Your data matches 5 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000293
(load all 3 compositions to match this statistic)
Mapping
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
 => [1]
 => 10 => 01 => 0
[[2],[]]
 => [2]
 => 100 => 011 => 0
[[1,1],[]]
 => [1,1]
 => 110 => 001 => 0
[[2,1],[1]]
 => [2,1]
 => 1010 => 0101 => 1
[[3],[]]
 => [3]
 => 1000 => 0111 => 0
[[2,1],[]]
 => [2,1]
 => 1010 => 0101 => 1
[[3,1],[1]]
 => [3,1]
 => 10010 => 01101 => 2
[[2,2],[1]]
 => [2,2]
 => 1100 => 0011 => 0
[[3,2],[2]]
 => [3,2]
 => 10100 => 01011 => 1
[[1,1,1],[]]
 => [1,1,1]
 => 1110 => 0001 => 0
[[2,2,1],[1,1]]
 => [2,2,1]
 => 11010 => 00101 => 1
[[2,1,1],[1]]
 => [2,1,1]
 => 10110 => 01001 => 2
[[3,2,1],[2,1]]
 => [3,2,1]
 => 101010 => 010101 => 3
[[4],[]]
 => [4]
 => 10000 => 01111 => 0
[[3,1],[]]
 => [3,1]
 => 10010 => 01101 => 2
[[4,1],[1]]
 => [4,1]
 => 100010 => 011101 => 3
[[2,2],[]]
 => [2,2]
 => 1100 => 0011 => 0
[[3,2],[1]]
 => [3,2]
 => 10100 => 01011 => 1
[[4,2],[2]]
 => [4,2]
 => 100100 => 011011 => 2
[[2,1,1],[]]
 => [2,1,1]
 => 10110 => 01001 => 2
[[3,2,1],[1,1]]
 => [3,2,1]
 => 101010 => 010101 => 3
[[3,1,1],[1]]
 => [3,1,1]
 => 100110 => 011001 => 4
[[4,2,1],[2,1]]
 => [4,2,1]
 => 1001010 => 0110101 => 5
[[3,3],[2]]
 => [3,3]
 => 11000 => 00111 => 0
[[4,3],[3]]
 => [4,3]
 => 101000 => 010111 => 1
[[2,2,1],[1]]
 => [2,2,1]
 => 11010 => 00101 => 1
[[3,3,1],[2,1]]
 => [3,3,1]
 => 110010 => 001101 => 2
[[3,2,1],[2]]
 => [3,2,1]
 => 101010 => 010101 => 3
[[4,3,1],[3,1]]
 => [4,3,1]
 => 1010010 => 0101101 => 4
[[2,2,2],[1,1]]
 => [2,2,2]
 => 11100 => 00011 => 0
[[3,3,2],[2,2]]
 => [3,3,2]
 => 110100 => 001011 => 1
[[3,2,2],[2,1]]
 => [3,2,2]
 => 101100 => 010011 => 2
[[4,3,2],[3,2]]
 => [4,3,2]
 => 1010100 => 0101011 => 3
[[1,1,1,1],[]]
 => [1,1,1,1]
 => 11110 => 00001 => 0
[[2,2,2,1],[1,1,1]]
 => [2,2,2,1]
 => 111010 => 000101 => 1
[[2,2,1,1],[1,1]]
 => [2,2,1,1]
 => 110110 => 001001 => 2
[[3,3,2,1],[2,2,1]]
 => [3,3,2,1]
 => 1101010 => 0010101 => 3
[[2,1,1,1],[1]]
 => [2,1,1,1]
 => 101110 => 010001 => 3
[[3,2,2,1],[2,1,1]]
 => [3,2,2,1]
 => 1011010 => 0100101 => 4
[[3,2,1,1],[2,1]]
 => [3,2,1,1]
 => 1010110 => 0101001 => 5
[[4,3,2,1],[3,2,1]]
 => [4,3,2,1]
 => 10101010 => 01010101 => 6
[[5],[]]
 => [5]
 => 100000 => 011111 => 0
[[4,1],[]]
 => [4,1]
 => 100010 => 011101 => 3
[[5,1],[1]]
 => [5,1]
 => 1000010 => 0111101 => 4
[[3,2],[]]
 => [3,2]
 => 10100 => 01011 => 1
[[4,2],[1]]
 => [4,2]
 => 100100 => 011011 => 2
[[5,2],[2]]
 => [5,2]
 => 1000100 => 0111011 => 3
[[3,1,1],[]]
 => [3,1,1]
 => 100110 => 011001 => 4
[[4,2,1],[1,1]]
 => [4,2,1]
 => 1001010 => 0110101 => 5
[[4,1,1],[1]]
 => [4,1,1]
 => 1000110 => 0111001 => 6
Description
The number of inversions of a binary word.
