- FindStat

Identifier

Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,2,3,4,5,6]] => [6] => 100000 => 1

[[1,3,4,5,6],[2]] => [1,5] => 110000 => 2

[[1,2,4,5,6],[3]] => [2,4] => 101000 => 1

[[1,2,3,5,6],[4]] => [3,3] => 100100 => 1

[[1,2,3,4,6],[5]] => [4,2] => 100010 => 1

[[1,2,3,4,5],[6]] => [5,1] => 100001 => 1

[[1,3,5,6],[2,4]] => [1,2,3] => 110100 => 2

[[1,2,5,6],[3,4]] => [2,4] => 101000 => 1

[[1,3,4,6],[2,5]] => [1,3,2] => 110010 => 2

[[1,2,4,6],[3,5]] => [2,2,2] => 101010 => 1

[[1,2,3,6],[4,5]] => [3,3] => 100100 => 1

[[1,3,4,5],[2,6]] => [1,4,1] => 110001 => 2

[[1,2,4,5],[3,6]] => [2,3,1] => 101001 => 1

[[1,2,3,5],[4,6]] => [3,2,1] => 100101 => 1

[[1,2,3,4],[5,6]] => [4,2] => 100010 => 1

[[1,4,5,6],[2],[3]] => [1,1,4] => 111000 => 3

[[1,3,5,6],[2],[4]] => [1,2,3] => 110100 => 2

[[1,2,5,6],[3],[4]] => [2,1,3] => 101100 => 1

[[1,3,4,6],[2],[5]] => [1,3,2] => 110010 => 2

[[1,2,4,6],[3],[5]] => [2,2,2] => 101010 => 1

[[1,2,3,6],[4],[5]] => [3,1,2] => 100110 => 1

[[1,3,4,5],[2],[6]] => [1,4,1] => 110001 => 2

[[1,2,4,5],[3],[6]] => [2,3,1] => 101001 => 1

[[1,2,3,5],[4],[6]] => [3,2,1] => 100101 => 1

[[1,2,3,4],[5],[6]] => [4,1,1] => 100011 => 1

[[1,3,5],[2,4,6]] => [1,2,2,1] => 110101 => 2

[[1,2,5],[3,4,6]] => [2,3,1] => 101001 => 1

[[1,3,4],[2,5,6]] => [1,3,2] => 110010 => 2

[[1,2,4],[3,5,6]] => [2,2,2] => 101010 => 1

[[1,2,3],[4,5,6]] => [3,3] => 100100 => 1

[[1,4,6],[2,5],[3]] => [1,1,2,2] => 111010 => 3

[[1,3,6],[2,5],[4]] => [1,2,3] => 110100 => 2

[[1,2,6],[3,5],[4]] => [2,1,3] => 101100 => 1

[[1,3,6],[2,4],[5]] => [1,2,1,2] => 110110 => 2

[[1,2,6],[3,4],[5]] => [2,2,2] => 101010 => 1

[[1,4,5],[2,6],[3]] => [1,1,3,1] => 111001 => 3

[[1,3,5],[2,6],[4]] => [1,2,2,1] => 110101 => 2

[[1,2,5],[3,6],[4]] => [2,1,2,1] => 101101 => 1

[[1,3,4],[2,6],[5]] => [1,3,2] => 110010 => 2

[[1,2,4],[3,6],[5]] => [2,2,2] => 101010 => 1

[[1,2,3],[4,6],[5]] => [3,1,2] => 100110 => 1

[[1,3,5],[2,4],[6]] => [1,2,2,1] => 110101 => 2

[[1,2,5],[3,4],[6]] => [2,3,1] => 101001 => 1

[[1,3,4],[2,5],[6]] => [1,3,1,1] => 110011 => 2

[[1,2,4],[3,5],[6]] => [2,2,1,1] => 101011 => 1

[[1,2,3],[4,5],[6]] => [3,2,1] => 100101 => 1

[[1,5,6],[2],[3],[4]] => [1,1,1,3] => 111100 => 4

[[1,4,6],[2],[3],[5]] => [1,1,2,2] => 111010 => 3

[[1,3,6],[2],[4],[5]] => [1,2,1,2] => 110110 => 2

[[1,2,6],[3],[4],[5]] => [2,1,1,2] => 101110 => 1

[[1,4,5],[2],[3],[6]] => [1,1,3,1] => 111001 => 3

[[1,3,5],[2],[4],[6]] => [1,2,2,1] => 110101 => 2

[[1,2,5],[3],[4],[6]] => [2,1,2,1] => 101101 => 1

[[1,3,4],[2],[5],[6]] => [1,3,1,1] => 110011 => 2

[[1,2,4],[3],[5],[6]] => [2,2,1,1] => 101011 => 1

[[1,2,3],[4],[5],[6]] => [3,1,1,1] => 100111 => 1

[[1,4],[2,5],[3,6]] => [1,1,2,2] => 111010 => 3

[[1,3],[2,5],[4,6]] => [1,2,2,1] => 110101 => 2

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$F_{1} = q$

$F_{2} = q + q^{2}$

$F_{3} = 2\ q + q^{2} + q^{3}$

$F_{4} = 5\ q + 3\ q^{2} + q^{3} + q^{4}$

$F_{5} = 13\ q + 8\ q^{2} + 3\ q^{3} + q^{4} + q^{5}$

$F_{6} = 38\ q + 24\ q^{2} + 9\ q^{3} + 3\ q^{4} + q^{5} + q^{6}$

$F_{7} = 116\ q + 74\ q^{2} + 28\ q^{3} + 9\ q^{4} + 3\ q^{5} + q^{6} + q^{7}$

$F_{8} = 382\ q + 246\ q^{2} + 93\ q^{3} + 29\ q^{4} + 9\ q^{5} + 3\ q^{6} + q^{7} + q^{8}$

Description

The number of leading ones in a binary word.

Map

to binary word

Description

Return the composition as a binary word, treating ones as separators.
Encoding a positive integer

$i$ as the word

$10\dots 0$ consisting of a one followed by

$i-1$ zeros, the binary word of a composition

$(i_1,\dots,i_k)$ is the concatenation of of words for

$i_1,\dots,i_k$ .
The image of this map contains precisely the words which do not begin with a

$0$ .

Map

horizontal strip sizes

Description

The composition of horizontal strip sizes.
We associate to a standard Young tableau

$T$ the composition

$(c_1,\dots,c_k)$ , such that

$k$ is minimal and the numbers

$c_1+\dots+c_i + 1,\dots,c_1+\dots+c_{i+1}$ form a horizontal strip in

$T$ for all

$i$ .