Identifier
Mp00045:
Integer partitions
—reading tableau⟶
Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Images
=>
Cc0002;cc-rep-0Cc0007;cc-rep-1
[2]=>[[1,2]]=>0=>[2]
[1,1]=>[[1],[2]]=>1=>[1,1]
[3]=>[[1,2,3]]=>00=>[3]
[2,1]=>[[1,3],[2]]=>10=>[1,2]
[1,1,1]=>[[1],[2],[3]]=>11=>[1,1,1]
[4]=>[[1,2,3,4]]=>000=>[4]
[3,1]=>[[1,3,4],[2]]=>100=>[1,3]
[2,2]=>[[1,2],[3,4]]=>010=>[2,2]
[2,1,1]=>[[1,4],[2],[3]]=>110=>[1,1,2]
[1,1,1,1]=>[[1],[2],[3],[4]]=>111=>[1,1,1,1]
[5]=>[[1,2,3,4,5]]=>0000=>[5]
[4,1]=>[[1,3,4,5],[2]]=>1000=>[1,4]
[3,2]=>[[1,2,5],[3,4]]=>0100=>[2,3]
[3,1,1]=>[[1,4,5],[2],[3]]=>1100=>[1,1,3]
[2,2,1]=>[[1,3],[2,5],[4]]=>1010=>[1,2,2]
[2,1,1,1]=>[[1,5],[2],[3],[4]]=>1110=>[1,1,1,2]
[1,1,1,1,1]=>[[1],[2],[3],[4],[5]]=>1111=>[1,1,1,1,1]
[6]=>[[1,2,3,4,5,6]]=>00000=>[6]
[5,1]=>[[1,3,4,5,6],[2]]=>10000=>[1,5]
[4,2]=>[[1,2,5,6],[3,4]]=>01000=>[2,4]
[4,1,1]=>[[1,4,5,6],[2],[3]]=>11000=>[1,1,4]
[3,3]=>[[1,2,3],[4,5,6]]=>00100=>[3,3]
[3,2,1]=>[[1,3,6],[2,5],[4]]=>10100=>[1,2,3]
[3,1,1,1]=>[[1,5,6],[2],[3],[4]]=>11100=>[1,1,1,3]
[2,2,2]=>[[1,2],[3,4],[5,6]]=>01010=>[2,2,2]
[2,2,1,1]=>[[1,4],[2,6],[3],[5]]=>11010=>[1,1,2,2]
[2,1,1,1,1]=>[[1,6],[2],[3],[4],[5]]=>11110=>[1,1,1,1,2]
[1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6]]=>11111=>[1,1,1,1,1,1]
[7]=>[[1,2,3,4,5,6,7]]=>000000=>[7]
[6,1]=>[[1,3,4,5,6,7],[2]]=>100000=>[1,6]
[5,2]=>[[1,2,5,6,7],[3,4]]=>010000=>[2,5]
[5,1,1]=>[[1,4,5,6,7],[2],[3]]=>110000=>[1,1,5]
[4,3]=>[[1,2,3,7],[4,5,6]]=>001000=>[3,4]
[4,2,1]=>[[1,3,6,7],[2,5],[4]]=>101000=>[1,2,4]
[4,1,1,1]=>[[1,5,6,7],[2],[3],[4]]=>111000=>[1,1,1,4]
[3,3,1]=>[[1,3,4],[2,6,7],[5]]=>100100=>[1,3,3]
[3,2,2]=>[[1,2,7],[3,4],[5,6]]=>010100=>[2,2,3]
[3,2,1,1]=>[[1,4,7],[2,6],[3],[5]]=>110100=>[1,1,2,3]
[3,1,1,1,1]=>[[1,6,7],[2],[3],[4],[5]]=>111100=>[1,1,1,1,3]
[2,2,2,1]=>[[1,3],[2,5],[4,7],[6]]=>101010=>[1,2,2,2]
[2,2,1,1,1]=>[[1,5],[2,7],[3],[4],[6]]=>111010=>[1,1,1,2,2]
[2,1,1,1,1,1]=>[[1,7],[2],[3],[4],[5],[6]]=>111110=>[1,1,1,1,1,2]
[1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7]]=>111111=>[1,1,1,1,1,1,1]
