Identifier
Mp00021: Cores —to bounded partition⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Images
=>
Cc0013;cc-rep-0Cc0002;cc-rep-1
([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.