Identifier
Mp00042:
Integer partitions
—initial 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]=>[[1,2],[3]]=>01=>[2,1]
[1,1,1]=>[[1],[2],[3]]=>11=>[1,1,1]
[4]=>[[1,2,3,4]]=>000=>[4]
[3,1]=>[[1,2,3],[4]]=>001=>[3,1]
[2,2]=>[[1,2],[3,4]]=>010=>[2,2]
[2,1,1]=>[[1,2],[3],[4]]=>011=>[2,1,1]
[1,1,1,1]=>[[1],[2],[3],[4]]=>111=>[1,1,1,1]
[5]=>[[1,2,3,4,5]]=>0000=>[5]
[4,1]=>[[1,2,3,4],[5]]=>0001=>[4,1]
[3,2]=>[[1,2,3],[4,5]]=>0010=>[3,2]
[3,1,1]=>[[1,2,3],[4],[5]]=>0011=>[3,1,1]
[2,2,1]=>[[1,2],[3,4],[5]]=>0101=>[2,2,1]
[2,1,1,1]=>[[1,2],[3],[4],[5]]=>0111=>[2,1,1,1]
[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]=>[[1,2,3,4,5],[6]]=>00001=>[5,1]
[4,2]=>[[1,2,3,4],[5,6]]=>00010=>[4,2]
[4,1,1]=>[[1,2,3,4],[5],[6]]=>00011=>[4,1,1]
[3,3]=>[[1,2,3],[4,5,6]]=>00100=>[3,3]
[3,2,1]=>[[1,2,3],[4,5],[6]]=>00101=>[3,2,1]
[3,1,1,1]=>[[1,2,3],[4],[5],[6]]=>00111=>[3,1,1,1]
[2,2,2]=>[[1,2],[3,4],[5,6]]=>01010=>[2,2,2]
[2,2,1,1]=>[[1,2],[3,4],[5],[6]]=>01011=>[2,2,1,1]
[2,1,1,1,1]=>[[1,2],[3],[4],[5],[6]]=>01111=>[2,1,1,1,1]
[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]=>[[1,2,3,4,5,6],[7]]=>000001=>[6,1]
[5,2]=>[[1,2,3,4,5],[6,7]]=>000010=>[5,2]
[5,1,1]=>[[1,2,3,4,5],[6],[7]]=>000011=>[5,1,1]
[4,3]=>[[1,2,3,4],[5,6,7]]=>000100=>[4,3]
[4,2,1]=>[[1,2,3,4],[5,6],[7]]=>000101=>[4,2,1]
[4,1,1,1]=>[[1,2,3,4],[5],[6],[7]]=>000111=>[4,1,1,1]
[3,3,1]=>[[1,2,3],[4,5,6],[7]]=>001001=>[3,3,1]
[3,2,2]=>[[1,2,3],[4,5],[6,7]]=>001010=>[3,2,2]
[3,2,1,1]=>[[1,2,3],[4,5],[6],[7]]=>001011=>[3,2,1,1]
[3,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7]]=>001111=>[3,1,1,1,1]
[2,2,2,1]=>[[1,2],[3,4],[5,6],[7]]=>010101=>[2,2,2,1]
[2,2,1,1,1]=>[[1,2],[3,4],[5],[6],[7]]=>010111=>[2,2,1,1,1]
[2,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7]]=>011111=>[2,1,1,1,1,1]
[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]=>[[1,2,3,4,5,6,7],[8]]=>0000001=>[7,1]
[6,2]=>[[1,2,3,4,5,6],[7,8]]=>0000010=>[6,2]
[6,1,1]=>[[1,2,3,4,5,6],[7],[8]]=>0000011=>[6,1,1]
[5,3]=>[[1,2,3,4,5],[6,7,8]]=>0000100=>[5,3]
[5,2,1]=>[[1,2,3,4,5],[6,7],[8]]=>0000101=>[5,2,1]
[5,1,1,1]=>[[1,2,3,4,5],[6],[7],[8]]=>0000111=>[5,1,1,1]
[4,4]=>[[1,2,3,4],[5,6,7,8]]=>0001000=>[4,4]
[4,3,1]=>[[1,2,3,4],[5,6,7],[8]]=>0001001=>[4,3,1]
[4,2,2]=>[[1,2,3,4],[5,6],[7,8]]=>0001010=>[4,2,2]
[4,2,1,1]=>[[1,2,3,4],[5,6],[7],[8]]=>0001011=>[4,2,1,1]
[4,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8]]=>0001111=>[4,1,1,1,1]
[3,3,2]=>[[1,2,3],[4,5,6],[7,8]]=>0010010=>[3,3,2]
[3,3,1,1]=>[[1,2,3],[4,5,6],[7],[8]]=>0010011=>[3,3,1,1]
[3,2,2,1]=>[[1,2,3],[4,5],[6,7],[8]]=>0010101=>[3,2,2,1]
[3,2,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8]]=>0010111=>[3,2,1,1,1]