Matching statistic: St001034
Mapping
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 90%
Values
[[1],[]]
 => [[1],[]]
 => []
 => []
 => 0
[[2],[]]
 => [[2],[]]
 => []
 => []
 => 0
[[1,1],[]]
 => [[1,1],[]]
 => []
 => []
 => 0
[[2,1],[1]]
 => [[2,1],[1]]
 => [1]
 => [1,0]
 => 1
[[3],[]]
 => [[3],[]]
 => []
 => []
 => 0
[[2,1],[]]
 => [[2,2],[1]]
 => [1]
 => [1,0]
 => 1
[[3,1],[1]]
 => [[3,2],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[2,2],[1]]
 => [[2,1],[]]
 => []
 => []
 => 0
[[3,2],[2]]
 => [[3,1],[1]]
 => [1]
 => [1,0]
 => 1
[[1,1,1],[]]
 => [[1,1,1],[]]
 => []
 => []
 => 0
[[2,2,1],[1,1]]
 => [[2,1,1],[1]]
 => [1]
 => [1,0]
 => 1
[[2,1,1],[1]]
 => [[2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[[3,2,1],[2,1]]
 => [[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[[4],[]]
 => [[4],[]]
 => []
 => []
 => 0
[[3,1],[]]
 => [[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[4,1],[1]]
 => [[4,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[2,2],[]]
 => [[2,2],[]]
 => []
 => []
 => 0
[[3,2],[1]]
 => [[3,2],[1]]
 => [1]
 => [1,0]
 => 1
[[4,2],[2]]
 => [[4,2],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[2,1,1],[]]
 => [[2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[[3,2,1],[1,1]]
 => [[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[[3,1,1],[1]]
 => [[3,3,2],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 4
[[4,2,1],[2,1]]
 => [[4,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 5
[[3,3],[2]]
 => [[3,1],[]]
 => []
 => []
 => 0
[[4,3],[3]]
 => [[4,1],[1]]
 => [1]
 => [1,0]
 => 1
[[2,2,1],[1]]
 => [[2,2,1],[1]]
 => [1]
 => [1,0]
 => 1
[[3,3,1],[2,1]]
 => [[3,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[3,2,1],[2]]
 => [[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[[4,3,1],[3,1]]
 => [[4,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 4
[[2,2,2],[1,1]]
 => [[2,1,1],[]]
 => []
 => []
 => 0
[[3,3,2],[2,2]]
 => [[3,1,1],[1]]
 => [1]
 => [1,0]
 => 1
[[3,2,2],[2,1]]
 => [[3,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[[4,3,2],[3,2]]
 => [[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[[1,1,1,1],[]]
 => [[1,1,1,1],[]]
 => []
 => []
 => 0
[[2,2,2,1],[1,1,1]]
 => [[2,1,1,1],[1]]
 => [1]
 => [1,0]
 => 1
[[2,2,1,1],[1,1]]
 => [[2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[[3,3,2,1],[2,2,1]]
 => [[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[[2,1,1,1],[1]]
 => [[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 3
[[3,2,2,1],[2,1,1]]
 => [[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 4
[[3,2,1,1],[2,1]]
 => [[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 5
[[4,3,2,1],[3,2,1]]
 => [[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 6
[[5],[]]
 => [[5],[]]
 => []
 => []
 => 0
[[4,1],[]]
 => [[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[5,1],[1]]
 => [[5,4],[4]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[[3,2],[]]
 => [[3,3],[1]]
 => [1]
 => [1,0]
 => 1
[[4,2],[1]]
 => [[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[5,2],[2]]
 => [[5,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[3,1,1],[]]
 => [[3,3,3],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 4
[[4,2,1],[1,1]]
 => [[4,3,3],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 5
[[4,1,1],[1]]
 => [[4,4,3],[3,3]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 6
[[6,4,4,3,2,1],[4,3,3,2,1]]
 => [[6,5,4,3,3,2],[5,4,3,2,2]]
 => [5,4,3,2,2]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
 => ? = 16
[[6,4,3,3,2,1],[4,3,2,2,1]]
 => [[6,5,4,4,3,2],[5,4,3,3,2]]
 => [5,4,3,3,2]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? = 17
[[6,4,3,2,2,1],[4,3,2,1,1]]
 => [[6,5,5,4,3,2],[5,4,4,3,2]]
 => [5,4,4,3,2]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
 => ? = 18
[[6,4,3,2,1,1],[4,3,2,1]]
 => [[6,6,5,4,3,2],[5,5,4,3,2]]
 => [5,5,4,3,2]
 => [1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
 => ? = 19
[[6,5,3,3,2,1],[5,3,2,2,1]]
 => [[6,5,4,4,3,1],[5,4,3,3,1]]
 => [5,4,3,3,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 16
[[6,5,3,2,2,1],[5,3,2,1,1]]
 => [[6,5,5,4,3,1],[5,4,4,3,1]]
 => [5,4,4,3,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
 => ? = 17
[[6,5,3,2,1,1],[5,3,2,1]]
 => [[6,6,5,4,3,1],[5,5,4,3,1]]
 => [5,5,4,3,1]
 => [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
 => ? = 18
[[6,5,4,2,2,1],[5,4,2,1,1]]
 => [[6,5,5,4,2,1],[5,4,4,2,1]]
 => [5,4,4,2,1]
 => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
 => ? = 16
[[6,5,4,2,1,1],[5,4,2,1]]
 => [[6,6,5,4,2,1],[5,5,4,2,1]]
 => [5,5,4,2,1]
 => [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
 => ? = 17
[[6,5,4,3,1,1],[5,4,3,1]]
 => [[6,6,5,3,2,1],[5,5,3,2,1]]
 => [5,5,3,2,1]
 => [1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 16
[[5,5,4,1],[]]
 => ?
 => ?
 => ?
 => ? = 5
[[5,5,5,1],[]]
 => ?
 => ?
 => ?
 => ? = 4
[[4,4,4,3,1],[]]
 => ?
 => ?
 => ?
 => ? = 4
[[3,3,3,3,3,1],[]]
 => ?
 => ?
 => ?
 => ? = 2
[[4,4,4,4,1],[]]
 => ?
 => ?
 => ?
 => ? = 3
[[2,2,1,1,1,1,1],[1]]
 => ?
 => ?
 => ?
 => ? = 5
[[2,2,2,1,1,1,1],[1,1]]
 => ?
