Identifier
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
Images
=>
Cc0002;cc-rep-0Cc0007;cc-rep-1
[1]=>[[1]]=>[1]=> [2]=>[[1,2]]=>[1,2]=>0 [1,1]=>[[1],[2]]=>[2,1]=>1 [3]=>[[1,2,3]]=>[1,2,3]=>00 [2,1]=>[[1,2],[3]]=>[3,1,2]=>10 [1,1,1]=>[[1],[2],[3]]=>[3,2,1]=>11 [4]=>[[1,2,3,4]]=>[1,2,3,4]=>000 [3,1]=>[[1,2,3],[4]]=>[4,1,2,3]=>100 [2,2]=>[[1,2],[3,4]]=>[3,4,1,2]=>010 [2,1,1]=>[[1,2],[3],[4]]=>[4,3,1,2]=>110 [1,1,1,1]=>[[1],[2],[3],[4]]=>[4,3,2,1]=>111 [5]=>[[1,2,3,4,5]]=>[1,2,3,4,5]=>0000 [4,1]=>[[1,2,3,4],[5]]=>[5,1,2,3,4]=>1000 [3,2]=>[[1,2,3],[4,5]]=>[4,5,1,2,3]=>0100 [3,1,1]=>[[1,2,3],[4],[5]]=>[5,4,1,2,3]=>1100 [2,2,1]=>[[1,2],[3,4],[5]]=>[5,3,4,1,2]=>1010 [2,1,1,1]=>[[1,2],[3],[4],[5]]=>[5,4,3,1,2]=>1110 [1,1,1,1,1]=>[[1],[2],[3],[4],[5]]=>[5,4,3,2,1]=>1111 [6]=>[[1,2,3,4,5,6]]=>[1,2,3,4,5,6]=>00000 [5,1]=>[[1,2,3,4,5],[6]]=>[6,1,2,3,4,5]=>10000 [4,2]=>[[1,2,3,4],[5,6]]=>[5,6,1,2,3,4]=>01000 [4,1,1]=>[[1,2,3,4],[5],[6]]=>[6,5,1,2,3,4]=>11000 [3,3]=>[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>00100 [3,2,1]=>[[1,2,3],[4,5],[6]]=>[6,4,5,1,2,3]=>10100 [3,1,1,1]=>[[1,2,3],[4],[5],[6]]=>[6,5,4,1,2,3]=>11100 [2,2,2]=>[[1,2],[3,4],[5,6]]=>[5,6,3,4,1,2]=>01010 [2,2,1,1]=>[[1,2],[3,4],[5],[6]]=>[6,5,3,4,1,2]=>11010 [2,1,1,1,1]=>[[1,2],[3],[4],[5],[6]]=>[6,5,4,3,1,2]=>11110 [1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1]=>11111 [7]=>[[1,2,3,4,5,6,7]]=>[1,2,3,4,5,6,7]=>000000 [6,1]=>[[1,2,3,4,5,6],[7]]=>[7,1,2,3,4,5,6]=>100000 [5,2]=>[[1,2,3,4,5],[6,7]]=>[6,7,1,2,3,4,5]=>010000 [5,1,1]=>[[1,2,3,4,5],[6],[7]]=>[7,6,1,2,3,4,5]=>110000 [4,3]=>[[1,2,3,4],[5,6,7]]=>[5,6,7,1,2,3,4]=>001000 [4,2,1]=>[[1,2,3,4],[5,6],[7]]=>[7,5,6,1,2,3,4]=>101000 [4,1,1,1]=>[[1,2,3,4],[5],[6],[7]]=>[7,6,5,1,2,3,4]=>111000 [3,3,1]=>[[1,2,3],[4,5,6],[7]]=>[7,4,5,6,1,2,3]=>100100 [3,2,2]=>[[1,2,3],[4,5],[6,7]]=>[6,7,4,5,1,2,3]=>010100 [3,2,1,1]=>[[1,2,3],[4,5],[6],[7]]=>[7,6,4,5,1,2,3]=>110100 [3,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7]]=>[7,6,5,4,1,2,3]=>111100 [2,2,2,1]=>[[1,2],[3,4],[5,6],[7]]=>[7,5,6,3,4,1,2]=>101010 [2,2,1,1,1]=>[[1,2],[3,4],[5],[6],[7]]=>[7,6,5,3,4,1,2]=>111010 [2,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,1,2]=>111110 [1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,2,1]=>111111 [8]=>[[1,2,3,4,5,6,7,8]]=>[1,2,3,4,5,6,7,8]=>0000000 [7,1]=>[[1,2,3,4,5,6,7],[8]]=>[8,1,2,3,4,5,6,7]=>1000000 [6,2]=>[[1,2,3,4,5,6],[7,8]]=>[7,8,1,2,3,4,5,6]=>0100000 [6,1,1]=>[[1,2,3,4,5,6],[7],[8]]=>[8,7,1,2,3,4,5,6]=>1100000 [5,3]=>[[1,2,3,4,5],[6,7,8]]=>[6,7,8,1,2,3,4,5]=>0010000 [5,2,1]=>[[1,2,3,4,5],[6,7],[8]]=>[8,6,7,1,2,3,4,5]=>1010000 [5,1,1,1]=>[[1,2,3,4,5],[6],[7],[8]]=>[8,7,6,1,2,3,4,5]=>1110000 [4,4]=>[[1,2,3,4],[5,6,7,8]]=>[5,6,7,8,1,2,3,4]=>0001000 [4,3,1]=>[[1,2,3,4],[5,6,7],[8]]=>[8,5,6,7,1,2,3,4]=>1001000 [4,2,2]=>[[1,2,3,4],[5,6],[7,8]]=>[7,8,5,6,1,2,3,4]=>0101000 [4,2,1,1]=>[[1,2,3,4],[5,6],[7],[8]]=>[8,7,5,6,1,2,3,4]=>1101000 [4,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8]]=>[8,7,6,5,1,2,3,4]=>1111000 [3,3,2]=>[[1,2,3],[4,5,6],[7,8]]=>[7,8,4,5,6,1,2,3]=>0100100 [3,3,1,1]=>[[1,2,3],[4,5,6],[7],[8]]=>[8,7,4,5,6,1,2,3]=>1100100 [3,2,2,1]=>[[1,2,3],[4,5],[6,7],[8]]=>[8,6,7,4,5,1,2,3]=>1010100 [3,2,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8]]=>[8,7,6,4,5,1,2,3]=>1110100 [3,1,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,1,2,3]=>1111100 [2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8]]=>[7,8,5,6,3,4,1,2]=>0101010 [2,2,2,1,1]=>[[1,2],[3,4],[5,6],[7],[8]]=>[8,7,5,6,3,4,1,2]=>1101010 [2,2,1,1,1,1]=>[[1,2],[3,4],[5],[6],[7],[8]]=>[8,7,6,5,3,4,1,2]=>1111010 [2,1,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,1,2]=>1111110 [1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,2,1]=>1111111 [9]=>[[1,2,3,4,5,6,7,8,9]]=>[1,2,3,4,5,6,7,8,9]=>00000000 [8,1]=>[[1,2,3,4,5,6,7,8],[9]]=>[9,1,2,3,4,5,6,7,8]=>10000000 [7,2]=>[[1,2,3,4,5,6,7],[8,9]]=>[8,9,1,2,3,4,5,6,7]=>01000000 [7,1,1]=>[[1,2,3,4,5,6,7],[8],[9]]=>[9,8,1,2,3,4,5,6,7]=>11000000 [6,3]=>[[1,2,3,4,5,6],[7,8,9]]=>[7,8,9,1,2,3,4,5,6]=>00100000 [6,2,1]=>[[1,2,3,4,5,6],[7,8],[9]]=>[9,7,8,1,2,3,4,5,6]=>10100000 [6,1,1,1]=>[[1,2,3,4,5,6],[7],[8],[9]]=>[9,8,7,1,2,3,4,5,6]=>11100000 [5,4]=>[[1,2,3,4,5],[6,7,8,9]]=>[6,7,8,9,1,2,3,4,5]=>00010000 [5,3,1]=>[[1,2,3,4,5],[6,7,8],[9]]=>[9,6,7,8,1,2,3,4,5]=>10010000 [5,2,2]=>[[1,2,3,4,5],[6,7],[8,9]]=>[8,9,6,7,1,2,3,4,5]=>01010000 [5,2,1,1]=>[[1,2,3,4,5],[6,7],[8],[9]]=>[9,8,6,7,1,2,3,4,5]=>11010000 [5,1,1,1,1]=>[[1,2,3,4,5],[6],[7],[8],[9]]=>[9,8,7,6,1,2,3,4,5]=>11110000 [4,4,1]=>[[1,2,3,4],[5,6,7,8],[9]]=>[9,5,6,7,8,1,2,3,4]=>10001000 [4,3,2]=>[[1,2,3,4],[5,6,7],[8,9]]=>[8,9,5,6,7,1,2,3,4]=>01001000 [4,3,1,1]=>[[1,2,3,4],[5,6,7],[8],[9]]=>[9,8,5,6,7,1,2,3,4]=>11001000 [4,2,2,1]=>[[1,2,3,4],[5,6],[7,8],[9]]=>[9,7,8,5,6,1,2,3,4]=>10101000 [4,2,1,1,1]=>[[1,2,3,4],[5,6],[7],[8],[9]]=>[9,8,7,5,6,1,2,3,4]=>11101000 [4,1,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8],[9]]=>[9,8,7,6,5,1,2,3,4]=>11111000 [3,3,3]=>[[1,2,3],[4,5,6],[7,8,9]]=>[7,8,9,4,5,6,1,2,3]=>00100100 [3,3,2,1]=>[[1,2,3],[4,5,6],[7,8],[9]]=>[9,7,8,4,5,6,1,2,3]=>10100100 [3,3,1,1,1]=>[[1,2,3],[4,5,6],[7],[8],[9]]=>[9,8,7,4,5,6,1,2,3]=>11100100 [3,2,2,2]=>[[1,2,3],[4,5],[6,7],[8,9]]=>[8,9,6,7,4,5,1,2,3]=>01010100 [3,2,2,1,1]=>[[1,2,3],[4,5],[6,7],[8],[9]]=>[9,8,6,7,4,5,1,2,3]=>11010100 [3,2,1,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8],[9]]=>[9,8,7,6,4,5,1,2,3]=>11110100 [3,1,1,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7],[8],[9]]=>[9,8,7,6,5,4,1,2,3]=>11111100 [2,2,2,2,1]=>[[1,2],[3,4],[5,6],[7,8],[9]]=>[9,7,8,5,6,3,4,1,2]=>10101010 [2,2,2,1,1,1]=>[[1,2],[3,4],[5,6],[7],[8],[9]]=>[9,8,7,5,6,3,4,1,2]=>11101010 [2,2,1,1,1,1,1]=>[[1,2],[3,4],[5],[6],[7],[8],[9]]=>[9,8,7,6,5,3,4,1,2]=>11111010 [2,1,1,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7],[8],[9]]=>[9,8,7,6,5,4,3,1,2]=>11111110 [1,1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8],[9]]=>[9,8,7,6,5,4,3,2,1]=>11111111 [10]=>[[1,2,3,4,5,6,7,8,9,10]]=>[1,2,3,4,5,6,7,8,9,10]=>000000000 [9,1]=>[[1,2,3,4,5,6,7,8,9],[10]]=>[10,1,2,3,4,5,6,7,8,9]=>100000000 [8,2]=>[[1,2,3,4,5,6,7,8],[9,10]]=>[9,10,1,2,3,4,5,6,7,8]=>010000000 [8,1,1]=>[[1,2,3,4,5,6,7,8],[9],[10]]=>[10,9,1,2,3,4,5,6,7,8]=>110000000 [7,3]=>[[1,2,3,4,5,6,7],[8,9,10]]=>[8,9,10,1,2,3,4,5,6,7]=>001000000 [7,2,1]=>[[1,2,3,4,5,6,7],[8,9],[10]]=>[10,8,9,1,2,3,4,5,6,7]=>101000000 [7,1,1,1]=>[[1,2,3,4,5,6,7],[8],[9],[10]]=>[10,9,8,1,2,3,4,5,6,7]=>111000000 [6,4]=>[[1,2,3,4,5,6],[7,8,9,10]]=>[7,8,9,10,1,2,3,4,5,6]=>000100000 [6,3,1]=>[[1,2,3,4,5,6],[7,8,9],[10]]=>[10,7,8,9,1,2,3,4,5,6]=>100100000 [6,2,2]=>[[1,2,3,4,5,6],[7,8],[9,10]]=>[9,10,7,8,1,2,3,4,5,6]=>010100000 [6,2,1,1]=>[[1,2,3,4,5,6],[7,8],[9],[10]]=>[10,9,7,8,1,2,3,4,5,6]=>110100000 [6,1,1,1,1]=>[[1,2,3,4,5,6],[7],[8],[9],[10]]=>[10,9,8,7,1,2,3,4,5,6]=>111100000 [5,5]=>[[1,2,3,4,5],[6,7,8,9,10]]=>[6,7,8,9,10,1,2,3,4,5]=>000010000 [5,4,1]=>[[1,2,3,4,5],[6,7,8,9],[10]]=>[10,6,7,8,9,1,2,3,4,5]=>100010000 [5,3,2]=>[[1,2,3,4,5],[6,7,8],[9,10]]=>[9,10,6,7,8,1,2,3,4,5]=>010010000 [5,3,1,1]=>[[1,2,3,4,5],[6,7,8],[9],[10]]=>[10,9,6,7,8,1,2,3,4,5]=>110010000 [5,2,2,1]=>[[1,2,3,4,5],[6,7],[8,9],[10]]=>[10,8,9,6,7,1,2,3,4,5]=>101010000 [5,2,1,1,1]=>[[1,2,3,4,5],[6,7],[8],[9],[10]]=>[10,9,8,6,7,1,2,3,4,5]=>111010000 [5,1,1,1,1,1]=>[[1,2,3,4,5],[6],[7],[8],[9],[10]]=>[10,9,8,7,6,1,2,3,4,5]=>111110000 [4,4,2]=>[[1,2,3,4],[5,6,7,8],[9,10]]=>[9,10,5,6,7,8,1,2,3,4]=>010001000 [4,4,1,1]=>[[1,2,3,4],[5,6,7,8],[9],[10]]=>[10,9,5,6,7,8,1,2,3,4]=>110001000 [4,3,3]=>[[1,2,3,4],[5,6,7],[8,9,10]]=>[8,9,10,5,6,7,1,2,3,4]=>001001000 [4,3,2,1]=>[[1,2,3,4],[5,6,7],[8,9],[10]]=>[10,8,9,5,6,7,1,2,3,4]=>101001000 [4,3,1,1,1]=>[[1,2,3,4],[5,6,7],[8],[9],[10]]=>[10,9,8,5,6,7,1,2,3,4]=>111001000 [4,2,2,2]=>[[1,2,3,4],[5,6],[7,8],[9,10]]=>[9,10,7,8,5,6,1,2,3,4]=>010101000 [4,2,2,1,1]=>[[1,2,3,4],[5,6],[7,8],[9],[10]]=>[10,9,7,8,5,6,1,2,3,4]=>110101000 [4,2,1,1,1,1]=>[[1,2,3,4],[5,6],[7],[8],[9],[10]]=>[10,9,8,7,5,6,1,2,3,4]=>111101000 [4,1,1,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8],[9],[10]]=>[10,9,8,7,6,5,1,2,3,4]=>111111000 [3,3,3,1]=>[[1,2,3],[4,5,6],[7,8,9],[10]]=>[10,7,8,9,4,5,6,1,2,3]=>100100100 [3,3,2,2]=>[[1,2,3],[4,5,6],[7,8],[9,10]]=>[9,10,7,8,4,5,6,1,2,3]=>010100100 [3,3,2,1,1]=>[[1,2,3],[4,5,6],[7,8],[9],[10]]=>[10,9,7,8,4,5,6,1,2,3]=>110100100 [3,3,1,1,1,1]=>[[1,2,3],[4,5,6],[7],[8],[9],[10]]=>[10,9,8,7,4,5,6,1,2,3]=>111100100 [3,2,2,2,1]=>[[1,2,3],[4,5],[6,7],[8,9],[10]]=>[10,8,9,6,7,4,5,1,2,3]=>101010100 [3,2,2,1,1,1]=>[[1,2,3],[4,5],[6,7],[8],[9],[10]]=>[10,9,8,6,7,4,5,1,2,3]=>111010100 [3,2,1,1,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8],[9],[10]]=>[10,9,8,7,6,4,5,1,2,3]=>111110100 [3,1,1,1,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7],[8],[9],[10]]=>[10,9,8,7,6,5,4,1,2,3]=>111111100 [2,2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8],[9,10]]=>[9,10,7,8,5,6,3,4,1,2]=>010101010 [2,2,2,2,1,1]=>[[1,2],[3,4],[5,6],[7,8],[9],[10]]=>[10,9,7,8,5,6,3,4,1,2]=>110101010 [2,2,2,1,1,1,1]=>[[1,2],[3,4],[5,6],[7],[8],[9],[10]]=>[10,9,8,7,5,6,3,4,1,2]=>111101010 [2,2,1,1,1,1,1,1]=>[[1,2],[3,4],[5],[6],[7],[8],[9],[10]]=>[10,9,8,7,6,5,3,4,1,2]=>111111010 [2,1,1,1,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]=>[10,9,8,7,6,5,4,3,1,2]=>111111110 [1,1,1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]=>[10,9,8,7,6,5,4,3,2,1]=>111111111 [12]=>[[1,2,3,4,5,6,7,8,9,10,11,12]]=>[1,2,3,4,5,6,7,8,9,10,11,12]=>00000000000 [6,6]=>[[1,2,3,4,5,6],[7,8,9,10,11,12]]=>[7,8,9,10,11,12,1,2,3,4,5,6]=>00000100000 [6,4,2]=>[[1,2,3,4,5,6],[7,8,9,10],[11,12]]=>[11,12,7,8,9,10,1,2,3,4,5,6]=>01000100000 [4,4,2,2]=>[[1,2,3,4],[5,6,7,8],[9,10],[11,12]]=>[11,12,9,10,5,6,7,8,1,2,3,4]=>01010001000 [4,2,2,2,2]=>[[1,2,3,4],[5,6],[7,8],[9,10],[11,12]]=>[11,12,9,10,7,8,5,6,1,2,3,4]=>01010101000 [2,2,2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]=>[11,12,9,10,7,8,5,6,3,4,1,2]=>01010101010 [1,1,1,1,1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]=>[12,11,10,9,8,7,6,5,4,3,2,1]=>11111111111
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.
Map
descent word
Description
The descent positions of a permutation as a binary word.
For a permutation $\pi$ of $n$ letters and each $1\leq i\leq n-1$ such that $\pi(i) > \pi(i+1)$ we set $w_i=1$, otherwise $w_i=0$.
Thus, the length of the word is one less the size of the permutation. In particular, the descent word is undefined for the empty permutation.