Identifier
-
Mp00317:
Integer partitions
—odd parts⟶
Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000307: 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) => 2
[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) => 2
[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) => 2
[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) => 3
[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
[1,1,1,1,1,1] => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 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) => 2
[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) => 2
[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) => 2
[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) => 3
[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) => 2
[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) => 2
[3,3,1,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,1,1,1,1,1] => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 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) => 2
[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) => 2
[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) => 2
[4,3,2] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 3
[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) => 2
[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) => 3
[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) => 2
[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) => 3
[5,3,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[5,3,1,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,1,1,1,1,1] => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 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) => 2
[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
[3,3,1,1,1,1] => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 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) => 2
[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) => 2
[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) => 2
[6,3,2] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 3
[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) => 2
[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
>>> Load all 334 entries. <<<
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The number of rowmotion orbits of a poset.
Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
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.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!