Identifier
-
Mp00317:
Integer partitions
—odd parts⟶
Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000717: Posets ⟶ ℤ
Values
[1] => 1 => ([(0,1)],2) => 2
[2] => 0 => ([(0,1)],2) => 2
[1,1] => 11 => ([(0,2),(2,1)],3) => 3
[3] => 1 => ([(0,1)],2) => 2
[2,1] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[1,1,1] => 111 => ([(0,3),(2,1),(3,2)],4) => 4
[4] => 0 => ([(0,1)],2) => 2
[3,1] => 11 => ([(0,2),(2,1)],3) => 3
[2,2] => 00 => ([(0,2),(2,1)],3) => 3
[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) => 5
[5] => 1 => ([(0,1)],2) => 2
[4,1] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[3,2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[3,1,1] => 111 => ([(0,3),(2,1),(3,2)],4) => 4
[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) => 6
[6] => 0 => ([(0,1)],2) => 2
[5,1] => 11 => ([(0,2),(2,1)],3) => 3
[4,2] => 00 => ([(0,2),(2,1)],3) => 3
[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) => 3
[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) => 5
[2,2,2] => 000 => ([(0,3),(2,1),(3,2)],4) => 4
[7] => 1 => ([(0,1)],2) => 2
[6,1] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[5,2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[5,1,1] => 111 => ([(0,3),(2,1),(3,2)],4) => 4
[4,3] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[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) => 4
[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) => 6
[8] => 0 => ([(0,1)],2) => 2
[7,1] => 11 => ([(0,2),(2,1)],3) => 3
[6,2] => 00 => ([(0,2),(2,1)],3) => 3
[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) => 3
[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) => 5
[4,4] => 00 => ([(0,2),(2,1)],3) => 3
[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) => 4
[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) => 5
[2,2,2,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[9] => 1 => ([(0,1)],2) => 2
[8,1] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[7,2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[7,1,1] => 111 => ([(0,3),(2,1),(3,2)],4) => 4
[6,3] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[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) => 3
[5,3,1] => 111 => ([(0,3),(2,1),(3,2)],4) => 4
[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) => 6
[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) => 4
[3,3,1,1,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[10] => 0 => ([(0,1)],2) => 2
[9,1] => 11 => ([(0,2),(2,1)],3) => 3
[8,2] => 00 => ([(0,2),(2,1)],3) => 3
[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) => 3
[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) => 5
[6,4] => 00 => ([(0,2),(2,1)],3) => 3
[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) => 4
[5,5] => 11 => ([(0,2),(2,1)],3) => 3
[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) => 5
[4,4,2] => 000 => ([(0,3),(2,1),(3,2)],4) => 4
[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) => 5
[3,3,3,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[2,2,2,2,2] => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[11] => 1 => ([(0,1)],2) => 2
[10,1] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[9,2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[9,1,1] => 111 => ([(0,3),(2,1),(3,2)],4) => 4
[8,3] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[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) => 3
[7,3,1] => 111 => ([(0,3),(2,1),(3,2)],4) => 4
[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) => 6
[6,5] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[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) => 4
[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) => 4
[5,3,1,1,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[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) => 6
[12] => 0 => ([(0,1)],2) => 2
[11,1] => 11 => ([(0,2),(2,1)],3) => 3
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Description
The number of ordinal summands of a poset.
The ordinal sum of two posets $P$ and $Q$ is the poset having elements $(p,0)$ and $(q,1)$ for $p\in P$ and $q\in Q$, and relations $(a,0) < (b,0)$ if $a < b$ in $P$, $(a,1) < (b,1)$ if $a < b$ in $Q$, and $(a,0) < (b,1)$.
This statistic is the length of the longest ordinal decomposition of a poset.
The ordinal sum of two posets $P$ and $Q$ is the poset having elements $(p,0)$ and $(q,1)$ for $p\in P$ and $q\in Q$, and relations $(a,0) < (b,0)$ if $a < b$ in $P$, $(a,1) < (b,1)$ if $a < b$ in $Q$, and $(a,0) < (b,1)$.
This statistic is the length of the longest ordinal decomposition of a poset.
Map
odd parts
Description
Return the binary word indicating which parts of the partition are odd.
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 $v$.
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