- FindStat

Identifier

Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000326: Binary words ⟶ ℤ

Values

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

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

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

[[1,2,3]] => [3] => 100 => 010 => 2

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

[[1,2],[3]] => [3] => 100 => 010 => 2

[[1],[2],[3]] => [3] => 100 => 010 => 2

[[1,2,3,4]] => [4] => 1000 => 0010 => 3

[[1,3,4],[2]] => [2,2] => 1010 => 1100 => 1

[[1,2,4],[3]] => [3,1] => 1001 => 0101 => 2

[[1,2,3],[4]] => [4] => 1000 => 0010 => 3

[[1,3],[2,4]] => [2,2] => 1010 => 1100 => 1

[[1,2],[3,4]] => [3,1] => 1001 => 0101 => 2

[[1,4],[2],[3]] => [3,1] => 1001 => 0101 => 2

[[1,3],[2],[4]] => [2,2] => 1010 => 1100 => 1

[[1,2],[3],[4]] => [4] => 1000 => 0010 => 3

[[1],[2],[3],[4]] => [4] => 1000 => 0010 => 3

[[1,2,3,4,5]] => [5] => 10000 => 00010 => 4

[[1,3,4,5],[2]] => [2,3] => 10100 => 01100 => 2

[[1,2,4,5],[3]] => [3,2] => 10010 => 10100 => 1

[[1,2,3,5],[4]] => [4,1] => 10001 => 00101 => 3

[[1,2,3,4],[5]] => [5] => 10000 => 00010 => 4

[[1,3,5],[2,4]] => [2,2,1] => 10101 => 11001 => 1

[[1,2,5],[3,4]] => [3,2] => 10010 => 10100 => 1

[[1,3,4],[2,5]] => [2,3] => 10100 => 01100 => 2

[[1,2,4],[3,5]] => [3,2] => 10010 => 10100 => 1

[[1,2,3],[4,5]] => [4,1] => 10001 => 00101 => 3

[[1,4,5],[2],[3]] => [3,2] => 10010 => 10100 => 1

[[1,3,5],[2],[4]] => [2,2,1] => 10101 => 11001 => 1

[[1,2,5],[3],[4]] => [4,1] => 10001 => 00101 => 3

[[1,3,4],[2],[5]] => [2,3] => 10100 => 01100 => 2

[[1,2,4],[3],[5]] => [3,2] => 10010 => 10100 => 1

[[1,2,3],[4],[5]] => [5] => 10000 => 00010 => 4

[[1,4],[2,5],[3]] => [3,2] => 10010 => 10100 => 1

[[1,3],[2,5],[4]] => [2,2,1] => 10101 => 11001 => 1

[[1,2],[3,5],[4]] => [4,1] => 10001 => 00101 => 3

[[1,3],[2,4],[5]] => [2,3] => 10100 => 01100 => 2

[[1,2],[3,4],[5]] => [3,2] => 10010 => 10100 => 1

[[1,5],[2],[3],[4]] => [4,1] => 10001 => 00101 => 3

[[1,4],[2],[3],[5]] => [3,2] => 10010 => 10100 => 1

[[1,3],[2],[4],[5]] => [2,3] => 10100 => 01100 => 2

[[1,2],[3],[4],[5]] => [5] => 10000 => 00010 => 4

[[1],[2],[3],[4],[5]] => [5] => 10000 => 00010 => 4

[[1,2,3,4,5,6]] => [6] => 100000 => 000010 => 5

[[1,3,4,5,6],[2]] => [2,4] => 101000 => 001100 => 3

[[1,2,4,5,6],[3]] => [3,3] => 100100 => 010100 => 2

[[1,2,3,5,6],[4]] => [4,2] => 100010 => 100100 => 1

[[1,2,3,4,6],[5]] => [5,1] => 100001 => 000101 => 4

[[1,2,3,4,5],[6]] => [6] => 100000 => 000010 => 5

[[1,3,5,6],[2,4]] => [2,2,2] => 101010 => 111000 => 1

[[1,2,5,6],[3,4]] => [3,3] => 100100 => 010100 => 2

[[1,3,4,6],[2,5]] => [2,3,1] => 101001 => 011001 => 2

[[1,2,4,6],[3,5]] => [3,2,1] => 100101 => 101001 => 1

[[1,2,3,6],[4,5]] => [4,2] => 100010 => 100100 => 1

[[1,3,4,5],[2,6]] => [2,4] => 101000 => 001100 => 3

[[1,2,4,5],[3,6]] => [3,3] => 100100 => 010100 => 2

[[1,2,3,5],[4,6]] => [4,2] => 100010 => 100100 => 1

[[1,2,3,4],[5,6]] => [5,1] => 100001 => 000101 => 4

[[1,4,5,6],[2],[3]] => [3,3] => 100100 => 010100 => 2

[[1,3,5,6],[2],[4]] => [2,2,2] => 101010 => 111000 => 1

[[1,2,5,6],[3],[4]] => [4,2] => 100010 => 100100 => 1

[[1,3,4,6],[2],[5]] => [2,3,1] => 101001 => 011001 => 2

[[1,2,4,6],[3],[5]] => [3,2,1] => 100101 => 101001 => 1

[[1,2,3,6],[4],[5]] => [5,1] => 100001 => 000101 => 4

[[1,3,4,5],[2],[6]] => [2,4] => 101000 => 001100 => 3

[[1,2,4,5],[3],[6]] => [3,3] => 100100 => 010100 => 2

[[1,2,3,5],[4],[6]] => [4,2] => 100010 => 100100 => 1

[[1,2,3,4],[5],[6]] => [6] => 100000 => 000010 => 5

[[1,3,5],[2,4,6]] => [2,2,2] => 101010 => 111000 => 1

[[1,2,5],[3,4,6]] => [3,3] => 100100 => 010100 => 2

[[1,3,4],[2,5,6]] => [2,3,1] => 101001 => 011001 => 2

