Identifier
Mp00021: Cores to bounded partitionInteger partitions
Mp00317: Integer partitions odd partsBinary words
Mp00316: Binary words inverse Foata bijectionBinary words
Images
([2],3) => [2] => 0 => 0
([1,1],3) => [1,1] => 11 => 11
([3,1],3) => [2,1] => 01 => 01
([2,1,1],3) => [1,1,1] => 111 => 111
([4,2],3) => [2,2] => 00 => 00
([3,1,1],3) => [2,1,1] => 011 => 011
([2,2,1,1],3) => [1,1,1,1] => 1111 => 1111
([5,3,1],3) => [2,2,1] => 001 => 001
([4,2,1,1],3) => [2,1,1,1] => 0111 => 0111
([3,2,2,1,1],3) => [1,1,1,1,1] => 11111 => 11111
([6,4,2],3) => [2,2,2] => 000 => 000
([5,3,1,1],3) => [2,2,1,1] => 0011 => 0011
([4,2,2,1,1],3) => [2,1,1,1,1] => 01111 => 01111
([3,3,2,2,1,1],3) => [1,1,1,1,1,1] => 111111 => 111111
([2],4) => [2] => 0 => 0
([1,1],4) => [1,1] => 11 => 11
([3],4) => [3] => 1 => 1
([2,1],4) => [2,1] => 01 => 01
([1,1,1],4) => [1,1,1] => 111 => 111
([4,1],4) => [3,1] => 11 => 11
([2,2],4) => [2,2] => 00 => 00
([3,1,1],4) => [2,1,1] => 011 => 011
([2,1,1,1],4) => [1,1,1,1] => 1111 => 1111
([5,2],4) => [3,2] => 10 => 10
([4,1,1],4) => [3,1,1] => 111 => 111
([3,2,1],4) => [2,2,1] => 001 => 001
([3,1,1,1],4) => [2,1,1,1] => 0111 => 0111
([2,2,1,1,1],4) => [1,1,1,1,1] => 11111 => 11111
([6,3],4) => [3,3] => 11 => 11
([5,2,1],4) => [3,2,1] => 101 => 101
([4,1,1,1],4) => [3,1,1,1] => 1111 => 1111
([4,2,2],4) => [2,2,2] => 000 => 000
([3,3,1,1],4) => [2,2,1,1] => 0011 => 0011
([3,2,1,1,1],4) => [2,1,1,1,1] => 01111 => 01111
([2,2,2,1,1,1],4) => [1,1,1,1,1,1] => 111111 => 111111
([2],5) => [2] => 0 => 0
([1,1],5) => [1,1] => 11 => 11
([3],5) => [3] => 1 => 1
([2,1],5) => [2,1] => 01 => 01
([1,1,1],5) => [1,1,1] => 111 => 111
([4],5) => [4] => 0 => 0
([3,1],5) => [3,1] => 11 => 11
([2,2],5) => [2,2] => 00 => 00
([2,1,1],5) => [2,1,1] => 011 => 011
([1,1,1,1],5) => [1,1,1,1] => 1111 => 1111
([5,1],5) => [4,1] => 01 => 01
([3,2],5) => [3,2] => 10 => 10
([4,1,1],5) => [3,1,1] => 111 => 111
([2,2,1],5) => [2,2,1] => 001 => 001
([3,1,1,1],5) => [2,1,1,1] => 0111 => 0111
([2,1,1,1,1],5) => [1,1,1,1,1] => 11111 => 11111
([6,2],5) => [4,2] => 00 => 00
([5,1,1],5) => [4,1,1] => 011 => 011
([3,3],5) => [3,3] => 11 => 11
([4,2,1],5) => [3,2,1] => 101 => 101
([4,1,1,1],5) => [3,1,1,1] => 1111 => 1111
([2,2,2],5) => [2,2,2] => 000 => 000
([3,2,1,1],5) => [2,2,1,1] => 0011 => 0011
([3,1,1,1,1],5) => [2,1,1,1,1] => 01111 => 01111
([2,2,1,1,1,1],5) => [1,1,1,1,1,1] => 111111 => 111111
([2],6) => [2] => 0 => 0
([1,1],6) => [1,1] => 11 => 11
([3],6) => [3] => 1 => 1
([2,1],6) => [2,1] => 01 => 01
([1,1,1],6) => [1,1,1] => 111 => 111
([4],6) => [4] => 0 => 0
([3,1],6) => [3,1] => 11 => 11
([2,2],6) => [2,2] => 00 => 00
([2,1,1],6) => [2,1,1] => 011 => 011
([1,1,1,1],6) => [1,1,1,1] => 1111 => 1111
([5],6) => [5] => 1 => 1
([4,1],6) => [4,1] => 01 => 01
([3,2],6) => [3,2] => 10 => 10
([3,1,1],6) => [3,1,1] => 111 => 111
([2,2,1],6) => [2,2,1] => 001 => 001
([2,1,1,1],6) => [2,1,1,1] => 0111 => 0111
([1,1,1,1,1],6) => [1,1,1,1,1] => 11111 => 11111
([6,1],6) => [5,1] => 11 => 11
([4,2],6) => [4,2] => 00 => 00
([5,1,1],6) => [4,1,1] => 011 => 011
([3,3],6) => [3,3] => 11 => 11
([3,2,1],6) => [3,2,1] => 101 => 101
([4,1,1,1],6) => [3,1,1,1] => 1111 => 1111
([2,2,2],6) => [2,2,2] => 000 => 000
([2,2,1,1],6) => [2,2,1,1] => 0011 => 0011
([3,1,1,1,1],6) => [2,1,1,1,1] => 01111 => 01111
([2,1,1,1,1,1],6) => [1,1,1,1,1,1] => 111111 => 111111
([7,2],6) => [5,2] => 10 => 10
([6,1,1],6) => [5,1,1] => 111 => 111
([4,3],6) => [4,3] => 01 => 01
([5,2,1],6) => [4,2,1] => 001 => 001
([5,1,1,1],6) => [4,1,1,1] => 0111 => 0111
([3,3,1],6) => [3,3,1] => 111 => 111
([3,2,2],6) => [3,2,2] => 100 => 010
([4,2,1,1],6) => [3,2,1,1] => 1011 => 1011
([4,1,1,1,1],6) => [3,1,1,1,1] => 11111 => 11111
([2,2,2,1],6) => [2,2,2,1] => 0001 => 0001
([3,2,1,1,1],6) => [2,2,1,1,1] => 00111 => 00111
([3,1,1,1,1,1],6) => [2,1,1,1,1,1] => 011111 => 011111
([2,2,1,1,1,1,1],6) => [1,1,1,1,1,1,1] => 1111111 => 1111111
Map
to bounded partition
Description
The (k-1)-bounded partition of a k-core.
Starting with a $k$-core, deleting all cells of hook length greater than or equal to $k$ yields a $(k-1)$-bounded partition [1, Theorem 7], see also [2, Section 1.2].
Map
odd parts
Description
Return the binary word indicating which parts of the partition are odd.
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00096Foata bijection.