 => ?
 => ?
 => ? = 4
[[3,2,1,1,1,1],[1]]
 => ?
 => ?
 => ?
 => ? = 9
[[2,2,2,2,1,1,1],[1,1,1]]
 => ?
 => ?
 => ?
 => ? = 3
[[3,2,2,1,1,1],[1,1]]
 => ?
 => ?
 => ?
 => ? = 8
[[3,3,2,1,1,1],[2,1]]
 => ?
 => ?
 => ?
 => ? = 7
[[4,3,1,1,1],[2]]
 => ?
 => ?
 => ?
 => ? = 10
[[2,2,2,2,2,1,1],[1,1,1,1]]
 => ?
 => ?
 => ?
 => ? = 2
[[3,2,2,2,1,1],[1,1,1]]
 => ?
 => ?
 => ?
 => ? = 7
[[3,3,2,2,1,1],[2,1,1]]
 => ?
 => ?
 => ?
 => ? = 6
[[3,3,3,2,1,1],[2,2,1]]
 => ?
 => ?
 => ?
 => ? = 5
[[4,3,2,1,1],[2,1]]
 => ?
 => ?
 => ?
 => ? = 9
[[4,4,2,1,1],[3,1]]
 => ?
 => ?
 => ?
 => ? = 8
[[4,3,3,1,1],[2,2]]
 => ?
 => ?
 => ?
 => ? = 8
[[4,4,3,1,1],[3,2]]
 => ?
 => ?
 => ?
 => ? = 7
[[5,3,1,1],[2]]
 => ?
 => ?
 => ?
 => ? = 10
[[2,2,2,2,2,2,1],[1,1,1,1,1]]
 => ?
 => ?
 => ?
 => ? = 1
[[3,2,2,2,2,1],[1,1,1,1]]
 => ?
 => ?
 => ?
 => ? = 6
[[3,3,2,2,2,1],[2,1,1,1]]
 => ?
 => ?
 => ?
 => ? = 5
[[3,3,3,2,2,1],[2,2,1,1]]
 => ?
 => ?
 => ?
 => ? = 4
[[4,3,2,2,1],[2,1,1]]
 => ?
 => ?
 => ?
 => ? = 8
[[4,4,2,2,1],[3,1,1]]
 => ?
 => ?
 => ?
 => ? = 7
[[3,3,3,3,2,1],[2,2,2,1]]
 => ?
 => ?
 => ?
 => ? = 3
[[4,3,3,2,1],[2,2,1]]
 => ?
 => ?
 => ?
 => ? = 7
[[4,4,3,2,1],[3,2,1]]
 => ?
 => ?
 => ?
 => ? = 6
[[5,3,2,1],[2,1]]
 => ?
 => ?
 => ?
 => ? = 9
[[4,4,4,2,1],[3,3,1]]
 => ?
 => ?
 => ?
 => ? = 5
[[5,4,2,1],[3,1]]
 => ?
 => ?
 => ?
 => ? = 8
[[4,3,3,3,1],[2,2,2]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,4,3,3,1],[3,2,2]]
 => ?
 => ?
 => ?
 => ? = 5
[[5,3,3,1],[2,2]]
 => ?
 => ?
 => ?
 => ? = 8
[[4,4,4,3,1],[3,3,2]]
 => ?
 => ?
 => ?
 => ? = 4
[[5,4,3,1],[3,2]]
 => ?
 => ?
 => ?
 => ? = 7
[[5,5,3,1],[4,2]]
 => ?
 => ?
 => ?
 => ? = 6
[[4,4,4,4,1],[3,3,3]]
 => ?
 => ?
 => ?
 => ? = 3
Description
The area of the parallelogram polyomino associated with the Dyck path. The (bivariate) generating function is given in [1].
Matching statistic: St000228
(load all 8 compositions to match this statistic)
Mapping
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 71%●distinct values known / distinct values provided: 52%
Values
[[1],[]]
 => [[1],[]]
 => []
 => 0
[[2],[]]
 => [[2],[]]
 => []
 => 0
[[1,1],[]]
 => [[1,1],[]]
 => []
 => 0
[[2,1],[1]]
 => [[2,1],[1]]
 => [1]
 => 1
[[3],[]]
 => [[3],[]]
 => []
 => 0
[[2,1],[]]
 => [[2,2],[1]]
 => [1]
 => 1
[[3,1],[1]]
 => [[3,2],[2]]
 => [2]
 => 2
[[2,2],[1]]
 => [[2,1],[]]
 => []
 => 0
[[3,2],[2]]
 => [[3,1],[1]]
 => [1]
 => 1
[[1,1,1],[]]
 => [[1,1,1],[]]
 => []
 => 0
[[2,2,1],[1,1]]
 => [[2,1,1],[1]]
 => [1]
 => 1
[[2,1,1],[1]]
 => [[2,2,1],[1,1]]
 => [1,1]
 => 2
[[3,2,1],[2,1]]
 => [[3,2,1],[2,1]]
 => [2,1]
 => 3
[[4],[]]
 => [[4],[]]
 => []
 => 0
[[3,1],[]]
 => [[3,3],[2]]
 => [2]
 => 2
[[4,1],[1]]
 => [[4,3],[3]]
 => [3]
 => 3
[[2,2],[]]
 => [[2,2],[]]
 => []
 => 0
[[3,2],[1]]
 => [[3,2],[1]]
 => [1]
 => 1
[[4,2],[2]]
 => [[4,2],[2]]
 => [2]
 => 2
[[2,1,1],[]]
 => [[2,2,2],[1,1]]
 => [1,1]
 => 2
[[3,2,1],[1,1]]
 => [[3,2,2],[2,1]]
 => [2,1]
 => 3
[[3,1,1],[1]]
 => [[3,3,2],[2,2]]
 => [2,2]
 => 4
[[4,2,1],[2,1]]
 => [[4,3,2],[3,2]]
 => [3,2]
 => 5
[[3,3],[2]]
 => [[3,1],[]]
 => []
 => 0
[[4,3],[3]]
 => [[4,1],[1]]
 => [1]
 => 1
[[2,2,1],[1]]
 => [[2,2,1],[1]]
 => [1]
 => 1
[[3,3,1],[2,1]]
 => [[3,2,1],[2]]
 => [2]
 => 2
[[3,2,1],[2]]
 => [[3,3,1],[2,1]]
 => [2,1]
 => 3
[[4,3,1],[3,1]]
 => [[4,3,1],[3,1]]
 => [3,1]
 => 4
[[2,2,2],[1,1]]
 => [[2,1,1],[]]
 => []
 => 0
[[3,3,2],[2,2]]
 => [[3,1,1],[1]]
 => [1]
 => 1
[[3,2,2],[2,1]]
 => [[3,2,1],[1,1]]
 => [1,1]
 => 2
[[4,3,2],[3,2]]
 => [[4,2,1],[2,1]]
 => [2,1]
 => 3
[[1,1,1,1],[]]
