- FindStat

Identifier

Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000068: Posets ⟶ ℤ

Values

[1] => [1] => 1 => ([(0,1)],2) => 1
[1,1] => [2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2] => [1,1] => 11 => ([(0,2),(2,1)],3) => 1
[1,1,1] => [3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[1,2] => [2,1] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 1
[2,1] => [1,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[3] => [1,1,1] => 111 => ([(0,3),(2,1),(3,2)],4) => 1
[1,1,1,1] => [4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => 1
[2,1,1] => [1,3] => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 1
[3,1] => [1,1,2] => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => 1
[4] => [1,1,1,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,1,1,1,1] => [5] => 10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10) => 1
[2,1,1,1] => [1,4] => 11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12) => 1
[3,1,1] => [1,1,3] => 11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12) => 1
[4,1] => [1,1,1,2] => 11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10) => 1
[5] => [1,1,1,1,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[6] => [1,1,1,1,1,1] => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
[7] => [1,1,1,1,1,1,1] => 1111111 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) => 1
[8] => [1,1,1,1,1,1,1,1] => 11111111 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9) => 1
[9] => [1,1,1,1,1,1,1,1,1] => 111111111 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => 1

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$F_{1} = q$

$F_{2} = 2\ q$

$F_{3} = 4\ q$

Description

The number of minimal elements in a poset.

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

poset of factors

Description

The poset of factors of a binary word.
This is the partial order on the set of distinct factors of a binary word, such that

$u < v$ if and only if

$u$ is a factor of

$v$ .

Map

complement

Description

The complement of a composition.
The complement of a composition

$I$ is defined as follows:
If

$I$ is the empty composition, then the complement is also the empty composition. Otherwise, let

$S$ be the descent set corresponding to

$I=(i_1,\dots,i_k)$ , that is, the subset

$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$
of

$\{ 1, 2, \ldots, |I|-1 \}$ . Then, the complement of

$I$ is the composition of the same size as

$I$ , whose descent set is

$\{ 1, 2, \ldots, |I|-1 \} \setminus S$ .
The complement of a composition

$I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to

$I$ .