[[1,2,4],[3,5,6]] => [3,2,1] => 100101 => 101001 => 1

[[1,2,3],[4,5,6]] => [4,2] => 100010 => 100100 => 1

[[1,4,6],[2,5],[3]] => [3,2,1] => 100101 => 101001 => 1

[[1,3,6],[2,5],[4]] => [2,2,2] => 101010 => 111000 => 1

[[1,2,6],[3,5],[4]] => [4,2] => 100010 => 100100 => 1

[[1,3,6],[2,4],[5]] => [2,3,1] => 101001 => 011001 => 2

[[1,2,6],[3,4],[5]] => [3,2,1] => 100101 => 101001 => 1

[[1,4,5],[2,6],[3]] => [3,3] => 100100 => 010100 => 2

[[1,3,5],[2,6],[4]] => [2,2,2] => 101010 => 111000 => 1

[[1,2,5],[3,6],[4]] => [4,2] => 100010 => 100100 => 1

[[1,3,4],[2,6],[5]] => [2,3,1] => 101001 => 011001 => 2

[[1,2,4],[3,6],[5]] => [3,2,1] => 100101 => 101001 => 1

[[1,2,3],[4,6],[5]] => [5,1] => 100001 => 000101 => 4

[[1,3,5],[2,4],[6]] => [2,2,2] => 101010 => 111000 => 1

[[1,2,5],[3,4],[6]] => [3,3] => 100100 => 010100 => 2

[[1,3,4],[2,5],[6]] => [2,4] => 101000 => 001100 => 3

[[1,2,4],[3,5],[6]] => [3,3] => 100100 => 010100 => 2

[[1,2,3],[4,5],[6]] => [4,2] => 100010 => 100100 => 1

[[1,5,6],[2],[3],[4]] => [4,2] => 100010 => 100100 => 1

[[1,4,6],[2],[3],[5]] => [3,2,1] => 100101 => 101001 => 1

[[1,3,6],[2],[4],[5]] => [2,3,1] => 101001 => 011001 => 2

[[1,2,6],[3],[4],[5]] => [5,1] => 100001 => 000101 => 4

[[1,4,5],[2],[3],[6]] => [3,3] => 100100 => 010100 => 2

[[1,3,5],[2],[4],[6]] => [2,2,2] => 101010 => 111000 => 1

[[1,2,5],[3],[4],[6]] => [4,2] => 100010 => 100100 => 1

[[1,3,4],[2],[5],[6]] => [2,4] => 101000 => 001100 => 3

[[1,2,4],[3],[5],[6]] => [3,3] => 100100 => 010100 => 2

[[1,2,3],[4],[5],[6]] => [6] => 100000 => 000010 => 5

[[1,4],[2,5],[3,6]] => [3,3] => 100100 => 010100 => 2

[[1,3],[2,5],[4,6]] => [2,2,2] => 101010 => 111000 => 1

>>> Load all 1434 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = q$

$F_{2} = 2\ q$

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

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

$F_{5} = 11\ q + 5\ q^{2} + 5\ q^{3} + 5\ q^{4}$

$F_{6} = 33\ q + 23\ q^{2} + 7\ q^{3} + 7\ q^{4} + 6\ q^{5}$

$F_{7} = 102\ q + 70\ q^{2} + 35\ q^{3} + 9\ q^{4} + 9\ q^{5} + 7\ q^{6}$

$F_{8} = 340\ q + 232\ q^{2} + 115\ q^{3} + 47\ q^{4} + 11\ q^{5} + 11\ q^{6} + 8\ q^{7}$

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

Foata bijection

Description

The Foata bijection

$\phi$ is a bijection on the set of words of given content (by a slight generalization of Section 2 in [1]).
Given a word

$w_1 w_2 ... w_n$ , compute the image inductively by starting with

$\phi(w_1) = w_1$ . At the

$i$ -th step, if

$\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$ , define

$\phi(w_1 w_2 ... w_i w_{i+1})$ by placing

$w_{i+1}$ on the end of the word

$v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.

If $w_{i+1} \geq v_i$ , place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$ .
If $w_{i+1} < v_i$ , place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$ .

In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
For instance, to compute

$\phi(4154223)$ , the sequence of words is

4,
|4|1 -- > 41,
|4|1|5 -- > 415,
|415|4 -- > 5414,
|5|4|14|2 -- > 54412,
|5441|2|2 -- > 154422,
|1|5442|2|3 -- > 1254423.

$\phi(4154223) = 1254423$ .

Map

valley composition

Description

The composition corresponding to the valley set of a standard tableau.
Let

$T$ be a standard tableau of size

$n$ .
An entry

$i$ of

$T$ is a descent if

$i+1$ is in a lower row (in English notation), otherwise

$i$ is an ascent.
An entry

$2 \leq i \leq n-1$ is a valley if

$i-1$ is a descent and

$i$ is an ascent.
This map returns the composition

$c_1,\dots,c_k$ of

$n$ such that

$\{c_1, c_1+c_2,\dots, c_1+\dots+c_k\}$ is the valley set of

$T$ .

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

$0$ .