- FindStat

Identifier

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000691: Binary words ⟶ ℤ

Values

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

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

[[1,2,3]] => [1,2,3] => 00 => 0

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

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

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

[[1,2,3,4]] => [1,2,3,4] => 000 => 0

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

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

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

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

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

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

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

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

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

[[1,2,3,4,5]] => [1,2,3,4,5] => 0000 => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,2,3,4,5,6]] => [1,2,3,4,5,6] => 00000 => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of changes of a binary word.
This is the number of indices $i$ such that $w_i \neq w_{i+1}$.

Map

reading word permutation

Description

Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.

Map

descent tops

Description

The descent tops of a permutation as a binary word.
Since 1 is never a descent top, it is omitted and the first letter of the word corresponds to the element 2.