[3,1,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7],[8]]=>0011111=>[3,1,1,1,1,1]
[2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8]]=>0101010=>[2,2,2,2]
[2,2,2,1,1]=>[[1,2],[3,4],[5,6],[7],[8]]=>0101011=>[2,2,2,1,1]
[2,2,1,1,1,1]=>[[1,2],[3,4],[5],[6],[7],[8]]=>0101111=>[2,2,1,1,1,1]
[2,1,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7],[8]]=>0111111=>[2,1,1,1,1,1,1]
[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]=>[[1,2,3,4,5,6,7,8],[9]]=>00000001=>[8,1]
[7,2]=>[[1,2,3,4,5,6,7],[8,9]]=>00000010=>[7,2]
[7,1,1]=>[[1,2,3,4,5,6,7],[8],[9]]=>00000011=>[7,1,1]
[6,3]=>[[1,2,3,4,5,6],[7,8,9]]=>00000100=>[6,3]
[6,2,1]=>[[1,2,3,4,5,6],[7,8],[9]]=>00000101=>[6,2,1]
[6,1,1,1]=>[[1,2,3,4,5,6],[7],[8],[9]]=>00000111=>[6,1,1,1]
[5,4]=>[[1,2,3,4,5],[6,7,8,9]]=>00001000=>[5,4]
[5,3,1]=>[[1,2,3,4,5],[6,7,8],[9]]=>00001001=>[5,3,1]
[5,2,2]=>[[1,2,3,4,5],[6,7],[8,9]]=>00001010=>[5,2,2]
[5,2,1,1]=>[[1,2,3,4,5],[6,7],[8],[9]]=>00001011=>[5,2,1,1]
[5,1,1,1,1]=>[[1,2,3,4,5],[6],[7],[8],[9]]=>00001111=>[5,1,1,1,1]
[4,4,1]=>[[1,2,3,4],[5,6,7,8],[9]]=>00010001=>[4,4,1]
[4,3,2]=>[[1,2,3,4],[5,6,7],[8,9]]=>00010010=>[4,3,2]
[4,3,1,1]=>[[1,2,3,4],[5,6,7],[8],[9]]=>00010011=>[4,3,1,1]
[4,2,2,1]=>[[1,2,3,4],[5,6],[7,8],[9]]=>00010101=>[4,2,2,1]
[4,2,1,1,1]=>[[1,2,3,4],[5,6],[7],[8],[9]]=>00010111=>[4,2,1,1,1]
[4,1,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8],[9]]=>00011111=>[4,1,1,1,1,1]
[3,3,3]=>[[1,2,3],[4,5,6],[7,8,9]]=>00100100=>[3,3,3]
[3,3,2,1]=>[[1,2,3],[4,5,6],[7,8],[9]]=>00100101=>[3,3,2,1]
[3,3,1,1,1]=>[[1,2,3],[4,5,6],[7],[8],[9]]=>00100111=>[3,3,1,1,1]
[3,2,2,2]=>[[1,2,3],[4,5],[6,7],[8,9]]=>00101010=>[3,2,2,2]
[3,2,2,1,1]=>[[1,2,3],[4,5],[6,7],[8],[9]]=>00101011=>[3,2,2,1,1]
[3,2,1,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8],[9]]=>00101111=>[3,2,1,1,1,1]
[3,1,1,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7],[8],[9]]=>00111111=>[3,1,1,1,1,1,1]
[2,2,2,2,1]=>[[1,2],[3,4],[5,6],[7,8],[9]]=>01010101=>[2,2,2,2,1]
[2,2,2,1,1,1]=>[[1,2],[3,4],[5,6],[7],[8],[9]]=>01010111=>[2,2,2,1,1,1]
[2,2,1,1,1,1,1]=>[[1,2],[3,4],[5],[6],[7],[8],[9]]=>01011111=>[2,2,1,1,1,1,1]
[2,1,1,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7],[8],[9]]=>01111111=>[2,1,1,1,1,1,1,1]
[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]=>[[1,2,3,4,5,6,7,8,9],[10]]=>000000001=>[9,1]
[8,2]=>[[1,2,3,4,5,6,7,8],[9,10]]=>000000010=>[8,2]
[8,1,1]=>[[1,2,3,4,5,6,7,8],[9],[10]]=>000000011=>[8,1,1]
[7,3]=>[[1,2,3,4,5,6,7],[8,9,10]]=>000000100=>[7,3]
[7,2,1]=>[[1,2,3,4,5,6,7],[8,9],[10]]=>000000101=>[7,2,1]
[7,1,1,1]=>[[1,2,3,4,5,6,7],[8],[9],[10]]=>000000111=>[7,1,1,1]