[8]=>[[1,2,3,4,5,6,7,8]]=>0000000=>[8]
[7,1]=>[[1,3,4,5,6,7,8],[2]]=>1000000=>[1,7]
[6,2]=>[[1,2,5,6,7,8],[3,4]]=>0100000=>[2,6]
[6,1,1]=>[[1,4,5,6,7,8],[2],[3]]=>1100000=>[1,1,6]
[5,3]=>[[1,2,3,7,8],[4,5,6]]=>0010000=>[3,5]
[5,2,1]=>[[1,3,6,7,8],[2,5],[4]]=>1010000=>[1,2,5]
[5,1,1,1]=>[[1,5,6,7,8],[2],[3],[4]]=>1110000=>[1,1,1,5]
[4,4]=>[[1,2,3,4],[5,6,7,8]]=>0001000=>[4,4]
[4,3,1]=>[[1,3,4,8],[2,6,7],[5]]=>1001000=>[1,3,4]
[4,2,2]=>[[1,2,7,8],[3,4],[5,6]]=>0101000=>[2,2,4]
[4,2,1,1]=>[[1,4,7,8],[2,6],[3],[5]]=>1101000=>[1,1,2,4]
[4,1,1,1,1]=>[[1,6,7,8],[2],[3],[4],[5]]=>1111000=>[1,1,1,1,4]
[3,3,2]=>[[1,2,5],[3,4,8],[6,7]]=>0100100=>[2,3,3]
[3,3,1,1]=>[[1,4,5],[2,7,8],[3],[6]]=>1100100=>[1,1,3,3]
[3,2,2,1]=>[[1,3,8],[2,5],[4,7],[6]]=>1010100=>[1,2,2,3]
[3,2,1,1,1]=>[[1,5,8],[2,7],[3],[4],[6]]=>1110100=>[1,1,1,2,3]
[3,1,1,1,1,1]=>[[1,7,8],[2],[3],[4],[5],[6]]=>1111100=>[1,1,1,1,1,3]
[2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8]]=>0101010=>[2,2,2,2]
[2,2,2,1,1]=>[[1,4],[2,6],[3,8],[5],[7]]=>1101010=>[1,1,2,2,2]
[2,2,1,1,1,1]=>[[1,6],[2,8],[3],[4],[5],[7]]=>1111010=>[1,1,1,1,2,2]
[2,1,1,1,1,1,1]=>[[1,8],[2],[3],[4],[5],[6],[7]]=>1111110=>[1,1,1,1,1,1,2]
[1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8]]=>1111111=>[1,1,1,1,1,1,1,1]
[9]=>[[1,2,3,4,5,6,7,8,9]]=>00000000=>[9]
[8,1]=>[[1,3,4,5,6,7,8,9],[2]]=>10000000=>[1,8]
[7,2]=>[[1,2,5,6,7,8,9],[3,4]]=>01000000=>[2,7]
[7,1,1]=>[[1,4,5,6,7,8,9],[2],[3]]=>11000000=>[1,1,7]
[6,3]=>[[1,2,3,7,8,9],[4,5,6]]=>00100000=>[3,6]
[6,2,1]=>[[1,3,6,7,8,9],[2,5],[4]]=>10100000=>[1,2,6]
[6,1,1,1]=>[[1,5,6,7,8,9],[2],[3],[4]]=>11100000=>[1,1,1,6]
[5,4]=>[[1,2,3,4,9],[5,6,7,8]]=>00010000=>[4,5]
[5,3,1]=>[[1,3,4,8,9],[2,6,7],[5]]=>10010000=>[1,3,5]
[5,2,2]=>[[1,2,7,8,9],[3,4],[5,6]]=>01010000=>[2,2,5]
[5,2,1,1]=>[[1,4,7,8,9],[2,6],[3],[5]]=>11010000=>[1,1,2,5]
[5,1,1,1,1]=>[[1,6,7,8,9],[2],[3],[4],[5]]=>11110000=>[1,1,1,1,5]
[4,4,1]=>[[1,3,4,5],[2,7,8,9],[6]]=>10001000=>[1,4,4]
[4,3,2]=>[[1,2,5,9],[3,4,8],[6,7]]=>01001000=>[2,3,4]
[4,3,1,1]=>[[1,4,5,9],[2,7,8],[3],[6]]=>11001000=>[1,1,3,4]
