Identifier
-
Mp00317:
Integer partitions
—odd parts⟶
Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000632: Posets ⟶ ℤ
Values
[1] => 1 => ([(0,1)],2) => 0
[2] => 0 => ([(0,1)],2) => 0
[1,1] => 11 => ([(0,2),(2,1)],3) => 0
[3] => 1 => ([(0,1)],2) => 0
[2,1] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,1,1] => 111 => ([(0,3),(2,1),(3,2)],4) => 0
[4] => 0 => ([(0,1)],2) => 0
[3,1] => 11 => ([(0,2),(2,1)],3) => 0
[2,2] => 00 => ([(0,2),(2,1)],3) => 0
[2,1,1] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[1,1,1,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[5] => 1 => ([(0,1)],2) => 0
[4,1] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[3,2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[3,1,1] => 111 => ([(0,3),(2,1),(3,2)],4) => 0
[2,2,1] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[1,1,1,1,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[6] => 0 => ([(0,1)],2) => 0
[5,1] => 11 => ([(0,2),(2,1)],3) => 0
[4,2] => 00 => ([(0,2),(2,1)],3) => 0
[4,1,1] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[3,3] => 11 => ([(0,2),(2,1)],3) => 0
[3,2,1] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 2
[3,1,1,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[2,2,2] => 000 => ([(0,3),(2,1),(3,2)],4) => 0
[1,1,1,1,1,1] => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 0
[7] => 1 => ([(0,1)],2) => 0
[6,1] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[5,2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[5,1,1] => 111 => ([(0,3),(2,1),(3,2)],4) => 0
[4,3] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[4,2,1] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[3,3,1] => 111 => ([(0,3),(2,1),(3,2)],4) => 0
[3,2,2] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[3,1,1,1,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[8] => 0 => ([(0,1)],2) => 0
[7,1] => 11 => ([(0,2),(2,1)],3) => 0
[6,2] => 00 => ([(0,2),(2,1)],3) => 0
[6,1,1] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[5,3] => 11 => ([(0,2),(2,1)],3) => 0
[5,2,1] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 2
[5,1,1,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[4,4] => 00 => ([(0,2),(2,1)],3) => 0
[4,3,1] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[4,2,2] => 000 => ([(0,3),(2,1),(3,2)],4) => 0
[3,3,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[3,3,1,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[3,1,1,1,1,1] => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 0
[2,2,2,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[9] => 1 => ([(0,1)],2) => 0
[8,1] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[7,2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[7,1,1] => 111 => ([(0,3),(2,1),(3,2)],4) => 0
[6,3] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[6,2,1] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[5,4] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[5,3,1] => 111 => ([(0,3),(2,1),(3,2)],4) => 0
[5,2,2] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[5,1,1,1,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[4,4,1] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[4,3,2] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 2
[3,3,3] => 111 => ([(0,3),(2,1),(3,2)],4) => 0
[3,3,1,1,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[10] => 0 => ([(0,1)],2) => 0
[9,1] => 11 => ([(0,2),(2,1)],3) => 0
[8,2] => 00 => ([(0,2),(2,1)],3) => 0
[8,1,1] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[7,3] => 11 => ([(0,2),(2,1)],3) => 0
[7,2,1] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 2
[7,1,1,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[6,4] => 00 => ([(0,2),(2,1)],3) => 0
[6,3,1] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[6,2,2] => 000 => ([(0,3),(2,1),(3,2)],4) => 0
[5,5] => 11 => ([(0,2),(2,1)],3) => 0
[5,4,1] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 2
[5,3,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[5,3,1,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[5,1,1,1,1,1] => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 0
[4,4,2] => 000 => ([(0,3),(2,1),(3,2)],4) => 0
[4,3,3] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[4,2,2,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[3,3,3,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[3,3,1,1,1,1] => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 0
[2,2,2,2,2] => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[11] => 1 => ([(0,1)],2) => 0
[10,1] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[9,2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[9,1,1] => 111 => ([(0,3),(2,1),(3,2)],4) => 0
[8,3] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[8,2,1] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[7,4] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[7,3,1] => 111 => ([(0,3),(2,1),(3,2)],4) => 0
[7,2,2] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[7,1,1,1,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[6,5] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[6,4,1] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[6,3,2] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 2
[5,5,1] => 111 => ([(0,3),(2,1),(3,2)],4) => 0
[5,4,2] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[5,3,3] => 111 => ([(0,3),(2,1),(3,2)],4) => 0
[5,3,1,1,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
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Description
The jump number of the poset.
A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.
A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.
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 $v$.
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.
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