Identifier
Mp00042: Integer partitions —initial 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]=>[[1,2],[3]]=>01=>10 [1,1,1]=>[[1],[2],[3]]=>11=>11 [4]=>[[1,2,3,4]]=>000=>000 [3,1]=>[[1,2,3],[4]]=>001=>100 [2,2]=>[[1,2],[3,4]]=>010=>010 [2,1,1]=>[[1,2],[3],[4]]=>011=>110 [1,1,1,1]=>[[1],[2],[3],[4]]=>111=>111 [5]=>[[1,2,3,4,5]]=>0000=>0000 [4,1]=>[[1,2,3,4],[5]]=>0001=>1000 [3,2]=>[[1,2,3],[4,5]]=>0010=>0100 [3,1,1]=>[[1,2,3],[4],[5]]=>0011=>1100 [2,2,1]=>[[1,2],[3,4],[5]]=>0101=>1010 [2,1,1,1]=>[[1,2],[3],[4],[5]]=>0111=>1110 [1,1,1,1,1]=>[[1],[2],[3],[4],[5]]=>1111=>1111 [6]=>[[1,2,3,4,5,6]]=>00000=>00000 [5,1]=>[[1,2,3,4,5],[6]]=>00001=>10000 [4,2]=>[[1,2,3,4],[5,6]]=>00010=>01000 [4,1,1]=>[[1,2,3,4],[5],[6]]=>00011=>11000 [3,3]=>[[1,2,3],[4,5,6]]=>00100=>00100 [3,2,1]=>[[1,2,3],[4,5],[6]]=>00101=>10100 [3,1,1,1]=>[[1,2,3],[4],[5],[6]]=>00111=>11100 [2,2,2]=>[[1,2],[3,4],[5,6]]=>01010=>01010 [2,2,1,1]=>[[1,2],[3,4],[5],[6]]=>01011=>11010 [2,1,1,1,1]=>[[1,2],[3],[4],[5],[6]]=>01111=>11110 [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]=>[[1,2,3,4,5,6],[7]]=>000001=>100000 [5,2]=>[[1,2,3,4,5],[6,7]]=>000010=>010000 [5,1,1]=>[[1,2,3,4,5],[6],[7]]=>000011=>110000 [4,3]=>[[1,2,3,4],[5,6,7]]=>000100=>001000 [4,2,1]=>[[1,2,3,4],[5,6],[7]]=>000101=>101000 [4,1,1,1]=>[[1,2,3,4],[5],[6],[7]]=>000111=>111000 [3,3,1]=>[[1,2,3],[4,5,6],[7]]=>001001=>100100 [3,2,2]=>[[1,2,3],[4,5],[6,7]]=>001010=>010100 [3,2,1,1]=>[[1,2,3],[4,5],[6],[7]]=>001011=>110100 [3,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7]]=>001111=>111100 [2,2,2,1]=>[[1,2],[3,4],[5,6],[7]]=>010101=>101010 [2,2,1,1,1]=>[[1,2],[3,4],[5],[6],[7]]=>010111=>111010 [2,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7]]=>011111=>111110 [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]=>[[1,2,3,4,5,6,7],[8]]=>0000001=>1000000 [6,2]=>[[1,2,3,4,5,6],[7,8]]=>0000010=>0100000 [6,1,1]=>[[1,2,3,4,5,6],[7],[8]]=>0000011=>1100000 [5,3]=>[[1,2,3,4,5],[6,7,8]]=>0000100=>0010000 [5,2,1]=>[[1,2,3,4,5],[6,7],[8]]=>0000101=>1010000 [5,1,1,1]=>[[1,2,3,4,5],[6],[7],[8]]=>0000111=>1110000 [4,4]=>[[1,2,3,4],[5,6,7,8]]=>0001000=>0001000 [4,3,1]=>[[1,2,3,4],[5,6,7],[8]]=>0001001=>1001000 [4,2,2]=>[[1,2,3,4],[5,6],[7,8]]=>0001010=>0101000 [4,2,1,1]=>[[1,2,3,4],[5,6],[7],[8]]=>0001011=>1101000 [4,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8]]=>0001111=>1111000 [3,3,2]=>[[1,2,3],[4,5,6],[7,8]]=>0010010=>0100100 [3,3,1,1]=>[[1,2,3],[4,5,6],[7],[8]]=>0010011=>1100100 [3,2,2,1]=>[[1,2,3],[4,5],[6,7],[8]]=>0010101=>1010100 [3,2,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8]]=>0010111=>1110100 [3,1,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7],[8]]=>0011111=>1111100 [2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8]]=>0101010=>0101010 [2,2,2,1,1]=>[[1,2],[3,4],[5,6],[7],[8]]=>0101011=>1101010 [2,2,1,1,1,1]=>[[1,2],[3,4],[5],[6],[7],[8]]=>0101111=>1111010 [2,1,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7],[8]]=>0111111=>1111110 [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]=>[[1,2,3,4,5,6,7,8],[9]]=>00000001=>10000000 [7,2]=>[[1,2,3,4,5,6,7],[8,9]]=>00000010=>01000000 [7,1,1]=>[[1,2,3,4,5,6,7],[8],[9]]=>00000011=>11000000 [6,3]=>[[1,2,3,4,5,6],[7,8,9]]=>00000100=>00100000 [6,2,1]=>[[1,2,3,4,5,6],[7,8],[9]]=>00000101=>10100000 [6,1,1,1]=>[[1,2,3,4,5,6],[7],[8],[9]]=>00000111=>11100000 [5,4]=>[[1,2,3,4,5],[6,7,8,9]]=>00001000=>00010000 [5,3,1]=>[[1,2,3,4,5],[6,7,8],[9]]=>00001001=>10010000 [5,2,2]=>[[1,2,3,4,5],[6,7],[8,9]]=>00001010=>01010000 [5,2,1,1]=>[[1,2,3,4,5],[6,7],[8],[9]]=>00001011=>11010000 [5,1,1,1,1]=>[[1,2,3,4,5],[6],[7],[8],[9]]=>00001111=>11110000 [4,4,1]=>[[1,2,3,4],[5,6,7,8],[9]]=>00010001=>10001000 [4,3,2]=>[[1,2,3,4],[5,6,7],[8,9]]=>00010010=>01001000 [4,3,1,1]=>[[1,2,3,4],[5,6,7],[8],[9]]=>00010011=>11001000 [4,2,2,1]=>[[1,2,3,4],[5,6],[7,8],[9]]=>00010101=>10101000 [4,2,1,1,1]=>[[1,2,3,4],[5,6],[7],[8],[9]]=>00010111=>11101000 [4,1,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8],[9]]=>00011111=>11111000 [3,3,3]=>[[1,2,3],[4,5,6],[7,8,9]]=>00100100=>00100100 [3,3,2,1]=>[[1,2,3],[4,5,6],[7,8],[9]]=>00100101=>10100100 [3,3,1,1,1]=>[[1,2,3],[4,5,6],[7],[8],[9]]=>00100111=>11100100 [3,2,2,2]=>[[1,2,3],[4,5],[6,7],[8,9]]=>00101010=>01010100 [3,2,2,1,1]=>[[1,2,3],[4,5],[6,7],[8],[9]]=>00101011=>11010100 [3,2,1,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8],[9]]=>00101111=>11110100 [3,1,1,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7],[8],[9]]=>00111111=>11111100 [2,2,2,2,1]=>[[1,2],[3,4],[5,6],[7,8],[9]]=>01010101=>10101010 [2,2,2,1,1,1]=>[[1,2],[3,4],[5,6],[7],[8],[9]]=>01010111=>11101010 [2,2,1,1,1,1,1]=>[[1,2],[3,4],[5],[6],[7],[8],[9]]=>01011111=>11111010 [2,1,1,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7],[8],[9]]=>01111111=>11111110 [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]=>[[1,2,3,4,5,6,7,8,9],[10]]=>000000001=>100000000 [8,2]=>[[1,2,3,4,5,6,7,8],[9,10]]=>000000010=>010000000 [8,1,1]=>[[1,2,3,4,5,6,7,8],[9],[10]]=>000000011=>110000000 [7,3]=>[[1,2,3,4,5,6,7],[8,9,10]]=>000000100=>001000000 [7,2,1]=>[[1,2,3,4,5,6,7],[8,9],[10]]=>000000101=>101000000 [7,1,1,1]=>[[1,2,3,4,5,6,7],[8],[9],[10]]=>000000111=>111000000 [6,4]=>[[1,2,3,4,5,6],[7,8,9,10]]=>000001000=>000100000 [6,3,1]=>[[1,2,3,4,5,6],[7,8,9],[10]]=>000001001=>100100000 [6,2,2]=>[[1,2,3,4,5,6],[7,8],[9,10]]=>000001010=>010100000 [6,2,1,1]=>[[1,2,3,4,5,6],[7,8],[9],[10]]=>000001011=>110100000 [6,1,1,1,1]=>[[1,2,3,4,5,6],[7],[8],[9],[10]]=>000001111=>111100000 [5,5]=>[[1,2,3,4,5],[6,7,8,9,10]]=>000010000=>000010000 [5,4,1]=>[[1,2,3,4,5],[6,7,8,9],[10]]=>000010001=>100010000 [5,3,2]=>[[1,2,3,4,5],[6,7,8],[9,10]]=>000010010=>010010000 [5,3,1,1]=>[[1,2,3,4,5],[6,7,8],[9],[10]]=>000010011=>110010000 [5,2,2,1]=>[[1,2,3,4,5],[6,7],[8,9],[10]]=>000010101=>101010000 [5,2,1,1,1]=>[[1,2,3,4,5],[6,7],[8],[9],[10]]=>000010111=>111010000 [5,1,1,1,1,1]=>[[1,2,3,4,5],[6],[7],[8],[9],[10]]=>000011111=>111110000 [4,4,2]=>[[1,2,3,4],[5,6,7,8],[9,10]]=>000100010=>010001000 [4,4,1,1]=>[[1,2,3,4],[5,6,7,8],[9],[10]]=>000100011=>110001000 [4,3,3]=>[[1,2,3,4],[5,6,7],[8,9,10]]=>000100100=>001001000 [4,3,2,1]=>[[1,2,3,4],[5,6,7],[8,9],[10]]=>000100101=>101001000 [4,3,1,1,1]=>[[1,2,3,4],[5,6,7],[8],[9],[10]]=>000100111=>111001000 [4,2,2,2]=>[[1,2,3,4],[5,6],[7,8],[9,10]]=>000101010=>010101000 [4,2,2,1,1]=>[[1,2,3,4],[5,6],[7,8],[9],[10]]=>000101011=>110101000 [4,2,1,1,1,1]=>[[1,2,3,4],[5,6],[7],[8],[9],[10]]=>000101111=>111101000 [4,1,1,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8],[9],[10]]=>000111111=>111111000 [3,3,3,1]=>[[1,2,3],[4,5,6],[7,8,9],[10]]=>001001001=>100100100 [3,3,2,2]=>[[1,2,3],[4,5,6],[7,8],[9,10]]=>001001010=>010100100 [3,3,2,1,1]=>[[1,2,3],[4,5,6],[7,8],[9],[10]]=>001001011=>110100100 [3,3,1,1,1,1]=>[[1,2,3],[4,5,6],[7],[8],[9],[10]]=>001001111=>111100100 [3,2,2,2,1]=>[[1,2,3],[4,5],[6,7],[8,9],[10]]=>001010101=>101010100 [3,2,2,1,1,1]=>[[1,2,3],[4,5],[6,7],[8],[9],[10]]=>001010111=>111010100 [3,2,1,1,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8],[9],[10]]=>001011111=>111110100 [3,1,1,1,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7],[8],[9],[10]]=>001111111=>111111100 [2,2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8],[9,10]]=>010101010=>010101010 [2,2,2,2,1,1]=>[[1,2],[3,4],[5,6],[7,8],[9],[10]]=>010101011=>110101010 [2,2,2,1,1,1,1]=>[[1,2],[3,4],[5,6],[7],[8],[9],[10]]=>010101111=>111101010 [2,2,1,1,1,1,1,1]=>[[1,2],[3,4],[5],[6],[7],[8],[9],[10]]=>010111111=>111111010 [2,1,1,1,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]=>011111111=>111111110 [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
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.
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.