 => [[1,1,1,1],[]]
 => []
 => 0
[[2,2,2,1],[1,1,1]]
 => [[2,1,1,1],[1]]
 => [1]
 => 1
[[2,2,1,1],[1,1]]
 => [[2,2,1,1],[1,1]]
 => [1,1]
 => 2
[[3,3,2,1],[2,2,1]]
 => [[3,2,1,1],[2,1]]
 => [2,1]
 => 3
[[2,1,1,1],[1]]
 => [[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 3
[[3,2,2,1],[2,1,1]]
 => [[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => 4
[[3,2,1,1],[2,1]]
 => [[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => 5
[[4,3,2,1],[3,2,1]]
 => [[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => 6
[[5],[]]
 => [[5],[]]
 => []
 => 0
[[4,1],[]]
 => [[4,4],[3]]
 => [3]
 => 3
[[5,1],[1]]
 => [[5,4],[4]]
 => [4]
 => 4
[[3,2],[]]
 => [[3,3],[1]]
 => [1]
 => 1
[[4,2],[1]]
 => [[4,3],[2]]
 => [2]
 => 2
[[5,2],[2]]
 => [[5,3],[3]]
 => [3]
 => 3
[[3,1,1],[]]
 => [[3,3,3],[2,2]]
 => [2,2]
 => 4
[[4,2,1],[1,1]]
 => [[4,3,3],[3,2]]
 => [3,2]
 => 5
[[4,1,1],[1]]
 => [[4,4,3],[3,3]]
 => [3,3]
 => 6
[[5,2,1,1],[2,1]]
 => [[5,5,4,3],[4,4,3]]
 => [4,4,3]
 => ? = 11
[[5,3,3,2,1],[3,2,2,1]]
 => [[5,4,3,3,2],[4,3,2,2]]
 => [4,3,2,2]
 => ? = 11
[[4,2,1,1,1],[2,1]]
 => [[4,4,4,3,2],[3,3,3,2]]
 => [3,3,3,2]
 => ? = 11
[[5,3,2,2,1],[3,2,1,1]]
 => [[5,4,4,3,2],[4,3,3,2]]
 => [4,3,3,2]
 => ? = 12
[[5,3,2,1,1],[3,2,1]]
 => [[5,5,4,3,2],[4,4,3,2]]
 => [4,4,3,2]
 => ? = 13
[[5,4,2,2,1],[4,2,1,1]]
 => [[5,4,4,3,1],[4,3,3,1]]
 => [4,3,3,1]
 => ? = 11
[[5,4,2,1,1],[4,2,1]]
 => [[5,5,4,3,1],[4,4,3,1]]
 => [4,4,3,1]
 => ? = 12
[[5,4,3,1,1],[4,3,1]]
 => [[5,5,4,2,1],[4,4,2,1]]
 => [4,4,2,1]
 => ? = 11
[[5,4,4,3,2,1],[4,3,3,2,1]]
 => [[5,4,3,2,2,1],[4,3,2,1,1]]
 => [4,3,2,1,1]
 => ? = 11
[[5,4,3,3,2,1],[4,3,2,2,1]]
 => [[5,4,3,3,2,1],[4,3,2,2,1]]
 => [4,3,2,2,1]
 => ? = 12
[[5,4,3,2,2,1],[4,3,2,1,1]]
 => [[5,4,4,3,2,1],[4,3,3,2,1]]
 => [4,3,3,2,1]
 => ? = 13
[[5,4,3,2,1,1],[4,3,2,1]]
 => [[5,5,4,3,2,1],[4,4,3,2,1]]
 => [4,4,3,2,1]
 => ? = 14
[[6,5,4,3,2,1],[5,4,3,2,1]]
 => [[6,5,4,3,2,1],[5,4,3,2,1]]
 => [5,4,3,2,1]
 => ? = 15
[[7,2,1],[2,1]]
 => [[7,6,5],[6,5]]
 => [6,5]
 => ? = 11
[[5,2,1,1],[1,1]]
 => [[5,5,4,4],[4,4,3]]
 => [4,4,3]
 => ? = 11
[[5,1,1,1],[1]]
 => [[5,5,5,4],[4,4,4]]
 => [4,4,4]
 => ? = 12
[[6,2,1,1],[2,1]]
 => [[6,6,5,4],[5,5,4]]
 => [5,5,4]
 => ? = 14
[[5,2,1,1],[2]]
 => [[5,5,5,3],[4,4,3]]
 => [4,4,3]
 => ? = 11
[[5,3,3,2,1],[2,2,2,1]]
 => [[5,4,3,3,3],[4,3,2,2]]
 => [4,3,2,2]
 => ? = 11
[[4,2,1,1,1],[1,1]]
 => [[4,4,4,3,3],[3,3,3,2]]
 => [3,3,3,2]
 => ? = 11
[[5,3,2,2,1],[2,2,1,1]]
 => [[5,4,4,3,3],[4,3,3,2]]
 => [4,3,3,2]
 => ? = 12
[[5,3,2,1,1],[2,2,1]]
 => [[5,5,4,3,3],[4,4,3,2]]
 => [4,4,3,2]
 => ? = 13
[[4,1,1,1,1],[1]]
 => [[4,4,4,4,3],[3,3,3,3]]
 => [3,3,3,3]
 => ? = 12
[[5,2,2,2,1],[2,1,1,1]]
 => [[5,4,4,4,3],[4,3,3,3]]
 => [4,3,3,3]
 => ? = 13
[[5,2,2,1,1],[2,1,1]]
 => [[5,5,4,4,3],[4,4,3,3]]
 => [4,4,3,3]
 => ? = 14
[[5,2,1,1,1],[2,1]]
 => [[5,5,5,4,3],[4,4,4,3]]
 => [4,4,4,3]
 => ? = 15
[[5,4,2,2,1],[3,2,1,1]]
 => [[5,4,4,3,2],[4,3,3,1]]
 => [4,3,3,1]
 => ? = 11
[[5,4,2,1,1],[3,2,1]]
 => [[5,5,4,3,2],[4,4,3,1]]
 => [4,4,3,1]
 => ? = 12
[[4,2,1,1,1],[2]]
 => [[4,4,4,4,2],[3,3,3,2]]
 => [3,3,3,2]
 => ? = 11
[[5,3,2,2,1],[3,1,1,1]]
 => [[5,4,4,4,2],[4,3,3,2]]
 => [4,3,3,2]
 => ? = 12
[[5,3,2,1,1],[3,1,1]]
 => [[5,5,4,4,2],[4,4,3,2]]
 => [4,4,3,2]
 => ? = 13
[[5,3,1,1,1],[3,1]]
 => [[5,5,5,4,2],[4,4,4,2]]
 => [4,4,4,2]
 => ? = 14
[[5,4,3,1,1],[3,3,1]]
 => [[5,5,4,2,2],[4,4,2,1]]
 => [4,4,2,1]
 => ? = 11
[[5,3,3,2,1],[3,2,1,1]]
 => [[5,4,4,3,2],[4,3,2,2]]
 => [4,3,2,2]
 => ? = 11
[[5,3,3,1,1],[3,2,1]]
 => [[5,5,4,3,2],[4,4,2,2]]