[6,4]=>[[1,2,3,4,5,6],[7,8,9,10]]=>000001000=>[6,4]
[6,3,1]=>[[1,2,3,4,5,6],[7,8,9],[10]]=>000001001=>[6,3,1]
[6,2,2]=>[[1,2,3,4,5,6],[7,8],[9,10]]=>000001010=>[6,2,2]
[6,2,1,1]=>[[1,2,3,4,5,6],[7,8],[9],[10]]=>000001011=>[6,2,1,1]
[6,1,1,1,1]=>[[1,2,3,4,5,6],[7],[8],[9],[10]]=>000001111=>[6,1,1,1,1]
[5,5]=>[[1,2,3,4,5],[6,7,8,9,10]]=>000010000=>[5,5]
[5,4,1]=>[[1,2,3,4,5],[6,7,8,9],[10]]=>000010001=>[5,4,1]
[5,3,2]=>[[1,2,3,4,5],[6,7,8],[9,10]]=>000010010=>[5,3,2]
[5,3,1,1]=>[[1,2,3,4,5],[6,7,8],[9],[10]]=>000010011=>[5,3,1,1]
[5,2,2,1]=>[[1,2,3,4,5],[6,7],[8,9],[10]]=>000010101=>[5,2,2,1]
[5,2,1,1,1]=>[[1,2,3,4,5],[6,7],[8],[9],[10]]=>000010111=>[5,2,1,1,1]
[5,1,1,1,1,1]=>[[1,2,3,4,5],[6],[7],[8],[9],[10]]=>000011111=>[5,1,1,1,1,1]
[4,4,2]=>[[1,2,3,4],[5,6,7,8],[9,10]]=>000100010=>[4,4,2]
[4,4,1,1]=>[[1,2,3,4],[5,6,7,8],[9],[10]]=>000100011=>[4,4,1,1]
[4,3,3]=>[[1,2,3,4],[5,6,7],[8,9,10]]=>000100100=>[4,3,3]
[4,3,2,1]=>[[1,2,3,4],[5,6,7],[8,9],[10]]=>000100101=>[4,3,2,1]
[4,3,1,1,1]=>[[1,2,3,4],[5,6,7],[8],[9],[10]]=>000100111=>[4,3,1,1,1]
[4,2,2,2]=>[[1,2,3,4],[5,6],[7,8],[9,10]]=>000101010=>[4,2,2,2]
[4,2,2,1,1]=>[[1,2,3,4],[5,6],[7,8],[9],[10]]=>000101011=>[4,2,2,1,1]
[4,2,1,1,1,1]=>[[1,2,3,4],[5,6],[7],[8],[9],[10]]=>000101111=>[4,2,1,1,1,1]
[4,1,1,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8],[9],[10]]=>000111111=>[4,1,1,1,1,1,1]
[3,3,3,1]=>[[1,2,3],[4,5,6],[7,8,9],[10]]=>001001001=>[3,3,3,1]
[3,3,2,2]=>[[1,2,3],[4,5,6],[7,8],[9,10]]=>001001010=>[3,3,2,2]
[3,3,2,1,1]=>[[1,2,3],[4,5,6],[7,8],[9],[10]]=>001001011=>[3,3,2,1,1]
[3,3,1,1,1,1]=>[[1,2,3],[4,5,6],[7],[8],[9],[10]]=>001001111=>[3,3,1,1,1,1]
[3,2,2,2,1]=>[[1,2,3],[4,5],[6,7],[8,9],[10]]=>001010101=>[3,2,2,2,1]
[3,2,2,1,1,1]=>[[1,2,3],[4,5],[6,7],[8],[9],[10]]=>001010111=>[3,2,2,1,1,1]
[3,2,1,1,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8],[9],[10]]=>001011111=>[3,2,1,1,1,1,1]
[3,1,1,1,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7],[8],[9],[10]]=>001111111=>[3,1,1,1,1,1,1,1]
[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]=>[[1,2],[3,4],[5,6],[7,8],[9],[10]]=>010101011=>[2,2,2,2,1,1]
[2,2,2,1,1,1,1]=>[[1,2],[3,4],[5,6],[7],[8],[9],[10]]=>010101111=>[2,2,2,1,1,1,1]
[2,2,1,1,1,1,1,1]=>[[1,2],[3,4],[5],[6],[7],[8],[9],[10]]=>010111111=>[2,2,1,1,1,1,1,1]
[2,1,1,1,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]=>011111111=>[2,1,1,1,1,1,1,1,1]
[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
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 $1\leq i < n$.
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
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