- FindStat

Identifier

Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000326: Binary words ⟶ ℤ

Values

[[1,2]] => [[1],[2]] => 1 => 1

[[1],[2]] => [[1,2]] => 0 => 2

[[1,2,3]] => [[1],[2],[3]] => 11 => 1

[[1,3],[2]] => [[1,2],[3]] => 01 => 2

[[1,2],[3]] => [[1,3],[2]] => 10 => 1

[[1],[2],[3]] => [[1,2,3]] => 00 => 3

[[1,2,3,4]] => [[1],[2],[3],[4]] => 111 => 1

[[1,3,4],[2]] => [[1,2],[3],[4]] => 011 => 2

[[1,2,4],[3]] => [[1,3],[2],[4]] => 101 => 1

[[1,2,3],[4]] => [[1,4],[2],[3]] => 110 => 1

[[1,3],[2,4]] => [[1,2],[3,4]] => 010 => 2

[[1,2],[3,4]] => [[1,3],[2,4]] => 101 => 1

[[1,4],[2],[3]] => [[1,2,3],[4]] => 001 => 3

[[1,3],[2],[4]] => [[1,2,4],[3]] => 010 => 2

[[1,2],[3],[4]] => [[1,3,4],[2]] => 100 => 1

[[1],[2],[3],[4]] => [[1,2,3,4]] => 000 => 4

[[1,2,3,4,5]] => [[1],[2],[3],[4],[5]] => 1111 => 1

[[1,3,4,5],[2]] => [[1,2],[3],[4],[5]] => 0111 => 2

[[1,2,4,5],[3]] => [[1,3],[2],[4],[5]] => 1011 => 1

[[1,2,3,5],[4]] => [[1,4],[2],[3],[5]] => 1101 => 1

[[1,2,3,4],[5]] => [[1,5],[2],[3],[4]] => 1110 => 1

[[1,3,5],[2,4]] => [[1,2],[3,4],[5]] => 0101 => 2

[[1,2,5],[3,4]] => [[1,3],[2,4],[5]] => 1011 => 1

[[1,3,4],[2,5]] => [[1,2],[3,5],[4]] => 0110 => 2

[[1,2,4],[3,5]] => [[1,3],[2,5],[4]] => 1010 => 1

[[1,2,3],[4,5]] => [[1,4],[2,5],[3]] => 1101 => 1

[[1,4,5],[2],[3]] => [[1,2,3],[4],[5]] => 0011 => 3

[[1,3,5],[2],[4]] => [[1,2,4],[3],[5]] => 0101 => 2

[[1,2,5],[3],[4]] => [[1,3,4],[2],[5]] => 1001 => 1

[[1,3,4],[2],[5]] => [[1,2,5],[3],[4]] => 0110 => 2

[[1,2,4],[3],[5]] => [[1,3,5],[2],[4]] => 1010 => 1

[[1,2,3],[4],[5]] => [[1,4,5],[2],[3]] => 1100 => 1

[[1,4],[2,5],[3]] => [[1,2,3],[4,5]] => 0010 => 3

[[1,3],[2,5],[4]] => [[1,2,4],[3,5]] => 0101 => 2

[[1,2],[3,5],[4]] => [[1,3,4],[2,5]] => 1001 => 1

[[1,3],[2,4],[5]] => [[1,2,5],[3,4]] => 0100 => 2

[[1,2],[3,4],[5]] => [[1,3,5],[2,4]] => 1010 => 1

[[1,5],[2],[3],[4]] => [[1,2,3,4],[5]] => 0001 => 4

[[1,4],[2],[3],[5]] => [[1,2,3,5],[4]] => 0010 => 3

[[1,3],[2],[4],[5]] => [[1,2,4,5],[3]] => 0100 => 2

[[1,2],[3],[4],[5]] => [[1,3,4,5],[2]] => 1000 => 1

[[1],[2],[3],[4],[5]] => [[1,2,3,4,5]] => 0000 => 5

[[1,2,3,4,5,6]] => [[1],[2],[3],[4],[5],[6]] => 11111 => 1

[[1,3,4,5,6],[2]] => [[1,2],[3],[4],[5],[6]] => 01111 => 2

[[1,2,4,5,6],[3]] => [[1,3],[2],[4],[5],[6]] => 10111 => 1

[[1,2,3,5,6],[4]] => [[1,4],[2],[3],[5],[6]] => 11011 => 1

[[1,2,3,4,6],[5]] => [[1,5],[2],[3],[4],[6]] => 11101 => 1

[[1,2,3,4,5],[6]] => [[1,6],[2],[3],[4],[5]] => 11110 => 1

[[1,3,5,6],[2,4]] => [[1,2],[3,4],[5],[6]] => 01011 => 2

[[1,2,5,6],[3,4]] => [[1,3],[2,4],[5],[6]] => 10111 => 1

[[1,3,4,6],[2,5]] => [[1,2],[3,5],[4],[6]] => 01101 => 2

[[1,2,4,6],[3,5]] => [[1,3],[2,5],[4],[6]] => 10101 => 1

[[1,2,3,6],[4,5]] => [[1,4],[2,5],[3],[6]] => 11011 => 1

[[1,3,4,5],[2,6]] => [[1,2],[3,6],[4],[5]] => 01110 => 2

[[1,2,4,5],[3,6]] => [[1,3],[2,6],[4],[5]] => 10110 => 1