 => [4,4,2,2]
 => ? = 12
[[5,3,2,1,1],[3,2]]
 => [[5,5,5,3,2],[4,4,3,2]]
 => [4,4,3,2]
 => ? = 13
[[5,3,3,2,1],[3,2,2]]
 => [[5,5,3,3,2],[4,3,2,2]]
 => [4,3,2,2]
 => ? = 11
[[5,3,2,2,1],[3,2,1]]
 => [[5,5,4,3,2],[4,3,3,2]]
 => [4,3,3,2]
 => ? = 12
[[5,4,4,3,2,1],[3,3,3,2,1]]
 => [[5,4,3,2,2,2],[4,3,2,1,1]]
 => [4,3,2,1,1]
 => ? = 11
[[5,4,3,3,2,1],[3,3,2,2,1]]
 => [[5,4,3,3,2,2],[4,3,2,2,1]]
 => [4,3,2,2,1]
 => ? = 12
[[5,4,3,2,2,1],[3,3,2,1,1]]
 => [[5,4,4,3,2,2],[4,3,3,2,1]]
 => [4,3,3,2,1]
 => ? = 13
[[5,4,3,2,1,1],[3,3,2,1]]
 => [[5,5,4,3,2,2],[4,4,3,2,1]]
 => [4,4,3,2,1]
 => ? = 14
[[6,5,4,3,2,1],[4,4,3,2,1]]
 => [[6,5,4,3,2,2],[5,4,3,2,1]]
 => [5,4,3,2,1]
 => ? = 15
[[6,4,4,3,2,1],[4,3,3,2,1]]
 => [[6,5,4,3,3,2],[5,4,3,2,2]]
 => [5,4,3,2,2]
 => ? = 16
[[4,2,1,1,1,1],[2,1]]
 => [[4,4,4,4,3,2],[3,3,3,3,2]]
 => [3,3,3,3,2]
 => ? = 14
[[6,4,3,3,2,1],[4,3,2,2,1]]
 => [[6,5,4,4,3,2],[5,4,3,3,2]]
 => [5,4,3,3,2]
 => ? = 17
[[6,4,3,2,2,1],[4,3,2,1,1]]
 => [[6,5,5,4,3,2],[5,4,4,3,2]]
 => [5,4,4,3,2]
 => ? = 18
[[6,4,3,2,1,1],[4,3,2,1]]
 => [[6,6,5,4,3,2],[5,5,4,3,2]]
 => [5,5,4,3,2]
 => ? = 19
[[5,4,2,2,1],[4,1,1,1]]
 => [[5,4,4,4,1],[4,3,3,1]]
 => [4,3,3,1]
 => ? = 11
[[5,4,2,1,1],[4,1,1]]
 => [[5,5,4,4,1],[4,4,3,1]]
 => [4,4,3,1]
 => ? = 12
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000698
Mapping
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 49%●distinct values known / distinct values provided: 33%
Values
[[1],[]]
 => [1]
 => [1,0]
 => []
 => ? = 0
[[2],[]]
 => [2]
 => [1,0,1,0]
 => [1]
 => ? = 0
[[1,1],[]]
 => [1,1]
 => [1,1,0,0]
 => []
 => ? = 0
[[2,1],[1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1]
 => 1
[[3],[]]
 => [3]
 => [1,0,1,0,1,0]
 => [2,1]
 => 0
[[2,1],[]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1]
 => 1
[[3,1],[1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 2
[[2,2],[1]]
 => [2,2]
 => [1,1,1,0,0,0]
 => []
 => ? = 0
[[3,2],[2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[[1,1,1],[]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1]
 => ? = 0
[[2,2,1],[1,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 1
[[2,1,1],[1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 2
[[3,2,1],[2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => 3
[[4],[]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 0
[[3,1],[]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 2
[[4,1],[1]]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => 3
[[2,2],[]]
 => [2,2]
 => [1,1,1,0,0,0]
 => []
 => ? = 0
[[3,2],[1]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[[4,2],[2]]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => 2
[[2,1,1],[]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 2
[[3,2,1],[1,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => 3
[[3,1,1],[1]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => 4
[[4,2,1],[2,1]]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2,1]
 => 5
[[3,3],[2]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => ? = 0
[[4,3],[3]]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => 1
[[2,2,1],[1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 1
[[3,3,1],[2,1]]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => 2
[[3,2,1],[2]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => 3
[[4,3,1],[3,1]]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,1,1]
 => 4
[[2,2,2],[1,1]]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? = 0
[[3,3,2],[2,2]]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => 1
