 4
[-2,1,4,-3] => [2,2]
=> [1,1,1,0,0,0]
=> 4
[-2,1,-4,3] => [2,2]
=> [1,1,1,0,0,0]
=> 4
[2,3,-1,4] => [3]
=> [1,0,1,0,1,0]
=> 3
[2,-3,1,4] => [3]
=> [1,0,1,0,1,0]
=> 3
[-2,3,1,4] => [3]
=> [1,0,1,0,1,0]
=> 3
[-2,-3,-1,4] => [3]
=> [1,0,1,0,1,0]
=> 3
[2,3,4,-1] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,3,-4,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,-3,4,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,-3,-4,-1] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[-2,3,4,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[-2,3,-4,-1] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[-2,-3,4,-1] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[-2,-3,-4,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,4,1,-3] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,4,-1,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,-4,1,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,-4,-1,-3] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[-2,4,1,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[-2,4,-1,-3] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[-2,-4,1,-3] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[-2,-4,-1,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,4,3,-1] => [3]
=> [1,0,1,0,1,0]
=> 3
[2,-4,3,1] => [3]
=> [1,0,1,0,1,0]
=> 3
[-2,4,3,1] => [3]
=> [1,0,1,0,1,0]
=> 3
[-2,-4,3,-1] => [3]
=> [1,0,1,0,1,0]
=> 3
Description
The area of the parallelogram polyomino associated with the Dyck path. The (bivariate) generating function is given in [1].
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
St001643: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 5 = 3 + 2
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 5 = 3 + 2
[-1,2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 5 = 3 + 2
[-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 5 = 3 + 2
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 5 = 3 + 2
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6 = 4 + 2
[1,3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[1,3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[1,-3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[1,-3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[1,4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[1,4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[1,-4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[1,-4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[2,-1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 6 = 4 + 2
[2,-1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 6 = 4 + 2
[-2,1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 6 = 4 + 2
[-2,1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 6 = 4 + 2
[2,3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[2,-3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[-2,3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[-2,-3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[2,3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 4 + 2
[2,3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 4 + 2
[2,-3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 4 + 2
[2,-3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 4 + 2
[-2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 4 + 2
[-2,3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 4 + 2
[-2,-3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 4 + 2
[-2,-3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 4 + 2
[2,4,1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 4 + 2
[2,4,-1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 4 + 2
[2,-4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 4 + 2
[2,-4,-1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 4 + 2
[-2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 4 + 2
[-2,4,-1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 4 + 2
[-2,-4,1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 4 + 2
[-2,-4,-1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 4 + 2
[2,4,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[2,-4,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[-2,4,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[-2,-4,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
Description
The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path.
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00095: Integer partitions to binary wordBinary words
St001838: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[-1,-2,-3] => [1,1,1]
=> 1110 => 5 = 3 + 2
[2,3,-1] => [3]
=> 1000 => 5 = 3 + 2
[2,-3,1] => [3]
=> 1000 => 5 = 3 + 2
[-2,3,1] => [3]
=> 1000 => 5 = 3 + 2
[-2,-3,-1] => [3]
=> 1000 => 5 = 3 + 2
[3,1,-2] => [3]
=> 1000 => 5 = 3 + 2
[3,-1,2] => [3]
=> 1000 => 5 = 3 + 2
[-3,1,2] => [3]
=> 1000 => 5 = 3 + 2
[-3,-1,-2] => [3]
=> 1000 => 5 = 3 + 2
[1,-2,-3,-4] => [1,1,1]
=> 1110 => 5 = 3 + 2
[-1,2,-3,-4] => [1,1,1]
=> 1110 => 5 = 3 + 2
[-1,-2,3,-4] => [1,1,1]
=> 1110 => 5 = 3 + 2
[-1,-2,-3,4] => [1,1,1]
=> 1110 => 5 = 3 + 2
[-1,-2,-3,-4] => [1,1,1,1]
=> 11110 => 6 = 4 + 2
[1,3,4,-2] => [3]
=> 1000 => 5 = 3 + 2
[1,3,-4,2] => [3]
=> 1000 => 5 = 3 + 2
[1,-3,4,2] => [3]
=> 1000 => 5 = 3 + 2
[1,-3,-4,-2] => [3]
=> 1000 => 5 = 3 + 2
[1,4,2,-3] => [3]
=> 1000 => 5 = 3 + 2
[1,4,-2,3] => [3]
=> 1000 => 5 = 3 + 2
[1,-4,2,3] => [3]
=> 1000 => 5 = 3 + 2
[1,-4,-2,-3] => [3]
=> 1000 => 5 = 3 + 2
[2,-1,4,-3] => [2,2]
=> 1100 => 6 = 4 + 2
[2,-1,-4,3] => [2,2]
=> 1100 => 6 = 4 + 2
[-2,1,4,-3] => [2,2]
=> 1100 => 6 = 4 + 2
[-2,1,-4,3] => [2,2]
=> 1100 => 6 = 4 + 2
[2,3,-1,4] => [3]
=> 1000 => 5 = 3 + 2
[2,-3,1,4] => [3]
=> 1000 => 5 = 3 + 2
[-2,3,1,4] => [3]
=> 1000 => 5 = 3 + 2
[-2,-3,-1,4] => [3]
=> 1000 => 5 = 3 + 2
[2,3,4,-1] => [4]
=> 10000 => 6 = 4 + 2
[2,3,-4,1] => [4]
=> 10000 => 6 = 4 + 2
[2,-3,4,1] => [4]
=> 10000 => 6 = 4 + 2
[2,-3,-4,-1] => [4]
=> 10000 => 6 = 4 + 2
[-2,3,4,1] => [4]
=> 10000 => 6 = 4 + 2
[-2,3,-4,-1] => [4]
=> 10000 => 6 = 4 + 2
[-2,-3,4,-1] => [4]
=> 10000 => 6 = 4 + 2
[-2,-3,-4,1] => [4]
=> 10000 => 6 = 4 + 2
[2,4,1,-3] => [4]
=> 10000 => 6 = 4 + 2
[2,4,-1,3] => [4]
=> 10000 => 6 = 4 + 2
[2,-4,1,3] => [4]
=> 10000 => 6 = 4 + 2
[2,-4,-1,-3] => [4]
=> 10000 => 6 = 4 + 2
[-2,4,1,3] => [4]
=> 10000 => 6 = 4 + 2
[-2,4,-1,-3] => [4]
=> 10000 => 6 = 4 + 2
[-2,-4,1,-3] => [4]
=> 10000 => 6 = 4 + 2
[-2,-4,-1,3] => [4]
=> 10000 => 6 = 4 + 2
[2,4,3,-1] => [3]
=> 1000 => 5 = 3 + 2
[2,-4,3,1] => [3]
=> 1000 => 5 = 3 + 2
[-2,4,3,1] => [3]
=> 1000 => 5 = 3 + 2
[-2,-4,3,-1] => [3]
=> 1000 => 5 = 3 + 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
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
St000018: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[-1,2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[1,3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[1,3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[1,-3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[1,-3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[1,4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[1,4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[1,-4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[1,-4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[2,-1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
