Identifier
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
Images
=>
Cc0002;cc-rep-0Cc0007;cc-rep-1
[1]=>[[1]]=>=> [2]=>[[1,2]]=>0=>0 [1,1]=>[[1],[2]]=>1=>1 [3]=>[[1,2,3]]=>00=>00 [2,1]=>[[1,3],[2]]=>10=>01 [1,1,1]=>[[1],[2],[3]]=>11=>11 [4]=>[[1,2,3,4]]=>000=>000 [3,1]=>[[1,3,4],[2]]=>100=>001 [2,2]=>[[1,2],[3,4]]=>010=>010 [2,1,1]=>[[1,4],[2],[3]]=>110=>011 [1,1,1,1]=>[[1],[2],[3],[4]]=>111=>111 [5]=>[[1,2,3,4,5]]=>0000=>0000 [4,1]=>[[1,3,4,5],[2]]=>1000=>0001 [3,2]=>[[1,2,5],[3,4]]=>0100=>0010 [3,1,1]=>[[1,4,5],[2],[3]]=>1100=>0011 [2,2,1]=>[[1,3],[2,5],[4]]=>1010=>0101 [2,1,1,1]=>[[1,5],[2],[3],[4]]=>1110=>0111 [1,1,1,1,1]=>[[1],[2],[3],[4],[5]]=>1111=>1111 [6]=>[[1,2,3,4,5,6]]=>00000=>00000 [5,1]=>[[1,3,4,5,6],[2]]=>10000=>00001 [4,2]=>[[1,2,5,6],[3,4]]=>01000=>00010 [4,1,1]=>[[1,4,5,6],[2],[3]]=>11000=>00011 [3,3]=>[[1,2,3],[4,5,6]]=>00100=>00100 [3,2,1]=>[[1,3,6],[2,5],[4]]=>10100=>00101 [3,1,1,1]=>[[1,5,6],[2],[3],[4]]=>11100=>00111 [2,2,2]=>[[1,2],[3,4],[5,6]]=>01010=>01010 [2,2,1,1]=>[[1,4],[2,6],[3],[5]]=>11010=>01011 [2,1,1,1,1]=>[[1,6],[2],[3],[4],[5]]=>11110=>01111 [1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6]]=>11111=>11111 [7]=>[[1,2,3,4,5,6,7]]=>000000=>000000 [6,1]=>[[1,3,4,5,6,7],[2]]=>100000=>000001 [5,2]=>[[1,2,5,6,7],[3,4]]=>010000=>000010 [5,1,1]=>[[1,4,5,6,7],[2],[3]]=>110000=>000011 [4,3]=>[[1,2,3,7],[4,5,6]]=>001000=>000100 [4,2,1]=>[[1,3,6,7],[2,5],[4]]=>101000=>000101 [4,1,1,1]=>[[1,5,6,7],[2],[3],[4]]=>111000=>000111 [3,3,1]=>[[1,3,4],[2,6,7],[5]]=>100100=>001001 [3,2,2]=>[[1,2,7],[3,4],[5,6]]=>010100=>001010 [3,2,1,1]=>[[1,4,7],[2,6],[3],[5]]=>110100=>001011 [3,1,1,1,1]=>[[1,6,7],[2],[3],[4],[5]]=>111100=>001111 [2,2,2,1]=>[[1,3],[2,5],[4,7],[6]]=>101010=>010101 [2,2,1,1,1]=>[[1,5],[2,7],[3],[4],[6]]=>111010=>010111 [2,1,1,1,1,1]=>[[1,7],[2],[3],[4],[5],[6]]=>111110=>011111 [1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7]]=>111111=>111111 [8]=>[[1,2,3,4,5,6,7,8]]=>0000000=>0000000 [7,1]=>[[1,3,4,5,6,7,8],[2]]=>1000000=>0000001 [6,2]=>[[1,2,5,6,7,8],[3,4]]=>0100000=>0000010 [6,1,1]=>[[1,4,5,6,7,8],[2],[3]]=>1100000=>0000011 [5,3]=>[[1,2,3,7,8],[4,5,6]]=>0010000=>0000100 [5,2,1]=>[[1,3,6,7,8],[2,5],[4]]=>1010000=>0000101 [5,1,1,1]=>[[1,5,6,7,8],[2],[3],[4]]=>1110000=>0000111 [4,4]=>[[1,2,3,4],[5,6,7,8]]=>0001000=>0001000 [4,3,1]=>[[1,3,4,8],[2,6,7],[5]]=>1001000=>0001001 [4,2,2]=>[[1,2,7,8],[3,4],[5,6]]=>0101000=>0001010 [4,2,1,1]=>[[1,4,7,8],[2,6],[3],[5]]=>1101000=>0001011 [4,1,1,1,1]=>[[1,6,7,8],[2],[3],[4],[5]]=>1111000=>0001111 [3,3,2]=>[[1,2,5],[3,4,8],[6,7]]=>0100100=>0010010 [3,3,1,1]=>[[1,4,5],[2,7,8],[3],[6]]=>1100100=>0010011 [3,2,2,1]=>[[1,3,8],[2,5],[4,7],[6]]=>1010100=>0010101 [3,2,1,1,1]=>[[1,5,8],[2,7],[3],[4],[6]]=>1110100=>0010111 [3,1,1,1,1,1]=>[[1,7,8],[2],[3],[4],[5],[6]]=>1111100=>0011111 [2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8]]=>0101010=>0101010 [2,2,2,1,1]=>[[1,4],[2,6],[3,8],[5],[7]]=>1101010=>0101011 [2,2,1,1,1,1]=>[[1,6],[2,8],[3],[4],[5],[7]]=>1111010=>0101111 [2,1,1,1,1,1,1]=>[[1,8],[2],[3],[4],[5],[6],[7]]=>1111110=>0111111 [1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8]]=>1111111=>1111111 [9]=>[[1,2,3,4,5,6,7,8,9]]=>00000000=>00000000 [8,1]=>[[1,3,4,5,6,7,8,9],[2]]=>10000000=>00000001 [7,2]=>[[1,2,5,6,7,8,9],[3,4]]=>01000000=>00000010 [7,1,1]=>[[1,4,5,6,7,8,9],[2],[3]]=>11000000=>00000011 [6,3]=>[[1,2,3,7,8,9],[4,5,6]]=>00100000=>00000100 [6,2,1]=>[[1,3,6,7,8,9],[2,5],[4]]=>10100000=>00000101 [6,1,1,1]=>[[1,5,6,7,8,9],[2],[3],[4]]=>11100000=>00000111 [5,4]=>[[1,2,3,4,9],[5,6,7,8]]=>00010000=>00001000 [5,3,1]=>[[1,3,4,8,9],[2,6,7],[5]]=>10010000=>00001001 [5,2,2]=>[[1,2,7,8,9],[3,4],[5,6]]=>01010000=>00001010 [5,2,1,1]=>[[1,4,7,8,9],[2,6],[3],[5]]=>11010000=>00001011 [5,1,1,1,1]=>[[1,6,7,8,9],[2],[3],[4],[5]]=>11110000=>00001111 [4,4,1]=>[[1,3,4,5],[2,7,8,9],[6]]=>10001000=>00010001 [4,3,2]=>[[1,2,5,9],[3,4,8],[6,7]]=>01001000=>00010010 [4,3,1,1]=>[[1,4,5,9],[2,7,8],[3],[6]]=>11001000=>00010011 [4,2,2,1]=>[[1,3,8,9],[2,5],[4,7],[6]]=>10101000=>00010101 [4,2,1,1,1]=>[[1,5,8,9],[2,7],[3],[4],[6]]=>11101000=>00010111 [4,1,1,1,1,1]=>[[1,7,8,9],[2],[3],[4],[5],[6]]=>11111000=>00011111 [3,3,3]=>[[1,2,3],[4,5,6],[7,8,9]]=>00100100=>00100100 [3,3,2,1]=>[[1,3,6],[2,5,9],[4,8],[7]]=>10100100=>00100101 [3,3,1,1,1]=>[[1,5,6],[2,8,9],[3],[4],[7]]=>11100100=>00100111 [3,2,2,2]=>[[1,2,9],[3,4],[5,6],[7,8]]=>01010100=>00101010 [3,2,2,1,1]=>[[1,4,9],[2,6],[3,8],[5],[7]]=>11010100=>00101011 [3,2,1,1,1,1]=>[[1,6,9],[2,8],[3],[4],[5],[7]]=>11110100=>00101111 [3,1,1,1,1,1,1]=>[[1,8,9],[2],[3],[4],[5],[6],[7]]=>11111100=>00111111 [2,2,2,2,1]=>[[1,3],[2,5],[4,7],[6,9],[8]]=>10101010=>01010101 [2,2,2,1,1,1]=>[[1,5],[2,7],[3,9],[4],[6],[8]]=>11101010=>01010111 [2,2,1,1,1,1,1]=>[[1,7],[2,9],[3],[4],[5],[6],[8]]=>11111010=>01011111 [2,1,1,1,1,1,1,1]=>[[1,9],[2],[3],[4],[5],[6],[7],[8]]=>11111110=>01111111 [1,1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8],[9]]=>11111111=>11111111 [10]=>[[1,2,3,4,5,6,7,8,9,10]]=>000000000=>000000000 [9,1]=>[[1,3,4,5,6,7,8,9,10],[2]]=>100000000=>000000001 [8,2]=>[[1,2,5,6,7,8,9,10],[3,4]]=>010000000=>000000010 [8,1,1]=>[[1,4,5,6,7,8,9,10],[2],[3]]=>110000000=>000000011 [7,3]=>[[1,2,3,7,8,9,10],[4,5,6]]=>001000000=>000000100 [7,2,1]=>[[1,3,6,7,8,9,10],[2,5],[4]]=>101000000=>000000101 [7,1,1,1]=>[[1,5,6,7,8,9,10],[2],[3],[4]]=>111000000=>000000111 [6,4]=>[[1,2,3,4,9,10],[5,6,7,8]]=>000100000=>000001000 [6,3,1]=>[[1,3,4,8,9,10],[2,6,7],[5]]=>100100000=>000001001 [6,2,2]=>[[1,2,7,8,9,10],[3,4],[5,6]]=>010100000=>000001010 [6,2,1,1]=>[[1,4,7,8,9,10],[2,6],[3],[5]]=>110100000=>000001011 [6,1,1,1,1]=>[[1,6,7,8,9,10],[2],[3],[4],[5]]=>111100000=>000001111 [5,5]=>[[1,2,3,4,5],[6,7,8,9,10]]=>000010000=>000010000 [5,4,1]=>[[1,3,4,5,10],[2,7,8,9],[6]]=>100010000=>000010001 [5,3,2]=>[[1,2,5,9,10],[3,4,8],[6,7]]=>010010000=>000010010 [5,3,1,1]=>[[1,4,5,9,10],[2,7,8],[3],[6]]=>110010000=>000010011 [5,2,2,1]=>[[1,3,8,9,10],[2,5],[4,7],[6]]=>101010000=>000010101 [5,2,1,1,1]=>[[1,5,8,9,10],[2,7],[3],[4],[6]]=>111010000=>000010111 [5,1,1,1,1,1]=>[[1,7,8,9,10],[2],[3],[4],[5],[6]]=>111110000=>000011111 [4,4,2]=>[[1,2,5,6],[3,4,9,10],[7,8]]=>010001000=>000100010 [4,4,1,1]=>[[1,4,5,6],[2,8,9,10],[3],[7]]=>110001000=>000100011 [4,3,3]=>[[1,2,3,10],[4,5,6],[7,8,9]]=>001001000=>000100100 [4,3,2,1]=>[[1,3,6,10],[2,5,9],[4,8],[7]]=>101001000=>000100101 [4,3,1,1,1]=>[[1,5,6,10],[2,8,9],[3],[4],[7]]=>111001000=>000100111 [4,2,2,2]=>[[1,2,9,10],[3,4],[5,6],[7,8]]=>010101000=>000101010 [4,2,2,1,1]=>[[1,4,9,10],[2,6],[3,8],[5],[7]]=>110101000=>000101011 [4,2,1,1,1,1]=>[[1,6,9,10],[2,8],[3],[4],[5],[7]]=>111101000=>000101111 [4,1,1,1,1,1,1]=>[[1,8,9,10],[2],[3],[4],[5],[6],[7]]=>111111000=>000111111 [3,3,3,1]=>[[1,3,4],[2,6,7],[5,9,10],[8]]=>100100100=>001001001 [3,3,2,2]=>[[1,2,7],[3,4,10],[5,6],[8,9]]=>010100100=>001001010 [3,3,2,1,1]=>[[1,4,7],[2,6,10],[3,9],[5],[8]]=>110100100=>001001011 [3,3,1,1,1,1]=>[[1,6,7],[2,9,10],[3],[4],[5],[8]]=>111100100=>001001111 [3,2,2,2,1]=>[[1,3,10],[2,5],[4,7],[6,9],[8]]=>101010100=>001010101 [3,2,2,1,1,1]=>[[1,5,10],[2,7],[3,9],[4],[6],[8]]=>111010100=>001010111 [3,2,1,1,1,1,1]=>[[1,7,10],[2,9],[3],[4],[5],[6],[8]]=>111110100=>001011111 [3,1,1,1,1,1,1,1]=>[[1,9,10],[2],[3],[4],[5],[6],[7],[8]]=>111111100=>001111111 [2,2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8],[9,10]]=>010101010=>010101010 [2,2,2,2,1,1]=>[[1,4],[2,6],[3,8],[5,10],[7],[9]]=>110101010=>010101011 [2,2,2,1,1,1,1]=>[[1,6],[2,8],[3,10],[4],[5],[7],[9]]=>111101010=>010101111 [2,2,1,1,1,1,1,1]=>[[1,8],[2,10],[3],[4],[5],[6],[7],[9]]=>111111010=>010111111 [2,1,1,1,1,1,1,1,1]=>[[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]=>111111110=>011111111 [1,1,1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]=>111111111=>111111111 [6,6]=>[[1,2,3,4,5,6],[7,8,9,10,11,12]]=>00000100000=>00000100000 [2,2,2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]=>01010101010=>01010101010
Map
reading tableau
Description
Return the RSK recording tableau of the reading word of the (standard) tableau $T$ labeled down (in English convention) each column to the shape of a partition.
Map
descent word
Description
The descent word of a standard Young tableau.
For a standard Young tableau of size $n$ we set $w_i=1$ if $i+1$ is in a lower row than $i$, and $0$ otherwise, for $1\leq i < n$.
Map
reverse
Description
Return the reversal of a binary word.