[[3,2,2],[2,1]]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 2
[[4,3,2],[3,2]]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1,1]
 => 3
[[1,1,1,1],[]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 0
[[2,2,2,1],[1,1,1]]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [3]
 => 1
[[2,2,1,1],[1,1]]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => 2
[[3,3,2,1],[2,2,1]]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [4,1,1]
 => 3
[[2,1,1,1],[1]]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => 3
[[3,2,2,1],[2,1,1]]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [4,1,1,1,1]
 => 4
[[3,2,1,1],[2,1]]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,1,1]
 => 5
[[4,3,2,1],[3,2,1]]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [5,2,2,1,1,1]
 => 6
[[5],[]]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => 0
[[4,1],[]]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => 3
[[5,1],[1]]
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => ? = 4
[[3,2],[]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[[4,2],[1]]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => 2
[[5,2],[2]]
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => 3
[[3,1,1],[]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => 4
[[4,2,1],[1,1]]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2,1]
 => 5
[[4,1,1],[1]]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => ? = 6
[[5,2,1],[2,1]]
 => [5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,3,3,3,2,1]
 => ? = 7
[[3,3],[1]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => ? = 0
[[4,3],[2]]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => 1
[[5,3],[3]]
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2,1]
 => 2
[[2,2,1],[]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 1
[[3,3,1],[1,1]]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => 2
[[3,2,1],[1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => 3
[[4,3,1],[2,1]]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,1,1]
 => 4
[[4,2,1],[2]]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2,1]
 => 5
[[5,3,1],[3,1]]
 => [5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [5,3,2,2,2,1]
 => ? = 6
[[3,2,2],[1,1]]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 2
[[4,3,2],[2,2]]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1,1]
 => 3
[[4,2,2],[2,1]]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => 4
[[5,3,2],[3,2]]
 => [5,3,2]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,3,2,2,2,1]
 => ? = 5
[[4,2,2,1],[2,1,1]]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [5,2,2,2,2,1]
 => ? = 7
[[4,2,1,1],[2,1]]
 => [4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [5,4,2,2,2,1]
 => ? = 8
[[5,3,2,1],[3,2,1]]
 => [5,3,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [6,3,3,2,2,2,1]
 => ? = 9
[[5,4,1],[4,1]]
 => [5,4,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [5,3,2,1,1,1]
 => ? = 5
[[2,2,2],[1]]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? = 0
[[4,3,1,1],[3,1]]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [5,4,2,1,1,1]
 => ? = 7
[[5,4,2,1],[4,2,1]]
 => [5,4,2,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => [6,3,3,2,1,1,1]
 => ? = 8
[[3,3,3],[2,2]]
 => [3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 0
[[5,4,3,1],[4,3,1]]
 => [5,4,3,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => [6,2,2,2,1,1,1]
 => ? = 7
[[2,2,2,2],[1,1,1]]
 => [2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => ? = 0
[[5,4,3,2],[4,3,2]]