[2,-1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
[-2,1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
[-2,1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
[2,3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[2,-3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[-2,3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[-2,-3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[2,3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[2,3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[2,-3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[2,-3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[-2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[-2,3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[-2,-3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[-2,-3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[2,4,1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[2,4,-1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[2,-4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[2,-4,-1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[-2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[-2,4,-1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[-2,-4,1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[-2,-4,-1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[2,4,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[2,-4,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[-2,4,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[-2,-4,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
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 $\pi$.
Matching statistic: St000110
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00119: Dyck paths to 321-avoiding permutation (Krattenthaler)Permutations
St000110: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 3
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 3
[-1,2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 3
[-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 3
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 3
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 4
[1,3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[1,3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[1,-3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[1,-3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[1,4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[1,4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[1,-4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[1,-4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[2,-1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[2,-1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[-2,1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[-2,1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[2,3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[2,-3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[-2,3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[-2,-3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[2,3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 4
[2,3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 4
[2,-3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 4
[2,-3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 4
[-2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 4
[-2,3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 4
[-2,-3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 4
[-2,-3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 4
[2,4,1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 4
[2,4,-1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 4
[2,-4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 4
[2,-4,-1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 4
[-2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 4
[-2,4,-1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 4
[-2,-4,1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 4
[-2,-4,-1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 4
[2,4,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[2,-4,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[-2,4,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[-2,-4,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
Description
The number of permutations less than or equal to a permutation in left weak order. This is the same as the number of permutations less than or equal to the given permutation in right weak order.
Matching statistic: St000246
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00023: Dyck paths to non-crossing permutationPermutations
St000246: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[-1,2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 4
[1,3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[1,3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[1,-3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[1,-3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[1,4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[1,4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[1,-4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[1,-4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[2,-1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[2,-1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[-2,1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[-2,1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[2,3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[2,-3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[-2,3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[-2,-3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[2,3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
[2,3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
[2,-3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
[2,-3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
[-2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
[-2,3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
[-2,-3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
[-2,-3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
[2,4,1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
[2,4,-1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
[2,-4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
[2,-4,-1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
[-2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
[-2,4,-1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
[-2,-4,1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
[-2,-4,-1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
[2,4,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[2,-4,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[-2,4,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[-2,-4,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
Description
The number of non-inversions of a permutation. For a permutation of $\{1,\ldots,n\}$, this is given by $\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$.
