- FindStat

Identifier

Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000008: Integer compositions ⟶ ℤ

Values

[[1]] => => [1] => [1] => 0

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

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

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

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

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

[[1],[2],[3]] => 11 => [1,1,1] => [3] => 0

[[1,2,3,4]] => 000 => [4] => [1,1,1,1] => 6

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

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

[[1,2,3],[4]] => 001 => [3,1] => [2,1,1] => 5

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

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

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

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

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

[[1],[2],[3],[4]] => 111 => [1,1,1,1] => [4] => 0

[[1,2,3,4,5]] => 0000 => [5] => [1,1,1,1,1] => 10

[[1,3,4,5],[2]] => 1000 => [1,4] => [1,1,1,2] => 6

[[1,2,4,5],[3]] => 0100 => [2,3] => [1,1,2,1] => 7

[[1,2,3,5],[4]] => 0010 => [3,2] => [1,2,1,1] => 8

[[1,2,3,4],[5]] => 0001 => [4,1] => [2,1,1,1] => 9

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

[[1,2,5],[3,4]] => 0100 => [2,3] => [1,1,2,1] => 7

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

[[1,2,4],[3,5]] => 0101 => [2,2,1] => [2,2,1] => 6

[[1,2,3],[4,5]] => 0010 => [3,2] => [1,2,1,1] => 8

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

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

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

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

[[1,2,4],[3],[5]] => 0101 => [2,2,1] => [2,2,1] => 6

[[1,2,3],[4],[5]] => 0011 => [3,1,1] => [3,1,1] => 7

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

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

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

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

[[1,2],[3,4],[5]] => 0101 => [2,2,1] => [2,2,1] => 6

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

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

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

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

[[1],[2],[3],[4],[5]] => 1111 => [1,1,1,1,1] => [5] => 0

[[1,2,3,4,5,6]] => 00000 => [6] => [1,1,1,1,1,1] => 15

[[1,3,4,5,6],[2]] => 10000 => [1,5] => [1,1,1,1,2] => 10

[[1,2,4,5,6],[3]] => 01000 => [2,4] => [1,1,1,2,1] => 11

[[1,2,3,5,6],[4]] => 00100 => [3,3] => [1,1,2,1,1] => 12

[[1,2,3,4,6],[5]] => 00010 => [4,2] => [1,2,1,1,1] => 13

[[1,2,3,4,5],[6]] => 00001 => [5,1] => [2,1,1,1,1] => 14

[[1,3,5,6],[2,4]] => 10100 => [1,2,3] => [1,1,2,2] => 7

[[1,2,5,6],[3,4]] => 01000 => [2,4] => [1,1,1,2,1] => 11

[[1,3,4,6],[2,5]] => 10010 => [1,3,2] => [1,2,1,2] => 8

[[1,2,4,6],[3,5]] => 01010 => [2,2,2] => [1,2,2,1] => 9

[[1,2,3,6],[4,5]] => 00100 => [3,3] => [1,1,2,1,1] => 12

[[1,3,4,5],[2,6]] => 10001 => [1,4,1] => [2,1,1,2] => 9

[[1,2,4,5],[3,6]] => 01001 => [2,3,1] => [2,1,2,1] => 10

[[1,2,3,5],[4,6]] => 00101 => [3,2,1] => [2,2,1,1] => 11

[[1,2,3,4],[5,6]] => 00010 => [4,2] => [1,2,1,1,1] => 13

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

[[1,3,5,6],[2],[4]] => 10100 => [1,2,3] => [1,1,2,2] => 7

[[1,2,5,6],[3],[4]] => 01100 => [2,1,3] => [1,1,3,1] => 8

[[1,3,4,6],[2],[5]] => 10010 => [1,3,2] => [1,2,1,2] => 8

[[1,2,4,6],[3],[5]] => 01010 => [2,2,2] => [1,2,2,1] => 9

[[1,2,3,6],[4],[5]] => 00110 => [3,1,2] => [1,3,1,1] => 10

[[1,3,4,5],[2],[6]] => 10001 => [1,4,1] => [2,1,1,2] => 9

[[1,2,4,5],[3],[6]] => 01001 => [2,3,1] => [2,1,2,1] => 10

[[1,2,3,5],[4],[6]] => 00101 => [3,2,1] => [2,2,1,1] => 11

[[1,2,3,4],[5],[6]] => 00011 => [4,1,1] => [3,1,1,1] => 12

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

[[1,2,5],[3,4,6]] => 01001 => [2,3,1] => [2,1,2,1] => 10

[[1,3,4],[2,5,6]] => 10010 => [1,3,2] => [1,2,1,2] => 8

[[1,2,4],[3,5,6]] => 01010 => [2,2,2] => [1,2,2,1] => 9

[[1,2,3],[4,5,6]] => 00100 => [3,3] => [1,1,2,1,1] => 12

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

[[1,3,6],[2,5],[4]] => 10100 => [1,2,3] => [1,1,2,2] => 7

[[1,2,6],[3,5],[4]] => 01100 => [2,1,3] => [1,1,3,1] => 8

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

[[1,2,6],[3,4],[5]] => 01010 => [2,2,2] => [1,2,2,1] => 9

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

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

[[1,2,5],[3,6],[4]] => 01101 => [2,1,2,1] => [2,3,1] => 7

[[1,3,4],[2,6],[5]] => 10010 => [1,3,2] => [1,2,1,2] => 8

[[1,2,4],[3,6],[5]] => 01010 => [2,2,2] => [1,2,2,1] => 9

[[1,2,3],[4,6],[5]] => 00110 => [3,1,2] => [1,3,1,1] => 10

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

[[1,2,5],[3,4],[6]] => 01001 => [2,3,1] => [2,1,2,1] => 10

[[1,3,4],[2,5],[6]] => 10011 => [1,3,1,1] => [3,1,2] => 7

[[1,2,4],[3,5],[6]] => 01011 => [2,2,1,1] => [3,2,1] => 8

[[1,2,3],[4,5],[6]] => 00101 => [3,2,1] => [2,2,1,1] => 11

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

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

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

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

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

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

[[1,2,5],[3],[4],[6]] => 01101 => [2,1,2,1] => [2,3,1] => 7

[[1,3,4],[2],[5],[6]] => 10011 => [1,3,1,1] => [3,1,2] => 7

[[1,2,4],[3],[5],[6]] => 01011 => [2,2,1,1] => [3,2,1] => 8

[[1,2,3],[4],[5],[6]] => 00111 => [3,1,1,1] => [4,1,1] => 9

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

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

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Description

The major index of the composition.
The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see Permutations/Descents-Major.

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

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.

Map

conjugate

Description

The conjugate of a composition.
The conjugate of a composition $C$ is defined as the complement (Mp00039complement) of the reversal (Mp00038reverse) of $C$.
Equivalently, the ribbon shape corresponding to the conjugate of $C$ is the conjugate of the ribbon shape of $C$.