 => [5,4,3,2]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => [4,2,2,2,1,1,1]
 => ? = 6
[[3,2,2,1,1],[2,1,1]]
 => [3,2,2,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [5,4,1,1,1,1]
 => ? = 6
[[4,3,3,2,1],[3,2,2,1]]
 => [4,3,3,2,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => [6,3,1,1,1,1,1]
 => ? = 7
[[3,2,1,1,1],[2,1]]
 => [3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [5,4,3,1,1,1]
 => ? = 7
[[4,3,2,2,1],[3,2,1,1]]
 => [4,3,2,2,1]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => [6,3,2,2,1,1,1]
 => ? = 8
[[4,3,2,1,1],[3,2,1]]
 => [4,3,2,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => [6,5,2,2,1,1,1]
 => ? = 9
[[5,4,3,2,1],[4,3,2,1]]
 => [5,4,3,2,1]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => [7,4,2,2,2,1,1,1]
 => ? = 10
[[6],[]]
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => ? = 0
[[5,1],[]]
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => ? = 4
[[6,1],[1]]
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3,2,1]
 => ? = 5
[[6,2],[2]]
 => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,2,1]
 => ? = 4
[[4,1,1],[]]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => ? = 6
[[5,2,1],[1,1]]
 => [5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,3,3,3,2,1]
 => ? = 7
[[5,1,1],[1]]
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3,2,1]
 => ? = 8
[[6,2,1],[2,1]]
 => [6,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [6,4,4,4,3,2,1]
 => ? = 9
[[3,3],[]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => ? = 0
[[6,3],[3]]
 => [6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [4,3,3,3,2,1]
 => ? = 3
[[5,3,1],[2,1]]
 => [5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [5,3,2,2,2,1]
 => ? = 6
[[5,2,1],[2]]
 => [5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,3,3,3,2,1]
 => ? = 7
[[6,3,1],[3,1]]
 => [6,3,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [6,4,3,3,3,2,1]
 => ? = 8
[[5,3,2],[2,2]]
 => [5,3,2]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,3,2,2,2,1]
 => ? = 5
[[5,2,2],[2,1]]
 => [5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,3,3,3,2,1]
 => ? = 6
[[6,3,2],[3,2]]
 => [6,3,2]
 => [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [4,4,3,3,3,2,1]
 => ? = 7
[[4,2,2,1],[1,1,1]]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [5,2,2,2,2,1]
 => ? = 7
[[4,2,1,1],[1,1]]
 => [4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [5,4,2,2,2,1]
 => ? = 8
[[5,3,2,1],[2,2,1]]
 => [5,3,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [6,3,3,2,2,2,1]
 => ? = 9
Description
The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$. This statistic counts the $2$-rim hooks that are removed in this process to obtain a $2$-core.
Matching statistic: St001438
(load all 3 compositions to match this statistic)
Mapping
Mp00189: Skew partitions —rotate⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 52%
Values
[[1],[]]
 => [[1],[]]
 => 0
[[2],[]]
 => [[2],[]]
 => 0
[[1,1],[]]
 => [[1,1],[]]
 => 0
[[2,1],[1]]
 => [[2,1],[1]]
 => 1
[[3],[]]
 => [[3],[]]
 => 0
[[2,1],[]]
 => [[2,2],[1]]
 => 1
[[3,1],[1]]
 => [[3,2],[2]]
 => 2
[[2,2],[1]]
 => [[2,1],[]]
 => 0
[[3,2],[2]]
 => [[3,1],[1]]
 => 1
[[1,1,1],[]]
 => [[1,1,1],[]]
 => 0
[[2,2,1],[1,1]]
 => [[2,1,1],[1]]
 => 1
[[2,1,1],[1]]
 => [[2,2,1],[1,1]]
 => 2
[[3,2,1],[2,1]]
 => [[3,2,1],[2,1]]
 => 3
[[4],[]]
 => [[4],[]]
 => 0
[[3,1],[]]
 => [[3,3],[2]]
 => 2
[[4,1],[1]]
 => [[4,3],[3]]
 => 3
[[2,2],[]]
 => [[2,2],[]]
 => 0
[[3,2],[1]]
 => [[3,2],[1]]
 => 1
[[4,2],[2]]
 => [[4,2],[2]]