Matching statistic: St000290
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00095: Integer partitions to binary wordBinary words
Mp00316: Binary words inverse Foata bijectionBinary words
St000290: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[-1,-2,-3] => [1,1,1]
=> 1110 => 1110 => 3
[2,3,-1] => [3]
=> 1000 => 0010 => 3
[2,-3,1] => [3]
=> 1000 => 0010 => 3
[-2,3,1] => [3]
=> 1000 => 0010 => 3
[-2,-3,-1] => [3]
=> 1000 => 0010 => 3
[3,1,-2] => [3]
=> 1000 => 0010 => 3
[3,-1,2] => [3]
=> 1000 => 0010 => 3
[-3,1,2] => [3]
=> 1000 => 0010 => 3
[-3,-1,-2] => [3]
=> 1000 => 0010 => 3
[1,-2,-3,-4] => [1,1,1]
=> 1110 => 1110 => 3
[-1,2,-3,-4] => [1,1,1]
=> 1110 => 1110 => 3
[-1,-2,3,-4] => [1,1,1]
=> 1110 => 1110 => 3
[-1,-2,-3,4] => [1,1,1]
=> 1110 => 1110 => 3
[-1,-2,-3,-4] => [1,1,1,1]
=> 11110 => 11110 => 4
[1,3,4,-2] => [3]
=> 1000 => 0010 => 3
[1,3,-4,2] => [3]
=> 1000 => 0010 => 3
[1,-3,4,2] => [3]
=> 1000 => 0010 => 3
[1,-3,-4,-2] => [3]
=> 1000 => 0010 => 3
[1,4,2,-3] => [3]
=> 1000 => 0010 => 3
[1,4,-2,3] => [3]
=> 1000 => 0010 => 3
[1,-4,2,3] => [3]
=> 1000 => 0010 => 3
[1,-4,-2,-3] => [3]
=> 1000 => 0010 => 3
[2,-1,4,-3] => [2,2]
=> 1100 => 1010 => 4
[2,-1,-4,3] => [2,2]
=> 1100 => 1010 => 4
[-2,1,4,-3] => [2,2]
=> 1100 => 1010 => 4
[-2,1,-4,3] => [2,2]
=> 1100 => 1010 => 4
[2,3,-1,4] => [3]
=> 1000 => 0010 => 3
[2,-3,1,4] => [3]
=> 1000 => 0010 => 3
[-2,3,1,4] => [3]
=> 1000 => 0010 => 3
[-2,-3,-1,4] => [3]
=> 1000 => 0010 => 3
[2,3,4,-1] => [4]
=> 10000 => 00010 => 4
[2,3,-4,1] => [4]
=> 10000 => 00010 => 4
[2,-3,4,1] => [4]
=> 10000 => 00010 => 4
[2,-3,-4,-1] => [4]
=> 10000 => 00010 => 4
[-2,3,4,1] => [4]
=> 10000 => 00010 => 4
[-2,3,-4,-1] => [4]
=> 10000 => 00010 => 4
[-2,-3,4,-1] => [4]
=> 10000 => 00010 => 4
[-2,-3,-4,1] => [4]
=> 10000 => 00010 => 4
[2,4,1,-3] => [4]
=> 10000 => 00010 => 4
[2,4,-1,3] => [4]
=> 10000 => 00010 => 4
[2,-4,1,3] => [4]
=> 10000 => 00010 => 4
[2,-4,-1,-3] => [4]
=> 10000 => 00010 => 4
[-2,4,1,3] => [4]
=> 10000 => 00010 => 4
[-2,4,-1,-3] => [4]
=> 10000 => 00010 => 4
[-2,-4,1,-3] => [4]
=> 10000 => 00010 => 4
[-2,-4,-1,3] => [4]
=> 10000 => 00010 => 4
[2,4,3,-1] => [3]
=> 1000 => 0010 => 3
[2,-4,3,1] => [3]
=> 1000 => 0010 => 3
[-2,4,3,1] => [3]
=> 1000 => 0010 => 3
[-2,-4,3,-1] => [3]
=> 1000 => 0010 => 3
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 $\binom{n}{a}_q$.
The following 53 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000395The sum of the heights of the peaks of a Dyck path. St000564The number of occurrences of the pattern {{1},{2}} in a set partition. St000883The number of longest increasing subsequences of a permutation. 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. 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. 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. St000064The number of one-box pattern of a permutation. St000189The number of elements in the poset. 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. St001717The largest size of an interval in a poset. St000060The greater neighbor of the maximum. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001911A descent variant minus the number of inversions. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. 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. St001759The Rajchgot index 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. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St001249Sum of the odd parts of a partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001488The number of corners of a skew partition. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000075The orbit size of a standard tableau under promotion. St001424The number of distinct squares in a binary word. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001884The number of borders of a binary word. St000391The sum of the positions of the ones in a binary word. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000043The number of crossings plus two-nestings of a perfect matching. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset.