Identifier
Mp00033:
Dyck paths
—to two-row standard 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
=>
Cc0005;cc-rep-0Cc0007;cc-rep-1
[1,0]=>[[1],[2]]=>1=>[1,1]
[1,0,1,0]=>[[1,3],[2,4]]=>101=>[1,2,1]
[1,1,0,0]=>[[1,2],[3,4]]=>010=>[2,2]
[1,0,1,0,1,0]=>[[1,3,5],[2,4,6]]=>10101=>[1,2,2,1]
[1,0,1,1,0,0]=>[[1,3,4],[2,5,6]]=>10010=>[1,3,2]
[1,1,0,0,1,0]=>[[1,2,5],[3,4,6]]=>01001=>[2,3,1]
[1,1,0,1,0,0]=>[[1,2,4],[3,5,6]]=>01010=>[2,2,2]
[1,1,1,0,0,0]=>[[1,2,3],[4,5,6]]=>00100=>[3,3]
[1,0,1,0,1,0,1,0]=>[[1,3,5,7],[2,4,6,8]]=>1010101=>[1,2,2,2,1]
[1,0,1,0,1,1,0,0]=>[[1,3,5,6],[2,4,7,8]]=>1010010=>[1,2,3,2]
[1,0,1,1,0,0,1,0]=>[[1,3,4,7],[2,5,6,8]]=>1001001=>[1,3,3,1]
[1,0,1,1,0,1,0,0]=>[[1,3,4,6],[2,5,7,8]]=>1001010=>[1,3,2,2]
[1,0,1,1,1,0,0,0]=>[[1,3,4,5],[2,6,7,8]]=>1000100=>[1,4,3]
[1,1,0,0,1,0,1,0]=>[[1,2,5,7],[3,4,6,8]]=>0100101=>[2,3,2,1]
[1,1,0,0,1,1,0,0]=>[[1,2,5,6],[3,4,7,8]]=>0100010=>[2,4,2]
[1,1,0,1,0,0,1,0]=>[[1,2,4,7],[3,5,6,8]]=>0101001=>[2,2,3,1]
[1,1,0,1,0,1,0,0]=>[[1,2,4,6],[3,5,7,8]]=>0101010=>[2,2,2,2]
[1,1,0,1,1,0,0,0]=>[[1,2,4,5],[3,6,7,8]]=>0100100=>[2,3,3]
[1,1,1,0,0,0,1,0]=>[[1,2,3,7],[4,5,6,8]]=>0010001=>[3,4,1]
[1,1,1,0,0,1,0,0]=>[[1,2,3,6],[4,5,7,8]]=>0010010=>[3,3,2]
[1,1,1,0,1,0,0,0]=>[[1,2,3,5],[4,6,7,8]]=>0010100=>[3,2,3]
[1,1,1,1,0,0,0,0]=>[[1,2,3,4],[5,6,7,8]]=>0001000=>[4,4]
[1,0,1,0,1,0,1,0,1,0]=>[[1,3,5,7,9],[2,4,6,8,10]]=>101010101=>[1,2,2,2,2,1]
[1,0,1,0,1,0,1,1,0,0]=>[[1,3,5,7,8],[2,4,6,9,10]]=>101010010=>[1,2,2,3,2]
[1,0,1,0,1,1,0,0,1,0]=>[[1,3,5,6,9],[2,4,7,8,10]]=>101001001=>[1,2,3,3,1]
[1,0,1,0,1,1,0,1,0,0]=>[[1,3,5,6,8],[2,4,7,9,10]]=>101001010=>[1,2,3,2,2]
[1,0,1,0,1,1,1,0,0,0]=>[[1,3,5,6,7],[2,4,8,9,10]]=>101000100=>[1,2,4,3]
[1,0,1,1,0,0,1,0,1,0]=>[[1,3,4,7,9],[2,5,6,8,10]]=>100100101=>[1,3,3,2,1]
[1,0,1,1,0,0,1,1,0,0]=>[[1,3,4,7,8],[2,5,6,9,10]]=>100100010=>[1,3,4,2]
[1,0,1,1,0,1,0,0,1,0]=>[[1,3,4,6,9],[2,5,7,8,10]]=>100101001=>[1,3,2,3,1]
[1,0,1,1,0,1,0,1,0,0]=>[[1,3,4,6,8],[2,5,7,9,10]]=>100101010=>[1,3,2,2,2]
[1,0,1,1,0,1,1,0,0,0]=>[[1,3,4,6,7],[2,5,8,9,10]]=>100100100=>[1,3,3,3]
[1,0,1,1,1,0,0,0,1,0]=>[[1,3,4,5,9],[2,6,7,8,10]]=>100010001=>[1,4,4,1]
[1,0,1,1,1,0,0,1,0,0]=>[[1,3,4,5,8],[2,6,7,9,10]]=>100010010=>[1,4,3,2]
[1,0,1,1,1,0,1,0,0,0]=>[[1,3,4,5,7],[2,6,8,9,10]]=>100010100=>[1,4,2,3]
[1,0,1,1,1,1,0,0,0,0]=>[[1,3,4,5,6],[2,7,8,9,10]]=>100001000=>[1,5,4]
[1,1,0,0,1,0,1,0,1,0]=>[[1,2,5,7,9],[3,4,6,8,10]]=>010010101=>[2,3,2,2,1]
[1,1,0,0,1,0,1,1,0,0]=>[[1,2,5,7,8],[3,4,6,9,10]]=>010010010=>[2,3,3,2]
