Identifier
Mp00021:
Cores
—to bounded partition⟶
Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary 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].
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.
See Mp00096Foata bijection.
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