Identifier
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
Images
=>
Cc0002;cc-rep-0Cc0007;cc-rep-1
[1]=>[[1]]=> [2]=>[[1,2]]=>0 [1,1]=>[[1],[2]]=>1 [3]=>[[1,2,3]]=>00 [2,1]=>[[1,3],[2]]=>10 [1,1,1]=>[[1],[2],[3]]=>11 [4]=>[[1,2,3,4]]=>000 [3,1]=>[[1,3,4],[2]]=>100 [2,2]=>[[1,2],[3,4]]=>010 [2,1,1]=>[[1,4],[2],[3]]=>110 [1,1,1,1]=>[[1],[2],[3],[4]]=>111 [5]=>[[1,2,3,4,5]]=>0000 [4,1]=>[[1,3,4,5],[2]]=>1000 [3,2]=>[[1,2,5],[3,4]]=>0100 [3,1,1]=>[[1,4,5],[2],[3]]=>1100 [2,2,1]=>[[1,3],[2,5],[4]]=>1010 [2,1,1,1]=>[[1,5],[2],[3],[4]]=>1110 [1,1,1,1,1]=>[[1],[2],[3],[4],[5]]=>1111 [6]=>[[1,2,3,4,5,6]]=>00000 [5,1]=>[[1,3,4,5,6],[2]]=>10000 [4,2]=>[[1,2,5,6],[3,4]]=>01000 [4,1,1]=>[[1,4,5,6],[2],[3]]=>11000 [3,3]=>[[1,2,3],[4,5,6]]=>00100 [3,2,1]=>[[1,3,6],[2,5],[4]]=>10100 [3,1,1,1]=>[[1,5,6],[2],[3],[4]]=>11100 [2,2,2]=>[[1,2],[3,4],[5,6]]=>01010 [2,2,1,1]=>[[1,4],[2,6],[3],[5]]=>11010 [2,1,1,1,1]=>[[1,6],[2],[3],[4],[5]]=>11110 [1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6]]=>11111 [7]=>[[1,2,3,4,5,6,7]]=>000000 [6,1]=>[[1,3,4,5,6,7],[2]]=>100000 [5,2]=>[[1,2,5,6,7],[3,4]]=>010000 [5,1,1]=>[[1,4,5,6,7],[2],[3]]=>110000 [4,3]=>[[1,2,3,7],[4,5,6]]=>001000 [4,2,1]=>[[1,3,6,7],[2,5],[4]]=>101000 [4,1,1,1]=>[[1,5,6,7],[2],[3],[4]]=>111000 [3,3,1]=>[[1,3,4],[2,6,7],[5]]=>100100 [3,2,2]=>[[1,2,7],[3,4],[5,6]]=>010100 [3,2,1,1]=>[[1,4,7],[2,6],[3],[5]]=>110100 [3,1,1,1,1]=>[[1,6,7],[2],[3],[4],[5]]=>111100 [2,2,2,1]=>[[1,3],[2,5],[4,7],[6]]=>101010 [2,2,1,1,1]=>[[1,5],[2,7],[3],[4],[6]]=>111010 [2,1,1,1,1,1]=>[[1,7],[2],[3],[4],[5],[6]]=>111110 [1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7]]=>111111 [8]=>[[1,2,3,4,5,6,7,8]]=>0000000 [7,1]=>[[1,3,4,5,6,7,8],[2]]=>1000000 [6,2]=>[[1,2,5,6,7,8],[3,4]]=>0100000 [6,1,1]=>[[1,4,5,6,7,8],[2],[3]]=>1100000 [5,3]=>[[1,2,3,7,8],[4,5,6]]=>0010000 [5,2,1]=>[[1,3,6,7,8],[2,5],[4]]=>1010000 [5,1,1,1]=>[[1,5,6,7,8],[2],[3],[4]]=>1110000 [4,4]=>[[1,2,3,4],[5,6,7,8]]=>0001000 [4,3,1]=>[[1,3,4,8],[2,6,7],[5]]=>1001000 [4,2,2]=>[[1,2,7,8],[3,4],[5,6]]=>0101000 [4,2,1,1]=>[[1,4,7,8],[2,6],[3],[5]]=>1101000 [4,1,1,1,1]=>[[1,6,7,8],[2],[3],[4],[5]]=>1111000 [3,3,2]=>[[1,2,5],[3,4,8],[6,7]]=>0100100 [3,3,1,1]=>[[1,4,5],[2,7,8],[3],[6]]=>1100100 [3,2,2,1]=>[[1,3,8],[2,5],[4,7],[6]]=>1010100 [3,2,1,1,1]=>[[1,5,8],[2,7],[3],[4],[6]]=>1110100 [3,1,1,1,1,1]=>[[1,7,8],[2],[3],[4],[5],[6]]=>1111100 [2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8]]=>0101010 [2,2,2,1,1]=>[[1,4],[2,6],[3,8],[5],[7]]=>1101010 [2,2,1,1,1,1]=>[[1,6],[2,8],[3],[4],[5],[7]]=>1111010 [2,1,1,1,1,1,1]=>[[1,8],[2],[3],[4],[5],[6],[7]]=>1111110 [1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8]]=>1111111 [9]=>[[1,2,3,4,5,6,7,8,9]]=>00000000 [8,1]=>[[1,3,4,5,6,7,8,9],[2]]=>10000000 [7,2]=>[[1,2,5,6,7,8,9],[3,4]]=>01000000 [7,1,1]=>[[1,4,5,6,7,8,9],[2],[3]]=>11000000 [6,3]=>[[1,2,3,7,8,9],[4,5,6]]=>00100000 [6,2,1]=>[[1,3,6,7,8,9],[2,5],[4]]=>10100000 [6,1,1,1]=>[[1,5,6,7,8,9],[2],[3],[4]]=>11100000 [5,4]=>[[1,2,3,4,9],[5,6,7,8]]=>00010000 [5,3,1]=>[[1,3,4,8,9],[2,6,7],[5]]=>10010000 [5,2,2]=>[[1,2,7,8,9],[3,4],[5,6]]=>01010000 [5,2,1,1]=>[[1,4,7,8,9],[2,6],[3],[5]]=>11010000 [5,1,1,1,1]=>[[1,6,7,8,9],[2],[3],[4],[5]]=>11110000 [4,4,1]=>[[1,3,4,5],[2,7,8,9],[6]]=>10001000 [4,3,2]=>[[1,2,5,9],[3,4,8],[6,7]]=>01001000 [4,3,1,1]=>[[1,4,5,9],[2,7,8],[3],[6]]=>11001000 [4,2,2,1]=>[[1,3,8,9],[2,5],[4,7],[6]]=>10101000 [4,2,1,1,1]=>[[1,5,8,9],[2,7],[3],[4],[6]]=>11101000 [4,1,1,1,1,1]=>[[1,7,8,9],[2],[3],[4],[5],[6]]=>11111000 [3,3,3]=>[[1,2,3],[4,5,6],[7,8,9]]=>00100100 [3,3,2,1]=>[[1,3,6],[2,5,9],[4,8],[7]]=>10100100 [3,3,1,1,1]=>[[1,5,6],[2,8,9],[3],[4],[7]]=>11100100 [3,2,2,2]=>[[1,2,9],[3,4],[5,6],[7,8]]=>01010100 [3,2,2,1,1]=>[[1,4,9],[2,6],[3,8],[5],[7]]=>11010100 [3,2,1,1,1,1]=>[[1,6,9],[2,8],[3],[4],[5],[7]]=>11110100 [3,1,1,1,1,1,1]=>[[1,8,9],[2],[3],[4],[5],[6],[7]]=>11111100 [2,2,2,2,1]=>[[1,3],[2,5],[4,7],[6,9],[8]]=>10101010 [2,2,2,1,1,1]=>[[1,5],[2,7],[3,9],[4],[6],[8]]=>11101010 [2,2,1,1,1,1,1]=>[[1,7],[2,9],[3],[4],[5],[6],[8]]=>11111010 [2,1,1,1,1,1,1,1]=>[[1,9],[2],[3],[4],[5],[6],[7],[8]]=>11111110 [1,1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8],[9]]=>11111111 [10]=>[[1,2,3,4,5,6,7,8,9,10]]=>000000000 [9,1]=>[[1,3,4,5,6,7,8,9,10],[2]]=>100000000 [8,2]=>[[1,2,5,6,7,8,9,10],[3,4]]=>010000000 [8,1,1]=>[[1,4,5,6,7,8,9,10],[2],[3]]=>110000000 [7,3]=>[[1,2,3,7,8,9,10],[4,5,6]]=>001000000 [7,2,1]=>[[1,3,6,7,8,9,10],[2,5],[4]]=>101000000 [7,1,1,1]=>[[1,5,6,7,8,9,10],[2],[3],[4]]=>111000000 [6,4]=>[[1,2,3,4,9,10],[5,6,7,8]]=>000100000 [6,3,1]=>[[1,3,4,8,9,10],[2,6,7],[5]]=>100100000 [6,2,2]=>[[1,2,7,8,9,10],[3,4],[5,6]]=>010100000 [6,2,1,1]=>[[1,4,7,8,9,10],[2,6],[3],[5]]=>110100000 [6,1,1,1,1]=>[[1,6,7,8,9,10],[2],[3],[4],[5]]=>111100000 [5,5]=>[[1,2,3,4,5],[6,7,8,9,10]]=>000010000 [5,4,1]=>[[1,3,4,5,10],[2,7,8,9],[6]]=>100010000 [5,3,2]=>[[1,2,5,9,10],[3,4,8],[6,7]]=>010010000 [5,3,1,1]=>[[1,4,5,9,10],[2,7,8],[3],[6]]=>110010000 [5,2,2,1]=>[[1,3,8,9,10],[2,5],[4,7],[6]]=>101010000 [5,2,1,1,1]=>[[1,5,8,9,10],[2,7],[3],[4],[6]]=>111010000 [5,1,1,1,1,1]=>[[1,7,8,9,10],[2],[3],[4],[5],[6]]=>111110000 [4,4,2]=>[[1,2,5,6],[3,4,9,10],[7,8]]=>010001000 [4,4,1,1]=>[[1,4,5,6],[2,8,9,10],[3],[7]]=>110001000 [4,3,3]=>[[1,2,3,10],[4,5,6],[7,8,9]]=>001001000 [4,3,2,1]=>[[1,3,6,10],[2,5,9],[4,8],[7]]=>101001000 [4,3,1,1,1]=>[[1,5,6,10],[2,8,9],[3],[4],[7]]=>111001000 [4,2,2,2]=>[[1,2,9,10],[3,4],[5,6],[7,8]]=>010101000 [4,2,2,1,1]=>[[1,4,9,10],[2,6],[3,8],[5],[7]]=>110101000 [4,2,1,1,1,1]=>[[1,6,9,10],[2,8],[3],[4],[5],[7]]=>111101000 [4,1,1,1,1,1,1]=>[[1,8,9,10],[2],[3],[4],[5],[6],[7]]=>111111000 [3,3,3,1]=>[[1,3,4],[2,6,7],[5,9,10],[8]]=>100100100 [3,3,2,2]=>[[1,2,7],[3,4,10],[5,6],[8,9]]=>010100100 [3,3,2,1,1]=>[[1,4,7],[2,6,10],[3,9],[5],[8]]=>110100100 [3,3,1,1,1,1]=>[[1,6,7],[2,9,10],[3],[4],[5],[8]]=>111100100 [3,2,2,2,1]=>[[1,3,10],[2,5],[4,7],[6,9],[8]]=>101010100 [3,2,2,1,1,1]=>[[1,5,10],[2,7],[3,9],[4],[6],[8]]=>111010100 [3,2,1,1,1,1,1]=>[[1,7,10],[2,9],[3],[4],[5],[6],[8]]=>111110100 [3,1,1,1,1,1,1,1]=>[[1,9,10],[2],[3],[4],[5],[6],[7],[8]]=>111111100 [2,2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8],[9,10]]=>010101010 [2,2,2,2,1,1]=>[[1,4],[2,6],[3,8],[5,10],[7],[9]]=>110101010 [2,2,2,1,1,1,1]=>[[1,6],[2,8],[3,10],[4],[5],[7],[9]]=>111101010 [2,2,1,1,1,1,1,1]=>[[1,8],[2,10],[3],[4],[5],[6],[7],[9]]=>111111010 [2,1,1,1,1,1,1,1,1]=>[[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]=>111111110 [1,1,1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]=>111111111 [6,6]=>[[1,2,3,4,5,6],[7,8,9,10,11,12]]=>00000100000 [2,2,2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]=>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 $1\leq i < n$.