- FindStat

Identifier

Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St001105: Posets ⟶ ℤ

Values

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

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

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

[3] => 1 => ([(0,1)],2) => 1

[2,1] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[1,1,1] => 111 => ([(0,3),(2,1),(3,2)],4) => 1

[4] => 0 => ([(0,1)],2) => 1

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

[2,2] => 00 => ([(0,2),(2,1)],3) => 1

[2,1,1] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3

[1,1,1,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[5] => 1 => ([(0,1)],2) => 1

[4,1] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[3,2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[3,1,1] => 111 => ([(0,3),(2,1),(3,2)],4) => 1

[2,2,1] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3

[1,1,1,1,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1

[6] => 0 => ([(0,1)],2) => 1

[5,1] => 11 => ([(0,2),(2,1)],3) => 1

[4,2] => 00 => ([(0,2),(2,1)],3) => 1

[4,1,1] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3

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

[3,2,1] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 4

[3,1,1,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[2,2,2] => 000 => ([(0,3),(2,1),(3,2)],4) => 1

[7] => 1 => ([(0,1)],2) => 1

[6,1] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[5,2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[5,1,1] => 111 => ([(0,3),(2,1),(3,2)],4) => 1

[4,3] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[4,2,1] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3

[3,3,1] => 111 => ([(0,3),(2,1),(3,2)],4) => 1

[3,2,2] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3

[3,1,1,1,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1

[8] => 0 => ([(0,1)],2) => 1

[7,1] => 11 => ([(0,2),(2,1)],3) => 1

[6,2] => 00 => ([(0,2),(2,1)],3) => 1

[6,1,1] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3

[5,3] => 11 => ([(0,2),(2,1)],3) => 1

[5,2,1] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 4

[5,1,1,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[4,4] => 00 => ([(0,2),(2,1)],3) => 1

[4,3,1] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3

[4,2,2] => 000 => ([(0,3),(2,1),(3,2)],4) => 1

[3,3,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3

[3,3,1,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[2,2,2,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[9] => 1 => ([(0,1)],2) => 1

[8,1] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[7,2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[7,1,1] => 111 => ([(0,3),(2,1),(3,2)],4) => 1

[6,3] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[6,2,1] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3

[5,4] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[5,3,1] => 111 => ([(0,3),(2,1),(3,2)],4) => 1

[5,2,2] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3

[5,1,1,1,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1

[4,4,1] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3

[4,3,2] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 4

[3,3,3] => 111 => ([(0,3),(2,1),(3,2)],4) => 1

[3,3,1,1,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1

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

[9,1] => 11 => ([(0,2),(2,1)],3) => 1

[8,2] => 00 => ([(0,2),(2,1)],3) => 1

[8,1,1] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3

[7,3] => 11 => ([(0,2),(2,1)],3) => 1

[7,2,1] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 4

[7,1,1,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[6,4] => 00 => ([(0,2),(2,1)],3) => 1

[6,3,1] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3

[6,2,2] => 000 => ([(0,3),(2,1),(3,2)],4) => 1

[5,5] => 11 => ([(0,2),(2,1)],3) => 1

[5,4,1] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 4

[5,3,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3

[5,3,1,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[4,4,2] => 000 => ([(0,3),(2,1),(3,2)],4) => 1

[4,3,3] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3

[4,2,2,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[3,3,3,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[2,2,2,2,2] => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1

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

[10,1] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[9,2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[9,1,1] => 111 => ([(0,3),(2,1),(3,2)],4) => 1

[8,3] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[8,2,1] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3

[7,4] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[7,3,1] => 111 => ([(0,3),(2,1),(3,2)],4) => 1

[7,2,2] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3

[7,1,1,1,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1

[6,5] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[6,4,1] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3

[6,3,2] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 4

[5,5,1] => 111 => ([(0,3),(2,1),(3,2)],4) => 1

[5,4,2] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3

[5,3,3] => 111 => ([(0,3),(2,1),(3,2)],4) => 1

[5,3,1,1,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1

[4,4,3] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3

[3,3,3,1,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1

[12] => 0 => ([(0,1)],2) => 1

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

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$F_{1} = q$

$F_{2} = 2\ q$

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

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

Description

The number of greedy linear extensions of a poset.
A linear extension of a poset

$P$ with elements

$\{x_1,\dots,x_n\}$ is greedy, if it can be obtained by the following algorithm:

Step 1. Choose a minimal element $x_1$ .
Step 2. Suppose $X=\{x_1,\dots,x_i\}$ have been chosen. If there is at least one minimal element of $P\setminus X$ which is greater than $x_i$ then choose $x_{i+1}$ to be any such minimal element; otherwise, choose $x_{i+1}$ to be any minimal element of $P\setminus X$ .

This statistic records the number of greedy linear extensions.

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

odd parts

Description

Return the binary word indicating which parts of the partition are odd.