- FindStat

Identifier

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000629: Binary words ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,1,0,0] => [1,1,2] => [2,1] => 101 => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,1,0,0] => [1,1,1,2] => [3,1] => 1001 => 0

[1,0,1,0,1,1,0,0,1,0] => [1,1,2,1] => [2,1,1] => 1011 => 0

[1,0,1,0,1,1,0,1,0,0] => [1,1,2,1] => [2,1,1] => 1011 => 0

[1,0,1,0,1,1,1,0,0,0] => [1,1,3] => [2,1] => 101 => 0

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

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

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

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

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

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

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

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

[1,0,1,1,1,1,0,0,0,0] => [1,4] => [1,1] => 11 => 0

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

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

[1,1,0,0,1,1,0,0,1,0] => [2,2,1] => [2,1] => 101 => 0

[1,1,0,0,1,1,0,1,0,0] => [2,2,1] => [2,1] => 101 => 0

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

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

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

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

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

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

[1,1,0,1,1,0,0,0,1,0] => [2,2,1] => [2,1] => 101 => 0

[1,1,0,1,1,0,0,1,0,0] => [2,2,1] => [2,1] => 101 => 0

[1,1,0,1,1,0,1,0,0,0] => [2,2,1] => [2,1] => 101 => 0

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

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

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

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

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

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

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

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

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

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

[1,1,1,1,0,0,0,0,1,0] => [4,1] => [1,1] => 11 => 0

[1,1,1,1,0,0,0,1,0,0] => [4,1] => [1,1] => 11 => 0

[1,1,1,1,0,0,1,0,0,0] => [4,1] => [1,1] => 11 => 0

[1,1,1,1,0,1,0,0,0,0] => [4,1] => [1,1] => 11 => 0

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

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

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,2] => [4,1] => 10001 => 0

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,2,1] => [3,1,1] => 10011 => 0

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,2,1] => [3,1,1] => 10011 => 0

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,3] => [3,1] => 1001 => 0

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,2,1,1] => [2,1,2] => 10110 => 0

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,2,2] => [2,2] => 1010 => 0

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,2,1,1] => [2,1,2] => 10110 => 0

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,2,1,1] => [2,1,2] => 10110 => 0

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,2,2] => [2,2] => 1010 => 0

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,3,1] => [2,1,1] => 1011 => 0

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,3,1] => [2,1,1] => 1011 => 0

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,3,1] => [2,1,1] => 1011 => 0

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,4] => [2,1] => 101 => 0

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

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,2,1,2] => [1,1,1,1] => 1111 => 0

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,2,2,1] => [1,2,1] => 1101 => 0

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,2,2,1] => [1,2,1] => 1101 => 0

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

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

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,2,1,2] => [1,1,1,1] => 1111 => 0

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

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

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,2,1,2] => [1,1,1,1] => 1111 => 0

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,2,2,1] => [1,2,1] => 1101 => 0

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,2,2,1] => [1,2,1] => 1101 => 0

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,2,2,1] => [1,2,1] => 1101 => 0

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 5$

$F_{4} = 14$

$F_{5} = 42$

$F_{6} = 132$

$F_{7} = 429$

$F_{8} = 1430$

Description

The defect of a binary word.
The defect of a finite word

$w$ is given by the difference between the maximum possible number and the actual number of palindromic factors contained in

$w$ . The maximum possible number of palindromic factors in a word

$w$ is

$|w|+1$ .

Map

delta morphism

Description

Apply the delta morphism to an integer composition.
The delta morphism of a composition

$C$ is the compositions composed of the lengths of consecutive runs of the same integer in

$C$ .

Map

rise composition

Description

Send a Dyck path to the composition of sizes of its rises.

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

$0$ .