[4,2,2,1]=>[[1,3,8,9],[2,5],[4,7],[6]]=>10101000=>[1,2,2,4]
[4,2,1,1,1]=>[[1,5,8,9],[2,7],[3],[4],[6]]=>11101000=>[1,1,1,2,4]
[4,1,1,1,1,1]=>[[1,7,8,9],[2],[3],[4],[5],[6]]=>11111000=>[1,1,1,1,1,4]
[3,3,3]=>[[1,2,3],[4,5,6],[7,8,9]]=>00100100=>[3,3,3]
[3,3,2,1]=>[[1,3,6],[2,5,9],[4,8],[7]]=>10100100=>[1,2,3,3]
[3,3,1,1,1]=>[[1,5,6],[2,8,9],[3],[4],[7]]=>11100100=>[1,1,1,3,3]
[3,2,2,2]=>[[1,2,9],[3,4],[5,6],[7,8]]=>01010100=>[2,2,2,3]
[3,2,2,1,1]=>[[1,4,9],[2,6],[3,8],[5],[7]]=>11010100=>[1,1,2,2,3]
[3,2,1,1,1,1]=>[[1,6,9],[2,8],[3],[4],[5],[7]]=>11110100=>[1,1,1,1,2,3]
[3,1,1,1,1,1,1]=>[[1,8,9],[2],[3],[4],[5],[6],[7]]=>11111100=>[1,1,1,1,1,1,3]
[2,2,2,2,1]=>[[1,3],[2,5],[4,7],[6,9],[8]]=>10101010=>[1,2,2,2,2]
[2,2,2,1,1,1]=>[[1,5],[2,7],[3,9],[4],[6],[8]]=>11101010=>[1,1,1,2,2,2]
[2,2,1,1,1,1,1]=>[[1,7],[2,9],[3],[4],[5],[6],[8]]=>11111010=>[1,1,1,1,1,2,2]
[2,1,1,1,1,1,1,1]=>[[1,9],[2],[3],[4],[5],[6],[7],[8]]=>11111110=>[1,1,1,1,1,1,1,2]
[1,1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8],[9]]=>11111111=>[1,1,1,1,1,1,1,1,1]
[10]=>[[1,2,3,4,5,6,7,8,9,10]]=>000000000=>[10]
[9,1]=>[[1,3,4,5,6,7,8,9,10],[2]]=>100000000=>[1,9]
[8,2]=>[[1,2,5,6,7,8,9,10],[3,4]]=>010000000=>[2,8]
[8,1,1]=>[[1,4,5,6,7,8,9,10],[2],[3]]=>110000000=>[1,1,8]
[7,3]=>[[1,2,3,7,8,9,10],[4,5,6]]=>001000000=>[3,7]
[7,2,1]=>[[1,3,6,7,8,9,10],[2,5],[4]]=>101000000=>[1,2,7]
[7,1,1,1]=>[[1,5,6,7,8,9,10],[2],[3],[4]]=>111000000=>[1,1,1,7]
[6,4]=>[[1,2,3,4,9,10],[5,6,7,8]]=>000100000=>[4,6]
[6,3,1]=>[[1,3,4,8,9,10],[2,6,7],[5]]=>100100000=>[1,3,6]
[6,2,2]=>[[1,2,7,8,9,10],[3,4],[5,6]]=>010100000=>[2,2,6]
[6,2,1,1]=>[[1,4,7,8,9,10],[2,6],[3],[5]]=>110100000=>[1,1,2,6]
[6,1,1,1,1]=>[[1,6,7,8,9,10],[2],[3],[4],[5]]=>111100000=>[1,1,1,1,6]
[5,5]=>[[1,2,3,4,5],[6,7,8,9,10]]=>000010000=>[5,5]
[5,4,1]=>[[1,3,4,5,10],[2,7,8,9],[6]]=>100010000=>[1,4,5]
[5,3,2]=>[[1,2,5,9,10],[3,4,8],[6,7]]=>010010000=>[2,3,5]
[5,3,1,1]=>[[1,4,5,9,10],[2,7,8],[3],[6]]=>110010000=>[1,1,3,5]
[5,2,2,1]=>[[1,3,8,9,10],[2,5],[4,7],[6]]=>101010000=>[1,2,2,5]
[5,2,1,1,1]=>[[1,5,8,9,10],[2,7],[3],[4],[6]]=>111010000=>[1,1,1,2,5]
