Identifier
Values
([2],3) => [2] => 100 => 2
([1,1],3) => [1,1] => 110 => 2
([3,1],3) => [3,1] => 10010 => 4
([2,1,1],3) => [2,1,1] => 10110 => 4
([4,2],3) => [4,2] => 100100 => 6
([3,1,1],3) => [3,1,1] => 100110 => 5
([2,2,1,1],3) => [2,2,1,1] => 110110 => 6
([5,3,1],3) => [5,3,1] => 10010010 => 9
([4,2,1,1],3) => [4,2,1,1] => 10010110 => 8
([3,2,2,1,1],3) => [3,2,2,1,1] => 10110110 => 9
([6,4,2],3) => [6,4,2] => 100100100 => 12
([5,3,1,1],3) => [5,3,1,1] => 100100110 => 10
([4,2,2,1,1],3) => [4,2,2,1,1] => 100110110 => 10
([3,3,2,2,1,1],3) => [3,3,2,2,1,1] => 110110110 => 12
([2],4) => [2] => 100 => 2
([1,1],4) => [1,1] => 110 => 2
([3],4) => [3] => 1000 => 3
([2,1],4) => [2,1] => 1010 => 3
([1,1,1],4) => [1,1,1] => 1110 => 3
([4,1],4) => [4,1] => 100010 => 5
([2,2],4) => [2,2] => 1100 => 4
([3,1,1],4) => [3,1,1] => 100110 => 5
([2,1,1,1],4) => [2,1,1,1] => 101110 => 5
([5,2],4) => [5,2] => 1000100 => 7
([4,1,1],4) => [4,1,1] => 1000110 => 6
([3,2,1],4) => [3,2,1] => 101010 => 6
([3,1,1,1],4) => [3,1,1,1] => 1001110 => 6
([2,2,1,1,1],4) => [2,2,1,1,1] => 1101110 => 7
([6,3],4) => [6,3] => 10001000 => 9
([5,2,1],4) => [5,2,1] => 10001010 => 8
([4,1,1,1],4) => [4,1,1,1] => 10001110 => 7
([4,2,2],4) => [4,2,2] => 1001100 => 8
([3,3,1,1],4) => [3,3,1,1] => 1100110 => 8
([3,2,1,1,1],4) => [3,2,1,1,1] => 10101110 => 8
([2,2,2,1,1,1],4) => [2,2,2,1,1,1] => 11101110 => 9
([2],5) => [2] => 100 => 2
([1,1],5) => [1,1] => 110 => 2
([3],5) => [3] => 1000 => 3
([2,1],5) => [2,1] => 1010 => 3
([1,1,1],5) => [1,1,1] => 1110 => 3
([4],5) => [4] => 10000 => 4
([3,1],5) => [3,1] => 10010 => 4
([2,2],5) => [2,2] => 1100 => 4
([2,1,1],5) => [2,1,1] => 10110 => 4
([1,1,1,1],5) => [1,1,1,1] => 11110 => 4
([5,1],5) => [5,1] => 1000010 => 6
([3,2],5) => [3,2] => 10100 => 5
([4,1,1],5) => [4,1,1] => 1000110 => 6
([2,2,1],5) => [2,2,1] => 11010 => 5
([3,1,1,1],5) => [3,1,1,1] => 1001110 => 6
([2,1,1,1,1],5) => [2,1,1,1,1] => 1011110 => 6
([6,2],5) => [6,2] => 10000100 => 8
([5,1,1],5) => [5,1,1] => 10000110 => 7
([3,3],5) => [3,3] => 11000 => 6
([4,2,1],5) => [4,2,1] => 1001010 => 7
([4,1,1,1],5) => [4,1,1,1] => 10001110 => 7
([2,2,2],5) => [2,2,2] => 11100 => 6
([3,2,1,1],5) => [3,2,1,1] => 1010110 => 7
([3,1,1,1,1],5) => [3,1,1,1,1] => 10011110 => 7
([2,2,1,1,1,1],5) => [2,2,1,1,1,1] => 11011110 => 8
([2],6) => [2] => 100 => 2
([1,1],6) => [1,1] => 110 => 2
([3],6) => [3] => 1000 => 3
([2,1],6) => [2,1] => 1010 => 3
([1,1,1],6) => [1,1,1] => 1110 => 3
([4],6) => [4] => 10000 => 4
([3,1],6) => [3,1] => 10010 => 4
([2,2],6) => [2,2] => 1100 => 4
([2,1,1],6) => [2,1,1] => 10110 => 4
([1,1,1,1],6) => [1,1,1,1] => 11110 => 4
([5],6) => [5] => 100000 => 5
([4,1],6) => [4,1] => 100010 => 5
([3,2],6) => [3,2] => 10100 => 5
([3,1,1],6) => [3,1,1] => 100110 => 5
([2,2,1],6) => [2,2,1] => 11010 => 5
([2,1,1,1],6) => [2,1,1,1] => 101110 => 5
([1,1,1,1,1],6) => [1,1,1,1,1] => 111110 => 5
([6,1],6) => [6,1] => 10000010 => 7
([4,2],6) => [4,2] => 100100 => 6
([5,1,1],6) => [5,1,1] => 10000110 => 7
([3,3],6) => [3,3] => 11000 => 6
([3,2,1],6) => [3,2,1] => 101010 => 6
([4,1,1,1],6) => [4,1,1,1] => 10001110 => 7
([2,2,2],6) => [2,2,2] => 11100 => 6
([2,2,1,1],6) => [2,2,1,1] => 110110 => 6
([3,1,1,1,1],6) => [3,1,1,1,1] => 10011110 => 7
([2,1,1,1,1,1],6) => [2,1,1,1,1,1] => 10111110 => 7
([7,2],6) => [7,2] => 100000100 => 9
([6,1,1],6) => [6,1,1] => 100000110 => 8
([4,3],6) => [4,3] => 101000 => 7
([5,2,1],6) => [5,2,1] => 10001010 => 8
([5,1,1,1],6) => [5,1,1,1] => 100001110 => 8
([3,3,1],6) => [3,3,1] => 110010 => 7
([3,2,2],6) => [3,2,2] => 101100 => 7
([4,2,1,1],6) => [4,2,1,1] => 10010110 => 8
([4,1,1,1,1],6) => [4,1,1,1,1] => 100011110 => 8
([2,2,2,1],6) => [2,2,2,1] => 111010 => 7
([3,2,1,1,1],6) => [3,2,1,1,1] => 10101110 => 8
([3,1,1,1,1,1],6) => [3,1,1,1,1,1] => 100111110 => 8
([2,2,1,1,1,1,1],6) => [2,2,1,1,1,1,1] => 110111110 => 9
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Description
The number of inversions of a binary word.
Map
to binary word
Description
Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.
Map
to partition
Description
Considers a core as a partition.
This embedding is graded and injective but not surjective on $k$-cores for a given parameter $k$, while it is surjective and neither graded nor injective on the collection of all cores.