 => 2
[[2,1,1],[]]
 => [[2,2,2],[1,1]]
 => 2
[[3,2,1],[1,1]]
 => [[3,2,2],[2,1]]
 => 3
[[3,1,1],[1]]
 => [[3,3,2],[2,2]]
 => 4
[[4,2,1],[2,1]]
 => [[4,3,2],[3,2]]
 => 5
[[3,3],[2]]
 => [[3,1],[]]
 => 0
[[4,3],[3]]
 => [[4,1],[1]]
 => 1
[[2,2,1],[1]]
 => [[2,2,1],[1]]
 => 1
[[3,3,1],[2,1]]
 => [[3,2,1],[2]]
 => 2
[[3,2,1],[2]]
 => [[3,3,1],[2,1]]
 => 3
[[4,3,1],[3,1]]
 => [[4,3,1],[3,1]]
 => 4
[[2,2,2],[1,1]]
 => [[2,1,1],[]]
 => 0
[[3,3,2],[2,2]]
 => [[3,1,1],[1]]
 => 1
[[3,2,2],[2,1]]
 => [[3,2,1],[1,1]]
 => 2
[[4,3,2],[3,2]]
 => [[4,2,1],[2,1]]
 => 3
[[1,1,1,1],[]]
 => [[1,1,1,1],[]]
 => 0
[[2,2,2,1],[1,1,1]]
 => [[2,1,1,1],[1]]
 => 1
[[2,2,1,1],[1,1]]
 => [[2,2,1,1],[1,1]]
 => 2
[[3,3,2,1],[2,2,1]]
 => [[3,2,1,1],[2,1]]
 => 3
[[2,1,1,1],[1]]
 => [[2,2,2,1],[1,1,1]]
 => 3
[[3,2,2,1],[2,1,1]]
 => [[3,2,2,1],[2,1,1]]
 => 4
[[3,2,1,1],[2,1]]
 => [[3,3,2,1],[2,2,1]]
 => 5
[[4,3,2,1],[3,2,1]]
 => [[4,3,2,1],[3,2,1]]
 => 6
[[5],[]]
 => [[5],[]]
 => 0
[[4,1],[]]
 => [[4,4],[3]]
 => 3
[[5,1],[1]]
 => [[5,4],[4]]
 => 4
[[3,2],[]]
 => [[3,3],[1]]
 => 1
[[4,2],[1]]
 => [[4,3],[2]]
 => 2
[[5,2],[2]]
 => [[5,3],[3]]
 => 3
[[3,1,1],[]]
 => [[3,3,3],[2,2]]
 => 4
[[4,2,1],[1,1]]
 => [[4,3,3],[3,2]]
 => 5
[[4,1,1],[1]]
 => [[4,4,3],[3,3]]
 => 6
[[6],[]]
 => [[6],[]]
 => ? = 0
[[5,1],[]]
 => [[5,5],[4]]
 => ? = 4
[[6,1],[1]]
 => [[6,5],[5]]
 => ? = 5
[[4,2],[]]
 => [[4,4],[2]]
 => ? = 2
[[5,2],[1]]
 => [[5,4],[3]]
 => ? = 3
[[6,2],[2]]
 => [[6,4],[4]]
 => ? = 4
[[4,1,1],[]]
 => [[4,4,4],[3,3]]
 => ? = 6
[[5,2,1],[1,1]]
 => [[5,4,4],[4,3]]
 => ? = 7
[[5,1,1],[1]]
 => [[5,5,4],[4,4]]
 => ? = 8
[[6,2,1],[2,1]]
 => [[6,5,4],[5,4]]
 => ? = 9
[[3,3],[]]
 => [[3,3],[]]
 => ? = 0
[[4,3],[1]]
 => [[4,3],[1]]
 => ? = 1
[[5,3],[2]]
 => [[5,3],[2]]
 => ? = 2
[[6,3],[3]]
 => [[6,3],[3]]
 => ? = 3
[[3,2,1],[]]
 => [[3,3,3],[2,1]]
 => ? = 3
[[4,3,1],[1,1]]
 => [[4,3,3],[3,1]]
 => ? = 4
[[4,2,1],[1]]
 => [[4,4,3],[3,2]]
 => ? = 5
[[5,3,1],[2,1]]
 => [[5,4,3],[4,2]]
 => ? = 6
[[5,2,1],[2]]
 => [[5,5,3],[4,3]]
 => ? = 7
[[6,3,1],[3,1]]
 => [[6,5,3],[5,3]]
 => ? = 8
[[4,2,2],[1,1]]
 => [[4,3,3],[2,2]]
 => ? = 4
[[5,3,2],[2,2]]
 => [[5,3,3],[3,2]]
 => ? = 5
[[5,2,2],[2,1]]
 => [[5,4,3],[3,3]]
 => ? = 6
[[6,3,2],[3,2]]
 => [[6,4,3],[4,3]]
 => ? = 7
[[3,1,1,1],[]]
 => [[3,3,3,3],[2,2,2]]
 => ? = 6
[[4,2,2,1],[1,1,1]]
 => [[4,3,3,3],[3,2,2]]
 => ? = 7
[[4,2,1,1],[1,1]]
 => [[4,4,3,3],[3,3,2]]
 => ? = 8
[[5,3,2,1],[2,2,1]]
 => [[5,4,3,3],[4,3,2]]
 => ? = 9
[[4,1,1,1],[1]]
 => [[4,4,4,3],[3,3,3]]
 => ? = 9
[[5,2,2,1],[2,1,1]]
 => [[5,4,4,3],[4,3,3]]
 => ? = 10
[[5,2,1,1],[2,1]]
 => [[5,5,4,3],[4,4,3]]
 => ? = 11
[[4,4],[2]]
 => [[4,2],[]]
 => ? = 0
[[5,4],[3]]
 => [[5,2],[1]]
 => ? = 1
[[6,4],[4]]
 => [[6,2],[2]]
 => ? = 2
[[3,3,1],[1]]
 => [[3,3,2],[2]]
 => ? = 2
[[4,4,1],[2,1]]
 => [[4,3,2],[3]]
 => ? = 3
[[4,3,1],[2]]
 => [[4,4,2],[3,1]]
 => ? = 4
[[5,4,1],[3,1]]
 => [[5,4,2],[4,1]]
 => ? = 5
[[5,3,1],[3]]
 => [[5,5,2],[4,2]]
 => ? = 6
[[6,4,1],[4,1]]
 => [[6,5,2],[5,2]]
 => ? = 7
[[2,2,2],[]]
 => [[2,2,2],[]]
 => ? = 0
[[3,3,2],[1,1]]
 => [[3,2,2],[1]]
 => ? = 1
[[4,4,2],[2,2]]
 => [[4,2,2],[2]]
 => ? = 2
[[3,2,2],[1]]
 => [[3,3,2],[1,1]]
 => ? = 2
[[4,3,2],[2,1]]
 => [[4,3,2],[2,1]]
 => ? = 3
[[5,4,2],[3,2]]
 => [[5,3,2],[3,1]]
 => ? = 4
[[4,2,2],[2]]
 => [[4,4,2],[2,2]]
 => ? = 4
[[5,3,2],[3,1]]
 => [[5,4,2],[3,2]]
 => ? = 5
[[6,4,2],[4,2]]
 => [[6,4,2],[4,2]]
 => ? = 6
[[2,2,1,1],[]]
 => [[2,2,2,2],[1,1]]
 => ? = 2
Description
The number of missing boxes of a skew partition.