[1,1,0,0,1,1,0,0,1,0]=>[[1,2,5,6,9],[3,4,7,8,10]]=>010001001=>[2,4,3,1]
[1,1,0,0,1,1,0,1,0,0]=>[[1,2,5,6,8],[3,4,7,9,10]]=>010001010=>[2,4,2,2]
[1,1,0,0,1,1,1,0,0,0]=>[[1,2,5,6,7],[3,4,8,9,10]]=>010000100=>[2,5,3]
[1,1,0,1,0,0,1,0,1,0]=>[[1,2,4,7,9],[3,5,6,8,10]]=>010100101=>[2,2,3,2,1]
[1,1,0,1,0,0,1,1,0,0]=>[[1,2,4,7,8],[3,5,6,9,10]]=>010100010=>[2,2,4,2]
[1,1,0,1,0,1,0,0,1,0]=>[[1,2,4,6,9],[3,5,7,8,10]]=>010101001=>[2,2,2,3,1]
[1,1,0,1,0,1,0,1,0,0]=>[[1,2,4,6,8],[3,5,7,9,10]]=>010101010=>[2,2,2,2,2]
[1,1,0,1,0,1,1,0,0,0]=>[[1,2,4,6,7],[3,5,8,9,10]]=>010100100=>[2,2,3,3]
[1,1,0,1,1,0,0,0,1,0]=>[[1,2,4,5,9],[3,6,7,8,10]]=>010010001=>[2,3,4,1]
[1,1,0,1,1,0,0,1,0,0]=>[[1,2,4,5,8],[3,6,7,9,10]]=>010010010=>[2,3,3,2]
[1,1,0,1,1,0,1,0,0,0]=>[[1,2,4,5,7],[3,6,8,9,10]]=>010010100=>[2,3,2,3]
[1,1,0,1,1,1,0,0,0,0]=>[[1,2,4,5,6],[3,7,8,9,10]]=>010001000=>[2,4,4]
[1,1,1,0,0,0,1,0,1,0]=>[[1,2,3,7,9],[4,5,6,8,10]]=>001000101=>[3,4,2,1]
[1,1,1,0,0,0,1,1,0,0]=>[[1,2,3,7,8],[4,5,6,9,10]]=>001000010=>[3,5,2]
[1,1,1,0,0,1,0,0,1,0]=>[[1,2,3,6,9],[4,5,7,8,10]]=>001001001=>[3,3,3,1]
[1,1,1,0,0,1,0,1,0,0]=>[[1,2,3,6,8],[4,5,7,9,10]]=>001001010=>[3,3,2,2]
[1,1,1,0,0,1,1,0,0,0]=>[[1,2,3,6,7],[4,5,8,9,10]]=>001000100=>[3,4,3]
[1,1,1,0,1,0,0,0,1,0]=>[[1,2,3,5,9],[4,6,7,8,10]]=>001010001=>[3,2,4,1]
[1,1,1,0,1,0,0,1,0,0]=>[[1,2,3,5,8],[4,6,7,9,10]]=>001010010=>[3,2,3,2]
[1,1,1,0,1,0,1,0,0,0]=>[[1,2,3,5,7],[4,6,8,9,10]]=>001010100=>[3,2,2,3]
[1,1,1,0,1,1,0,0,0,0]=>[[1,2,3,5,6],[4,7,8,9,10]]=>001001000=>[3,3,4]
[1,1,1,1,0,0,0,0,1,0]=>[[1,2,3,4,9],[5,6,7,8,10]]=>000100001=>[4,5,1]
[1,1,1,1,0,0,0,1,0,0]=>[[1,2,3,4,8],[5,6,7,9,10]]=>000100010=>[4,4,2]
[1,1,1,1,0,0,1,0,0,0]=>[[1,2,3,4,7],[5,6,8,9,10]]=>000100100=>[4,3,3]
[1,1,1,1,0,1,0,0,0,0]=>[[1,2,3,4,6],[5,7,8,9,10]]=>000101000=>[4,2,4]
[1,1,1,1,1,0,0,0,0,0]=>[[1,2,3,4,5],[6,7,8,9,10]]=>000010000=>[5,5]
[1,0,1,0,1,0,1,0,1,0,1,0]=>[[1,3,5,7,9,11],[2,4,6,8,10,12]]=>10101010101=>[1,2,2,2,2,2,1]
Map
to two-row standard tableau
Description
Return a standard tableau of shape $(n,n)$ where $n$ is the semilength of the Dyck path.
Given a Dyck path $D$, its image is given by recording the positions of the up-steps in the first row and the positions of the down-steps in the second row.
Given a Dyck path $D$, its image is given by recording the positions of the up-steps in the first row and the positions of the down-steps in the second 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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