[5,1,1,1,1,1]=>[[1,7,8,9,10],[2],[3],[4],[5],[6]]=>111110000=>[1,1,1,1,1,5]
[4,4,2]=>[[1,2,5,6],[3,4,9,10],[7,8]]=>010001000=>[2,4,4]
[4,4,1,1]=>[[1,4,5,6],[2,8,9,10],[3],[7]]=>110001000=>[1,1,4,4]
[4,3,3]=>[[1,2,3,10],[4,5,6],[7,8,9]]=>001001000=>[3,3,4]
[4,3,2,1]=>[[1,3,6,10],[2,5,9],[4,8],[7]]=>101001000=>[1,2,3,4]
[4,3,1,1,1]=>[[1,5,6,10],[2,8,9],[3],[4],[7]]=>111001000=>[1,1,1,3,4]
[4,2,2,2]=>[[1,2,9,10],[3,4],[5,6],[7,8]]=>010101000=>[2,2,2,4]
[4,2,2,1,1]=>[[1,4,9,10],[2,6],[3,8],[5],[7]]=>110101000=>[1,1,2,2,4]
[4,2,1,1,1,1]=>[[1,6,9,10],[2,8],[3],[4],[5],[7]]=>111101000=>[1,1,1,1,2,4]
[4,1,1,1,1,1,1]=>[[1,8,9,10],[2],[3],[4],[5],[6],[7]]=>111111000=>[1,1,1,1,1,1,4]
[3,3,3,1]=>[[1,3,4],[2,6,7],[5,9,10],[8]]=>100100100=>[1,3,3,3]
[3,3,2,2]=>[[1,2,7],[3,4,10],[5,6],[8,9]]=>010100100=>[2,2,3,3]
[3,3,2,1,1]=>[[1,4,7],[2,6,10],[3,9],[5],[8]]=>110100100=>[1,1,2,3,3]
[3,3,1,1,1,1]=>[[1,6,7],[2,9,10],[3],[4],[5],[8]]=>111100100=>[1,1,1,1,3,3]
[3,2,2,2,1]=>[[1,3,10],[2,5],[4,7],[6,9],[8]]=>101010100=>[1,2,2,2,3]
[3,2,2,1,1,1]=>[[1,5,10],[2,7],[3,9],[4],[6],[8]]=>111010100=>[1,1,1,2,2,3]
[3,2,1,1,1,1,1]=>[[1,7,10],[2,9],[3],[4],[5],[6],[8]]=>111110100=>[1,1,1,1,1,2,3]
[3,1,1,1,1,1,1,1]=>[[1,9,10],[2],[3],[4],[5],[6],[7],[8]]=>111111100=>[1,1,1,1,1,1,1,3]
[2,2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8],[9,10]]=>010101010=>[2,2,2,2,2]
[2,2,2,2,1,1]=>[[1,4],[2,6],[3,8],[5,10],[7],[9]]=>110101010=>[1,1,2,2,2,2]
[2,2,2,1,1,1,1]=>[[1,6],[2,8],[3,10],[4],[5],[7],[9]]=>111101010=>[1,1,1,1,2,2,2]
[2,2,1,1,1,1,1,1]=>[[1,8],[2,10],[3],[4],[5],[6],[7],[9]]=>111111010=>[1,1,1,1,1,1,2,2]
[2,1,1,1,1,1,1,1,1]=>[[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]=>111111110=>[1,1,1,1,1,1,1,1,2]
[1,1,1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]=>111111111=>[1,1,1,1,1,1,1,1,1,1]
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 $1\leq i < n$.
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
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.
searching the database
Sorry, this map was not found in the database.