[[1,2,3,5],[4,6]] => [[1,4],[2,6],[3],[5]] => 11010 => 1

[[1,2,3,4],[5,6]] => [[1,5],[2,6],[3],[4]] => 11101 => 1

[[1,4,5,6],[2],[3]] => [[1,2,3],[4],[5],[6]] => 00111 => 3

[[1,3,5,6],[2],[4]] => [[1,2,4],[3],[5],[6]] => 01011 => 2

[[1,2,5,6],[3],[4]] => [[1,3,4],[2],[5],[6]] => 10011 => 1

[[1,3,4,6],[2],[5]] => [[1,2,5],[3],[4],[6]] => 01101 => 2

[[1,2,4,6],[3],[5]] => [[1,3,5],[2],[4],[6]] => 10101 => 1

[[1,2,3,6],[4],[5]] => [[1,4,5],[2],[3],[6]] => 11001 => 1

[[1,3,4,5],[2],[6]] => [[1,2,6],[3],[4],[5]] => 01110 => 2

[[1,2,4,5],[3],[6]] => [[1,3,6],[2],[4],[5]] => 10110 => 1

[[1,2,3,5],[4],[6]] => [[1,4,6],[2],[3],[5]] => 11010 => 1

[[1,2,3,4],[5],[6]] => [[1,5,6],[2],[3],[4]] => 11100 => 1

[[1,3,5],[2,4,6]] => [[1,2],[3,4],[5,6]] => 01010 => 2

[[1,2,5],[3,4,6]] => [[1,3],[2,4],[5,6]] => 10110 => 1

[[1,3,4],[2,5,6]] => [[1,2],[3,5],[4,6]] => 01101 => 2

[[1,2,4],[3,5,6]] => [[1,3],[2,5],[4,6]] => 10101 => 1

[[1,2,3],[4,5,6]] => [[1,4],[2,5],[3,6]] => 11011 => 1

[[1,4,6],[2,5],[3]] => [[1,2,3],[4,5],[6]] => 00101 => 3

[[1,3,6],[2,5],[4]] => [[1,2,4],[3,5],[6]] => 01011 => 2

[[1,2,6],[3,5],[4]] => [[1,3,4],[2,5],[6]] => 10011 => 1

[[1,3,6],[2,4],[5]] => [[1,2,5],[3,4],[6]] => 01001 => 2

[[1,2,6],[3,4],[5]] => [[1,3,5],[2,4],[6]] => 10101 => 1

[[1,4,5],[2,6],[3]] => [[1,2,3],[4,6],[5]] => 00110 => 3

[[1,3,5],[2,6],[4]] => [[1,2,4],[3,6],[5]] => 01010 => 2

[[1,2,5],[3,6],[4]] => [[1,3,4],[2,6],[5]] => 10010 => 1

[[1,3,4],[2,6],[5]] => [[1,2,5],[3,6],[4]] => 01101 => 2

[[1,2,4],[3,6],[5]] => [[1,3,5],[2,6],[4]] => 10101 => 1

[[1,2,3],[4,6],[5]] => [[1,4,5],[2,6],[3]] => 11001 => 1

[[1,3,5],[2,4],[6]] => [[1,2,6],[3,4],[5]] => 01010 => 2

[[1,2,5],[3,4],[6]] => [[1,3,6],[2,4],[5]] => 10110 => 1

[[1,3,4],[2,5],[6]] => [[1,2,6],[3,5],[4]] => 01100 => 2

[[1,2,4],[3,5],[6]] => [[1,3,6],[2,5],[4]] => 10100 => 1

[[1,2,3],[4,5],[6]] => [[1,4,6],[2,5],[3]] => 11010 => 1

[[1,5,6],[2],[3],[4]] => [[1,2,3,4],[5],[6]] => 00011 => 4

[[1,4,6],[2],[3],[5]] => [[1,2,3,5],[4],[6]] => 00101 => 3

[[1,3,6],[2],[4],[5]] => [[1,2,4,5],[3],[6]] => 01001 => 2

[[1,2,6],[3],[4],[5]] => [[1,3,4,5],[2],[6]] => 10001 => 1

[[1,4,5],[2],[3],[6]] => [[1,2,3,6],[4],[5]] => 00110 => 3

[[1,3,5],[2],[4],[6]] => [[1,2,4,6],[3],[5]] => 01010 => 2

[[1,2,5],[3],[4],[6]] => [[1,3,4,6],[2],[5]] => 10010 => 1

[[1,3,4],[2],[5],[6]] => [[1,2,5,6],[3],[4]] => 01100 => 2

[[1,2,4],[3],[5],[6]] => [[1,3,5,6],[2],[4]] => 10100 => 1

[[1,2,3],[4],[5],[6]] => [[1,4,5,6],[2],[3]] => 11000 => 1

[[1,4],[2,5],[3,6]] => [[1,2,3],[4,5,6]] => 00100 => 3

[[1,3],[2,5],[4,6]] => [[1,2,4],[3,5,6]] => 01010 => 2

[[1,2],[3,5],[4,6]] => [[1,3,4],[2,5,6]] => 10010 => 1

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Description

The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.

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 $1\leq i < n$.

Map

conjugate

Description

Sends a standard tableau to its conjugate tableau.