- FindStat

Identifier

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St001437: Binary words ⟶ ℤ

Values

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

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

[1,1] => [1,1,0,0] => [2] => 10 => 2

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

[2,1] => [1,0,1,1,0,0] => [1,2] => 110 => 3

[1,1,1] => [1,1,0,1,0,0] => [3] => 100 => 3

[4] => [1,0,1,0,1,0,1,0] => [1,1,1,1] => 1111 => 4

[3,1] => [1,0,1,0,1,1,0,0] => [1,1,2] => 1110 => 4

[2,2] => [1,1,1,0,0,0] => [3] => 100 => 3

[2,1,1] => [1,0,1,1,0,1,0,0] => [1,3] => 1100 => 4

[1,1,1,1] => [1,1,0,1,0,1,0,0] => [4] => 1000 => 4

[5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1] => 11111 => 5

[4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,2] => 11110 => 5

[3,2] => [1,0,1,1,1,0,0,0] => [1,3] => 1100 => 4

[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,3] => 11100 => 5

[2,2,1] => [1,1,1,0,0,1,0,0] => [4] => 1000 => 4

[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,4] => 11000 => 5

[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [5] => 10000 => 5

[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1] => 111111 => 6

[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,2] => 111110 => 6

[4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,1,3] => 11100 => 5

[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,3] => 111100 => 6

[3,3] => [1,1,1,0,1,0,0,0] => [4] => 1000 => 4

[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,4] => 11000 => 5

[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,4] => 111000 => 6

[2,2,2] => [1,1,1,1,0,0,0,0] => [4] => 1000 => 4

[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [5] => 10000 => 5

[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,5] => 110000 => 6

[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [6] => 100000 => 6

[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1] => 1111111 => 7

[6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,2] => 1111110 => 7

[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,3] => 111100 => 6

[5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,3] => 1111100 => 7

[4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,4] => 11000 => 5

[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,4] => 111000 => 6

[4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,4] => 1111000 => 7

[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [5] => 10000 => 5

[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,4] => 11000 => 5

[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,5] => 110000 => 6

[3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,5] => 1110000 => 7

[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [5] => 10000 => 5

[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [6] => 100000 => 6

[2,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,6] => 1100000 => 7

[1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [7] => 1000000 => 7

[8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1,1] => 11111111 => 8

[7,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,1,2] => 11111110 => 8

[6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,3] => 1111100 => 7

[6,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,3] => 11111100 => 8

[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,4] => 111000 => 6

[5,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,4] => 1111000 => 7

[5,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,4] => 11111000 => 8

[4,4] => [1,1,1,0,1,0,1,0,0,0] => [5] => 10000 => 5

[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,5] => 110000 => 6

[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,4] => 111000 => 6

[4,2,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,5] => 1110000 => 7

[4,1,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,1,5] => 11110000 => 8

[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [5] => 10000 => 5

[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [6] => 100000 => 6

[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,5] => 110000 => 6

[3,2,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [1,6] => 1100000 => 7

[3,1,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,1,6] => 11100000 => 8

[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [5] => 10000 => 5

[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [6] => 100000 => 6

[2,2,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [7] => 1000000 => 7

[2,1,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [1,7] => 11000000 => 8

[1,1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [8] => 10000000 => 8

[9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1,1,1] => 111111111 => 9

[8,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,1,1,2] => 111111110 => 9

[7,2] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,1,3] => 11111100 => 8

[7,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,1,3] => 111111100 => 9

[6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,4] => 1111000 => 7

[6,2,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,1,4] => 11111000 => 8

[6,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,1,4] => 111111000 => 9

[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,5] => 110000 => 6

[5,3,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,5] => 1110000 => 7

[5,2,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,4] => 1111000 => 7

[5,2,1,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,1,5] => 11110000 => 8

[5,1,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,5] => 111110000 => 9

[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [6] => 100000 => 6

[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,5] => 110000 => 6

[4,3,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0] => [1,6] => 1100000 => 7

[4,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,5] => 1110000 => 7

[4,2,1,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [1,1,6] => 11100000 => 8

[4,1,1,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,1,1,6] => 111100000 => 9

[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [5] => 10000 => 5

[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [6] => 100000 => 6

[3,3,1,1,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0] => [7] => 1000000 => 7

[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,5] => 110000 => 6

[3,2,2,1,1] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0] => [1,6] => 1100000 => 7

[3,2,1,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [1,7] => 11000000 => 8

[3,1,1,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [1,1,7] => 111000000 => 9

[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [6] => 100000 => 6

[2,2,2,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [7] => 1000000 => 7

[2,2,1,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0] => [8] => 10000000 => 8

[2,1,1,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [1,8] => 110000000 => 9

[1,1,1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [9] => 100000000 => 9

[9,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,1,1,1,2] => 1111111110 => 10

[8,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,1,1,3] => 111111100 => 9

[7,3] => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,4] => 11111000 => 8

[7,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,1,1,4] => 111111000 => 9

[6,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,5] => 1110000 => 7

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$F_{1} = q$

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

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

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

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

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

$F_{7} = 4\ q^{5} + 4\ q^{6} + 7\ q^{7}$

$F_{8} = 3\ q^{5} + 6\ q^{6} + 5\ q^{7} + 8\ q^{8}$

$F_{9} = q^{5} + 6\ q^{6} + 8\ q^{7} + 6\ q^{8} + 9\ q^{9}$

Description

The flex of a binary word.
This is the product of the lex statistic (St001436The index of a given binary word in the lex-order among all its cyclic shifts., augmented by 1) and its frequency (St000627The exponent of a binary word.), see [1, §8].

Map

touch composition

Description

Sends a Dyck path to its touch composition given by the composition of lengths of its touch points.

Map

parallelogram polyomino

Description

Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.

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

$0$ .